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Small +dsmall@wharton.upenn.edu +Department of Statistics +The Wharton School, University of Pennsylvania +Philadelphia, PA 19104 +Zijian Guo +zijguo@stat.rutgers.edu +Department of Statistics +Rutgers University +Piscataway, NJ 08854 +Abstract +We present R software packages RobustIV and controlfunctionIV for causal inference +with possibly invalid instrumental variables. RobustIV focuses on the linear outcome model. +It implements the two-stage hard thresholding method to select valid instrumental variables +from a set of candidate instrumental variables and make inferences for the causal effect in +both low- and high-dimensional settings. Furthermore, RobustIV implements the high- +dimensional endogeneity test and the searching and sampling method, a uniformly valid +inference method robust to errors in instrumental variable selection. controlfunctionIV +considers the nonlinear outcome model and makes inferences about the causal effect based +on the control function method. Our packages are demonstrated using two publicly avail- +able economic data sets together with applications to the Framingham Heart Study. +Keywords: Instrumental variable selection, confidence interval, nonlinear outcome model, +control function, maximum clique +©2023 Taehyeon Koo, Youjin Lee, Dylan S. Small, and Zijian Guo. + +Koo, Lee, Small, and Guo +1. Introduction +A common problem in making causal inferences from observational studies is that there may +be unmeasured confounders. The instrumental variable (IV) method is one of the most use- +ful methods to estimate the causal effect when there might exist unmeasured confounding. +The validity of IV methods relies on that the constructed IVs satisfy the following three +assumptions simultaneously (Wooldridge, 2010, e.g.): conditioning the measured covariates, +(A1) the IVs are associated with the treatment; +(A2) the IVs are independent with the unmeasured confounders; +(A3) the IVs have no direct effect on the outcome. +The main challenge of applying IV-based methods in practice is that the proposed IVs +might not satisfy the above assumptions (A1)-(A3). For example, in studying the causal +effect of education on earning, the proximity of school (Angrist and Krueger, 1991; Card, +1999) has been used as an instrumental variable. However, this instrument might be re- +lated to other factors, such as socioeconomic status, which could affect one’s earnings. +Also, there might be other advantages due to the proximity; for instance, people living +close to college could be more likely to be exposed to vocational programs linked to col- +leges. So, the instrument could have a direct effect on earnings. In addition, the problem +of IVs not satisfying assumptions (A1) to (A3) is a fundamental problem in Mendelian +Randomization (MR), whose goal is to estimate the causal effect of exposure on the disease +by using genetic variants as instruments. +These genetic variants might violate assump- +tions (A2) and (A3) due to pleiotropic effects (Bowden, Davey Smith, and Burgess, 2015; +Bowden, Davey Smith, Haycock, and Burgess, 2016; Kang, Zhang, Cai, and Small, 2016). +This paper presents the R packages RobustIV and controlfunctionIV, implementing +robust causal inference approaches proposed in Guo, Kang, Cai, and Small (2018a,b); Guo +(2021); Guo and Small (2016); Li and Guo (2020). +The implemented inference methods +choose the valid IVs among a set of candidate IVs that may violate the assumptions (A2) +and (A3). The proposed methods target both linear and nonlinear causal effects. We also +include the algorithm implementation for settings with high-dimensional covariates and IVs. +In the package RobustIV, we implement robust and high-dimensional IV algorithms for +models assuming a constant and linear treatment effect. +We implement the Two Stage +Hard Thresholding (TSHT) proposed in Guo et al. (2018b), which selects valid IVs based on +a voting method. The selected IVs are then used to infer the linear treatment effect. Addi- +tionally, RobustIV implements uniformly valid confidence intervals proposed in Guo (2021), +which guarantees valid coverage even if there are errors in selecting valid IVs. RobustIV also +contains the high-dimensional endogeneity test proposed in Guo et al. (2018a), generalizing +the Durbin-Wu-Hausman test (Durbin, 1954; Wu, 1973; Hausman, 1978). +2 + +RobustIV and controlfunctionIV +We implement several control function methods in the package controlfunctionIV +to infer causal effects under nonlinear outcome models. We implement the control func- +tion for the continuous outcome variable by showing it as the two-stage least squares +(TSLS) estimator with an augmented set of IVs (Guo and Small, 2016). We further follow +Guo and Small (2016) to test the validity of the augmented set of IVs and construct the +pretest estimator by comparing the control function estimator and the TSLS estimator. We +implement the probit control function method for the binary outcome and make inferences +for the conditional average treatment effect (CATE) with possibly invalid IVs. Moreover, +the controlfunctionIV package implements the SpotIV method proposed in Li and Guo +(2020) for the semi-parametric outcome model with possibly invalid IVs. +In R, there are well-developed IV methods when all IVs are assumed to satisfy the +assumptions (A1) to (A3), such as AER by Kleiber and Zeileis (2008) and ivmodel by +Kang, Jiang, Zhao, and Small (2020). The main difference in our packages RobustIV and +controlfunctionIV is that we allow for invalid IVs and leverage the multiple IVs to +learn the validity of the candidate IVs. We shall mention other R packages implementing +causal inference approaches with possibly invalid IVs: sisvive implemented the method +proposed in Kang et al. (2016) to estimate the treatment effect under the majority rule; +CIIV by Windmeijer, Liang, Hartwig, and Bowden (2021) considered the causal inference +approaches with possibly invalid IVs for the low-dimensional linear outcome model. In con- +trast, our packages RobustIV and controlfunctionIV are designed under a broader frame- +work by allowing for linear and nonlinear outcome models with low- and high-dimensional +IVs and covariates. Moreover, our package RobustIV provides a uniformly valid confidence +interval robust to the errors in separating valid and invalid IVs. +The GitHub repository at https://github.com/bluosun/MR-GENIUS implemented the +MR Genius method (Tchetgen, Sun, and Walter, 2021), generalizing the method in Lewbel +(2012) and leveraging the heteroscedastic regression errors in the treatment model to identify +the causal parameter. The R package TSCI implemented the two-stage curvature identifica- +tion method proposed in Guo and B¨uhlmann (2022), which leveraged the machine learning +methods to capture the nonlinearity in the treatment and identify the treatment effect +with possibly invalid instruments. In contrast, our packages use the different identification +conditions from Tchetgen et al. (2021); Lewbel (2012); Guo and B¨uhlmann (2022). +The paper is organized as follows. In Section 2, we review the methods implemented in +RobustIV under the linear outcome model; in Section 3, we discuss the inference approaches +for the nonlinear outcome models implemented in controlfunctionIV. In Section 4, we +demonstrate the usage of RobustIV and controlfunctionIV by analyzing economics data +sets from Angrist and Krueger (1991) and Mroz data. In Section 5, we demonstrate our +packages with an MR application to analyze the data from Framingham Heart Study (FHS). +3 + +Koo, Lee, Small, and Guo +Notation. +Let Rp be the set of real numbers with dimension p. For any vector v ∈ Rp, +vj denotes its jth element, v−j denotes whole v except for j-th index, and ∥v∥0 denotes +the number of non-zero elements in v. For any n × p matrix M, denote the (i, j) entry by +Mij, the ith row by Mi· , the jth column by M·j, and the transpose of M by MT; also, +MIJ denotes the submatrix of M consisting of rows specified by the set I ⊂ {1, ..., n} and +columns specified by the set J ⊂ {1, ..., p}, MI· denotes the submatrix of M consisting of +rows indexed by the set I and all columns, and M·J denotes the submatrix of M consisting of +columns specified by the set J and all rows. Ip denotes p × p identity matrix. +1 denotes the +indicator function. Φ denotes the CDF of the standard normal distribution. For a sequence +of random variable Xn, we use Xn +d→ X to denote that Xn converges to X in distribution. +2. Linear outcome models +Throughout the paper, we consider n i.i.d. +observations. +For 1 ≤ i ≤ n, let Yi ∈ R, +Di ∈ R, Zi· ∈ Rpz, and Xi· ∈ Rpx denote the outcome, the treatment, the instruments, and +the baseline covariates, respectively. This section reviews the robust instrumental variable +approaches in Guo et al. (2018a,b); Guo (2021), which are implemented in the RobustIV +package. We demonstrate the usage of RobustIV in Sections 4.1 and 4.2. +2.1 Model assumption +We assume the following outcome model with possibly invalid IVs (Small, 2007; Kang et al., +2016; Guo et al., 2018a; Windmeijer et al., 2021) +Yi = Diβ + ZT +i·π + XT +i·φ + ǫi, +E[ǫiZi·] = 0, E[ǫiXi·] = 0. +(1) +This is the linear structural model in econometrics (Wooldridge, 2010). Here, we aim to +estimate the constant causal effect β ∈ R. If Di is correlated with ǫi in the model (1), we +say it is an endogenous variable, and we cannot use popular estimators such as the OLS +estimator. We also assume the linear association model for the treatment +Di = ZT +i·γ + XT +i·ψ + δi, +E[δiZi·] = 0, E[δiXi·] = 0. +(2) +As a remark, the errors in (1) and (2) are allowed to be heteroscedastic. In (1) and (2), +πj = 0 if j-th IV satisfies the exclusion restriction conditions (A2) and (A3), and γj ̸= 0 if +it satisfies the strong IV assumption (A1). +We discuss the causal interpretation of the above model (1) using the potential outcome +framework (Small, 2007; Kang et al., 2016). Let Y (d,z) +i +be the potential outcome if individual +i were to receive the treatment d and the instruments z. For two possible values of the +treatment d′, d and instruments z′, z, if we assume the following potential outcomes model +Y (d′,z′) +i +− Y (d,z) +i += (d′ − d)β + (z′ − z)Tκ, +E[Y (0,0) +i +|Zi·, Xi·] = XT +i·φ + ZT +i·η, +(3) +4 + +RobustIV and controlfunctionIV +and define π = κ + η, and ǫi = Y (0,0) +i +− E[Y (0,0) +i +|Zi·, Xi·], we obtain the model (1). +By combining (1) and (2), we obtain the reduced form models of Y and D as +Yi = ZT +i·Γ + XT +i·Ψ + ξi, +E[ξiZi·] = 0, E[ξiXi·] = 0, +(4) +Di = ZT +i·γ + XT +i·ψ + δi, +E[δiZi·] = 0, E[δiXi·] = 0. +(5) +Here, Γ = βγ +π, Ψ = βψ +φ are reduced form parameters and ξi = βδi +ǫi is the reduced +form error term. +We introduce identifiability conditions for models (4) and (5). +Let S be the set of +relevant IVs, i.e., S = {1 ≤ j ≤ pz : γj ̸= 0} and V be the set of relevant and valid IVs, i.e., +V = {j ∈ S : πj = 0}. The set S contains all candidate IVs that are strongly associated +with the treatment. The set V is a subset of S, which contains all candidate IVs satisfying +all classical IV assumptions. The main challenge is that the set V is not known a priori in +the data analysis. Additional identifiability conditions are needed for identifying the causal +effect without any prior knowledge of V. The majority rule is introduced to identify causal +effects with invalid IVs (Bowden et al., 2016; Kang et al., 2016). +Condition 1 (Majority Rule). More than half of the relevant IVs are valid: |V| > |S|/2. +The following plurality rule is a weaker identification condition than the majority rule +(Hartwig, Davey Smith, and Bowden, 2017; Guo, Kang, Cai, and Small, 2018b). +Condition 2 (Plurality Rule). The valid instruments form a plurality compared to the +invalid instruments: |V| > maxc̸=0 |{j ∈ S : πj/γj = c}|. +We present two inference methods for β utilizing the majority and plurality: two stage +hard thresholding in Section 2.2 and searching and sampling in Section 2.3. To present the +methods, we consider the reduced form estimators (�Γ⊺, �γ⊺)⊺ satisfying +√n +���Γ +�γ +� +− +� +Γ +γ +�� +d→ N2pz +� +02pz, +� +VΓ +C +CT +Vγ +�� +. +(6) +We use �VΓ, �C, and �Vγ to denote consistent estimators of asymptotic covariance matrix +terms. In low dimensions, we estimate the reduced form (Γ⊺, γ⊺)⊺ by the OLS estimator +(�Γ⊺, �γ⊺)⊺ and estimate the variance covariance matrices by sandwich estimators; see the +detailed construction in Section 2 of Guo (2021). In high-dimensional settings, we can con- +struct (�Γ⊺, �γ⊺)⊺ as the debiased Lasso estimator (Belloni, Chernozhukov, and Wang, 2011; +Javanmard and Montanari, 2014; Guo, Kang, Cai, and Small, 2018b); see more details in +Section 4.1 of Guo et al. (2018b). +5 + +Koo, Lee, Small, and Guo +2.2 Two stage hard thresholding (TSHT) +The TSHT consists of two steps: the first step is to screen out the weak IVs, and the +second step is to screen out invalid IVs. Specifically, the first step of TSHT is to estimate +the set S of relevant IVs by �S = +� +1 ≤ j ≤ pz : |�γj| ≥ λ1 +� +�Vγ +jj/n +� +, where λ1 > 0 is a tuning +parameter adjusting the testing multiplicity. +The second thresholding step estimates the set V of valid instruments. Our main strategy +is to assume that one IV is valid and evaluate whether the other IVs are valid from the +point of view of that IV. Particularly, for j ∈ �S, we assume the j-th IV to be valid (i.e., +πj = 0) and construct an estimator �π−j of π−j using the equation π−j = Γ−j − β[j]γ−j +with β[j] = Γj/γj, and get the standard error of the estimator. We test whether π−j = 0 +by comparing �π−j to a threshold, calculated as multiplying the standard error of �π−j by +a tuning parameter λ2, which is a Bonferroni correction adjusting for testing multiplicity. +Using the above test procedures, we construct a voting matrix ˜Π ∈ R| �S|×| � +S| where ˜Πj,k = 1 +indicates that the k-th and j-th IVs agree with each other to be valid. Finally, we get a +symmetric voting matrix �Π by setting �Πj,k = min{˜Πj,k, ˜Πk,j}. +Once we get �Π, we estimate V by two options. Let VMk denote the number of votes +that the kth IV, with k ∈ �S, received from other candidates of IVs. First, we define �V by +the set of IVs that receive a majority and a plurality of votes (Guo et al., 2018b) +�VMP := {k ∈ �S : VMk > | �S|/2} ∪ {k ∈ �S : VMk = max +l∈ � +S +VMl}. +(7) +The next method is to estimate V by the maximum clique method. We can generate a graph +G with indexes belonging to �S and the adjacency matrix as �Π. That is, the indexes j, k ∈ �S +are connected if and only if �Πj,k = 1. Then as suggested in Windmeijer et al. (2021), we can +estimate �VMC as the maximum clique of the graph G, which is the largest fully connected sub- +graph of G (Csardi and Nepusz, 2006). Note that there might be several maximum cliques. +In this case, each maximum clique forms an estimator of V and our proposal reports several +causal effect estimators based on each maximum clique. +We further illustrate the definitions of �VMP and �VMC using the following example. Consider +pz = 8 with {z1, z2, z3, z4} being valid and {z5, z6, z7} being invalid with the same invalidity +level, and z8 being invalid IV with a different invalidity level. +The left side of Table 1 +corresponds to an ideal setting where the valid and invalid IVs are well separated and the +valid IVs {z1, z2, z3, z4} only vote for each other. In this case, �VMC = �VMP = {z1, z2, z3, z4}. +On the right side of Table 1, we consider the setting that the invalidity level of z5 might be +mild and the IV z5 receives the votes from three valid IVs {z2, z3, z4}. In this case, �VMP = +{z2, z3, z4, z5}. In contrast, there are two maximum cliques {z1, z2, z3, z4} and {z2, z3, z4, z5} +and �VMC can be either of these two. +6 + +RobustIV and controlfunctionIV +z1 +z2 +z3 +z4 +z5 +z6 +z7 +z8 +z1 +✓ +✓ +✓ +✓ +X +X +X +X +z2 +✓ +✓ +✓ +✓ +X +X +X +X +z3 +✓ +✓ +✓ +✓ +X +X +X +X +z4 +✓ +✓ +✓ +✓ +X +X +X +X +z5 +X +X +X +X +✓ +✓ +✓ +X +z6 +X +X +X +X +✓ +✓ +✓ +X +z7 +X +X +X +X +✓ +✓ +✓ +X +z8 +X +X +X +X +X +X +X +✓ +Votes +4 +4 +4 +4 +3 +3 +3 +1 +z1 +z2 +z3 +z4 +z5 +z6 +z7 +z8 +z1 +✓ +✓ +✓ +✓ +X +X +X +X +z2 +✓ +✓ +✓ +✓ +✓ +X +X +X +z3 +✓ +✓ +✓ +✓ +✓ +X +X +X +z4 +✓ +✓ +✓ +✓ +✓ +X +X +X +z5 +X +✓ +✓ +✓ +✓ +✓ +✓ +X +z6 +X +X +X +X +✓ +✓ +✓ +X +z7 +X +X +X +X +✓ +✓ +✓ +X +z8 +X +X +X +X +X +X +X +✓ +Votes +4 +5 +5 +5 +6 +3 +3 +1 +Table 1: The left voting matrix �Π denotes that all valid IVs {z1, z2, z3, z4} vote each other +but not any other invalid IV. The right voting matrix �Π denotes that the locally +invalid IV z5 receives votes from valid IVs {z2, z3, z4} and invalid IVs {z6, z7}. +Once we have �V, we can construct an efficient point estimator �β for β in a low- +dimensional setting via one-step iteration as follows. First, we construct an initial estimator +˜β = +�γT +�V +˜ +A�Γ�V +�γT +�V +˜ +A�γ�V , where ˜A = �Σ�V,�V −�Σ�V,�Vc �Σ−1 +�Vc,�Vc �Σ�Vc,�V, �Σ = 1 +n +�n +i=1 Wi·WT +i· , and Wi· = (ZT +i·, XT +i·)T. +Next, we get a point estimator �β by one-step iteration (Holland and Welsch, 1977) +�β = +�γT +�V �A�Γ�V +�γT +�V �A�γ�V +, +where +�A = [( �VΓ − 2˜β �C + ˜β2 �Vγ)�V,�V]−1. +(8) +Finally, the 1 − α confidence interval for β is +(�β − z1−α/2 � +SE, �β + z1−α/2 � +SE) +where +� +SE = +� +� +� +��γT +�V �A( �VΓ − 2�β �C + �β2 �Vγ)�V,�V �A�γ�V +n(�γT +�V �A�γ�V)2 +. +(9) +As a remark, �VΓ, �Vγ, and �C are heteroscedasticity-robust covariance estimators and hence +(9) is also robust to heteroscedastic errors in a low-dimensional setting. In a high-dimensional +setting, we set �A = I in (8), and �VΓ, �Vγ, and �C are constructed under the homoscedastic +error assumptions; see more details in Guo et al. (2018b). +2.3 Searching and Sampling +We now review the searching and sampling method proposed in Guo (2021), which provides +uniformly valid conference intervals even if there are errors in separating valid and invalid +IVs. The right-hand side of Table 1 illustrates an example of the invalid IVs not being +separated from valid IVs in finite samples. In the following, we review the idea of searching +7 + +Koo, Lee, Small, and Guo +and sampling under the majority rule and the more general method with the plurality rule +can be found in Guo (2021). +Let α ∈ (0, 1) denote the pre-specified significance level. Given β ∈ R and the reduced +form estimator �Γ and �γ, we estimate πj with j ∈ �S by +�πj(β) = (�Γj − β�γj)1(|�Γj − β�γj| ≥ �ρj(β, α)), +(10) +where �ρj(β, α) = Φ−1 � +1 − +α +2| � +S| +� +� +SE(�Γj − β�γj) with � +SE(�Γj − β�γj) denoting a consistent +estimator of the standard error of �Γj − β�γj. We search for the value of β leading to enough +valid IVs and construct the searching confidence interval as +CIsearch = +� +β ∈ R : +���π � +S(β) +�� +0 < | �S|/2 +� +, +(11) +which collects all β values such that more than half of IVs in �S are selected as valid. +Based on the searching method, Guo (2021) proposed a sampling confidence interval, +which retains the uniform coverage property and improves the precision of the confidence +interval. In particular, we sample +��Γ[m] +�γ[m] +� +iid +∼ N +���Γ +�γ +� +, 1 +n +� �VΓ +�C +�CT +�Vγ +�� +, +for +1 ≤ m ≤ M. +For 1 ≤ m ≤ M and j ∈ �S, we modify (10) and define +�π[m] +j +(β, λ) = (�Γ[m] +j +− β�γ[m] +j +)1(|�Γ[m] +j +− β�γ[m] +j +| ≥ λ · �ρj(β, α)) +with the shrinkage parameter λ ≍ (log n/M) +1 +2| � +S| . A data-dependent way of choosing λ can +be found in Remark 3 of Guo (2021). For each 1 ≤ m ≤ M, we construct a searching +interval (β[m] +min, β[m] +max) where +β[m] +min = min +β∈B[m] +λ +β +and +β[m] +max = max +β∈B[m] +λ +β +with B[m] +λ += +� +β ∈ R : +����π[m] +� +S (β, λ) +��� +0 < | �S|/2 +� +. Then the sampling CI is defined as +CIsample = +� +min +m∈M β[m] +min, max +m∈M β[m] +max +� +. +(12) +with M = {1 ≤ m ≤ M : (β[m] +min, β[m] +max) ̸= ∅}. The sampling confidence intervals in general +improve the precision of the searching confidence intervals. But both intervals can provide +uniformly valid coverage robust to the errors in separating valid and invalid IVs. +8 + +RobustIV and controlfunctionIV +2.4 Endogeneity test in high dimensions +We review the high-dimensional endogeneity test proposed in Guo et al. (2018a). We focus +on the homoscedastic error setting by writing Θ11 = Var[ξi|Zi·, Xi·], Θ22 = Var[δi|Zi·, Xi·], +and Θ12 = Cov[ξi, δi|Zi·, Xi·] for the reduced form models (4) and (5). +With the same +estimators from TSHT in the high-dimensional settings in Section 2.2, we can estimate the +covariance σ12 by �σ12 = �Θ12 − �β �Θ22 where �β = +� +j∈ �V �γj�Γj +� +j∈ �V �γ2 +j +and �Θ12 and �Θ22 are consistent +estimators of the reduced form covariance Θ22 and Θ12. We establish the asymptotic nor- +mality of �σ12 − σ12 in Guo et al. (2018a) and propose a testing procedure for H0 : σ12 = 0. +3. Nonlinear outcome models +This section reviews the control function IV methods (Guo and Small, 2016; Li and Guo, +2020) implemented in the controlfunctionIV package, whose usage is demonstrated in +Sections 4.3 and 4.4. +3.1 Control function and pretest estimators +We consider the following nonlinear outcome and treatment models: +Yi = G(Di)Tβ + XT +i·φ + ui, E[uiZi·] = E[uiXi·] = 0, +(13) +Di = H(Zi·)Tγ + XT +i·ψ + vi, E[viZi·] = E[viXi·] = 0, +(14) +where G(Di) = (Di, g2(Di), ..., gk(Di))T, H(Zi·) = (Zi·, h2(Zi·), ..., hk(Zi·))T with {gj(·)}2≤j≤k +and {hj(·)}2≤j≤k denoting the known nonlinear transformations. Under the models (13) +and (14), the IVs are assumed to be valid and the causal effect of increasing the value of D +from d2 to d1 is defined as G(d1)Tβ − G(d2)Tβ. +The control function (CF) method is a two-stage procedure. In the first stage, regress D +on H(Z) and X, and obtain the predicted value �D and its associated residual �v = D− �D. In +the second stage, we use �v as the proxies for the unmeasured confounders and regress Y on +G(D), X, and �v. We use �βCF to denote the estimated regression coefficient corresponding +to D. Guo and Small (2016) showed that �βCF is equivalent to the TSLS estimator with +the augmented set of IVs. +Even if all IVs satisfy the classical assumptions (A1)-(A3), +there is no guarantee of the validity of the augmented IVs generated by the CF estimator. +Guo and Small (2016) applied the Hausman test to assess the validity of the augmented set +of IVs generated by the CF estimator. The test statistic is defined as +H(�βCF, �βTSLS) = (�βCF − �βTSLS)T[Cov(�βTSLS) − Cov(�βCF)]−(�βCF − �βTSLS), +(15) +where �βTSLS is the two stage least square estimator, Cov(�βTSLS) and Cov(�βCF) are the +covariance matrices of �βTSLS and �βCF, and A− denote the Moore-Penrose pseudoinverse. +9 + +Koo, Lee, Small, and Guo +If the p-value P +� +χ2 +1 ≥ H(�βCF, �βTSLS) +� +is less than α = 0.05, then we define the level α +pretest estimator �βPretest as �βTSLS; otherwise, �βCF defined above (Guo and Small, 2016). +3.2 Probit CF and SpotIV +We now consider the binary outcome model and continuous treatment model, +E [Yi|Di = d, Wi· = w, ui = u] = +1(dβ + wTκ + u > 0), +and +Di = WT +i· γ + vi, +(16) +where Wi· = (ZT +i·, XT +i·)T, the errors (ui, vi)⊺ are bivariate normal random variables with +zero means and independent of Wi·, κ = (κT +z , κT +x)T is the coefficient vector of the IVs and +measured covariates, and γ = (γT +z , γT +x)T is a parameter representing the association between +Di and Wi·. When κz ̸= 0, the instruments are invalid. Since ui and vi are bivariate normal, +we write ui = ρvi +ei. The model (16) implies E [Yi|Wi·, vi] = Φ(Diβ∗ +W ⊺ +i·Γ∗ +ρ∗vi) where +β∗ = β/σe, Γ∗ = κ/σe + β∗ · γ, and ρ∗ = ρ/σe + β∗ with σe denoting the standard error +of ei = ui − ρvi. That is, the conditional outcome model of Yi given Wi,· and vi is a probit +regression model. +Our goal is to estimate the conditional average treatment effect (CATE) from d2 to d1 +CATE(d1, d2|w) := E[Yi|Di = d1, Wi· = w] − E[Yi|Di = d2, Wi· = w]. +(17) +We first construct the OLS estimator �γ of γ. We compute its residual �v = D − W�γ and +define �Σ = 1 +n +�n +i=1 Wi·WT +i· . We estimate S = {1 ≤ j ≤ pz : (γz)j ̸= 0} by +�S = +� +1 ≤ j ≤ pz : |�γj| ≥ �σv +� +2{�Σ−1}j,j log n/n +� +(18) +with �σ2 +v = �n +i=1 �v2 +i /n. Next, as CF in Section 3.1, we use �v as the proxy for unmeasured con- +founders and implement the probit regression Y on W and �v. We use �Γ and �ρ to denote the +probit regression coefficients of W of �v respectively. We apply the majority rule and compute +�β as the median of (�Γj/�γj)j∈ �S. We then estimate �κ = �Γ − �γ �β. Finally, we estimate CATE +defined in (17) by the partial mean method (Newey, 1994; Mammen, Rothe, and Schienle, +2012), +1 +n +n +� +i=1 +� +Φ(d1 �β + wT�κ + �vi�ρ) +� +− 1 +n +n +� +i=1 +� +Φ(d2 �β + wT�κ + �vi�ρ) +� +and construct the confidence interval by bootstrap (Li and Guo, 2020). +Li and Guo (2020) has proposed a more general methodology, named SpotIV, to con- +duct robust causal inference with possibly invalid IVs. The model considered in Li and Guo +(2020) includes the probit outcome model in (16) as a special case. In particular, Li and Guo +(2020) replaced the known probit transformation in (16) with the more general non-parametric +function, which is possibly unknown. Moreover, Li and Guo (2020) allows some instruments +to be correlated with the unmeasured confounders ui in the outcome model. +10 + +RobustIV and controlfunctionIV +4. RobustIV and controlfunctonIV Usage +In this section, we illustrate the basic usage of RobustIV with the data set from Angrist and Krueger +(1991) and simulated high-dimensional data. Also, we use the Mroz data set from Wooldridge +(2010) to demonstrate the usage of controlfunctionIV. +4.1 TSHT and SearchingSampling +In the following, we introduce usages of the R functions TSHT and SearchingSampling +with the data used in Angrist and Krueger (1991). +Angrist and Krueger (1991) studied +the causal effect of the years of education (EDUC) on the log weekly earnings (LWKLYWGE). +Following Angrist and Krueger (1991), we take 30 interactions (QTR120-QTR129, QTR220- +QTR229, QTR320-QTR329) between three quarter-of-birth dummies (QTR1-QTR3) and ten year- +of-birth dummies (YR20-YR29) as the instruments Z. For example, QTR120 is element-wise +product of QTR1 and YR20. Here, the quarter-of-birth dummies are the indicators of whether +the observed person was born in the first, second, and third quarter of the year respectively, +and the year-of-birth dummies are indicators of which year the subject was born from 1940 +to 1949 respectively. We also include the following baseline covariates X: 9 year-of-birth +dummies (YR20-YR28), a race dummy (RACE), a marital status dummy (MARRIED), a dummy +for residence in an SMSA (SMSA), and eight region-of-residence dummies (NEWENG, MIDATL, +ENOCENT, WNOCENT, SOATL, ESOCENT, WSOCENT, MT). We first apply the function TSHT. +R> Y <- as.vector(LWKLYWGE); D <- as.vector(EDUC) +R> Z <- sapply(paste0("QTR", c(seq(120,129), seq(220,229), seq(320,329))), +function(x){get(x)}) +R> X <- cbind(sapply(paste0("YR",seq(20,28)),function(x){get(x)}),RACE, MARRIED, +SMSA, NEWENG, MIDATL, ENOCENT, WNOCENT, SOATL, ESOCENT, WSOCENT, MT) +R> pz <- ncol(Z) +R> out.TSHT <- TSHT(Y=Y,D=D,Z=Z,X=X, +tuning.1st = sqrt(2.01*log(pz)), tuning.2nd = sqrt(2.01*log(pz))) +R> summary(out.TSHT) +betaHat Std.Error CI(2.5%) CI(97.5%) Valid IVs +0.0874 +0.019 +0.0502 +0.1247 +QTR120 QTR121 QTR122 QTR220 QTR222 QTR227 QTR322 +_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ +Detected invalid IVs: QTR126 QTR226 +Here, tuning.1st and tuning.2nd are tuning parameters λ1 and λ2 used for the thresholds +to get �S and �V in Section 2.2 respectively. The default values for these parameters are +√log n in the low-dimensional setting. However, in theory, any value above √2 log p and +diverging to infinity would suffice. Since the data has 486926 observations, we choose the +tuning parameters as √2.01 log pz to avoid too conservative threshold levels due to the +huge sample. Once TSHT is implemented, we can call summary to see the outputs of TSHT +11 + +Koo, Lee, Small, and Guo +including the point estimator, its standard error, confidence interval, and valid IVs as we +discussed in Section 2.2. +The above result shows that TSHT selected QTR120, QTR121, QTR122, QTR220, QTR222, +QTR227, and QTR322 as valid IVs. Thus, valid IVs are interactions with the first quarter +of birth and dummies representing births in 1940, 1941, and 1942, interactions with the +second quarter of birth and dummies representing births in 1940, 1942, and 1947, and +finally the interaction between the third quarter-of-birth and dummy representing births in +1942. On the other hand, it is reported that QTR126 and QTR226 are invalid IVs. That is, +interactions with the first and the second quarter of birth and dummy representing births +in 1946 are relevant but invalid IVs. The remaining IVs have been screened out of the +first-stage selection as individually weak IVs. +The detection of invalid IVs implies that using whole Z as valid IVs can cause the +estimate to be biased. In Angrist and Krueger (1991), the TSLS estimate by using whole Z +as valid IVs is 0.0393. In contrast, our procedure is more robust to the existence of possibly +invalid IVs, giving the causal estimate as 0.0874. Our 95% confidence interval is above zero, +indicating a positive effect of education on earning. +In addition to the above output, the class object TSHT has other values that are not +reported by summary, for example, whether the majority rule is satisfied or not, and the +voting matrix to construct �V in Section 2.2. These can be checked by directly calling TSHT. +As discussed in Section 2.2, there are different voting options to get �V, where the default +option voting = ’MaxClique’ stands for �VMC and voting = ’MP’ stands for �VMP in (7). +If there are several maximum cliques, summary returns results corresponding to each maxi- +mum clique. Furthermore, since the default argument for which estimator to use is method = +’OLS’, one can choose other estimators by method = ’DeLasso’ for the debiased Lasso esti- +mator with SIHR R package (Rakshit, Cai, and Guo, 2021) and method = ’Fast.DeLasso’ +for the fast computation of the debiased Lasso estimator (Javanmard and Montanari, 2014). +The above methods are useful in a high-dimensional setting. +Next, we implement the uniformly valid confidence intervals by calling the function +SearchingSampling. We start with the searching CI defined in (11) with the argument +Sampling = FALSE. +R> out1 = SearchingSampling(Y=Y, D=D, Z=Z, X=X, Sampling=FALSE, +tuning.1st = sqrt(2.01*log(pz)), tuning.2nd = sqrt(2.01*log(pz))) +R> summary(out1) +Confidence Interval for Causal Effect: [-0.0964,0.2274] +With the default argument Sampling = TRUE, one can use the following code to implement +the more efficient sampling CI in (12). +12 + +RobustIV and controlfunctionIV +R> set.seed(1) +R> out.SS = SearchingSampling(Y=Y, D=D, Z=Z, X=X, +tuning.1st = sqrt(2.01*log(pz)), tuning.2nd = sqrt(2.01*log(pz))) +R> summary(out.SS) +Confidence Interval for Causal Effect: [0.0135,0.1775] +The SearchingSampling confidence intervals are generally wider than that of the TSHT +since they are robust to the IV selection errors. The function summary displays confidence +interval for β, which are discussed in Section 2.3. As in TSHT, one can use the argument +method to employ the high-dimensional debiased estimators instead of OLS. +4.2 endo.test +In the following, we show the usage of endo.test, a function for the endogeneity test in high +dimension with a simulated example. The corresponding model and method are presented +in Section 2.4. We consider the models (1) and (2) and set pz = 600 with only the first +10 IVs being relevant. Among these 10 IVs, the first 3 IVs are invalid but the remaining +IVs are valid. Moreover, we set Corr (ǫi, δi) = 0.8, which indicates a level of endogeneity. +The function endo.test generates a class object with same arguments in TSHT. The class +object from endo.test can be used by calling summary function, which enable us to see a +brief result of ento.test. +R> set.seed(5) +R> n = 500; L = 600; s = 3; k = 10; px = 10; epsilonSigma = matrix(c(1,0.8,0.8,1),2,2) +R> beta = 1; gamma = c(rep(1,k),rep(0,L-k)) +R> phi = (1/px)*seq(1,px)+0.5; psi = (1/px)*seq(1,px)+1 +R> Z = matrix(rnorm(n*L),n,L); X = matrix(rnorm(n*px),n,px); +R> epsilon = MASS::mvrnorm(n,rep(0,2),epsilonSigma) +R> D = 0.5 + Z %*% gamma + X %*% psi + epsilon[,1] +R> Y = -0.5 + Z %*% c(rep(1,s),rep(0,L-s)) + D * beta + X %*% phi + epsilon[,2] +R> endo.test.model <- endo.test(Y,D,Z,X, invalid = TRUE) +R> summary(endo.test.model) +P-value Test +Valid IVs +0 +H0 rejected Z4 Z5 Z6 Z7 Z8 Z9 Z10 +_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ +Detected invalid IVs: Z1 Z2 Z3 +When we call summary function, p-value, it reports the test result with significance level +α (default alpha = 0.05), the valid IVs, and detected invalid IVs. H0 rejected means +that the treatment is endogenous, otherwise not. Since we set invalid = TRUE, ento.test +allows some of IVs to be invalid and conducts the endogeneity test with the selected �V +defined in Section 2.2. With invalid = FALSE, the function assumes that all IVs are valid. +13 + +Koo, Lee, Small, and Guo +As in the above sections, one can use method argument to employ other estimators both in +the low and high dimensions. +4.3 cf and pretest +In this section, we introduce usages of cf and pretest in the package controlfunctionIV. +The Mroz data was introduced in Mroz (1987) and then used in various works of literature +including Wooldridge (2010), which has n = 428 individuals after removing the data with +NA. Following Wooldridge (2010), we estimate the causal effect of education on the log +earnings of married working women. The data is available in the Wooldridge package. +Here, the outcome Y is log earnings (lwage), and the exposure D is years of schooling +(educ). Moreover, there are other variables such as the father’s education (fatheduc), the +mother’s education (motheduc), the husband’s education (huseduc), actual labor market +experience (exper), its square (expersq), and the women’s age (age). +Following Example 5.3 in Wooldridge (2010), we assume motheduc, fatheduc, and +huseduc to be valid IVs, denoted as Zi = (Zi1, Zi2, Zi3)T; we use and exper, expersq, +and age as baseline covariates, denoted as Xi = (Xi1, Xi2, Xi3)T. Also assume that the +outcome and treatment models are (13) and (14) respectively with G(Di) = (Di, D2 +i )T and +H(Zi·) = (Zi1, Zi2, Zi3, Z2 +i1, Z2 +i2, Z2 +i3)T. +Then we can implement the cf function by inputting a formula object, which has the +same form as that of ivreg in AER package. The function summary gives us information on +coefficients of the control function estimators, including the point estimator, its standard +error, t value, and p value. +R> library(wooldridge); library(controlfunctionIV); data(mroz); mroz <- na.exclude(mroz) +R> Y <- mroz[,"lwage"]; D <- mroz[,"educ"] +R> Z <- as.matrix(mroz[,c("motheduc","fatheduc","huseduc")]) +R> X <- as.matrix(mroz[,c("exper","expersq","age")]) +R> cf.model <- cf(Y~D+I(D^2)+X|Z+I(Z^2)+X) +R> summary(cf.model) +_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ +Coefficients of the control function estimators: +Estimate +Std.Error t value Pr(>|t|) +(Intercept) +1.2573907 +0.7871438 +1.597 0.055457 . +D +-0.1434395 +0.1102058 +1.302 0.096884 . +I(D^2) +0.0086426 +0.0041004 +2.108 0.017817 * +Xexper +0.0438690 +0.0131574 +3.334 0.000465 *** +Xexpersq +-0.0008713 +0.0003984 +2.187 0.014631 * +Xage +-0.0011636 +0.0048634 +0.239 0.405511 +--- +Signif. codes: +0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 +14 + +RobustIV and controlfunctionIV +The following code infers the causal effect G(d1)Tβ−G(d2)Tβ by changing the treatment +level from d2 to d1 = d2 + 1. Since the second and third coefficients are related to D, we +use the second and third index to get the causal effect and its standard error. +R> d2 = median(D); d1 = median(D)+1; +R> D.diff <- c(d1,d1^2)-c(d2,d2^2); CE <- (D.diff)%*%cf.model$coefficients[c(2,3)] +R> CE.sd <-sqrt(D.diff%*%cf.model$vcov[c(2,3),c(2,3)]%*%D.diff) +R> CE.ci <- c(CE-qnorm(0.975)*CE.sd,CE+qnorm(0.975)*CE.sd) +R> cmat <- cbind(CE,CE.sd,CE.ci[1],CE.ci[2]) +R> colnames(cmat)<-c("Estimate","Std.Error","CI(2.5%)","CI(97.5%)"); rownames(cmat)<- "CE" +R> +print(cmat, digits = 4) +Estimate Std.Error CI(2.5%) CI(97.5%) +CE +0.07263 +0.02171 +0.03007 +0.1152 +The function pretest can be used to choose between the TSLS or the control function +method. If we run pretest with the same argument above and call summary, it will output +the following result: +R> pretest.model <- pretest(Y~D+I(D^2)+X|Z+I(Z^2)+X) +R> summary(pretest.model) +Level 0.05 pretest estimator is control function estimator. +_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ +Coefficients of the pretest estimators: +Estimate +Std.Error t value Pr(>|t|) +(Intercept) +1.2573907 +0.7871438 +1.597 0.055457 . +D +-0.1434395 +0.1102058 +1.302 0.096884 . +I(D^2) +0.0086426 +0.0041004 +2.108 0.017817 * +Xexper +0.0438690 +0.0131574 +3.334 0.000465 *** +Xexpersq +-0.0008713 +0.0003984 +2.187 0.014631 * +Xage +-0.0011636 +0.0048634 +0.239 0.405511 +--- +Signif. codes: +0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 +The first section of the output of summary reports which estimator is chosen after the +pretesting step. The second section lists brief information on coefficients of pretest estima- +tors including the point estimator, its standard error, t value, and p-value, similar to cf. +Since the pretest estimator is the control function estimator, the second section of summary +is the same as that of summary(cf.model). +4.4 Probit.cf +Finally, we conclude the usage part by looking at the usage of Probit.cf, which is designed +for the binary outcome with unmeasured confounders and possibly invalid IVs. For illus- +15 + +Koo, Lee, Small, and Guo +tration, we use the Mroz data and define the binary outcome variable Y0 to take the value +1 if the continuous outcome Y is greater than the median of Y and 0 otherwise. We use the +same treatment variable D as in the cf example. Contrary to the cf example, we set the +candidates of IVs Z as motheduc, fatheduc, huseduc, exper, and expersq, and assume +that we have covariates X as age. +We implement the Probit.cf function to estimate the CATE by increasing the treat- +ment value from the median of D to the median plus one. We can call summary to see the +result of Probit.cf. The function summary provides information on the valid IVs �V, the +point estimator, standard error, and 95% confidence interval for β in (16), and the point +estimator, the standard error, and 95% confidence interval of CATE. +R> Z <- as.matrix(mroz[,c("motheduc","fatheduc","huseduc","exper","expersq")]) +R> Y0 <- as.numeric((Y>median(Y))) +R> d2 = median(D); d1 = d2+1; w0 = apply(cbind(Z,X)[which(D == d2),], 2, mean) +R> Probit.model <- Probit.cf(Y0,D,Z,X,d1 = d1,d2 = d2,w0 = w0) +R> summary(Probit.model) +Estimate Std.Error CI(2.5%) CI(97.5%) Valid IVs +Beta 0.2119 +0.092 +0.0316 +0.3922 +motheduc fatheduc huseduc +CATE 0.0844 +0.033 +0.0198 +0.1489 +motheduc fatheduc huseduc +_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ +No invalid IV is detected +With the option invalid = TRUE, we allow invalid IVs and choose the valid IVs among all +provided IVs. If one wants to assume all IVs are valid, one can set invalid = FALSE. +5. Application to Framingham Heart Study +We analyze the Framingham Heart Study (FHS) data and illustrate our package using ge- +netic variants as IVs. The FHS is an ongoing cohort study of participants from the town of +Framingham, Massachusetts, that has grown over the years to include five cohorts with a +total sample of over 15,000. The FHS, initiated in 1948, is among the most critical sources +of data on cardiovascular epidemiology (Sytkowski, Kannel, and D’Agostino, 1990; Kannel, +2000; Mahmood, Levy, Vasan, and Wang, 2014). Since the late 1980s, researchers across +human health-related fields have used genetic factors underlying cardiovascular diseases +and other disorders. Over the last two decades, DNA has been collected from blood sam- +ples and immortalized cell lines from members of the Original Cohort, the Offspring Cohort, +and the Third Generation Cohort (Govindaraju et al., 2008). Several large-scale genotyping +projects and genome-wide linkage analysis have been conducted, and several other recent +collaborative projects have completed thousands of SNP genotypes for candidate gene re- +gions in subsets of FHS subjects with available DNA. The FHS has recently been used for +Mendelian Randomization to determine causal relationships even in the presence of unmea- +16 + +RobustIV and controlfunctionIV +sured confounding thanks to the availability of genotype and phenotype data (Holmes et al., +2014; Dalbeth et al., 2015; Hughes et al., 2014). As candidate IVs, we will use genotype +data from the FHS associated with the phenotype of interest and apply the proposed meth- +ods described above. +We apply the RobustIV package to investigate the effect of low-density lipoprotein (LDL- +C) on globulin levels among individuals in the Framingham Heart Study (FHS) Offspring +Cohort, as was studied in Kang et al. (2020). +We use eight SNP genotypes (rs646776, +rs693, rs2228671, rs2075650, rs4299376, rs3764261, rs12916, rs2000999) that are known to be +significantly associated with LDL-C measured in mg/dL as candidate IVs (Kathiresan et al., +2007; Ma et al., 2010; Smith et al., 2014). See Table 2 for details. The outcome of interest +Yi is a continuous globulin level (g/L) and the exposure variable Di is the LDL-C level. +Globulin is known to play a crucial role in liver function, clotting, and the immune system. +We also use the age and sex of the subjects as covariates Xi·. The study includes n = 1445 +subjects, with an average globulin level of 27.27 (SD: 3.74) and an average LDL-C of 1.55 +(SD: 0.50). An average age is 35.58 (SD: 9.74) and 54.95% are males. +Zj +SNP +Position +lm(D ∼ Z) +lm(Y ∼ Z) +Estimate (Std. Error) +t-statistic (p-value) +Estimate (Std. Error) +t-statistic (p-value) +Z1 +rs646776 +chr1:109275908 +-5.160 (1.610) +-3.205 (0.001) +-0.001 (0.170) +-0.007 (0.994) +Z2 +rs693 +chr2:21009323 +-3.600 (1.286) +-2.799 (0.005) +0.318 (0.135) +2.349 (0.019) +Z3 +rs2228671 +chr19:11100236 +7.138 (2.029) +3.518 (<0.001) +0.529 (0.214) +2.474 (0.014) +Z4 +rs2075650 +chr19:44892362 +8.451 (2.021) +4.183 (<0.001) +0.471 (0.213) +2.208 (0.027) +Z5 +rs4299376 +chr2:43845437 +3.847 (1.387) +2.773 (0.006) +0.110 (0.146) +0.752(0.452) +Z6 +rs3764261 +chr16:56959412 +3.651 (1.429) +2.555 (0.011) +0.275 (0.151) +1.829 (0.067) +Z7 +rs12916 +chr5:75360714 +3.363 (1.365) +2.463 (0.014) +-0.195 (0.144) +-1.357 (0.175) +Z8 +rs2000999 +chr16:72074194 +-2.961 (1.629) +-1.818 (0.069) +-0.119 (0.172) +-0.691 (0.489) +Table 2: Summary of the relationship between the single nucleotide polymorphisms (SNPs) +and low-density lipoprotein. The point estimator, its standard error, t value, and +p-value are summary statistics from running a marginal regression model specified +in the column title. Position refers to the position of the SNP in the chromosome, +denoted as chr. +By applying endo.test, we detect one invalid IV and observe the evidence for the +existence of unmeasured confounders since the null hypothesis H0 : σ12 = 0 is rejected. +R> pz <- ncol(Z) +R> globulin.endo2 <- endo.test(Y,D,Z,X, invalid = TRUE, +tuning.1st = sqrt(2.01*log(pz)), tuning.2nd = sqrt(2.01*log(pz))) +R> summary(globulin.endo2) +P-value Test +Valid IVs +0.0091 +H0 rejected Z.1 Z.3 Z.4 Z.5 Z.6 Z.8 +17 + +Koo, Lee, Small, and Guo +_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ +Detected invalid IVs: Z.2 +Next, we implement TSHT with the default method of "OLS" under the low-dimensional +setting. Again, the same invalid IV is detected and the confidence interval is above zero, +indicating a positive effect of LDL on the glucose level. +R> pz <- ncol(Z) +R> TSHT2 <- TSHT(Y, D, Z, X, +tuning.1st = sqrt(2.01*log(pz)), tuning.2nd = sqrt(2.01*log(pz))) +R> summary(TSHT2) +betaHat Std.Error CI(2.5%) CI(97.5%) Valid IVs +0.0529 +0.0146 +0.0243 +0.0814 +Z.1 Z.3 Z.4 Z.5 Z.6 Z.8 +_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ +Detected invalid IVs: Z.2 +We also constructed the confidence interval using the searching method, which provides +robustness to the IV selection errors. +R> SS1 <- SearchingSampling(Y, D, Z, X, tuning.1st = sqrt(2.01*log(pz)), +tuning.2nd = sqrt(2.01*log(pz)), Sampling = FALSE) +R> summary(SS1) +Confidence Interval for Causal Effect: [-0.2427,0.1894] +We further implement the sampling method, which leads to a shorter uniformly valid CI +than the searching method. +R> SS2 <- SearchingSampling(Y, D, Z, X, tuning.1st = sqrt(2.01*log(pz)), +tuning.2nd = sqrt(2.01*log(pz)), Sampling = TRUE) +R> summary(SS2) +Confidence Interval for Causal Effect: [-0.0521,0.1259] +In the following, we study nonlinear causal relationships using the controlfunctionIV +package. Burgess, Davies, and Thompson (2014) investigated a nonlinear causal relation- +ship between BMI and diverse cardiovascular risk factors. Here we examine BMI’s possibly +nonlinear causal effect on the insulin level. Among n = 3733 subjects, we excluded 618 +subjects with missing information on insulin level, and 50 subjects whose insulin level is +greater than 300pmol/L and whose BMI is greater than 45kg/m2. We use log-transformed +insulin as the outcome of interest Yi measured at Exam 2. The exposure Di denotes the BMI +measures at Exam 1. The covariates Xi· that we adjusted for are age and sex. As valid IVs +Zi·, we propose using four SNP genotypes known to be significantly associated with obesity. +In our analysis, we include I(D^2) and I(X^2) to account for quadratic effects of BMI, age, +and sex on the outcome. We also include I(Z^2) to account for possible quadratic effects +of SNPs on the exposure. The result from the pretest estimator is as follows: +18 + +RobustIV and controlfunctionIV +R> insulin.pretest = pretest( Y ~ D + I(D^2) + X ++ I(X^2) | Z + I(Z^2) + X + I(X^2)) +R> summary(insulin.pretest) +Level 0.05 pretest estimator is control function estimator. +_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ +Coefficients of Pretest Estimators: +Estimate +Std.Err t value Pr(>|t|) +(Intercept) +2.674e+00 +6.006e-01 +4.453 4.39e-06 *** +D +8.295e-02 +2.828e-02 +2.933 0.001690 ** +I(D^2) +-7.784e-04 +2.742e-04 +2.839 0.002276 ** +X1 +-1.780e-02 +7.816e-03 +2.277 0.011427 * +I(X1^2) +2.954e-04 +8.852e-05 +3.337 0.000428 *** +X2 +-1.361e-01 +5.654e-02 +2.406 0.008087 ** +--- +Signif. codes: +0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 +The pretest estimator chooses the control function over the standard TSLS. The results also +show that BMI has a positive linear effect on the outcome but a negative quadratic effect +on the outcome. +Acknowledgement +The research of T. Koo was supported in part by NIH grants R01GM140463 and R01LM013614. +The research of D. Small was supported in part by NIH grant 5R01AG065276-02.The re- +search of Z. Guo was partly supported by the NSF grants DMS 1811857 and 2015373 and +NIH grants R01GM140463 and R01LM013614. Z. Guo is grateful to Dr. Frank Windmeijer +for bringing up the maximum clique method. +The Framingham Heart Study is conducted and supported by the National Heart, Lung, +and Blood Institute (NHLBI) in collaboration with Boston University (Contract No. N01- +HC-25195, HHSN268201500001I, and 75N92019D00031). This manuscript was not prepared +in collaboration with investigators of the Framingham Heart Study and does not necessarily +reflect the opinions or views of the Framingham Heart Study, Boston University, or NHLBI. +Funding for SHARe Affymetrix genotyping was provided by NHLBI Contract N02-HL64278. +SHARe Illumina genotyping was provided under an agreement between Illumina and Boston +University. 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Econometrica, 41(4):733–750, 1973. +23 + diff --git a/49E3T4oBgHgl3EQfQQn6/content/tmp_files/load_file.txt b/49E3T4oBgHgl3EQfQQn6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8109c4f58625511729879d4bb6b83eb90069873e --- /dev/null +++ b/49E3T4oBgHgl3EQfQQn6/content/tmp_files/load_file.txt @@ -0,0 +1,1111 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf,len=1110 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='04412v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='ME] 11 Jan 2023 Observational Studies (2023) Submitted ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Published RobustIV and controlfunctionIV: Causal Inference for Linear and Nonlinear Models with Invalid Instrumental Variables Taehyeon Koo tk587@stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='rutgers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='edu Department of Statistics Rutgers University Piscataway, NJ 08854 Youjin Lee youjin lee@brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='edu Department of Biostatistics Brown University Providence, RI 02912 Dylan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Small dsmall@wharton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='edu Department of Statistics The Wharton School, University of Pennsylvania Philadelphia, PA 19104 Zijian Guo zijguo@stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='rutgers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='edu Department of Statistics Rutgers University Piscataway, NJ 08854 Abstract We present R software packages RobustIV and controlfunctionIV for causal inference with possibly invalid instrumental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' RobustIV focuses on the linear outcome model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' It implements the two-stage hard thresholding method to select valid instrumental variables from a set of candidate instrumental variables and make inferences for the causal effect in both low- and high-dimensional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Furthermore, RobustIV implements the high- dimensional endogeneity test and the searching and sampling method, a uniformly valid inference method robust to errors in instrumental variable selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' controlfunctionIV considers the nonlinear outcome model and makes inferences about the causal effect based on the control function method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Our packages are demonstrated using two publicly avail- able economic data sets together with applications to the Framingham Heart Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Keywords: Instrumental variable selection, confidence interval, nonlinear outcome model, control function, maximum clique ©2023 Taehyeon Koo, Youjin Lee, Dylan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Small, and Zijian Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Koo, Lee, Small, and Guo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Introduction A common problem in making causal inferences from observational studies is that there may be unmeasured confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The instrumental variable (IV) method is one of the most use- ful methods to estimate the causal effect when there might exist unmeasured confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The validity of IV methods relies on that the constructed IVs satisfy the following three assumptions simultaneously (Wooldridge, 2010, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' ): conditioning the measured covariates, (A1) the IVs are associated with the treatment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (A2) the IVs are independent with the unmeasured confounders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (A3) the IVs have no direct effect on the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The main challenge of applying IV-based methods in practice is that the proposed IVs might not satisfy the above assumptions (A1)-(A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' For example, in studying the causal effect of education on earning, the proximity of school (Angrist and Krueger, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Card, 1999) has been used as an instrumental variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' However, this instrument might be re- lated to other factors, such as socioeconomic status, which could affect one’s earnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Also, there might be other advantages due to the proximity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' for instance, people living close to college could be more likely to be exposed to vocational programs linked to col- leges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' So, the instrument could have a direct effect on earnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In addition, the problem of IVs not satisfying assumptions (A1) to (A3) is a fundamental problem in Mendelian Randomization (MR), whose goal is to estimate the causal effect of exposure on the disease by using genetic variants as instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' These genetic variants might violate assump- tions (A2) and (A3) due to pleiotropic effects (Bowden, Davey Smith, and Burgess, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Bowden, Davey Smith, Haycock, and Burgess, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Kang, Zhang, Cai, and Small, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' This paper presents the R packages RobustIV and controlfunctionIV, implementing robust causal inference approaches proposed in Guo, Kang, Cai, and Small (2018a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Guo (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Guo and Small (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Li and Guo (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The implemented inference methods choose the valid IVs among a set of candidate IVs that may violate the assumptions (A2) and (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The proposed methods target both linear and nonlinear causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We also include the algorithm implementation for settings with high-dimensional covariates and IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In the package RobustIV, we implement robust and high-dimensional IV algorithms for models assuming a constant and linear treatment effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We implement the Two Stage Hard Thresholding (TSHT) proposed in Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2018b), which selects valid IVs based on a voting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The selected IVs are then used to infer the linear treatment effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Addi- tionally, RobustIV implements uniformly valid confidence intervals proposed in Guo (2021), which guarantees valid coverage even if there are errors in selecting valid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' RobustIV also contains the high-dimensional endogeneity test proposed in Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2018a), generalizing the Durbin-Wu-Hausman test (Durbin, 1954;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Wu, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Hausman, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 2 RobustIV and controlfunctionIV We implement several control function methods in the package controlfunctionIV to infer causal effects under nonlinear outcome models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We implement the control func- tion for the continuous outcome variable by showing it as the two-stage least squares (TSLS) estimator with an augmented set of IVs (Guo and Small, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We further follow Guo and Small (2016) to test the validity of the augmented set of IVs and construct the pretest estimator by comparing the control function estimator and the TSLS estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We implement the probit control function method for the binary outcome and make inferences for the conditional average treatment effect (CATE) with possibly invalid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Moreover, the controlfunctionIV package implements the SpotIV method proposed in Li and Guo (2020) for the semi-parametric outcome model with possibly invalid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In R, there are well-developed IV methods when all IVs are assumed to satisfy the assumptions (A1) to (A3), such as AER by Kleiber and Zeileis (2008) and ivmodel by Kang, Jiang, Zhao, and Small (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The main difference in our packages RobustIV and controlfunctionIV is that we allow for invalid IVs and leverage the multiple IVs to learn the validity of the candidate IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We shall mention other R packages implementing causal inference approaches with possibly invalid IVs: sisvive implemented the method proposed in Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2016) to estimate the treatment effect under the majority rule;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' CIIV by Windmeijer, Liang, Hartwig, and Bowden (2021) considered the causal inference approaches with possibly invalid IVs for the low-dimensional linear outcome model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In con- trast, our packages RobustIV and controlfunctionIV are designed under a broader frame- work by allowing for linear and nonlinear outcome models with low- and high-dimensional IVs and covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Moreover, our package RobustIV provides a uniformly valid confidence interval robust to the errors in separating valid and invalid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The GitHub repository at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='com/bluosun/MR-GENIUS implemented the MR Genius method (Tchetgen, Sun, and Walter, 2021), generalizing the method in Lewbel (2012) and leveraging the heteroscedastic regression errors in the treatment model to identify the causal parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The R package TSCI implemented the two-stage curvature identifica- tion method proposed in Guo and B¨uhlmann (2022), which leveraged the machine learning methods to capture the nonlinearity in the treatment and identify the treatment effect with possibly invalid instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In contrast, our packages use the different identification conditions from Tchetgen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Lewbel (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Guo and B¨uhlmann (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In Section 2, we review the methods implemented in RobustIV under the linear outcome model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' in Section 3, we discuss the inference approaches for the nonlinear outcome models implemented in controlfunctionIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In Section 4, we demonstrate the usage of RobustIV and controlfunctionIV by analyzing economics data sets from Angrist and Krueger (1991) and Mroz data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In Section 5, we demonstrate our packages with an MR application to analyze the data from Framingham Heart Study (FHS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 3 Koo, Lee, Small, and Guo Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Let Rp be the set of real numbers with dimension p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' For any vector v ∈ Rp, vj denotes its jth element, v−j denotes whole v except for j-th index, and ∥v∥0 denotes the number of non-zero elements in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' For any n × p matrix M, denote the (i, j) entry by Mij, the ith row by Mi· , the jth column by M·j, and the transpose of M by MT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' also, MIJ denotes the submatrix of M consisting of rows specified by the set I ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', n} and columns specified by the set J ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', p}, MI· denotes the submatrix of M consisting of rows indexed by the set I and all columns, and M·J denotes the submatrix of M consisting of columns specified by the set J and all rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Ip denotes p × p identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 1 denotes the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Φ denotes the CDF of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' For a sequence of random variable Xn, we use Xn d→ X to denote that Xn converges to X in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Linear outcome models Throughout the paper, we consider n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' For 1 ≤ i ≤ n, let Yi ∈ R, Di ∈ R, Zi· ∈ Rpz, and Xi· ∈ Rpx denote the outcome, the treatment, the instruments, and the baseline covariates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' This section reviews the robust instrumental variable approaches in Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2018a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Guo (2021), which are implemented in the RobustIV package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We demonstrate the usage of RobustIV in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1 Model assumption We assume the following outcome model with possibly invalid IVs (Small, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Windmeijer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2021) Yi = Diβ + ZT i·π + XT i·φ + ǫi, E[ǫiZi·] = 0, E[ǫiXi·] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (1) This is the linear structural model in econometrics (Wooldridge, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Here, we aim to estimate the constant causal effect β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' If Di is correlated with ǫi in the model (1), we say it is an endogenous variable, and we cannot use popular estimators such as the OLS estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We also assume the linear association model for the treatment Di = ZT i·γ + XT i·ψ + δi, E[δiZi·] = 0, E[δiXi·] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2) As a remark, the errors in (1) and (2) are allowed to be heteroscedastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In (1) and (2), πj = 0 if j-th IV satisfies the exclusion restriction conditions (A2) and (A3), and γj ̸= 0 if it satisfies the strong IV assumption (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We discuss the causal interpretation of the above model (1) using the potential outcome framework (Small, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Let Y (d,z) i be the potential outcome if individual i were to receive the treatment d and the instruments z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' For two possible values of the treatment d′, d and instruments z′, z, if we assume the following potential outcomes model Y (d′,z′) i − Y (d,z) i = (d′ − d)β + (z′ − z)Tκ, E[Y (0,0) i |Zi·, Xi·] = XT i·φ + ZT i·η, (3) 4 RobustIV and controlfunctionIV and define π = κ + η, and ǫi = Y (0,0) i − E[Y (0,0) i |Zi·, Xi·], we obtain the model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' By combining (1) and (2), we obtain the reduced form models of Y and D as Yi = ZT i·Γ + XT i·Ψ + ξi, E[ξiZi·] = 0, E[ξiXi·] = 0, (4) Di = ZT i·γ + XT i·ψ + δi, E[δiZi·] = 0, E[δiXi·] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (5) Here, Γ = βγ +π, Ψ = βψ +φ are reduced form parameters and ξi = βδi +ǫi is the reduced form error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We introduce identifiability conditions for models (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Let S be the set of relevant IVs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', S = {1 ≤ j ≤ pz : γj ̸= 0} and V be the set of relevant and valid IVs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', V = {j ∈ S : πj = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The set S contains all candidate IVs that are strongly associated with the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The set V is a subset of S, which contains all candidate IVs satisfying all classical IV assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The main challenge is that the set V is not known a priori in the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Additional identifiability conditions are needed for identifying the causal effect without any prior knowledge of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The majority rule is introduced to identify causal effects with invalid IVs (Bowden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Condition 1 (Majority Rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' More than half of the relevant IVs are valid: |V| > |S|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The following plurality rule is a weaker identification condition than the majority rule (Hartwig, Davey Smith, and Bowden, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Guo, Kang, Cai, and Small, 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Condition 2 (Plurality Rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The valid instruments form a plurality compared to the invalid instruments: |V| > maxc̸=0 |{j ∈ S : πj/γj = c}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We present two inference methods for β utilizing the majority and plurality: two stage hard thresholding in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2 and searching and sampling in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' To present the methods, we consider the reduced form estimators (�Γ⊺, �γ⊺)⊺ satisfying √n ���Γ �γ � − � Γ γ �� d→ N2pz � 02pz, � VΓ C CT Vγ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (6) We use �VΓ, �C, and �Vγ to denote consistent estimators of asymptotic covariance matrix terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In low dimensions, we estimate the reduced form (Γ⊺, γ⊺)⊺ by the OLS estimator (�Γ⊺, �γ⊺)⊺ and estimate the variance covariance matrices by sandwich estimators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' see the detailed construction in Section 2 of Guo (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In high-dimensional settings, we can con- struct (�Γ⊺, �γ⊺)⊺ as the debiased Lasso estimator (Belloni, Chernozhukov, and Wang, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Javanmard and Montanari, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Guo, Kang, Cai, and Small, 2018b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' see more details in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1 of Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 5 Koo, Lee, Small, and Guo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2 Two stage hard thresholding (TSHT) The TSHT consists of two steps: the first step is to screen out the weak IVs, and the second step is to screen out invalid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Specifically, the first step of TSHT is to estimate the set S of relevant IVs by �S = � 1 ≤ j ≤ pz : |�γj| ≥ λ1 � �Vγ jj/n � , where λ1 > 0 is a tuning parameter adjusting the testing multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The second thresholding step estimates the set V of valid instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Our main strategy is to assume that one IV is valid and evaluate whether the other IVs are valid from the point of view of that IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Particularly, for j ∈ �S, we assume the j-th IV to be valid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', πj = 0) and construct an estimator �π−j of π−j using the equation π−j = Γ−j − β[j]γ−j with β[j] = Γj/γj, and get the standard error of the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We test whether π−j = 0 by comparing �π−j to a threshold, calculated as multiplying the standard error of �π−j by a tuning parameter λ2, which is a Bonferroni correction adjusting for testing multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Using the above test procedures, we construct a voting matrix ˜Π ∈ R| �S|×| � S| where ˜Πj,k = 1 indicates that the k-th and j-th IVs agree with each other to be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Finally, we get a symmetric voting matrix �Π by setting �Πj,k = min{˜Πj,k, ˜Πk,j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Once we get �Π, we estimate V by two options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Let VMk denote the number of votes that the kth IV, with k ∈ �S, received from other candidates of IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' First, we define �V by the set of IVs that receive a majority and a plurality of votes (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2018b) �VMP := {k ∈ �S : VMk > | �S|/2} ∪ {k ∈ �S : VMk = max l∈ � S VMl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (7) The next method is to estimate V by the maximum clique method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We can generate a graph G with indexes belonging to �S and the adjacency matrix as �Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' That is, the indexes j, k ∈ �S are connected if and only if �Πj,k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Then as suggested in Windmeijer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2021), we can estimate �VMC as the maximum clique of the graph G, which is the largest fully connected sub- graph of G (Csardi and Nepusz, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Note that there might be several maximum cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In this case, each maximum clique forms an estimator of V and our proposal reports several causal effect estimators based on each maximum clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We further illustrate the definitions of �VMP and �VMC using the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Consider pz = 8 with {z1, z2, z3, z4} being valid and {z5, z6, z7} being invalid with the same invalidity level, and z8 being invalid IV with a different invalidity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The left side of Table 1 corresponds to an ideal setting where the valid and invalid IVs are well separated and the valid IVs {z1, z2, z3, z4} only vote for each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In this case, �VMC = �VMP = {z1, z2, z3, z4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' On the right side of Table 1, we consider the setting that the invalidity level of z5 might be mild and the IV z5 receives the votes from three valid IVs {z2, z3, z4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In this case, �VMP = {z2, z3, z4, z5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In contrast, there are two maximum cliques {z1, z2, z3, z4} and {z2, z3, z4, z5} and �VMC can be either of these two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='RobustIV and controlfunctionIV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='z1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='z2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='z7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='z8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='Votes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='Table 1: The left voting matrix �Π denotes that all valid IVs {z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' z3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' z4} vote each other but not any other invalid IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The right voting matrix �Π denotes that the locally invalid IV z5 receives votes from valid IVs {z2, z3, z4} and invalid IVs {z6, z7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Once we have �V, we can construct an efficient point estimator �β for β in a low- dimensional setting via one-step iteration as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' First, we construct an initial estimator ˜β = �γT �V ˜ A�Γ�V �γT �V ˜ A�γ�V , where ˜A = �Σ�V,�V −�Σ�V,�Vc �Σ−1 �Vc,�Vc �Σ�Vc,�V, �Σ = 1 n �n i=1 Wi·WT i· , and Wi· = (ZT i·, XT i·)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Next, we get a point estimator �β by one-step iteration (Holland and Welsch, 1977) �β = �γT �V �A�Γ�V �γT �V �A�γ�V , where �A = [( �VΓ − 2˜β �C + ˜β2 �Vγ)�V,�V]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (8) Finally, the 1 − α confidence interval for β is (�β − z1−α/2 � SE, �β + z1−α/2 � SE) where � SE = � � � ��γT �V �A( �VΓ − 2�β �C + �β2 �Vγ)�V,�V �A�γ�V n(�γT �V �A�γ�V)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (9) As a remark, �VΓ, �Vγ, and �C are heteroscedasticity-robust covariance estimators and hence (9) is also robust to heteroscedastic errors in a low-dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In a high-dimensional setting, we set �A = I in (8), and �VΓ, �Vγ, and �C are constructed under the homoscedastic error assumptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' see more details in Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='3 Searching and Sampling We now review the searching and sampling method proposed in Guo (2021), which provides uniformly valid conference intervals even if there are errors in separating valid and invalid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The right-hand side of Table 1 illustrates an example of the invalid IVs not being separated from valid IVs in finite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In the following, we review the idea of searching 7 Koo, Lee, Small, and Guo and sampling under the majority rule and the more general method with the plurality rule can be found in Guo (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Let α ∈ (0, 1) denote the pre-specified significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Given β ∈ R and the reduced form estimator �Γ and �γ, we estimate πj with j ∈ �S by �πj(β) = (�Γj − β�γj)1(|�Γj − β�γj| ≥ �ρj(β, α)), (10) where �ρj(β, α) = Φ−1 � 1 − α 2| � S| � � SE(�Γj − β�γj) with � SE(�Γj − β�γj) denoting a consistent estimator of the standard error of �Γj − β�γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We search for the value of β leading to enough valid IVs and construct the searching confidence interval as CIsearch = � β ∈ R : ���π � S(β) �� 0 < | �S|/2 � , (11) which collects all β values such that more than half of IVs in �S are selected as valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Based on the searching method, Guo (2021) proposed a sampling confidence interval, which retains the uniform coverage property and improves the precision of the confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In particular, we sample ��Γ[m] �γ[m] � iid ∼ N ���Γ �γ � , 1 n � �VΓ �C �CT �Vγ �� , for 1 ≤ m ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' For 1 ≤ m ≤ M and j ∈ �S, we modify (10) and define �π[m] j (β, λ) = (�Γ[m] j − β�γ[m] j )1(|�Γ[m] j − β�γ[m] j | ≥ λ · �ρj(β, α)) with the shrinkage parameter λ ≍ (log n/M) 1 2| � S| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' A data-dependent way of choosing λ can be found in Remark 3 of Guo (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' For each 1 ≤ m ≤ M, we construct a searching interval (β[m] min, β[m] max) where β[m] min = min β∈B[m] λ β and β[m] max = max β∈B[m] λ β with B[m] λ = � β ∈ R : ����π[m] � S (β, λ) ��� 0 < | �S|/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Then the sampling CI is defined as CIsample = � min m∈M β[m] min, max m∈M β[m] max � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (12) with M = {1 ≤ m ≤ M : (β[m] min, β[m] max) ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The sampling confidence intervals in general improve the precision of the searching confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' But both intervals can provide uniformly valid coverage robust to the errors in separating valid and invalid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 8 RobustIV and controlfunctionIV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='4 Endogeneity test in high dimensions We review the high-dimensional endogeneity test proposed in Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We focus on the homoscedastic error setting by writing Θ11 = Var[ξi|Zi·, Xi·], Θ22 = Var[δi|Zi·, Xi·], and Θ12 = Cov[ξi, δi|Zi·, Xi·] for the reduced form models (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' With the same estimators from TSHT in the high-dimensional settings in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2, we can estimate the covariance σ12 by �σ12 = �Θ12 − �β �Θ22 where �β = � j∈ �V �γj�Γj � j∈ �V �γ2 j and �Θ12 and �Θ22 are consistent estimators of the reduced form covariance Θ22 and Θ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We establish the asymptotic nor- mality of �σ12 − σ12 in Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2018a) and propose a testing procedure for H0 : σ12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Nonlinear outcome models This section reviews the control function IV methods (Guo and Small, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Li and Guo, 2020) implemented in the controlfunctionIV package, whose usage is demonstrated in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1 Control function and pretest estimators We consider the following nonlinear outcome and treatment models: Yi = G(Di)Tβ + XT i·φ + ui, E[uiZi·] = E[uiXi·] = 0, (13) Di = H(Zi·)Tγ + XT i·ψ + vi, E[viZi·] = E[viXi·] = 0, (14) where G(Di) = (Di, g2(Di), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', gk(Di))T, H(Zi·) = (Zi·, h2(Zi·), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', hk(Zi·))T with {gj(·)}2≤j≤k and {hj(·)}2≤j≤k denoting the known nonlinear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Under the models (13) and (14), the IVs are assumed to be valid and the causal effect of increasing the value of D from d2 to d1 is defined as G(d1)Tβ − G(d2)Tβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The control function (CF) method is a two-stage procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In the first stage, regress D on H(Z) and X, and obtain the predicted value �D and its associated residual �v = D− �D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In the second stage, we use �v as the proxies for the unmeasured confounders and regress Y on G(D), X, and �v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We use �βCF to denote the estimated regression coefficient corresponding to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Guo and Small (2016) showed that �βCF is equivalent to the TSLS estimator with the augmented set of IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Even if all IVs satisfy the classical assumptions (A1)-(A3), there is no guarantee of the validity of the augmented IVs generated by the CF estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Guo and Small (2016) applied the Hausman test to assess the validity of the augmented set of IVs generated by the CF estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The test statistic is defined as H(�βCF, �βTSLS) = (�βCF − �βTSLS)T[Cov(�βTSLS) − Cov(�βCF)]−(�βCF − �βTSLS), (15) where �βTSLS is the two stage least square estimator, Cov(�βTSLS) and Cov(�βCF) are the covariance matrices of �βTSLS and �βCF, and A− denote the Moore-Penrose pseudoinverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 9 Koo, Lee, Small, and Guo If the p-value P � χ2 1 ≥ H(�βCF, �βTSLS) � is less than α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='05, then we define the level α pretest estimator �βPretest as �βTSLS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' otherwise, �βCF defined above (Guo and Small, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2 Probit CF and SpotIV We now consider the binary outcome model and continuous treatment model, E [Yi|Di = d, Wi· = w, ui = u] = 1(dβ + wTκ + u > 0), and Di = WT i· γ + vi, (16) where Wi· = (ZT i·, XT i·)T, the errors (ui, vi)⊺ are bivariate normal random variables with zero means and independent of Wi·, κ = (κT z , κT x)T is the coefficient vector of the IVs and measured covariates, and γ = (γT z , γT x)T is a parameter representing the association between Di and Wi·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' When κz ̸= 0, the instruments are invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Since ui and vi are bivariate normal, we write ui = ρvi +ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The model (16) implies E [Yi|Wi·, vi] = Φ(Diβ∗ +W ⊺ i·Γ∗ +ρ∗vi) where β∗ = β/σe, Γ∗ = κ/σe + β∗ · γ, and ρ∗ = ρ/σe + β∗ with σe denoting the standard error of ei = ui − ρvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' That is, the conditional outcome model of Yi given Wi,· and vi is a probit regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Our goal is to estimate the conditional average treatment effect (CATE) from d2 to d1 CATE(d1, d2|w) := E[Yi|Di = d1, Wi· = w] − E[Yi|Di = d2, Wi· = w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (17) We first construct the OLS estimator �γ of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We compute its residual �v = D − W�γ and define �Σ = 1 n �n i=1 Wi·WT i· .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We estimate S = {1 ≤ j ≤ pz : (γz)j ̸= 0} by �S = � 1 ≤ j ≤ pz : |�γj| ≥ �σv � 2{�Σ−1}j,j log n/n � (18) with �σ2 v = �n i=1 �v2 i /n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Next, as CF in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1, we use �v as the proxy for unmeasured con- founders and implement the probit regression Y on W and �v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We use �Γ and �ρ to denote the probit regression coefficients of W of �v respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We apply the majority rule and compute �β as the median of (�Γj/�γj)j∈ �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We then estimate �κ = �Γ − �γ �β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Finally, we estimate CATE defined in (17) by the partial mean method (Newey, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Mammen, Rothe, and Schienle, 2012), 1 n n � i=1 � Φ(d1 �β + wT�κ + �vi�ρ) � − 1 n n � i=1 � Φ(d2 �β + wT�κ + �vi�ρ) � and construct the confidence interval by bootstrap (Li and Guo, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Li and Guo (2020) has proposed a more general methodology, named SpotIV, to con- duct robust causal inference with possibly invalid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The model considered in Li and Guo (2020) includes the probit outcome model in (16) as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In particular, Li and Guo (2020) replaced the known probit transformation in (16) with the more general non-parametric function, which is possibly unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Moreover, Li and Guo (2020) allows some instruments to be correlated with the unmeasured confounders ui in the outcome model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 10 RobustIV and controlfunctionIV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' RobustIV and controlfunctonIV Usage In this section, we illustrate the basic usage of RobustIV with the data set from Angrist and Krueger (1991) and simulated high-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Also, we use the Mroz data set from Wooldridge (2010) to demonstrate the usage of controlfunctionIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1 TSHT and SearchingSampling In the following, we introduce usages of the R functions TSHT and SearchingSampling with the data used in Angrist and Krueger (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Angrist and Krueger (1991) studied the causal effect of the years of education (EDUC) on the log weekly earnings (LWKLYWGE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Following Angrist and Krueger (1991), we take 30 interactions (QTR120-QTR129, QTR220- QTR229, QTR320-QTR329) between three quarter-of-birth dummies (QTR1-QTR3) and ten year- of-birth dummies (YR20-YR29) as the instruments Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' For example, QTR120 is element-wise product of QTR1 and YR20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Here, the quarter-of-birth dummies are the indicators of whether the observed person was born in the first, second, and third quarter of the year respectively, and the year-of-birth dummies are indicators of which year the subject was born from 1940 to 1949 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We also include the following baseline covariates X: 9 year-of-birth dummies (YR20-YR28), a race dummy (RACE), a marital status dummy (MARRIED), a dummy for residence in an SMSA (SMSA), and eight region-of-residence dummies (NEWENG, MIDATL, ENOCENT, WNOCENT, SOATL, ESOCENT, WSOCENT, MT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We first apply the function TSHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> Y <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='vector(LWKLYWGE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' D <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='vector(EDUC) R> Z <- sapply(paste0("QTR", c(seq(120,129), seq(220,229), seq(320,329))), function(x){get(x)}) R> X <- cbind(sapply(paste0("YR",seq(20,28)),function(x){get(x)}),RACE, MARRIED, SMSA, NEWENG, MIDATL, ENOCENT, WNOCENT, SOATL, ESOCENT, WSOCENT, MT) R> pz <- ncol(Z) R> out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='TSHT <- TSHT(Y=Y,D=D,Z=Z,X=X, tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1st = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz)), tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2nd = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz))) R> summary(out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='TSHT) betaHat Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='Error CI(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5%) CI(97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5%) Valid IVs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1247 QTR120 QTR121 QTR122 QTR220 QTR222 QTR227 QTR322 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Detected invalid IVs: QTR126 QTR226 Here, tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1st and tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2nd are tuning parameters λ1 and λ2 used for the thresholds to get �S and �V in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The default values for these parameters are √log n in the low-dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' However, in theory, any value above √2 log p and diverging to infinity would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Since the data has 486926 observations, we choose the tuning parameters as √2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01 log pz to avoid too conservative threshold levels due to the huge sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Once TSHT is implemented, we can call summary to see the outputs of TSHT 11 Koo, Lee, Small, and Guo including the point estimator, its standard error, confidence interval, and valid IVs as we discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The above result shows that TSHT selected QTR120, QTR121, QTR122, QTR220, QTR222, QTR227, and QTR322 as valid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Thus, valid IVs are interactions with the first quarter of birth and dummies representing births in 1940, 1941, and 1942, interactions with the second quarter of birth and dummies representing births in 1940, 1942, and 1947, and finally the interaction between the third quarter-of-birth and dummy representing births in 1942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' On the other hand, it is reported that QTR126 and QTR226 are invalid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' That is, interactions with the first and the second quarter of birth and dummy representing births in 1946 are relevant but invalid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The remaining IVs have been screened out of the first-stage selection as individually weak IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The detection of invalid IVs implies that using whole Z as valid IVs can cause the estimate to be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In Angrist and Krueger (1991), the TSLS estimate by using whole Z as valid IVs is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In contrast, our procedure is more robust to the existence of possibly invalid IVs, giving the causal estimate as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Our 95% confidence interval is above zero, indicating a positive effect of education on earning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In addition to the above output, the class object TSHT has other values that are not reported by summary, for example, whether the majority rule is satisfied or not, and the voting matrix to construct �V in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' These can be checked by directly calling TSHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2, there are different voting options to get �V, where the default option voting = ’MaxClique’ stands for �VMC and voting = ’MP’ stands for �VMP in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' If there are several maximum cliques, summary returns results corresponding to each maxi- mum clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Furthermore, since the default argument for which estimator to use is method = ’OLS’, one can choose other estimators by method = ’DeLasso’ for the debiased Lasso esti- mator with SIHR R package (Rakshit, Cai, and Guo, 2021) and method = ’Fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='DeLasso’ for the fast computation of the debiased Lasso estimator (Javanmard and Montanari, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The above methods are useful in a high-dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Next, we implement the uniformly valid confidence intervals by calling the function SearchingSampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We start with the searching CI defined in (11) with the argument Sampling = FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> out1 = SearchingSampling(Y=Y, D=D, Z=Z, X=X, Sampling=FALSE, tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1st = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz)), tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2nd = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz))) R> summary(out1) Confidence Interval for Causal Effect: [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0964,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2274] With the default argument Sampling = TRUE, one can use the following code to implement the more efficient sampling CI in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 12 RobustIV and controlfunctionIV R> set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='seed(1) R> out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='SS = SearchingSampling(Y=Y, D=D, Z=Z, X=X, tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1st = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz)), tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2nd = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz))) R> summary(out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='SS) Confidence Interval for Causal Effect: [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0135,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1775] The SearchingSampling confidence intervals are generally wider than that of the TSHT since they are robust to the IV selection errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The function summary displays confidence interval for β, which are discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' As in TSHT, one can use the argument method to employ the high-dimensional debiased estimators instead of OLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2 endo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='test In the following, we show the usage of endo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='test, a function for the endogeneity test in high dimension with a simulated example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The corresponding model and method are presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We consider the models (1) and (2) and set pz = 600 with only the first 10 IVs being relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Among these 10 IVs, the first 3 IVs are invalid but the remaining IVs are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Moreover, we set Corr (ǫi, δi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='8, which indicates a level of endogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The function endo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='test generates a class object with same arguments in TSHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The class object from endo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='test can be used by calling summary function, which enable us to see a brief result of ento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='seed(5) R> n = 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' L = 600;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' s = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' k = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' px = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' epsilonSigma = matrix(c(1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='8,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='8,1),2,2) R> beta = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' gamma = c(rep(1,k),rep(0,L-k)) R> phi = (1/px)*seq(1,px)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' psi = (1/px)*seq(1,px)+1 R> Z = matrix(rnorm(n*L),n,L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' X = matrix(rnorm(n*px),n,px);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> epsilon = MASS::mvrnorm(n,rep(0,2),epsilonSigma) R> D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5 + Z %*% gamma + X %*% psi + epsilon[,1] R> Y = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5 + Z %*% c(rep(1,s),rep(0,L-s)) + D * beta + X %*% phi + epsilon[,2] R> endo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='model <- endo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='test(Y,D,Z,X, invalid = TRUE) R> summary(endo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='model) P-value Test Valid IVs 0 H0 rejected Z4 Z5 Z6 Z7 Z8 Z9 Z10 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Detected invalid IVs: Z1 Z2 Z3 When we call summary function, p-value, it reports the test result with significance level α (default alpha = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='05), the valid IVs, and detected invalid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' H0 rejected means that the treatment is endogenous, otherwise not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Since we set invalid = TRUE, ento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='test allows some of IVs to be invalid and conducts the endogeneity test with the selected �V defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' With invalid = FALSE, the function assumes that all IVs are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 13 Koo, Lee, Small, and Guo As in the above sections, one can use method argument to employ other estimators both in the low and high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='3 cf and pretest In this section, we introduce usages of cf and pretest in the package controlfunctionIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The Mroz data was introduced in Mroz (1987) and then used in various works of literature including Wooldridge (2010), which has n = 428 individuals after removing the data with NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Following Wooldridge (2010), we estimate the causal effect of education on the log earnings of married working women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The data is available in the Wooldridge package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Here, the outcome Y is log earnings (lwage), and the exposure D is years of schooling (educ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Moreover, there are other variables such as the father’s education (fatheduc), the mother’s education (motheduc), the husband’s education (huseduc), actual labor market experience (exper), its square (expersq), and the women’s age (age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Following Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='3 in Wooldridge (2010), we assume motheduc, fatheduc, and huseduc to be valid IVs, denoted as Zi = (Zi1, Zi2, Zi3)T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' we use and exper, expersq, and age as baseline covariates, denoted as Xi = (Xi1, Xi2, Xi3)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Also assume that the outcome and treatment models are (13) and (14) respectively with G(Di) = (Di, D2 i )T and H(Zi·) = (Zi1, Zi2, Zi3, Z2 i1, Z2 i2, Z2 i3)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Then we can implement the cf function by inputting a formula object, which has the same form as that of ivreg in AER package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The function summary gives us information on coefficients of the control function estimators, including the point estimator, its standard error, t value, and p value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> library(wooldridge);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' library(controlfunctionIV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' data(mroz);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' mroz <- na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='exclude(mroz) R> Y <- mroz[,"lwage"];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' D <- mroz[,"educ"] R> Z <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='matrix(mroz[,c("motheduc","fatheduc","huseduc")]) R> X <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='matrix(mroz[,c("exper","expersq","age")]) R> cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='model <- cf(Y~D+I(D^2)+X|Z+I(Z^2)+X) R> summary(cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='model) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Coefficients of the control function estimators: Estimate Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='Error t value Pr(>|t|) (Intercept) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2573907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='7871438 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='055457 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1434395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1102058 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='096884 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' I(D^2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0086426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0041004 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='017817 * Xexper 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0438690 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0131574 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='000465 *** Xexpersq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0008713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0003984 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='014631 * Xage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0011636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0048634 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='405511 --- Signif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' codes: 0 ‘***’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='001 ‘**’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01 ‘*’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='05 ‘.’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1 ‘ ’ 1 14 RobustIV and controlfunctionIV The following code infers the causal effect G(d1)Tβ−G(d2)Tβ by changing the treatment level from d2 to d1 = d2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Since the second and third coefficients are related to D, we use the second and third index to get the causal effect and its standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> d2 = median(D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' d1 = median(D)+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='diff <- c(d1,d1^2)-c(d2,d2^2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' CE <- (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='diff)%*%cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='model$coefficients[c(2,3)] R> CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='sd <-sqrt(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='diff%*%cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='model$vcov[c(2,3),c(2,3)]%*%D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='diff) R> CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='ci <- c(CE-qnorm(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='975)*CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='sd,CE+qnorm(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='975)*CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='sd) R> cmat <- cbind(CE,CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='sd,CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='ci[1],CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='ci[2]) R> colnames(cmat)<-c("Estimate","Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='Error","CI(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5%)","CI(97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5%)");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' rownames(cmat)<- "CE" R> print(cmat, digits = 4) Estimate Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='Error CI(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5%) CI(97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5%) CE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='07263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='02171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='03007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1152 The function pretest can be used to choose between the TSLS or the control function method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' If we run pretest with the same argument above and call summary, it will output the following result: R> pretest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='model <- pretest(Y~D+I(D^2)+X|Z+I(Z^2)+X) R> summary(pretest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='model) Level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='05 pretest estimator is control function estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Coefficients of the pretest estimators: Estimate Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='Error t value Pr(>|t|) (Intercept) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2573907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='7871438 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='055457 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1434395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1102058 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='096884 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' I(D^2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0086426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0041004 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='017817 * Xexper 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0438690 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0131574 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='000465 *** Xexpersq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0008713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0003984 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='014631 * Xage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0011636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0048634 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='405511 --- Signif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' codes: 0 ‘***’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='001 ‘**’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01 ‘*’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='05 ‘.’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1 ‘ ’ 1 The first section of the output of summary reports which estimator is chosen after the pretesting step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The second section lists brief information on coefficients of pretest estima- tors including the point estimator, its standard error, t value, and p-value, similar to cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Since the pretest estimator is the control function estimator, the second section of summary is the same as that of summary(cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='4 Probit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='cf Finally, we conclude the usage part by looking at the usage of Probit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='cf, which is designed for the binary outcome with unmeasured confounders and possibly invalid IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' For illus- 15 Koo, Lee, Small, and Guo tration, we use the Mroz data and define the binary outcome variable Y0 to take the value 1 if the continuous outcome Y is greater than the median of Y and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We use the same treatment variable D as in the cf example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Contrary to the cf example, we set the candidates of IVs Z as motheduc, fatheduc, huseduc, exper, and expersq, and assume that we have covariates X as age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We implement the Probit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='cf function to estimate the CATE by increasing the treat- ment value from the median of D to the median plus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We can call summary to see the result of Probit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The function summary provides information on the valid IVs �V, the point estimator, standard error, and 95% confidence interval for β in (16), and the point estimator, the standard error, and 95% confidence interval of CATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> Z <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='matrix(mroz[,c("motheduc","fatheduc","huseduc","exper","expersq")]) R> Y0 <- as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='numeric((Y>median(Y))) R> d2 = median(D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' d1 = d2+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' w0 = apply(cbind(Z,X)[which(D == d2),], 2, mean) R> Probit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='model <- Probit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='cf(Y0,D,Z,X,d1 = d1,d2 = d2,w0 = w0) R> summary(Probit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='model) Estimate Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='Error CI(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5%) CI(97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5%) Valid IVs Beta 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='3922 motheduc fatheduc huseduc CATE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1489 motheduc fatheduc huseduc _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ No invalid IV is detected With the option invalid = TRUE, we allow invalid IVs and choose the valid IVs among all provided IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' If one wants to assume all IVs are valid, one can set invalid = FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Application to Framingham Heart Study We analyze the Framingham Heart Study (FHS) data and illustrate our package using ge- netic variants as IVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The FHS is an ongoing cohort study of participants from the town of Framingham, Massachusetts, that has grown over the years to include five cohorts with a total sample of over 15,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The FHS, initiated in 1948, is among the most critical sources of data on cardiovascular epidemiology (Sytkowski, Kannel, and D’Agostino, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Kannel, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Mahmood, Levy, Vasan, and Wang, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Since the late 1980s, researchers across human health-related fields have used genetic factors underlying cardiovascular diseases and other disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Over the last two decades, DNA has been collected from blood sam- ples and immortalized cell lines from members of the Original Cohort, the Offspring Cohort, and the Third Generation Cohort (Govindaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Several large-scale genotyping projects and genome-wide linkage analysis have been conducted, and several other recent collaborative projects have completed thousands of SNP genotypes for candidate gene re- gions in subsets of FHS subjects with available DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The FHS has recently been used for Mendelian Randomization to determine causal relationships even in the presence of unmea- 16 RobustIV and controlfunctionIV sured confounding thanks to the availability of genotype and phenotype data (Holmes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Dalbeth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' As candidate IVs, we will use genotype data from the FHS associated with the phenotype of interest and apply the proposed meth- ods described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We apply the RobustIV package to investigate the effect of low-density lipoprotein (LDL- C) on globulin levels among individuals in the Framingham Heart Study (FHS) Offspring Cohort, as was studied in Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We use eight SNP genotypes (rs646776, rs693, rs2228671, rs2075650, rs4299376, rs3764261, rs12916, rs2000999) that are known to be significantly associated with LDL-C measured in mg/dL as candidate IVs (Kathiresan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' See Table 2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The outcome of interest Yi is a continuous globulin level (g/L) and the exposure variable Di is the LDL-C level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Globulin is known to play a crucial role in liver function, clotting, and the immune system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We also use the age and sex of the subjects as covariates Xi·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The study includes n = 1445 subjects, with an average globulin level of 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='27 (SD: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='74) and an average LDL-C of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='55 (SD: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' An average age is 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='58 (SD: 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='74) and 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='95% are males.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Zj SNP Position lm(D ∼ Z) lm(Y ∼ Z) Estimate (Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Error) t-statistic (p-value) Estimate (Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Error) t-statistic (p-value) Z1 rs646776 chr1:109275908 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='160 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='610) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='205 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='001) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='170) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='007 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='994) Z2 rs693 chr2:21009323 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='600 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='286) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='799 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='005) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='318 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='135) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='349 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='019) Z3 rs2228671 chr19:11100236 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='138 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='029) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='518 (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='001) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='529 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='214) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='474 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='014) Z4 rs2075650 chr19:44892362 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='451 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='021) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='183 (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='001) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='471 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='213) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='208 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='027) Z5 rs4299376 chr2:43845437 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='847 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='387) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='773 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='006) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='110 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='146) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='752(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='452) Z6 rs3764261 chr16:56959412 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='651 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='429) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='555 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='011) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='275 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='151) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='829 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='067) Z7 rs12916 chr5:75360714 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='363 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='365) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='463 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='014) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='195 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='144) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='357 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='175) Z8 rs2000999 chr16:72074194 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='961 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='629) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='818 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='069) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='119 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='172) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='691 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='489) Table 2: Summary of the relationship between the single nucleotide polymorphisms (SNPs) and low-density lipoprotein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The point estimator, its standard error, t value, and p-value are summary statistics from running a marginal regression model specified in the column title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Position refers to the position of the SNP in the chromosome, denoted as chr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' By applying endo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='test, we detect one invalid IV and observe the evidence for the existence of unmeasured confounders since the null hypothesis H0 : σ12 = 0 is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> pz <- ncol(Z) R> globulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='endo2 <- endo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='test(Y,D,Z,X, invalid = TRUE, tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1st = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz)), tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2nd = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz))) R> summary(globulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='endo2) P-value Test Valid IVs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0091 H0 rejected Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='3 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='4 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='6 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='8 17 Koo, Lee, Small, and Guo _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Detected invalid IVs: Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2 Next, we implement TSHT with the default method of "OLS" under the low-dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Again, the same invalid IV is detected and the confidence interval is above zero, indicating a positive effect of LDL on the glucose level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> pz <- ncol(Z) R> TSHT2 <- TSHT(Y, D, Z, X, tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1st = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz)), tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2nd = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz))) R> summary(TSHT2) betaHat Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='Error CI(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5%) CI(97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5%) Valid IVs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0529 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0814 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='3 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='4 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='5 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='6 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='8 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Detected invalid IVs: Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2 We also constructed the confidence interval using the searching method, which provides robustness to the IV selection errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> SS1 <- SearchingSampling(Y, D, Z, X, tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1st = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz)), tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2nd = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz)), Sampling = FALSE) R> summary(SS1) Confidence Interval for Causal Effect: [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2427,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1894] We further implement the sampling method, which leads to a shorter uniformly valid CI than the searching method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' R> SS2 <- SearchingSampling(Y, D, Z, X, tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1st = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz)), tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='2nd = sqrt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='01*log(pz)), Sampling = TRUE) R> summary(SS2) Confidence Interval for Causal Effect: [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='0521,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1259] In the following, we study nonlinear causal relationships using the controlfunctionIV package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Burgess, Davies, and Thompson (2014) investigated a nonlinear causal relation- ship between BMI and diverse cardiovascular risk factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Here we examine BMI’s possibly nonlinear causal effect on the insulin level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Among n = 3733 subjects, we excluded 618 subjects with missing information on insulin level, and 50 subjects whose insulin level is greater than 300pmol/L and whose BMI is greater than 45kg/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We use log-transformed insulin as the outcome of interest Yi measured at Exam 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The exposure Di denotes the BMI measures at Exam 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The covariates Xi· that we adjusted for are age and sex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' As valid IVs Zi·, we propose using four SNP genotypes known to be significantly associated with obesity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' In our analysis, we include I(D^2) and I(X^2) to account for quadratic effects of BMI, age, and sex on the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' We also include I(Z^2) to account for possible quadratic effects of SNPs on the exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The result from the pretest estimator is as follows: 18 RobustIV and controlfunctionIV R> insulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='pretest = pretest( Y ~ D + I(D^2) + X + I(X^2) | Z + I(Z^2) + X + I(X^2)) R> summary(insulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='pretest) Level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='05 pretest estimator is control function estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Coefficients of Pretest Estimators: Estimate Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='Err t value Pr(>|t|) (Intercept) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='674e+00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='006e-01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='453 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='39e-06 *** D 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='295e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='828e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='001690 ** I(D^2) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='784e-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='742e-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='002276 ** X1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='780e-02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='05 ‘.’ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='1 ‘ ’ 1 The pretest estimator chooses the control function over the standard TSLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The results also show that BMI has a positive linear effect on the outcome but a negative quadratic effect on the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Acknowledgement The research of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Koo was supported in part by NIH grants R01GM140463 and R01LM013614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The research of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Small was supported in part by NIH grant 5R01AG065276-02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content='The re- search of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Guo was partly supported by the NSF grants DMS 1811857 and 2015373 and NIH grants R01GM140463 and R01LM013614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Guo is grateful to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' Frank Windmeijer for bringing up the maximum clique method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' The Framingham Heart Study is conducted and supported by the National Heart, Lung, and Blood Institute (NHLBI) in collaboration with Boston University (Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' N01- HC-25195, HHSN268201500001I, and 75N92019D00031).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E3T4oBgHgl3EQfQQn6/content/2301.04412v1.pdf'} +page_content=' This manuscript was not prepared in collaboration with investigators of the Framingham Heart Study and does not necessarily reflect the opinions 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b/49FJT4oBgHgl3EQfkSwi/content/tmp_files/2301.11578v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ac845d24f5aa67b910dbc1a145c27ce1561f83c --- /dev/null +++ b/49FJT4oBgHgl3EQfkSwi/content/tmp_files/2301.11578v1.pdf.txt @@ -0,0 +1,2297 @@ +Preprint. Under review. +LEARNING TO UNLEARN: INSTANCE-WISE UNLEARN- +ING FOR PRE-TRAINED CLASSIFIERS +Sungmin Cha1,2*, Sungjun Cho1*, Dasol Hwang1*, Honglak Lee1,4, Taesup Moon2, and Moontae Lee1,3 +1LG AI Research +2Seoul National University +3University of Illinois Chicago +4University of Michigan +sungmin.cha@snu.ac.kr, {sungjun.cho, dasol.hwang, honglak.lee}@lgresearch.ai, +tsmoon@snu.ac.kr, moontae.lee@lgresearch.ai +* denotes equal contribution +ABSTRACT +Since the recent advent of regulations for data protection (e.g., the General Data +Protection Regulation), there has been increasing demand in deleting information +learned from sensitive data in pre-trained models without retraining from scratch. +The inherent vulnerability of neural networks towards adversarial attacks and un- +fairness also calls for a robust method to remove or correct information in an +instance-wise fashion, while retaining the predictive performance across remain- +ing data. To this end, we define instance-wise unlearning, of which the goal is to +delete information on a set of instances from a pre-trained model, by either mis- +classifying each instance away from its original prediction or relabeling the in- +stance to a different label. We also propose two methods that reduce forgetting on +the remaining data: 1) utilizing adversarial examples to overcome forgetting at the +representation-level and 2) leveraging weight importance metrics to pinpoint net- +work parameters guilty of propagating unwanted information. Both methods only +require the pre-trained model and data instances to forget, allowing painless ap- +plication to real-life settings where the entire training set is unavailable. Through +extensive experimentation on various image classification benchmarks, we show +that our approach effectively preserves knowledge of remaining data while un- +learning given instances in both single-task and continual unlearning scenarios. +1 +INTRODUCTION +Humans remember and forget: efficiently learning useful knowledge yet regulating privately sensi- +tive information and protecting from malicious attacks. Recent advances in large-scale pre-training +enable models to memorize massive information for intelligent operations (Radford et al., 2019), +but there is a cost. Language models trained on indiscriminately collected data often disclose pri- +vate information such as occupations, phone numbers, and family background during text genera- +tion (Heikkil¨a, 2022). Vision models trained on numerous image data sometimes misclassify natu- +rally adversarial or adversarially attacked examples with high-confidence (Hendrycks et al., 2021). +A na¨ıve solution is to retrain these models from scratch after refining or reweighting their training +datasets (Lison et al., 2021; Zemel et al., 2013; Lahoti et al., 2020). However, such post-hoc process- +ing is impractical due to growing volumes of data and substantial cost of large-scale training: while +exercising the Right to be Forgotten (Rosen, 2011; Villaronga et al., 2018) may be straightforward +to humans, it is not so straightforward in the context of machine learning. This has sparked the field +of machine unlearning, in which the main goal is to efficiently delete information while preserving +information on the remaining data. +While many machine unlearning approaches have shown promising results deleting data from tradi- +tional machine learning algorithms (Mahadevan & Mathioudakis, 2021; Ginart et al., 2019; Brophy +& Lowd, 2021) as well as DNN-based classifiers (Tarun et al., 2021; Chundawat et al., 2022; Ye +et al., 2022; Yoon et al., 2022; Golatkar et al., 2020; Kim & Woo, 2022; Mehta et al., 2022), existing +work are built upon assumptions far too restrictive compared to real-life scenarios. First off, many +approaches assume a class-wise unlearning setup, where the task is to delete information from all +data points that belong to a particular class or set of classes. However, data deletion requests are +1 +arXiv:2301.11578v1 [cs.LG] 27 Jan 2023 + +Preprint. Under review. +practically received at a per-instance basis, potentially resulting in a set of data points with a mix- +ture of class labels (Heikkil¨a, 2022; Mehrabi et al., 2021). Another widely used assumption is that +at least a subset of the original training data is available at the time of unlearning. In real settings, +however, loading the original dataset may not be an option due to data expiration policies or lack +of storage for large amounts of data. Lastly, many approaches consider the main objective as re- +moving the previous effect of the deleting data during training. While this is indeed the ideal case, +recent work have shown that fulfilling the objective can still lead to information leakage (Suriyaku- +mar & Wilson, 2022), and unlearning mechanisms must explicitly enforce misprediction for tighter +security (Graves et al., 2021). +In light of aforementioned limitations, we propose a framework for instance-wise unlearning that +deletes information with access only to the pre-trained model and the data points requested for +unlearning. Instead of undoing the previous influence of deleting data, we pursue a stronger goal +where all requested data points are misclassified, preventing collection of information via interpo- +lation of nearby data points. Inspired by work in the continual learning literature (Ebrahimi et al., +2020; Aljundi et al., 2018), we propose two regularization methods that minimize loss in predic- +tive performance on the remaining data, while completely forgetting information on deleting data. +Specifically, we 1) generate adversarial examples by attacking each deleting data point with the +pre-trained model and retrain on these examples to prevent representation-level forgetting and 2) +use weight importance measures from unlearning instances to focus gradient updates more towards +parameters responsible for the originally correct classification of such instances. Extensive experi- +ments on CIFAR-10/-100 (Krizhevsky et al., 2009) and ImageNet-1K (Deng et al., 2009) datasets +show that our proposed method effectively preserves overall predictive performance, while com- +pletely misclassifying images chosen for deletion. Our qualitative analyses also reveal interesting +insights, including lack of any discernible pattern in misclassification that may be exploited by +adversaries, preservation of the previously learned decision boundary, and forgetting of high-level +features within deleted images. In summary, our main contributions are as follows: +• We propose instance-wise unlearning through intended misclassification, under the as- +sumption that only the pre-trained model and data to forget are available at hand. +• We present two model-agnostic regularization methods that reduce forgetting on remaining +data while misclassifying data for deletion. +• Empirical evaluations on well-known image classification benchmarks show that our pro- +posed method significantly boosts predictive performance after unlearning. +2 +RELATED WORK +Machine unlearning. +Machine unlearning (Cao & Yang, 2015) is a field that makes a pre-trained +model forget information learned from a specified subset of data. For this, the existing studies have +taken an approach that deletes the influence of unwanted data points (denoted as Df) from the model +while retaining the predictive performance on the rest of the data (denoted as Dr). Mahadevan & +Mathioudakis (2021); Ginart et al. (2019); Brophy & Lowd (2021) proposed unlearning methods for +a linear/logistic regression, k-means clustering, and random forests, respectively. These methods are +specifically designed for simple machine learning models, not for neural networks. +Table 1: Comparison between existing unlearning methods. +Methods +Unit +Goal +Dr +Df +Tarun et al. (2021) +class +undo + + +Chundawat et al. (2022) +class +undo + + +Ye et al. (2022) +class +undo + + +Yoon et al. (2022) +class +undo + + +Golatkar et al. (2020) +instance +undo + + +Kim & Woo (2022) +instance +undo + + +Mehta et al. (2022) +instance +undo + + +Our methods +instance +misclassify + + +Recently, the machine unlearning for +neural networks have been studied in +different settings, shown in Table 1. +These methods can be categorized +into two approaches: class-wise and +instance-wise unlearning. The class- +wise unlearning is to forget a cer- +tain class (e.g., 9-th class of CIFAR- +10) while retaining the performance +on the remaining class (Tarun et al., +2021; Chundawat et al., 2022; Ye +et al., 2022; Yoon et al., 2022). On +the other hand, the instance-wise un- +learning is designed to delete instance-wise information (e.g., several images of CIFAR-10) from +2 + +Preprint. Under review. +the pre-trained model (Golatkar et al., 2020; Kim & Woo, 2022; Mehta et al., 2022). In other words, +only instances that are requested to be forgotten should be deleted and the others from the same class +should be remembered. +The goal of the existing methods is to make the already trained model identical to the model trained +on the dataset with unwanted instances removed (denote as undo). Unfortunately, even if the model +is trained on the removed dataset, the interpolation capabilities of the neural networks may correctly +predict even that we want to erase. This does not lead to complete unlearning in practical applica- +tions. Therefore, we define the goal of unlearning as to make the already trained model completely +misclassifies the set of instances that should be forgotten (denote as misclassify). +Also, the existing methods have different access level to the unlearning data Df and the rest Dr. The +existing solutions for instance-wise unlearning require access to the entire dataset (i.e., Dr ∪ Df). +These methods which rely on the availability of the entire data are very far from real-world scenarios. +On the other hand, our proposed methods only need to the unlearning dataset (i.e., Df). +Adversarial examples. +Since the vulnerability of neural networks has been revealed (Szegedy +et al., 2013), various methods have been proposed to generate adversarial examples that can de- +ceive neural networks (Goodfellow et al., 2014; Kurakin et al., 2016; Madry et al., 2017; Carlini & +Wagner, 2017). In the case of white-box attack, an adversarial example can be generated by adding +a hardly visible perturbation on a given image based on the gradient information from the model, +making the model classify the image to a wrong class. The injected noise of the example is hard +to distinguish visually but it causes a serious misclassification of the model. Recently, (Ilyas et al., +2019) experimentally demonstrates that those noise is not meaningless but it rather contains (attack) +target label’s features for the model. +Weight importance. +Weight importance is a measure of how important each weight is when the +model predicts an output for a given input data, and it has been used for different purposes, such +as weight pruning (Molchanov et al., 2019; Liu et al., 2017; Wen et al., 2016; Alvarez & Salz- +mann, 2016; Li et al., 2016) and regularization-based continual learning (Kirkpatrick et al., 2017; +Aljundi et al., 2018; Chaudhry et al., 2018; Jung et al., 2020; Aljundi et al., 2019). Among them, +regularization-based continual learning has actively proposed various methods for measuring the +weight importance. For overcoming catastrophic forgetting of previous tasks, the weight importance +is utilized as the strength of the L2 regularization between a current model’s weight and the model’s +weight trained up to the previous task. Most methods estimate the weight-level importance based on +a gradient of a given input data (Kirkpatrick et al., 2017; Aljundi et al., 2018). +3 +METHOD +3.1 +PRELIMINARIES AND NOTATIONS +Dataset and pre-trained model. +Let Dtrain be the entire training dataset used to pre-train a clas- +sification model gθ : X → Y. We denote Df ⊂ Dtrain as the unlearning dataset that we want to +intentionally forget from the pre-trained model and Dr as the remaining dataset on which we wish +to maintain predictive accuracy (Dr := Dtrain \ Df). We denote a pair of an input image x ∈ X +and its ground-truth label y ∈ Y from Dtrain as (x, y) ∼ Dtrain, similarly (xf, yf) ∼ Df and +(xr, yr) ∼ Dr. Also, Dtest denotes the test dataset used for evaluation. Note that our approaches +assumes access to only the pre-trained model gθ and the unlearning dataset Df during unlearning. +Adversarial examples. +The goal of an adversarial attack on an input (x, y) is to generate an +adversarial example x′ that is similar to x, but leads to misclassification (gθ(x′) ̸= y) when fed +to the pre-trained model gθ. In the case of targeted adversarial attack, it makes the model predict a +specific class different from the true class (gθ(x′) = ¯y). The typical optimization form of generating +adversarial examples in targeted attack is denoted as +x′ = +arg min +z:∥z−x∥p≤ϵ +LCE(gθ(z), ¯y; θ) +(1) +where LCE stands for the cross-entropy loss. The ∥z − x∥p ≤ ϵ condition requires that the Lp- +norm is less than a perturbation budget ϵ. The optimization above is intractable in general, and +3 + +Preprint. Under review. +Figure 1: Illustrations of our approaches that reduce forgetting on the remaining data. (Top) Aug- +menting adversarial examples from unlearning data provides support for preserving the overall de- +cision boundary. (Bottom) Weight importance measures allow us to pinpoint weights we should +change to induce misclassification while maintaining other weights to mitigate forgetting. +thus several papers have proposed approximations that can generate adversarial examples without +directly solving it (Goodfellow et al., 2014; Kurakin et al., 2016; Carlini & Wagner, 2017; Madry +et al., 2017). In this paper, we make use of L2-PGD targeted attacks Madry et al. (2017) for all +experiments. +Measuring weight importance with MAS. +To measure weight importance Ω, we consider +MAS (Aljundi et al., 2018), an algorithm that estimates weight importance by finding parameters +that brings a significant change in the output when perturbed slightly. It estimates the weight impor- +tance via a sum of gradients on the L2 norm of the outputs: +Ωi = 1 +N +N +� +n=1 +���� +∂∥gθ(x(n); θ)∥2 +2 +∂θi +���� +(2) +where i stands for the index of network parameter weights and x(n) denotes n-th input image from +a total of N numbers of images. Note that each Ωi can be interpreted as a measure of influence or +importance of θi in producing the output of given N input images. +3.2 +INSTANCE-WISE UNLEARNING FOR PRE-TRAINED CLASSIFIERS +Definition of instance-wise unlearning. +Let ˆgθ denote the model after unlearning. We consider +two types of goals for instance-wise unlearning: (i) misclassifying all data points in Df, (i.e., +ˆgθ(xf) ̸= yf). (ii) relabeling (or correcting) the predictions of Df (i.e., ˆgθ(xf) = y∗ +f) where +y∗ +f ̸= yf is chosen individually for each input xf. Let LUL denote a loss function used for unlearn- +ing on a classification model. The above two goals can be realized with the following loss functions: +LMS +UL(Df; θ) = −LCE(gθ(xf), yf; θ) +(3) +LCor +UL (Df; θ) = LCE(gθ(xf), y∗ +f; θ) +(4) +When unlearning solely based on the two loss functions above, the model is likely to suffer from +significant forgetting on Dr. Therefore, a crucial objective shared across both unlearning goals is to +overcome forgetting of previously learning knowledge, and maintain as much classification accuracy +as possible on Dr. +When both Df and Dr are available, we can easily obtain an oracle model that satisfies the objec- +tive by re-training the model with the following loss function: Loracle(Df, Dr; θ) = LUL(Df; θ) + +LCE(Dr; θ). However in real-settings, access to Dr may not be an option due to high cost in data +4 + +Unlearning dataset D +Adversarial examples D +★:(★,★) +Remaining dataset DrPreprint. Under review. +Algorithm 1 Generate adversarial examples +Input: Forgetting data Df, Model gθ +Output: Adversarial examples ¯Dr +1: ¯Dr ← ∅ +2: for i in range Nf do +3: +(x(i), y(i)) ∼ Df +4: +Randomly sample ¯y(i) ̸= y(i) +5: +for j in range Nadv do +6: +x′(j) +f +← L2-PGD(x(i), ¯y(i)) (Eq. 1) +7: +¯Dr ← ¯Dr ∪ {(x′(j) +f +, ¯y(j))} +8: +end for +9: end for +10: return ¯Dr +Algorithm 2 Measure weight importance +Input: Forgetting data Df, Model gθ +Output: Weight importance ¯Ω +1: ¯Ω ← {0} +2: Ω ← weight importances(Df, gθ) (Eq. 2) +3: for l in range L do +4: +Get importance of l-th layer Ωl ← Ω +5: +Normalize Ωl ← +Ωl − Min(Ωl) +Max(Ωl) − Min(Ωl) +6: +Update ¯Ωl ← {1 − Ωl} +7: end for +8: return ¯Ω +storage. To tackle this limitation, we define regularization-based unlearning for which the goal is to +achieve the goal above without explicit use of Dr: +LRegUL(Df; θ) = LUL(Df; θ) + R(Df, gθ) +(5) +Here, R(·) is the regularization term used to overcome forgetting of knowledge on the remaining +data Dr. In the following subsections, we introduce two novel regularization methods designed to +overcome representation- and weight-level forgetting during the unlearning process. +Regularization using adversarial examples. +The motivation of using adversarial examples stems +from the work of Ilyas et al. (2019), which showed that perturbations added to x to generate an ad- +versarial example x′ contain class-specific features of the attack target label ¯y ̸= y. Based on this +finding, we utilize generated adversarial examples as part of regularization R(·) to preserve class- +specific knowledge previously learned by the model, overcoming forgetting during unlearning at the +representation-level. Let Df be a set of Nf images: {(x(i) +f , y(i) +f )}Nf +i=1. Prior to the unlearning pro- +cess, we generate adversarial examples x′ +f using the targeted PGD attack with a randomly selected +attack target label ¯y ̸= yf. We generate Nadv adversarial examples per input xf. Then, we have +¯Df = {(x′(k) +f +, ¯y(k) +f )} +¯ +Nf +k=1 where ¯Nf = Nf × Nadv. During unlearning, we add LCE( ¯Df; θ) as a +regularization term with adversarial examples: +LAdv +UL (Df; θ) = LUL(Df; θ) + RAdv(Df, gθ) += LUL(Df; θ) + LCE( ¯Df; θ) +(6) +An intuitive illustration of this approach in the representation-level is shown in Figure 1. The gen- +erated adversarial examples ¯Df mimic the remaining dataset Dr, providing information of the pre- +trained decision boundary within the representation space. As a result, by adding LCE( ¯Df; θ) as a +regularizer to the unlearning process, the model can learn a new decision boundary that minimizes +LUL (in Eq. 3 and 4) while simultaneously attempting to keep the decision boundary of the original +model. The pseudocode for generating adversarial examples is in Algorithm 1. +Regularization with weight importance. +We also propose a regularization using weight impor- +tance to overcome forgetting at the weight-level. As depicted in Figure 1, our approach is to maintain +the weights that were less important for Df prediction as much as possible, while allowing changes +in weights that are considered important for correctly predicting Df. That is, it is to prevent the +weight-level forgetting by penalizing weights that were less important when predicting Df. +For this, we calculate the weight importance with MAS before unlearning given gθ and Df, and +normalize the measured importances Ωl within each l-th layer to lie within [0, 1]. Note that this +normalized importance Ωl assigns large values to weights important for Df. Therefore, we define +¯Ωl = 1 − Ωl as the weight importance for the regularization used for unlearning, so that more +important weights are updated more. The objective including weight importance regularization in +5 + +Preprint. Under review. +Table 2: Evaluation results before and after unlearning k instances from ResNet-50 pretrained on +respective image classification datasets. While using negative gradients only loses significant infor- +mation on Dr, our proposed methods ADV and ADV+IMP retain predictive performance on Dr as +well as Dtest, while completely forgetting instances in Df. +CIFAR-10 +CIFAR-100 +ImageNet-1K +k = 4 +k = 16 +k = 64 +k = 128 +k = 4 +k = 16 +k = 64 +k = 128 +k = 4 +k = 16 +k = 64 +k = 128 +Df (↓) +BEFORE +100.0 +100.0 +99.38 +99.53 +100.0 +100.0 +100.0 +100.0 +91.66 +87.50 +84.90 +86.72 +NEGGRAD +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +ADV +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +ADV+IMP +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +Dr (↑) +BEFORE +99.60 +99.60 +99.60 +99.60 +99.98 +99.98 +99.98 +99.98 +87.42 +87.42 +87.42 +87.42 +NEGGRAD +38.44 +15.79 +9.22 +7.11 +99.71 +66.97 +26.20 +11.64 +83.34 +61.18 +40.50 +30.16 +ADV +79.40 +69.70 +66.97 +53.49 +83.90 +89.18 +81.07 +76.28 +74.13 +81.09 +76.02 +69.01 +ADV+IMP +82.95 +85.75 +72.77 +54.51 +83.89 +89.91 +89.48 +82.86 +74.16 +81.77 +79.36 +75.33 +Dtest (↑) +BEFORE +92.59 +92.59 +92.59 +92.59 +77.10 +77.10 +77.10 +77.10 +76.01 +76.01 +76.01 +76.01 +NEGGRAD +36.56 +15.87 +9.28 +7.11 +74.54 +48.07 +21.11 +10.19 +72.53 +53.30 +35.61 +26.73 +ADV +74.34 +65.14 +62.23 +49.47 +60.00 +63.17 +57.43 +53.89 +62.12 +70.42 +65.89 +59.73 +ADV+IMP +77.53 +79.65 +67.08 +50.82 +60.50 +63.69 +62.83 +58.44 +65.15 +70.97 +68.72 +65.09 +addition to regularization via adversarial examples can be written as: +LAdv+Imp +UL +(Df; θ) = LAdv +UL (Df; θ) + RImp(Df, gθ) += LAdv +UL (Df; θ) + +� +i +¯Ωi(θi − ˜θi)2 +(7) +where i is the index of each weight and ˜θ is the initial weight of the pre-trained classifier before +unlearning. The pseudocode of measuring weight importance is shown in Algorithm 2. Throughout +various experiments, we observe that applying the regularization using adversarial examples is al- +ready effective to overcome the forgetting for knowledge of Dr, and the additional regularization +with weight importance further enhances performance even further, especially in more harder sce- +narios such as continual unlearning. The pseudocode of the overall unlearning pipeline is shown in +the supplementary material. +4 +EXPERIMENTS +In this section, we evaluate our proposed instance-wise unlearning methods in various image clas- +sification benchmarks. We first describe our experimental setup, including datasets, baselines and +experimental details. We then show that our methods effectively preserves knowledge of remaining +data while unlearning instances that should be forgotten in both single-task and continual unlearning +scenarios. Lastly, we offer qualitative analyses on three parts: prediction patterns, decision boundary +and layer-wise representations in unlearning. +4.1 +SETUP +Datasets and baselines. +We evaluate our unlearning methods on three different image classifi- +cation datasets: CIFAR-10, CIFAR-100 (Krizhevsky et al., 2009), and ImageNet-1K (Deng et al., +2009). Also, we use the ResNet-50 (He et al., 2016) as a base model. The experimental results of +various base models are available in the appendix. The compared methods are as follows: BEFORE, +the pre-trained model before unlearning; NEGGRAD (Golatkar et al., 2020), fine-tuning on Df using +negative gradients (i.e. LMS +UL); CORRECT, fine-tuning using LCor +UL ; ADV is our proposed method using +adversarial examples (i.e. LAdv +UL ); ADV+IMP, our unlearning method using both adversarial examples +and the weight importance regularization (i.e. LAdv+Imp +UL +). +Experimental details. +For each dataset, we randomly pick k ∈ {4, 16, 64, 128} images from the +entire training dataset as the unlearning data Df and consider the remaining as Dr. For the unlearn- +ing, we use a SGD optimizer with a learning rate of 1e-3, weight decay of 1e-5, and momentum +of 0.9 across all experiments. We take early stopping when the model attains zero accuracy from +the unlearning data Df. For generating adversarial examples from Df, we use L2-PGD targeted +attack (Madry et al., 2017) with a learning rate of 1e-1, attack iterations of 100 and ϵ = 0.4. It gen- +erates 20 adversarial examples for CIFAR-10 and 200 examples for CIFAR-100 and ImageNet-1K. +For the weight importance regularization, we set regularization strength λ = 1 in Eq. 5. +6 + +Preprint. Under review. +Table 3: Results analogous to Table 6, but with unlearning via relabeling each image in Df to an +arbitrarily chosen class. We see a similar trend where CORRECT loses significant information on +Dr, while our proposed methods retain predictive performance on Dr as well as Dtest. +CIFAR-10 +CIFAR-100 +ImageNet-1K +k = 4 +k = 16 +k = 64 +k = 128 +k = 4 +k = 16 +k = 64 +k = 128 +k = 4 +k = 16 +k = 64 +k = 128 +Df (↑) +BEFORE +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +CORRECT +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +99.84 +100.0 +100.0 +100.0 +100.0 +ADV +95.0 +100.0 +99.375 +98.28 +90.0 +100.0 +100.0 +98.28 +100.0 +100.0 +87.5 +71.32 +ADV+IMP +90.0 +100.0 +53.75 +50.16 +80.0 +86.25 +20.63 +15.16 +100.0 +100.0 +8.59 +4.30 +Dr (↑) +BEFORE +99.60 +99.60 +99.60 +99.60 +99.98 +99.98 +99.98 +99.98 +87.42 +87.42 +87.42 +87.42 +CORRECT +28.39 +11.75 +12.33 +9.71 +96.14 +74.84 +31.79 +18.64 +84.34 +82.94 +76.21 +68.03 +ADV +81.43 +85.53 +83.36 +81.06 +69.55 +92.94 +94.64 +96.32 +70.05 +83.09 +84.75 +84.54 +ADV+IMP +83.43 +91.15 +94.76 +90.57 +68.77 +90.73 +96.68 +96.44 +74.18 +83.34 +83.27 +80.15 +Dtest (↑) +BEFORE +92.59 +92.59 +92.59 +92.59 +77.10 +77.10 +77.10 +77.10 +76.01 +76.01 +76.01 +76.01 +CORRECT +27.62 +11.79 +12.16 +9.80 +69.82 +53.11 +24.37 +14.64 +73.26 +71.90 +65.68 +58.25 +ADV +76.35 +79.15 +76.95 +74.61 +51.23 +65.62 +66.79 +68.56 +64.81 +72.02 +73.41 +73.32 +ADV+IMP +78.08 +84.24 +86.92 +82.82 +50.60 +64.28 +69.15 +68.60 +64.94 +72.20 +71.82 +68.92 +4.2 +MAIN RESULTS +Results on various datasets. +Table 6 shows evaluation results before and after unlearning k in- +stances from ResNet-50 models pre-trained on each of three different datasets. With respect to +accuracies on Df, we find that ResNet-50 can completely forget up to k = 128 instances with +consistently zero post-unlearning accuracies. On CIFAR-10, using negative gradients only results +in significant loss of accuracy on the remaining data (i.e. Dr and Dtest), performing worse than +random-choice when the number of forgetting instances is as large as 128. Meanwhile, adding regu- +larization with adversarial examples boosts the accuracy by more than 40% depending on the number +of instances to forget. Incorporating weight importances from MAS provides further improvement. +Results from CIFAR-100 and ImageNet-1K show a similar trend except when k = 4, where adding +our regularization approaches deteriorates performance. This well aligns with our intuition as the +model can easily misclassify a small number of examples by tweaking a small number of model +parameters, hence forgetting Df without losing much information on Dr and Dtest despite lack +of regularization. The benefit of using adversarial examples is also small when k is small as the +diversity amongst images in Dadv is limited by the number of instances to forget. +Table 7 shows results analogous to Table 6, but with the goal of relabeling data points in Df to +arbitrarily chosen labels rather than misclassifying. We find that a similar trend, where ADV attains +significantly less forgetting in Dr and Dtest compared to CORRECT, while succesfully relabeling all +points in most cases. While ADV+IMP show even less forgetting, it loses accuracy in relabeling Df, +showing that regularization via weight importance focuses too much on retaining previous knowl- +edge rather than adapting to corrections provided in Df. An intuitive explanation on why this occurs +particularly in relabeling is that while misclassifying can be done easily by driving the input to its +closest decision boundary, relabeling can be difficult if the new class is far from the original class in +the representation space. The difficulty rises even more when the size of Df is large, in which case +more parameters in the network are discouraged from being updated during unlearning. +Correcting natural adversarial examples. +Leveraging the ImageNet-A (Hendrycks et al., 2021) +dataset consisting of natural images that are misclassified with high-confidence by strong classifiers, +we test whether our method can make corrections on these adversarial examples, while preserving +knowledge from the original training data. For this experiment, we consider Df to consist k adver- +sarial images from ImageNet-A, and adjust a ResNet-50 model pre-trained on ImageNet-1K to cor- +rectly classify Df via our unlearning framework. Table 4 shows the results for k = {16, 32, 64, 128}. +We find that correcting predictions of a small number of images (e.g. k = 16), finetuning the model +na¨ıvely with cross-entropy only attains the best accuracy in both Dr and Dtest. When correcting +larger number of images, however, the absence of regularization terms results in larger forgetting in +Dr compared to ADV and ADV+IMP, with a performance gap that consistently increases with the +number of adversarial images. Another takeaway is that regularization via weight importance does +not help in this scenario, even showing a significant drop in Df accuracy when a large number of +adversarial images are introduced. This implies that using weight importances imposes too strong +a regularzation that correcting predictions for Df itself becomes non-trivial. We conjecture that the +aggregation of important parameters for predictions in Df cover a large proportion of the network +with large k, and that careful search for the Pareto optimal between accuracies on Df and on Dr is +required. +7 + +Preprint. Under review. +Table 4: Correcting adversarial images from +ImageNet-A. ADV achieves the least forgetting, +while ADV+IMP fails to correct large number of +predictions due to strong regularization. +ImageNet-A +k = 16 +k = 32 +k = 64 +k = 128 +Df (↑) +BEFORE +0.0 +0.0 +0.0 +0.0 +CORRECT +100.0 +100.0 +100.0 +100.0 +ADV +100.0 +100.0 +95.31 +83.44 +ADV+IMP +100.0 +100.0 +10.94 +9.38 +Dr (↑) +BEFORE +87.46 +87.46 +87.46 +87.46 +CORRECT +84.41 +83.29 +80.79 +77.38 +ADV +81.75 +83.80 +83.74 +83.44 +ADV+IMP +81.82 +83.73 +83.53 +82.86 +Dtest (↑) +BEFORE +76.15 +76.15 +76.15 +76.15 +CORRECT +73.21 +72.04 +69.91 +66.73 +ADV +70.89 +72.58 +72.68 +72.36 +ADV+IMP +70.98 +72.51 +72.39 +71.68 +Table 5: Unlearning instances continually by in- +crements of kCL = 8 images per step. Our meth- +ods outperform NEGGRAD in the continual un- +learning scenario as well. +CIFAR-100 (kCL = 8) +k = 8 +k = 16 +k = 64 +k = 128 +Df (↓) +BEFORE +100.0 +100.0 +100.0 +100.0 +NEGGRAD +0.0 +0.0 +0.0 +0.52 +ADV +0.0 +0.0 +1.04 +0.0 +ADV+IMP +0.0 +0.0 +0.0 +1.04 +Dr (↑) +BEFORE +99.98 +99.98 +99.98 +99.98 +NEGGRAD +80.58 +31.85 +6.60 +1.89 +ADV +80.33 +70.54 +59.67 +38.16 +ADV+IMP +81.46 +72.78 +62.30 +47.14 +Dtest (↑) +BEFORE +77.10 +77.10 +77.10 +77.10 +NEGGRAD +58.20 +24.48 +5.73 +1.22 +ADV +57.56 +50.43 +43.48 +30.10 +ADV+IMP +58.33 +51.97 +45.09 +36.17 +Continual unlearning. +In real-world scenarios, it is likely that data removal requests come as +a stream, rather than all at once. Ultimately, despite continual unlearning requests, we need the +unlearning method that can delete the requested data while maintaining performance for the rest data. +Thus, we consider the setting of deleting k = {8, 16, 64, 128} data by repeating the procedure of +continually unlearning Df in small fragments of size kCL = 8. Table 5 shows the results of continual +unlearning in the model trained with ResNet-50 on CIFAR-100. We observe that NEGGRAD suffers +from large forgetting as the iteration of unlearning procedure increases. On the other hand, our +proposed method shows significantly less forgetting while effectively deleting for Df even after +multiple iterations of unlearning. +4.3 +QUALITATIVE ANALYSIS +Through further analysis, we gather insight on the following questions: Q1. Is there any particular +pattern in how the model unlearns a set of instances (i.e. does the model use any particular label as +a retainer for deleted data)? Q2. How does the model isolate out instances in Df from its previous +decision boundary? Q3. How do layer-wise representations of data points in Df and Dr change +before and after unlearning? For interpretable visualizations, we perform the following analysis on +a ResNet-18 model pre-trained on CIFAR-10. +(a) NEGGRAD +(b) ADV +(c) ADV+IMP +Figure 2: Confusion matrices showing average +pairwise frequencies of pre- (Y-axis) and post- +unlearning (X-axis) prediction labels from Df. A +hue closer to blue indicates higher frequency. Our +unlearning framework does not produce any dis- +cernible correlation in misclassification. +A1. Our method shows no pattern in mis- +classification. +We first check whether the un- +learned model classifies all instances in Df to +a particular set of labels. The model exhibit- +ing no correlation between true labels and new +misclassified labels is crucial with respect to +data privacy, as it indicates that the unlearn- +ing process avoids the so-called Streisand ef- +fect where data instances being forgotten unin- +tentionally becomes more noticeable (Golatkar +et al., 2020). Figure 2 shows the confusion ma- +trices of (pre-unlearning label, post-unlearning +label) pairs from Df for k = 512. We find no +distinguishable pattern when unlearning with our methods as well as NegGrad, which shows that no +specific label is used as a retainer, which adds another layer of security against adversaries in search +of unlearned data points. +A2. Our method effectively preserves the decision boundary. +We check whether the adversar- +ial examples generated from forgetting data help in preserving the decision boundary in the feature +space. Figure 3 shows t-SNE (Van der Maaten & Hinton, 2008) visualizations of final-layer acti- +vations from examples in Dr and Df before and after unlearning. We find that unlearning through +only negative gradient significantly distorts the previous decision boundary, leading to poor predic- +8 + +NegGrad +0.5 +9 +80 +0.4 +7 +6 +labels +0.3 +5 +4 +0.2 +m +2 +0.1 +1 +0.0 +0 +2 +7 +Predicted labelsOurs (Adv) +0.5 +9 +8 +0.4 +7 +6 +labels +0.3 +5 +4 +0.2 +m +2 +0.1 +1 +1 +0.0 +0 +2 +7 +Predicted labelsOurs（Adv+Imp） +0.5 +9 +8 +0.4 +7 +6 +labels +0.3 +5 +4 +0.2 +m +2 +0.1 +1 +1 +1 +0.0 +0 +2 +3 +4 +> +9 +Predicted labelsPreprint. Under review. +(a) BEFORE +(b) NEGGRAD +(c) ADV +(d) ADV+IMP +Figure 3: t-SNE plots of CIFAR-10 datapoints in Df (triangles) and Dr (dots) before and after +unlearning. Colors indicate true labels for all plots. Regularization with adversarial examples and +weight importance effectively preserves the decision boundary while migrating instances in Df +towards the class boundary to induce misclassification. +tive performance across Dr. However, when we incorporate adversarial samples from instances in +Df, the decision boundary is well-preserved with unlearned examples being inferred as boundary +cases in-between multiple classes. Even for examples that lie far from the decision boundary be- +fore unlearning, our method successfully relocates the corresponding representations towards the +decision boundary, while keeping each class cluster intact. +(a) NEGGRAD +(b) ADV +(c) ADV+IMP +Figure 4: Layer-wise CKA correlations on Df +(top row) and Dr (bottom row) between repre- +sentations before (X-axis) and after (Y-axis) un- +learning. Brighter color indicates higher CKA cor- +relation. NEGGRAD results in large forgetting of +high-level features in not only Df, but also Dr. +Our approaches, on the other hand, selectively for- +get high-level features only in Df. +A3. Our method unlearns data by forgetting +high-level features. +Lastly, we compare the +representations at each layer of the model be- +fore and after unlearning to identify where the +intended forgetting occurs. For this analysis, we +leverage CKA (Kornblith et al., 2019) which +measures correlations between representations +given two distinct models. Figure 4 shows the +CKA correlation heatmaps between the origi- +nal ResNet-18 model pre-trained on CIFAR-10 +and the same model after unlearning. Results +show that for examples in Df, representations +are no longer aligned starting from the 10-th +layer while the representations before that layer +still resemble those from the original model. +This indicates that the model forgets examples +by forgetting high-level features, while simi- +larly recognizing low-level features in images +as the original model. This insight is consis- +tent with previous observations in the contin- +ual learning literature that more forgettable ex- +amples exhibit peculiarities in high-level fea- +tures (Toneva et al., 2018). +5 +CONCLUDING REMARKS +We propose an instance-wise unlearning framework that deletes information from a pre-trained +model given a set of data instances with mixed labels. Rather than undoing the influence of given +instances during the pre-training, we aim for a stronger form of unlearning via intended misclas- +sification. We develop two regularization techniques that reduce forgetting on the remaining data, +one utilizing adversarial examples of deleting instances and another leveraging weight importances +to focus updates to parameters responsible for propagating information we wish to forget. Both ap- +proaches are agnostic to the choice of architecture, and requires access only to the pre-trained model +and instances requested for deletion. Experiments on various image classification datasets showed +that our methods effectively mitigates forgetting on remaining data, while completely misclassify- +ing deletion data. Further qualitative analyses show that our unlearning framework does not show +any pattern in misclassification (i.e. the Streisand effect), preserves the decision boundary with the +help of adversarial examples, and unlearns by forgetting high-level features of deleting data. These +9 + +Residual Data (D_r) +1.0 +16 +14 +0.8 +Case 3: -CE(D_f) + CE(adv) + Reg(importance) +12 +0.6 +10 +0.4 +4 - +0.2 +2 +Fo +0.0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +Before UnlearningForgettingData(D_f) +1.0 +16 +14 - +F0.8 +12 +0.6 +10 +Case 1: - CE(D_f) +8 + 0.4 +6 +4 - +0.2 +2 +0 +0.0 +2 +4 +6 +8 +10 +12 +14 +16 +Before UnlearningForgettingData(D_f) +1.0 +16 +14 +0.8 +12 +Case 2: -CE(D_f) + CE(adv) +0.6 +10 +8 +0.4 +6 +4 - +0.2 +2 +0 +0.0 +2 +4 +6 +8 +10 +12 +14 +16 +Before UnlearningForgettingData(D_f) +1.0 +16 +14 + 0.8 +Case 3: -CE(D_f) + CE(adv) + Reg(importance) +12 +0.6 +10 +8 +0.4 +6 +4 +0.2 +2 +0.0 +2 +4 +6 +8 +10 +12 +14 +16 +Before UnlearningResidual Data (D_r) +1.0 +16 +14 +0.8 +12 + 0.6 +Case 1: - CE(D_f) +10 + 0.4 +9 +4 - +0.2 +2 +0 +0.0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +Before UnlearningResidual Data (D_r) +1.0 +16 +14 +0.8 +12 +Case 2: -CE(D_f) + CE(adv) +0.6 +10 +8 +0.4 +6 +4 - +0.2 +2 +0.0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +Before UnlearningPreprint. Under review. +observations shed light towards future work evaluating the utility our approach as a defense mech- +anism against membership inference attacks that predict whether a data point was included in the +training set by using posterior confidence (Shokri et al., 2017; Salem et al., 2018; Yeom et al., 2018; +Sablayrolles et al., 2019) or its distance to nearby decision boundaries (Choquette-Choo et al., 2021; +Li & Zhang, 2021). Removing harmful information that lead to socially unfair and biased predic- +tions based upon sensitive traits such as race, gender, and religion (Mehrabi et al., 2021) is another +potential contribution from this work. +REFERENCES +Rahaf Aljundi, Francesca Babiloni, Mohamed Elhoseiny, Marcus Rohrbach, and Tinne Tuytelaars. +Memory aware synapses: Learning what (not) to forget. In Proceedings of the European Confer- +ence on Computer Vision (ECCV), pp. 139–154, 2018. +Rahaf Aljundi, Marcus Rohrbach, and Tinne Tuytelaars. Selfless sequential learning. 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Under review. +A +APPENDIX +A.1 +PSEUDO CODE OF OVERALL UNLEARNING PROCESS +Algorithm 3 The pseudo code of overall unlearning process the case of using LMS +UL. +1: UNLEARNACC = 100 +2: MAXEP = 100 +3: EP = 0 +4: ¯Dr ← Generate adversarial examples with Algorithm 1 +5: ¯Ω ← Measure weight importance with Algorithm 2 +6: ˜θ ← θ +7: while UNLEARNACC ̸= 0 do +8: +Minimize Eqn (6) and (7) +9: +UNLEARNACC = GetAccuracy(Df, gθ) +10: +if EP > MAXEP then +11: +break +12: +EP += 1 +13: +end if +14: end while +15: return ˆθ +Algorithm 4 The pseudo code of overall unlearning process the case of using LCor +UL . +1: UNLEARNACC = 0 +2: MAXEP = 100 +3: EP = 0 +4: ¯Dr ← Generate adversarial examples with Algorithm 1 +5: ¯Ω ← Measure weight importance with Algorithm 2 +6: ˜θ ← θ +7: while UNLEARNACC ̸= 100 do +8: +Minimize Eqn (6) and (7) +9: +UNLEARNACC = GetAccuracy(Df, gθ) +10: +if EP > MAXEP then +11: +break +12: +EP += 1 +13: +end if +14: end while +15: return ˆθ +A.2 +ADDITIONAL RESULTS ON VARIOUS MODELS +Results on various models. +Figure 5 shows unlearning results on CIFAR-100, but with different +model architectures. We find that our methods effectively preserve knowledge outside the forgetting +data, resulting in up to 40% boost in accuracy. NegGrad again outperforms our methods when k = 4, +but soon breaks down when unlearning more instances. Interestingly, SqueezeNet and MobileNetv2 +suffer from larger forgetting in Dr and Dtest than ResNet-50, possibly due to the width being nar- +rower as previously investigated by Mirzadeh et al. (2022). ViT also suffers from large forgetting, an +observation consistent with previous work which showed that ViT suffers more catastrophic forget- +ting compared to other CNN-based methods in continual learning due to Transformer architectures +requiring large amounts of data. We also evaluate the results of unlearning on ImageNet-1K with +varying k in Figure 6. Our proposed methods prevent forgetting knowledge about the rest data Dr +better than NegGrad in all cases where k is greater than 8. At the same time, the methods effectively +delete information about Df. +A.3 +SUPPLEMENTARY MATERIALS FOR REBUTTAL +14 + +Preprint. Under review. +Table 6: Evaluation results before and after unlearning k instances from ResNet-50 pretrained on +respective image classification datasets. While using negative gradients only loses significant infor- +mation on Dr, our proposed methods ADV and ADV+IMP retain predictive performance on Dr as +well as Dtest, while completely forgetting instances in Df. +CIFAR-10 +CIFAR-100 +k = 4 +k = 16 +k = 64 +k = 128 +k = 4 +k = 16 +k = 64 +k = 128 +Df (↓) +BEFORE +100.0 +100.0 +99.38 +99.53 +100.0 +100.0 +100.0 +100.0 +ORACLE +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +NEGGRAD +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +ADV +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +ADV+IMP +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +Dr (↑) +BEFORE +99.60 +99.60 +99.60 +99.60 +99.98 +99.98 +99.98 +99.9 +ORACLE +93.43 +98.74 +99.72 +98.97 +99.68 +99.96 +96.17 +96.74 +NEGGRAD +38.44 +15.79 +9.22 +7.11 +99.71 +66.97 +26.20 +11.64 +ADV +79.40 +69.70 +66.97 +53.49 +83.90 +89.18 +81.07 +76.28 +ADV+IMP +82.95 +85.75 +72.77 +54.51 +83.89 +89.91 +89.48 +82.86 +Dtest (↑) +BEFORE +92.59 +92.59 +92.59 +92.59 +77.10 +77.10 +77.10 +77.10 +ORACLE +86.28 +90.21 +91.01 +89.44 +77.49 +64.41 +67.06 +66.88 +NEGGRAD +36.56 +15.87 +9.28 +7.11 +74.54 +48.07 +21.11 +10.19 +ADV +74.34 +65.14 +62.23 +49.47 +60.00 +63.17 +57.43 +53.89 +ADV+IMP +77.53 +79.65 +67.08 +50.82 +60.50 +63.69 +62.83 +58.44 +Table 7: Results analogous to Table 6, but with unlearning via relabeling each image in Df to an +arbitrarily chosen class. We see a similar trend where CORRECT loses significant information on +Dr, while our proposed methods retain predictive performance on Dr as well as Dtest. +CIFAR-10 +CIFAR-100 +ImageNet-1K +k = 4 +k = 16 +k = 64 +k = 128 +k = 4 +k = 16 +k = 64 +k = 128 +k = 4 +k = 16 +k = 64 +k = 128 +Df (↑) +BEFORE +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +ORACLE +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +CORRECT +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +99.84 +100.0 +100.0 +100.0 +100.0 +ADV +95.0 +100.0 +99.375 +98.28 +90.0 +100.0 +100.0 +98.28 +100.0 +100.0 +87.5 +71.32 +ADV+IMP +90.0 +100.0 +53.75 +50.16 +80.0 +86.25 +20.63 +15.16 +100.0 +100.0 +8.59 +4.30 +Dr (↑) +BEFORE +99.60 +99.60 +99.60 +99.60 +99.98 +99.98 +99.98 +99.98 +87.42 +87.42 +87.42 +87.42 +ORACLE +94.90 +99.73 +99.94 +99.90 +97.94 +99.90 +99.97 +99.79 +CORRECT +28.39 +11.75 +12.33 +9.71 +96.14 +74.84 +31.79 +18.64 +84.34 +82.94 +76.21 +68.03 +ADV +81.43 +85.53 +83.36 +81.06 +69.55 +92.94 +94.64 +96.32 +70.05 +83.09 +84.75 +84.54 +ADV+IMP +83.43 +91.15 +94.76 +90.57 +68.77 +90.73 +96.68 +96.44 +74.18 +83.34 +83.27 +80.15 +Dtest (↑) +BEFORE +92.59 +92.59 +92.59 +92.59 +77.10 +77.10 +77.10 +77.10 +76.01 +76.01 +76.01 +76.01 +ORACLE +87.33 +91.65 +91.99 +91.57 +71.56 +74.05 +74.93 +74.15 +CORRECT +27.62 +11.79 +12.16 +9.80 +69.82 +53.11 +24.37 +14.64 +73.26 +71.90 +65.68 +58.25 +ADV +76.35 +79.15 +76.95 +74.61 +51.23 +65.62 +66.79 +68.56 +64.81 +72.02 +73.41 +73.32 +ADV+IMP +78.08 +84.24 +86.92 +82.82 +50.60 +64.28 +69.15 +68.60 +64.94 +72.20 +71.82 +68.92 +15 + +Preprint. Under review. +(a) MobileNetv2 (Sandler et al., 2018) +(b) SqueezeNet (Iandola et al., 2016) +(c) ViT (Dosovitskiy et al., 2020) +Figure 5: Experimental results before and after unlearning varying k instances from various models +on CIFAR-100. +16 + +100 +Original +NegGrad +Ours (Adv) +80 +Ours (Adv+Imp) +Acc. +60 +40 +20 +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +Original +NegGrad +Ours (Adv) +80 +Ours (Adv+Imp) +Acc. +60 +40 +20 +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 + Acc. +Original +60 +NegGrad +Ours (Adv) +40 +Ours (Adv+Imp) +20 +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearnina dataset (Df)100 +80 +Dr Acc. +60 +40 +Original +20 +NegGrad +Ours (Adv) +Ours (Adv+Imp) +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +Df Acc. +60 +.★-Original +-. NegGrad +Ours (Adv) +40 +Ours (Adv+Imp) +20 +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +Dr Acc. +60 +40 +Original +20 +NegGrad +Ours (Adv) +Ours (Adv+Imp) +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +Df Acc. +60 +★:Original +NegGrad +Ours (Adv) +40 +Ours (Adv+Imp) +20 +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +Dr Acc. +Original +60 +NegGrad +Ours (Adv) +40 +Ours (Adv+Imp) +20 +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +Dr Acc. +Original +60 +NegGrad +Ours (Adv) +40 +Ours (Adv+Imp) +20 +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)Preprint. Under review. +(a) MobileNet v2 +(b) ResNet34 +(c) DenseNet121 +Figure 6: Experimental results before and after unlearning varying k instances from various models +on ImageNet-1K. +(a) Analysis for entropy-accuracy +(b) Analysis for a forgotten label +Figure 7: Experimental analysis with CIFAR-10 dataset using ResNet-18. We randomly select single +image (k = 1) for unlearning and unlearn it with NegGrad. All experiments are conducted with 100 +seeds. Each class number denotes a specific label, such as {airplane : 0, automobile : 1, bird : 2, cat +: 3, deer : 4, dog : 5, frog : 6, horse : 7, ship : 8, truck : 9}. +17 + +100 +Original +NegGrad +Ours (Adv) +80 +Ours (Adv+Imp) +Acc. +60 +40 +20 +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +Acc. +60 +40 +Original +20 +NegGrad +Ours (Adv) +Ours (Adv+Imp) +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +Acc. +60 +40 +Original +20 +NegGrad +Ours (Adv) +Ours (Adv+Imp) +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +r Acc. +60 +D +40 +Original +20 +NegGrad +Ours (Adv) +Ours (Adv+Imp) +0 +L +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +Df Acc. +60 +★:Original +NegGrad +Ours (Adv) +40 +Ours (Adv+Imp) +20 +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +r Acc. +60 +D +40 +Original +20 +NegGrad +Ours (Adv) +Ours (Adv+Imp) +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +Df Acc. +60 +★:Original +NegGrad +Ours (Adv) +40 +Ours (Adv+Imp) +20 +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +Dr Acc. +60 +40 +Original +20 +NegGrad +Ours (Adv) +Ours (Adv+Imp) +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df)100 +80 +Df Acc. +60 +Original +NegGrad +Ours (Adv) +40 +Ours (Adv+Imp) +20 +0 +1 +2 +4 +8 +16 +32 +64 +128 +256 +Number of unlearning dataset (Df) \ No newline at end of file diff --git a/49FJT4oBgHgl3EQfkSwi/content/tmp_files/load_file.txt b/49FJT4oBgHgl3EQfkSwi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..db312d6afb7bbd60cb3e3d84f44f0a93438e1d06 --- /dev/null +++ b/49FJT4oBgHgl3EQfkSwi/content/tmp_files/load_file.txt @@ -0,0 +1,1429 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf,len=1428 +page_content='Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' LEARNING TO UNLEARN: INSTANCE-WISE UNLEARN- ING FOR PRE-TRAINED CLASSIFIERS Sungmin Cha1,2*, Sungjun Cho1*, Dasol Hwang1*, Honglak Lee1,4, Taesup Moon2, and Moontae Lee1,3 1LG AI Research 2Seoul National University 3University of Illinois Chicago 4University of Michigan sungmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='cha@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='kr, {sungjun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='cho, dasol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='hwang, honglak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='lee}@lgresearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='ai, tsmoon@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='kr, moontae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='lee@lgresearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='ai denotes equal contribution ABSTRACT Since the recent advent of regulations for data protection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', the General Data Protection Regulation), there has been increasing demand in deleting information learned from sensitive data in pre-trained models without retraining from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The inherent vulnerability of neural networks towards adversarial attacks and un- fairness also calls for a robust method to remove or correct information in an instance-wise fashion, while retaining the predictive performance across remain- ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' To this end, we define instance-wise unlearning, of which the goal is to delete information on a set of instances from a pre-trained model, by either mis- classifying each instance away from its original prediction or relabeling the in- stance to a different label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We also propose two methods that reduce forgetting on the remaining data: 1) utilizing adversarial examples to overcome forgetting at the representation-level and 2) leveraging weight importance metrics to pinpoint net- work parameters guilty of propagating unwanted information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Both methods only require the pre-trained model and data instances to forget, allowing painless ap- plication to real-life settings where the entire training set is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Through extensive experimentation on various image classification benchmarks, we show that our approach effectively preserves knowledge of remaining data while un- learning given instances in both single-task and continual unlearning scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 1 INTRODUCTION Humans remember and forget: efficiently learning useful knowledge yet regulating privately sensi- tive information and protecting from malicious attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Recent advances in large-scale pre-training enable models to memorize massive information for intelligent operations (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2019), but there is a cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Language models trained on indiscriminately collected data often disclose pri- vate information such as occupations, phone numbers, and family background during text genera- tion (Heikkil¨a, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Vision models trained on numerous image data sometimes misclassify natu- rally adversarial or adversarially attacked examples with high-confidence (Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' A na¨ıve solution is to retrain these models from scratch after refining or reweighting their training datasets (Lison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Zemel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Lahoti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' However, such post-hoc process- ing is impractical due to growing volumes of data and substantial cost of large-scale training: while exercising the Right to be Forgotten (Rosen, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Villaronga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2018) may be straightforward to humans, it is not so straightforward in the context of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' This has sparked the field of machine unlearning, in which the main goal is to efficiently delete information while preserving information on the remaining data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' While many machine unlearning approaches have shown promising results deleting data from tradi- tional machine learning algorithms (Mahadevan & Mathioudakis, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Ginart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Brophy & Lowd, 2021) as well as DNN-based classifiers (Tarun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Chundawat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Golatkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Kim & Woo, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2022), existing work are built upon assumptions far too restrictive compared to real-life scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' First off, many approaches assume a class-wise unlearning setup, where the task is to delete information from all data points that belong to a particular class or set of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' However, data deletion requests are 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='11578v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='LG] 27 Jan 2023 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' practically received at a per-instance basis, potentially resulting in a set of data points with a mix- ture of class labels (Heikkil¨a, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Mehrabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Another widely used assumption is that at least a subset of the original training data is available at the time of unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' In real settings, however, loading the original dataset may not be an option due to data expiration policies or lack of storage for large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Lastly, many approaches consider the main objective as re- moving the previous effect of the deleting data during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' While this is indeed the ideal case, recent work have shown that fulfilling the objective can still lead to information leakage (Suriyaku- mar & Wilson, 2022), and unlearning mechanisms must explicitly enforce misprediction for tighter security (Graves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' In light of aforementioned limitations, we propose a framework for instance-wise unlearning that deletes information with access only to the pre-trained model and the data points requested for unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Instead of undoing the previous influence of deleting data, we pursue a stronger goal where all requested data points are misclassified, preventing collection of information via interpo- lation of nearby data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Inspired by work in the continual learning literature (Ebrahimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2018), we propose two regularization methods that minimize loss in predic- tive performance on the remaining data, while completely forgetting information on deleting data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Specifically, we 1) generate adversarial examples by attacking each deleting data point with the pre-trained model and retrain on these examples to prevent representation-level forgetting and 2) use weight importance measures from unlearning instances to focus gradient updates more towards parameters responsible for the originally correct classification of such instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Extensive experi- ments on CIFAR-10/-100 (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2009) and ImageNet-1K (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2009) datasets show that our proposed method effectively preserves overall predictive performance, while com- pletely misclassifying images chosen for deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Our qualitative analyses also reveal interesting insights, including lack of any discernible pattern in misclassification that may be exploited by adversaries, preservation of the previously learned decision boundary, and forgetting of high-level features within deleted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' In summary, our main contributions are as follows: We propose instance-wise unlearning through intended misclassification, under the as- sumption that only the pre-trained model and data to forget are available at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We present two model-agnostic regularization methods that reduce forgetting on remaining data while misclassifying data for deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Empirical evaluations on well-known image classification benchmarks show that our pro- posed method significantly boosts predictive performance after unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 2 RELATED WORK Machine unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Machine unlearning (Cao & Yang, 2015) is a field that makes a pre-trained model forget information learned from a specified subset of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' For this, the existing studies have taken an approach that deletes the influence of unwanted data points (denoted as Df) from the model while retaining the predictive performance on the rest of the data (denoted as Dr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Mahadevan & Mathioudakis (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Ginart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Brophy & Lowd (2021) proposed unlearning methods for a linear/logistic regression, k-means clustering, and random forests, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' These methods are specifically designed for simple machine learning models, not for neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Table 1: Comparison between existing unlearning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Methods Unit Goal Dr Df Tarun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (2021) class undo \x13 \x17 Chundawat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (2022) class undo \x17 \x17 Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (2022) class undo \x17 \x13 Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (2022) class undo \x17 \x13 Golatkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (2020) instance undo \x13 \x13 Kim & Woo (2022) instance undo \x13 \x13 Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (2022) instance undo \x13 \x13 Our methods instance misclassify \x17 \x13 Recently, the machine unlearning for neural networks have been studied in different settings, shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' These methods can be categorized into two approaches: class-wise and instance-wise unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The class- wise unlearning is to forget a cer- tain class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 9-th class of CIFAR- 10) while retaining the performance on the remaining class (Tarun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Chundawat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' On the other hand, the instance-wise un- learning is designed to delete instance-wise information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', several images of CIFAR-10) from 2 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' the pre-trained model (Golatkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Kim & Woo, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' In other words, only instances that are requested to be forgotten should be deleted and the others from the same class should be remembered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The goal of the existing methods is to make the already trained model identical to the model trained on the dataset with unwanted instances removed (denote as undo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Unfortunately, even if the model is trained on the removed dataset, the interpolation capabilities of the neural networks may correctly predict even that we want to erase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' This does not lead to complete unlearning in practical applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Therefore, we define the goal of unlearning as to make the already trained model completely misclassifies the set of instances that should be forgotten (denote as misclassify).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Also, the existing methods have different access level to the unlearning data Df and the rest Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The existing solutions for instance-wise unlearning require access to the entire dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', Dr ∪ Df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' These methods which rely on the availability of the entire data are very far from real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' On the other hand, our proposed methods only need to the unlearning dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', Df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Since the vulnerability of neural networks has been revealed (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2013), various methods have been proposed to generate adversarial examples that can de- ceive neural networks (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Kurakin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Carlini & Wagner, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' In the case of white-box attack, an adversarial example can be generated by adding a hardly visible perturbation on a given image based on the gradient information from the model, making the model classify the image to a wrong class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The injected noise of the example is hard to distinguish visually but it causes a serious misclassification of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Recently, (Ilyas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2019) experimentally demonstrates that those noise is not meaningless but it rather contains (attack) target label’s features for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Weight importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Weight importance is a measure of how important each weight is when the model predicts an output for a given input data, and it has been used for different purposes, such as weight pruning (Molchanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Alvarez & Salz- mann, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2016) and regularization-based continual learning (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Among them, regularization-based continual learning has actively proposed various methods for measuring the weight importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' For overcoming catastrophic forgetting of previous tasks, the weight importance is utilized as the strength of the L2 regularization between a current model’s weight and the model’s weight trained up to the previous task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Most methods estimate the weight-level importance based on a gradient of a given input data (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 3 METHOD 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='1 PRELIMINARIES AND NOTATIONS Dataset and pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Let Dtrain be the entire training dataset used to pre-train a clas- sification model gθ : X → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We denote Df ⊂ Dtrain as the unlearning dataset that we want to intentionally forget from the pre-trained model and Dr as the remaining dataset on which we wish to maintain predictive accuracy (Dr := Dtrain \\ Df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We denote a pair of an input image x ∈ X and its ground-truth label y ∈ Y from Dtrain as (x, y) ∼ Dtrain, similarly (xf, yf) ∼ Df and (xr, yr) ∼ Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Also, Dtest denotes the test dataset used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Note that our approaches assumes access to only the pre-trained model gθ and the unlearning dataset Df during unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The goal of an adversarial attack on an input (x, y) is to generate an adversarial example x′ that is similar to x, but leads to misclassification (gθ(x′) ̸= y) when fed to the pre-trained model gθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' In the case of targeted adversarial attack, it makes the model predict a specific class different from the true class (gθ(x′) = ¯y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The typical optimization form of generating adversarial examples in targeted attack is denoted as x′ = arg min z:∥z−x∥p≤ϵ LCE(gθ(z), ¯y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) (1) where LCE stands for the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The ∥z − x∥p ≤ ϵ condition requires that the Lp- norm is less than a perturbation budget ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The optimization above is intractable in general, and 3 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Figure 1: Illustrations of our approaches that reduce forgetting on the remaining data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (Top) Aug- menting adversarial examples from unlearning data provides support for preserving the overall de- cision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (Bottom) Weight importance measures allow us to pinpoint weights we should change to induce misclassification while maintaining other weights to mitigate forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' thus several papers have proposed approximations that can generate adversarial examples without directly solving it (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Kurakin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Carlini & Wagner, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' In this paper, we make use of L2-PGD targeted attacks Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (2017) for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Measuring weight importance with MAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' To measure weight importance Ω, we consider MAS (Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2018), an algorithm that estimates weight importance by finding parameters that brings a significant change in the output when perturbed slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' It estimates the weight impor- tance via a sum of gradients on the L2 norm of the outputs: Ωi = 1 N N � n=1 ���� ∂∥gθ(x(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ)∥2 2 ∂θi ���� (2) where i stands for the index of network parameter weights and x(n) denotes n-th input image from a total of N numbers of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Note that each Ωi can be interpreted as a measure of influence or importance of θi in producing the output of given N input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 INSTANCE-WISE UNLEARNING FOR PRE-TRAINED CLASSIFIERS Definition of instance-wise unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Let ˆgθ denote the model after unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We consider two types of goals for instance-wise unlearning: (i) misclassifying all data points in Df, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', ˆgθ(xf) ̸= yf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (ii) relabeling (or correcting) the predictions of Df (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', ˆgθ(xf) = y∗ f) where y∗ f ̸= yf is chosen individually for each input xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Let LUL denote a loss function used for unlearn- ing on a classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The above two goals can be realized with the following loss functions: LMS UL(Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) = −LCE(gθ(xf), yf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) (3) LCor UL (Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) = LCE(gθ(xf), y∗ f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) (4) When unlearning solely based on the two loss functions above, the model is likely to suffer from significant forgetting on Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Therefore, a crucial objective shared across both unlearning goals is to overcome forgetting of previously learning knowledge, and maintain as much classification accuracy as possible on Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' When both Df and Dr are available, we can easily obtain an oracle model that satisfies the objec- tive by re-training the model with the following loss function: Loracle(Df, Dr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) = LUL(Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) + LCE(Dr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' However in real-settings, access to Dr may not be an option due to high cost in data 4 Unlearning dataset D Adversarial examples D ★:(★,★) Remaining dataset DrPreprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Algorithm 1 Generate adversarial examples Input: Forgetting data Df, Model gθ Output: Adversarial examples ¯Dr 1: ¯Dr ← ∅ 2: for i in range Nf do 3: (x(i), y(i)) ∼ Df 4: Randomly sample ¯y(i) ̸= y(i) 5: for j in range Nadv do 6: x′(j) f ← L2-PGD(x(i), ¯y(i)) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 1) 7: ¯Dr ← ¯Dr ∪ {(x′(j) f , ¯y(j))} 8: end for 9: end for 10: return ¯Dr Algorithm 2 Measure weight importance Input: Forgetting data Df, Model gθ Output: Weight importance ¯Ω 1: ¯Ω ← {0} 2: Ω ← weight importances(Df, gθ) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 2) 3: for l in range L do 4: Get importance of l-th layer Ωl ← Ω 5: Normalize Ωl ← Ωl − Min(Ωl) Max(Ωl) − Min(Ωl) 6: Update ¯Ωl ← {1 − Ωl} 7: end for 8: return ¯Ω storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' To tackle this limitation, we define regularization-based unlearning for which the goal is to achieve the goal above without explicit use of Dr: LRegUL(Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) = LUL(Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) + R(Df, gθ) (5) Here, R(·) is the regularization term used to overcome forgetting of knowledge on the remaining data Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' In the following subsections, we introduce two novel regularization methods designed to overcome representation- and weight-level forgetting during the unlearning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Regularization using adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The motivation of using adversarial examples stems from the work of Ilyas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (2019), which showed that perturbations added to x to generate an ad- versarial example x′ contain class-specific features of the attack target label ¯y ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Based on this finding, we utilize generated adversarial examples as part of regularization R(·) to preserve class- specific knowledge previously learned by the model, overcoming forgetting during unlearning at the representation-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Let Df be a set of Nf images: {(x(i) f , y(i) f )}Nf i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Prior to the unlearning pro- cess, we generate adversarial examples x′ f using the targeted PGD attack with a randomly selected attack target label ¯y ̸= yf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We generate Nadv adversarial examples per input xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Then, we have ¯Df = {(x′(k) f , ¯y(k) f )} ¯ Nf k=1 where ¯Nf = Nf × Nadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' During unlearning, we add LCE( ¯Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) as a regularization term with adversarial examples: LAdv UL (Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) = LUL(Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) + RAdv(Df, gθ) = LUL(Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) + LCE( ¯Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) (6) An intuitive illustration of this approach in the representation-level is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The gen- erated adversarial examples ¯Df mimic the remaining dataset Dr, providing information of the pre- trained decision boundary within the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' As a result, by adding LCE( ¯Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) as a regularizer to the unlearning process, the model can learn a new decision boundary that minimizes LUL (in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 3 and 4) while simultaneously attempting to keep the decision boundary of the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The pseudocode for generating adversarial examples is in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Regularization with weight importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We also propose a regularization using weight impor- tance to overcome forgetting at the weight-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' As depicted in Figure 1, our approach is to maintain the weights that were less important for Df prediction as much as possible, while allowing changes in weights that are considered important for correctly predicting Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' That is, it is to prevent the weight-level forgetting by penalizing weights that were less important when predicting Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' For this, we calculate the weight importance with MAS before unlearning given gθ and Df, and normalize the measured importances Ωl within each l-th layer to lie within [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Note that this normalized importance Ωl assigns large values to weights important for Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Therefore, we define ¯Ωl = 1 − Ωl as the weight importance for the regularization used for unlearning, so that more important weights are updated more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The objective including weight importance regularization in 5 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Table 2: Evaluation results before and after unlearning k instances from ResNet-50 pretrained on respective image classification datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' While using negative gradients only loses significant infor- mation on Dr, our proposed methods ADV and ADV+IMP retain predictive performance on Dr as well as Dtest, while completely forgetting instances in Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' CIFAR-10 CIFAR-100 ImageNet-1K k = 4 k = 16 k = 64 k = 128 k = 4 k = 16 k = 64 k = 128 k = 4 k = 16 k = 64 k = 128 Df (↓) BEFORE 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='10 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='01 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='01 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='01 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='01 NEGGRAD 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='56 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='87 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='28 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='11 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='54 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='07 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='11 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='19 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='53 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='30 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='61 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='73 ADV 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='34 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='14 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='23 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='47 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='00 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='17 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='43 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='89 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='12 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='42 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='89 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='73 ADV+IMP 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='53 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='65 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='08 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='82 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='50 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='69 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='83 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='44 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='15 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='97 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='72 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='09 addition to regularization via adversarial examples can be written as: LAdv+Imp UL (Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) = LAdv UL (Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) + RImp(Df, gθ) = LAdv UL (Df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' θ) + � i ¯Ωi(θi − ˜θi)2 (7) where i is the index of each weight and ˜θ is the initial weight of the pre-trained classifier before unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The pseudocode of measuring weight importance is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Throughout various experiments, we observe that applying the regularization using adversarial examples is al- ready effective to overcome the forgetting for knowledge of Dr, and the additional regularization with weight importance further enhances performance even further, especially in more harder sce- narios such as continual unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The pseudocode of the overall unlearning pipeline is shown in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 4 EXPERIMENTS In this section, we evaluate our proposed instance-wise unlearning methods in various image clas- sification benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We first describe our experimental setup, including datasets, baselines and experimental details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We then show that our methods effectively preserves knowledge of remaining data while unlearning instances that should be forgotten in both single-task and continual unlearning scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Lastly, we offer qualitative analyses on three parts: prediction patterns, decision boundary and layer-wise representations in unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='1 SETUP Datasets and baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We evaluate our unlearning methods on three different image classifi- cation datasets: CIFAR-10, CIFAR-100 (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2009), and ImageNet-1K (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Also, we use the ResNet-50 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2016) as a base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The experimental results of various base models are available in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The compared methods are as follows: BEFORE, the pre-trained model before unlearning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' NEGGRAD (Golatkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2020), fine-tuning on Df using negative gradients (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' LMS UL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' CORRECT, fine-tuning using LCor UL ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' ADV is our proposed method using adversarial examples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' LAdv UL );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' ADV+IMP, our unlearning method using both adversarial examples and the weight importance regularization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' LAdv+Imp UL ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Experimental details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' For each dataset, we randomly pick k ∈ {4, 16, 64, 128} images from the entire training dataset as the unlearning data Df and consider the remaining as Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' For the unlearn- ing, we use a SGD optimizer with a learning rate of 1e-3, weight decay of 1e-5, and momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='9 across all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We take early stopping when the model attains zero accuracy from the unlearning data Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' For generating adversarial examples from Df, we use L2-PGD targeted attack (Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2017) with a learning rate of 1e-1, attack iterations of 100 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' It gen- erates 20 adversarial examples for CIFAR-10 and 200 examples for CIFAR-100 and ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' For the weight importance regularization, we set regularization strength λ = 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 6 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Table 3: Results analogous to Table 6, but with unlearning via relabeling each image in Df to an arbitrarily chosen class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We see a similar trend where CORRECT loses significant information on Dr, while our proposed methods retain predictive performance on Dr as well as Dtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' CIFAR-10 CIFAR-100 ImageNet-1K k = 4 k = 16 k = 64 k = 128 k = 4 k = 16 k = 64 k = 128 k = 4 k = 16 k = 64 k = 128 Df (↑) BEFORE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='08 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='24 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='92 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='82 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='60 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='28 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='15 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='60 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='94 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='20 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='82 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='92 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 MAIN RESULTS Results on various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Table 6 shows evaluation results before and after unlearning k in- stances from ResNet-50 models pre-trained on each of three different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' With respect to accuracies on Df, we find that ResNet-50 can completely forget up to k = 128 instances with consistently zero post-unlearning accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' On CIFAR-10, using negative gradients only results in significant loss of accuracy on the remaining data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Dr and Dtest), performing worse than random-choice when the number of forgetting instances is as large as 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Meanwhile, adding regu- larization with adversarial examples boosts the accuracy by more than 40% depending on the number of instances to forget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Incorporating weight importances from MAS provides further improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Results from CIFAR-100 and ImageNet-1K show a similar trend except when k = 4, where adding our regularization approaches deteriorates performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' This well aligns with our intuition as the model can easily misclassify a small number of examples by tweaking a small number of model parameters, hence forgetting Df without losing much information on Dr and Dtest despite lack of regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The benefit of using adversarial examples is also small when k is small as the diversity amongst images in Dadv is limited by the number of instances to forget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Table 7 shows results analogous to Table 6, but with the goal of relabeling data points in Df to arbitrarily chosen labels rather than misclassifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We find that a similar trend, where ADV attains significantly less forgetting in Dr and Dtest compared to CORRECT, while succesfully relabeling all points in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' While ADV+IMP show even less forgetting, it loses accuracy in relabeling Df, showing that regularization via weight importance focuses too much on retaining previous knowl- edge rather than adapting to corrections provided in Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' An intuitive explanation on why this occurs particularly in relabeling is that while misclassifying can be done easily by driving the input to its closest decision boundary, relabeling can be difficult if the new class is far from the original class in the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The difficulty rises even more when the size of Df is large, in which case more parameters in the network are discouraged from being updated during unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Correcting natural adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Leveraging the ImageNet-A (Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2021) dataset consisting of natural images that are misclassified with high-confidence by strong classifiers, we test whether our method can make corrections on these adversarial examples, while preserving knowledge from the original training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' For this experiment, we consider Df to consist k adver- sarial images from ImageNet-A, and adjust a ResNet-50 model pre-trained on ImageNet-1K to cor- rectly classify Df via our unlearning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Table 4 shows the results for k = {16, 32, 64, 128}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We find that correcting predictions of a small number of images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' k = 16), finetuning the model na¨ıvely with cross-entropy only attains the best accuracy in both Dr and Dtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' When correcting larger number of images, however, the absence of regularization terms results in larger forgetting in Dr compared to ADV and ADV+IMP, with a performance gap that consistently increases with the number of adversarial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Another takeaway is that regularization via weight importance does not help in this scenario, even showing a significant drop in Df accuracy when a large number of adversarial images are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' This implies that using weight importances imposes too strong a regularzation that correcting predictions for Df itself becomes non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We conjecture that the aggregation of important parameters for predictions in Df cover a large proportion of the network with large k, and that careful search for the Pareto optimal between accuracies on Df and on Dr is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 7 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Table 4: Correcting adversarial images from ImageNet-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' ADV achieves the least forgetting, while ADV+IMP fails to correct large number of predictions due to strong regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' ImageNet-A k = 16 k = 32 k = 64 k = 128 Df (↑) BEFORE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 CORRECT 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 ADV 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='31 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='44 ADV+IMP 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='94 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='38 Dr (↑) BEFORE 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='46 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='46 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='46 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='46 CORRECT 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='41 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='29 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='79 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='38 ADV 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='75 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='80 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='74 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='44 ADV+IMP 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='82 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='73 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='53 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='86 Dtest (↑) BEFORE 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='15 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='15 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='15 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='15 CORRECT 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='21 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='04 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='91 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='73 ADV 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='89 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='58 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='68 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='36 ADV+IMP 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='98 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='51 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='39 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='68 Table 5: Unlearning instances continually by in- crements of kCL = 8 images per step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Our meth- ods outperform NEGGRAD in the continual un- learning scenario as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' CIFAR-100 (kCL = 8) k = 8 k = 16 k = 64 k = 128 Df (↓) BEFORE 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 NEGGRAD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='52 ADV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 ADV+IMP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='04 Dr (↑) BEFORE 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='98 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='98 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='98 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='98 NEGGRAD 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='58 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='85 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='89 ADV 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='33 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='54 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='67 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='16 ADV+IMP 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='46 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='78 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='30 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='14 Dtest (↑) BEFORE 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='10 NEGGRAD 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='20 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='22 ADV 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='56 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='43 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='48 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='10 ADV+IMP 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='33 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='97 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='09 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='17 Continual unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' In real-world scenarios, it is likely that data removal requests come as a stream, rather than all at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Ultimately, despite continual unlearning requests, we need the unlearning method that can delete the requested data while maintaining performance for the rest data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Thus, we consider the setting of deleting k = {8, 16, 64, 128} data by repeating the procedure of continually unlearning Df in small fragments of size kCL = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Table 5 shows the results of continual unlearning in the model trained with ResNet-50 on CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We observe that NEGGRAD suffers from large forgetting as the iteration of unlearning procedure increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' On the other hand, our proposed method shows significantly less forgetting while effectively deleting for Df even after multiple iterations of unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='3 QUALITATIVE ANALYSIS Through further analysis, we gather insight on the following questions: Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Is there any particular pattern in how the model unlearns a set of instances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' does the model use any particular label as a retainer for deleted data)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' How does the model isolate out instances in Df from its previous decision boundary?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' How do layer-wise representations of data points in Df and Dr change before and after unlearning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' For interpretable visualizations, we perform the following analysis on a ResNet-18 model pre-trained on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (a) NEGGRAD (b) ADV (c) ADV+IMP Figure 2: Confusion matrices showing average pairwise frequencies of pre- (Y-axis) and post- unlearning (X-axis) prediction labels from Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' A hue closer to blue indicates higher frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Our unlearning framework does not produce any dis- cernible correlation in misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Our method shows no pattern in mis- classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We first check whether the un- learned model classifies all instances in Df to a particular set of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' The model exhibit- ing no correlation between true labels and new misclassified labels is crucial with respect to data privacy, as it indicates that the unlearn- ing process avoids the so-called Streisand ef- fect where data instances being forgotten unin- tentionally becomes more noticeable (Golatkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Figure 2 shows the confusion ma- trices of (pre-unlearning label, post-unlearning label) pairs from Df for k = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We find no distinguishable pattern when unlearning with our methods as well as NegGrad, which shows that no specific label is used as a retainer, which adds another layer of security against adversaries in search of unlearned data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Our method effectively preserves the decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We check whether the adversar- ial examples generated from forgetting data help in preserving the decision boundary in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Figure 3 shows t-SNE (Van der Maaten & Hinton, 2008) visualizations of final-layer acti- vations from examples in Dr and Df before and after unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We find that unlearning through only negative gradient significantly distorts the previous decision boundary, leading to poor predic- 8 NegGrad 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='5 9 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='4 7 6 labels 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='3 5 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 m 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0 2 7 Predicted labelsOurs (Adv) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='5 9 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='4 7 6 labels 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='3 5 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 m 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0 2 7 Predicted labelsOurs（Adv+Imp） 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='5 9 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='4 7 6 labels 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='3 5 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 m 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='1 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0 2 3 4 > 9 Predicted labelsPreprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (a) BEFORE (b) NEGGRAD (c) ADV (d) ADV+IMP Figure 3: t-SNE plots of CIFAR-10 datapoints in Df (triangles) and Dr (dots) before and after unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Colors indicate true labels for all plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Regularization with adversarial examples and weight importance effectively preserves the decision boundary while migrating instances in Df towards the class boundary to induce misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' tive performance across Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' However, when we incorporate adversarial samples from instances in Df, the decision boundary is well-preserved with unlearned examples being inferred as boundary cases in-between multiple classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Even for examples that lie far from the decision boundary be- fore unlearning, our method successfully relocates the corresponding representations towards the decision boundary, while keeping each class cluster intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (a) NEGGRAD (b) ADV (c) ADV+IMP Figure 4: Layer-wise CKA correlations on Df (top row) and Dr (bottom row) between repre- sentations before (X-axis) and after (Y-axis) un- learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Brighter color indicates higher CKA cor- relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' NEGGRAD results in large forgetting of high-level features in not only Df, but also Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Our approaches, on the other hand, selectively for- get high-level features only in Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Our method unlearns data by forgetting high-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Lastly, we compare the representations at each layer of the model be- fore and after unlearning to identify where the intended forgetting occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' For this analysis, we leverage CKA (Kornblith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2019) which measures correlations between representations given two distinct models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Figure 4 shows the CKA correlation heatmaps between the origi- nal ResNet-18 model pre-trained on CIFAR-10 and the same model after unlearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Results show that for examples in Df, representations are no longer aligned starting from the 10-th layer while the representations before that layer still resemble those from the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' This indicates that the model forgets examples by forgetting high-level features, while simi- larly recognizing low-level features in images as the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' This insight is consis- tent with previous observations in the contin- ual learning literature that more forgettable ex- amples exhibit peculiarities in high-level fea- tures (Toneva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 5 CONCLUDING REMARKS We propose an instance-wise unlearning framework that deletes information from a pre-trained model given a set of data instances with mixed labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Rather than undoing the influence of given instances during the pre-training, we aim for a stronger form of unlearning via intended misclas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We develop two regularization techniques that reduce forgetting on the remaining data, one utilizing adversarial examples of deleting instances and another leveraging weight importances to focus updates to parameters responsible for propagating information we wish to forget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Both ap- proaches are agnostic to the choice of architecture, and requires access only to the pre-trained model and instances requested for deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Experiments on various image classification datasets showed that our methods effectively mitigates forgetting on remaining data, while completely misclassify- ing deletion data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Further qualitative analyses show that our unlearning framework does not show any pattern in misclassification (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' the Streisand effect), preserves the decision boundary with the help of adversarial examples, and unlearns by forgetting high-level features of deleting data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' These 9 Residual Data (D_r) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 16 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='8 Case 3: -CE(D_f) + CE(adv) + Reg(importance) 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='4 4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 2 Fo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0 2 4 6 8 10 12 14 16 Before UnlearningForgettingData(D_f) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 16 14 - F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='8 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='6 10 Case 1: - CE(D_f) 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='4 6 4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 2 4 6 8 10 12 14 16 Before UnlearningForgettingData(D_f) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 16 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='8 12 Case 2: -CE(D_f) + CE(adv) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='6 10 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='4 6 4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 2 4 6 8 10 12 14 16 Before UnlearningForgettingData(D_f) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 16 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='8 Case 3: -CE(D_f) + CE(adv) + Reg(importance) 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='6 10 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='4 6 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 2 4 6 8 10 12 14 16 Before UnlearningResidual Data (D_r) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 16 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='8 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='6 Case 1: - CE(D_f) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='4 9 4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0 2 4 6 8 10 12 14 16 Before UnlearningResidual Data (D_r) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 16 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='8 12 Case 2: -CE(D_f) + CE(adv) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='6 10 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='4 6 4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 0 2 4 6 8 10 12 14 16 Before UnlearningPreprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' observations shed light towards future work evaluating the utility our approach as a defense mech- anism against membership inference attacks that predict whether a data point was included in the training set by using posterior confidence (Shokri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Yeom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Sablayrolles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2019) or its distance to nearby decision boundaries (Choquette-Choo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Li & Zhang, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Removing harmful information that lead to socially unfair and biased predic- tions based upon sensitive traits such as race, gender, and religion (Mehrabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2021) is another potential contribution from this work.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 13 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='1 PSEUDO CODE OF OVERALL UNLEARNING PROCESS Algorithm 3 The pseudo code of overall unlearning process the case of using LMS UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 1: UNLEARNACC = 100 2: MAXEP = 100 3: EP = 0 4: ¯Dr ← Generate adversarial examples with Algorithm 1 5: ¯Ω ← Measure weight importance with Algorithm 2 6: ˜θ ← θ 7: while UNLEARNACC ̸= 0 do 8: Minimize Eqn (6) and (7) 9: UNLEARNACC = GetAccuracy(Df, gθ) 10: if EP > MAXEP then 11: break 12: EP += 1 13: end if 14: end while 15: return ˆθ Algorithm 4 The pseudo code of overall unlearning process the case of using LCor UL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 1: UNLEARNACC = 0 2: MAXEP = 100 3: EP = 0 4: ¯Dr ← Generate adversarial examples with Algorithm 1 5: ¯Ω ← Measure weight importance with Algorithm 2 6: ˜θ ← θ 7: while UNLEARNACC ̸= 100 do 8: Minimize Eqn (6) and (7) 9: UNLEARNACC = GetAccuracy(Df, gθ) 10: if EP > MAXEP then 11: break 12: EP += 1 13: end if 14: end while 15: return ˆθ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='2 ADDITIONAL RESULTS ON VARIOUS MODELS Results on various models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Figure 5 shows unlearning results on CIFAR-100, but with different model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We find that our methods effectively preserve knowledge outside the forgetting data, resulting in up to 40% boost in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' NegGrad again outperforms our methods when k = 4, but soon breaks down when unlearning more instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Interestingly, SqueezeNet and MobileNetv2 suffer from larger forgetting in Dr and Dtest than ResNet-50, possibly due to the width being nar- rower as previously investigated by Mirzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' ViT also suffers from large forgetting, an observation consistent with previous work which showed that ViT suffers more catastrophic forget- ting compared to other CNN-based methods in continual learning due to Transformer architectures requiring large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We also evaluate the results of unlearning on ImageNet-1K with varying k in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Our proposed methods prevent forgetting knowledge about the rest data Dr better than NegGrad in all cases where k is greater than 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' At the same time, the methods effectively delete information about Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='3 SUPPLEMENTARY MATERIALS FOR REBUTTAL 14 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Table 6: Evaluation results before and after unlearning k instances from ResNet-50 pretrained on respective image classification datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' While using negative gradients only loses significant infor- mation on Dr, our proposed methods ADV and ADV+IMP retain predictive performance on Dr as well as Dtest, while completely forgetting instances in Df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' CIFAR-10 CIFAR-100 k = 4 k = 16 k = 64 k = 128 k = 4 k = 16 k = 64 k = 128 Df (↓) BEFORE 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2016) (c) ViT (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=', 2020) Figure 5: Experimental results before and after unlearning varying k instances from various models on CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 16 100 Original NegGrad Ours (Adv) 80 Ours (Adv+Imp) Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 40 20 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 Original NegGrad Ours (Adv) 80 Ours (Adv+Imp) Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 40 20 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Original 60 NegGrad Ours (Adv) 40 Ours (Adv+Imp) 20 0 1 2 4 8 16 32 64 128 256 Number of unlearnina dataset (Df)100 80 Dr Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 40 Original 20 NegGrad Ours (Adv) Ours (Adv+Imp) 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Df Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content='★-Original .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' NegGrad Ours (Adv) 40 Ours (Adv+Imp) 20 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Dr Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 40 Original 20 NegGrad Ours (Adv) Ours (Adv+Imp) 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Df Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 ★:Original NegGrad Ours (Adv) 40 Ours (Adv+Imp) 20 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Dr Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Original 60 NegGrad Ours (Adv) 40 Ours (Adv+Imp) 20 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Dr Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Original 60 NegGrad Ours (Adv) 40 Ours (Adv+Imp) 20 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (a) MobileNet v2 (b) ResNet34 (c) DenseNet121 Figure 6: Experimental results before and after unlearning varying k instances from various models on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' (a) Analysis for entropy-accuracy (b) Analysis for a forgotten label Figure 7: Experimental analysis with CIFAR-10 dataset using ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' We randomly select single image (k = 1) for unlearning and unlearn it with NegGrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' All experiments are conducted with 100 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' Each class number denotes a specific label, such as {airplane : 0, automobile : 1, bird : 2, cat : 3, deer : 4, dog : 5, frog : 6, horse : 7, ship : 8, truck : 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 17 100 Original NegGrad Ours (Adv) 80 Ours (Adv+Imp) Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 40 20 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 40 Original 20 NegGrad Ours (Adv) Ours (Adv+Imp) 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 40 Original 20 NegGrad Ours (Adv) Ours (Adv+Imp) 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 r Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 D 40 Original 20 NegGrad Ours (Adv) Ours (Adv+Imp) 0 L 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Df Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 ★:Original NegGrad Ours (Adv) 40 Ours (Adv+Imp) 20 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 r Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 D 40 Original 20 NegGrad Ours (Adv) Ours (Adv+Imp) 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Df Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 ★:Original NegGrad Ours (Adv) 40 Ours (Adv+Imp) 20 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Dr Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 40 Original 20 NegGrad Ours (Adv) Ours (Adv+Imp) 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)100 80 Df Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} +page_content=' 60 Original NegGrad Ours (Adv) 40 Ours (Adv+Imp) 20 0 1 2 4 8 16 32 64 128 256 Number of unlearning dataset (Df)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FJT4oBgHgl3EQfkSwi/content/2301.11578v1.pdf'} diff --git a/8NE1T4oBgHgl3EQfngRF/content/tmp_files/2301.03309v1.pdf.txt b/8NE1T4oBgHgl3EQfngRF/content/tmp_files/2301.03309v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d2931c21317eb1370884c343fe67d051b07d406 --- /dev/null +++ b/8NE1T4oBgHgl3EQfngRF/content/tmp_files/2301.03309v1.pdf.txt @@ -0,0 +1,1059 @@ +Mapping Charge-Transfer Excitations in +Bacteriochlorophyll Dimers from First Principles +Zohreh Hashemi1, Matthias Knodt1, Mario R. G. Marques1, Linn +Leppert1,2 +1)Institute of Physics, University of Bayreuth, Bayreuth 95440, Germany, +2)MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The +Netherlands +E-mail: l.leppert@utwente.nl +Abstract. +Photoinduced charge-transfer excitations are key to understand the primary +processes of natural photosynthesis and for designing photovoltaic and photocatalytic devices. +In this paper, we use Bacteriochlorophyll dimers extracted from the light harvesting apparatus +and reaction center of a photosynthetic purple bacterium as model systems to study such +excitations using first-principles numerical simulation methods. We distinguish four different +regimes of intermolecular coupling, ranging from very weakly coupled to strongly coupled, +and identify the factors that determine the energy and character of charge-transfer excitations +in each case. +We also construct an artificial dimer to systematically study the effects of +intermolecular distance and orientation on charge-transfer excitations, as well as the impact of +molecular vibrations on these excitations. Our results provide design rules for tailoring charge- +transfer excitations in Bacteriochloropylls and related photoactive molecules, and highlight +the importance of including charge-transfer excitations in accurate models of the excited-state +structure and dynamics of Bacteriochlorophyll aggregates. +arXiv:2301.03309v1 [physics.chem-ph] 9 Jan 2023 + +2 +1. Introduction +Photoinduced charge-transfer excitations are of central importance to the primary processes +of natural photosynthesis and for photovoltaic and photocatalytic applications [1, 2]. +In +organic semiconductors, charge-transfer excitations are believed to be important intermediates +between excited states localized on donor molecules and charge-separated electron-hole states +on acceptor and donor units, respectively, even though the exact mechanism of charge- +separation is debated [3–12]. +In photosynthesis, the efficient conversion of solar energy +into chemical energy is achieved by structurally complex aggregates of Bacteriochlorophylls +(BCL), Chlorophylls, and other pigment molecules embedded in transmembrane proteins +that modulate their structure and function. These pigment-protein complexes form light- +harvesting complexes and reaction centers that are responsible for photon absorption, +excitation-energy transfer, and charge-separation. Their main operating principles are well- +understood due to a wealth of crystallographic and spectroscopic studies complemented by +numerical modelling using semi-empirical and first-principles approaches [13–22]. +Figure 1. Crystal structure of BCL aggregates in the reaction center (RC) and light-harvesting +II (LHII) complex of the purple bacteria Rhodobacter sphaeroides and Rhodoblastus +acidophilus, respectively. Dimers of BCLs are highlighted in color using (a) pink for the +special pair PA – PB, (b) orange for the A branch dimer PA – BA, (c) red for a dimer from +the B800 and blue for a dimer from the B850 ring of the LHII complex. Hydrogen atoms are +omitted for clarity. +In purple bacteria, charge separation occurs in reaction centers (RCs) comprising a +hexameric aggregate of four BCLs and two Bacteriopheophytins, tightly surrounded by +several protein chains [23–25]. The primary four BCL molecules of this reaction center are +shown in Figure 1a, highlighting the so-called special pair (SP), a strongly-coupled dimer of +BCLs called PA – PB in the following. Charge separation in the bacterial RC is initiated by a +series of energy- and charge-transfer excitations that involve the SP and proceed along the A +branch, the photoactive of the two pseudo-symmetric branches the RC consists of [19,26–28]. +In Figure 1b, we have highlighted the A-branch dimer PA – BA that has been speculated to be +involved in the primary charge-separation step, although this assignment is debated in the +literature [29–32]. Excitation energy reaches the RC through a cascade of excitation-energy + +c +B +B +A +B850 +B +B +A +B8003 +transfer processes that are initiated in the light harvesting II (LHII) complex, consisting of +two rings of BCL molecules dubbed B850 and B800, respectively, and shown in Figure 1c. +Neighboring BCLs in the B800 ring are only weakly coupled and excitation-energy transfer +is well-described by Förster dipole-dipole coupling [33]. In the B850 ring, neighboring BCL +molecules are closer and intermediate between the weakly coupled B800 and the strongly +coupled special pair BCLs. +The excited states that are believed to be responsible for excitation energy transfer +in and between the light-harvesting complexes and the RC, are commonly thought of as +Frenkel-like excitons that are spatially relatively localized on one or two BCL molecules [34]. +Semi-empirical models based on Frenkel-excitons Hamiltonians have played an important +role in modelling the excitation-energy and charge-transfer dynamics in large photosynthetic +pigment-protein complexes [35–37]. However, for a reliable and predictive representation +of the electronic coupling between adjacent pigments, charge-transfer excitations need to +be included in these model Hamiltonians [36, 38–40], calling for accurate first-principles +calculations of such excitations. +For computationally efficient first-principles methods such as time-dependent density +functional theory (TDDFT), charge-transfer excitations were long considered a major +challenge due to their inherently nonlocal nature, i.e., the spatial separation of the occupied +and virtual orbitals contributing to these excitations [41]. +TDDFT with optimally-tuned +range-separated hybrid functionals is a viable solution to this problem, and has been used +to predict excited states of molecular systems and solids with great success [42–48]. In +these exchange-correlation functionals, the presence of long-range exact exchange leads to +asymptotically correct potentials. Additionally, a parameter controlling the range-separation +of exact and semilocal exchange can be used to tune the energies of the highest occupied and +the lowest unoccupied orbitals to correspond to the negative of the ionization potentials and +the electron affinity, respectively, within the conceptual framework of generalized Kohn-Sham +[49]. Both conditions are crucial for accurately capturing charge-transfer excitations within +linear-response TDDFT [50] and have been extended to solvated molecular systems [46, 51] +and extended solids [48]. +An alternative approach for calculating charge-transfer excitations of molecules and +solids is the GW+Bethe-Salpeter Equation (GW+BSE) approach [52,53]. While this method +was initially primarily applied to solids, recent years have witnessed a multitude of studies +that have demonstrated the accuracy and predictive power of the GW+BSE method for small +molecules [54–56] and larger molecular complexes [57–61]. +In particular, we [62] and +others [61] benchmarked the accuracy of the GW+BSE approach against experiment and +wavefunction-based methods and found excellent agreement for the Qy and Qx excitations +of a range of BCL and Chlorophyll molecules. +We showed that both eigenvalue self- +consistent GW calculations and one-shot G0W0 calculations where the zeroth-order single- +particle Green’s function G0 and screened Coulomb interaction W0 were constructed from a +DFT eigensystem obtained with an optimally-tuned range-separated hybrid functional lead +to the best results. TDDFT with an optimally-tuned hybrid-functional performed slightly +worse and tended to overestimate the energy of the Qy excitations, in agreement with previous + +4 +studies [57]. +In this article, we report a systematic first-principles study of charge-transfer excitations +in BCL dimers - the smallest structural units in which excitations with intermolecular charge- +transfer character can be observed. These BCL dimers, extracted from the LHII complex and +RC of purple bacteria, constitute our model systems. Our goal is to elucidate the factors that +determine the energy and character of these excitations, in particular their mixing with the +coupled Qy and Qx excitations of the dimers. We treat these dimers as representative of four +different regimes of intermolecular coupling resulting in distinct charge-transfer properties: +1. The B800 dimer is very weakly coupled with Qy and Qx excitations resembling those +of the monomeric units and high-energy charge-transfer excitations due to vanishing orbital +overlap. 2. The A-branch dimer is more strongly coupled and exhibits one charge-transfer +excitation corresponding to electron transfer from PA to BA. We use the notation P+ +A B− +A to +indicate the direction of charge-transfer in the following. +This charge-transfer excitation +is ∼0.4 eV higher in energy than the coupled Qx excitations. 3. The B850 dimer is even +more strongly coupled. The lowest-energy charge-transfer excitation mixes with the coupled +Qx excitations and another charge-transfer state appears at higher energies. 4. Finally, the +special pair SP is the most strongly coupled case with three charge-transfer excitations mixing +with the coupled Qx excitations. Additionally, we construct an artificial BCL dimer and +systematically study the effects of intermolecular distance and orientation on charge-transfer +excitations. We also estimate the effect of molecular vibrations on charge-transfer excitations. +We do this by calculating the vibrational normal modes of a dimeric system and determining +the renormalization of excitation energies for structures distorted along normal modes. This +allows us to identify vibrational modes with pronounced effects on charge-transfer excitations. +Finally, we comment on differences and similarities between TDDFT with an optimally-tuned +range separated hybrid functional and the GW+BSE approach. +2. Computational Methods +2.1. First-Principles Methods and Computational Details +For all calculations reported in this article, we used TDDFT as implemented in TURBOMOLE +version 7.5 [63] and the GW+BSE approach as implemented in MOLGW version 3.0 [64]. +Briefly, in the linear-response formulation of both methods the excitation energies Ωn can be +obtained by solving the matrix eigenvalue equation CZ = Ω2 +nZ, where C is +Cijσ,klτ = (εiσ −εjσ)2δi jδ jlδστ +2�εiσ −εjσ +√εkτ −εlτKi jσ,klτ +(1) +and the indices i,k refer to occupied, j,l to virtual orbitals and σ,τ to spin-indices. +Differences between TDDFT and the GW+BSE approach enter Equation 1 in two distinct +ways: 1. Through the differences between virtual and occupied orbital energies εiσ − εjσ +which are obtained from a (generalized) Kohn-Sham calculation in TDDFT and from the GW +approach in GW+BSE. 2. Through the kernel matrix element Ki jσ,klτ, which depends on +the exchange-correlation kernel fxc,σ - the functional derivative of the exchange-correlation + +5 +potential - in TDDFT, and on the screened Coulomb interaction W, typically evaluated in the +random phase approximation and at zero frequency, in the BSE approach [65–68]. +Here we use the optimally-tuned range-separated hybrid functional ωPBE for our +TDDFT calculations. +We use a range-separation parameter ω=0.171 a−1 +0 , which we +determined previously for a single BCL a molecule [62]. +The optimal-tuning procedure +follows the recipe by Stein et al. and ensures that the HOMO eigenvalue corresponds to +the ionization potential and the LUMO eigenvalue corresponds to the electron affinity of the +molecule [69]. We do not perform a new tuning procedure for the dimers for general reasons: +Using the same ω for each dimer allows us to compare the electronic and excited state +structure of these systems on the same footing. Furthermore, optimal tuning of conjugated +systems of increasing size leads to artificially low values of ω and, thus, a dominance +of semilocal exchange at long range, which deteriorates the description of charge-transfer +excitations [46,70]. +For our GW+BSE calculations we use a "one-shot" G0W0 approach in which we +construct the zeroth-order single-particle Green’s function G0 and the screened Coulomb +interaction W0 from DFT eigenvalues and eigenfunctions calculated using the same ωPBE +as described above. This approach leads to excellent agreement with experimental excitation +energies and reference values from wavefunction-based methods for a range of BCL and +Chlorophyll molecules [62]. Range-separated hybrid functionals have been shown to lead +to accurate charge-transfer excitations for larger molecular complexes as well [57,71]. In all +calculations we used a def2-TZVP basis set, and the frozen core and resolution-of-the-identity +approximations (with the DeMon auxiliary basis set [72]). We did not apply the Tamm- +Dancoff approximation in any of the results reported in this paper. In our G0W0 calculations, +we used the optimized virtual subspace method by Bruneval with an aug-cc-pVDZ basis set +for the reduced virtual orbital subspace [73]. With these settings, our excitation energies +are converged to within 40 meV. Further details on our convergence tests can be found in +Section 2.2 and in the Supplemental Material (SM). +For evaluating the character of the excited states, we calculated their transition densities. +Since the transition density vanishes for charge-transfer states, we calculated the difference +density ∆ni = ni − n0 between the excited (ni) and the ground-state density (n0) for every +excitation i. The excited-state density ni is calculated as the diagonal part of the excited +state density matrix γii(r,r′) = N +� Ψi(r,r2,r3,...,rn)Ψi(r′,r2,r3,...,rn)dr2...drn, where N is +the number of electrons and Ψi is the generalized Kohn-Sham excited-state wavefunction, that +consists of a sum of Slater determinants of generalized Kohn-Sham orbitals with coefficients +obtained from TDDFT [74]. To quantify the magnitude of charge transfer we integrated over +subsystem difference densities. For this purpose, we subdivided the volume containing the +difference densities of the dimer into subsystem volumes, each containing one pigment. Our +aim is to assign each grid point of the difference-density grid to its closest pigment molecule. +For achieving this, we used the distances between grid points and each molecule’s atomic +coordinates (including hydrogen atoms), as previously done in Ref. [75]. +Finally, to obtain a mode-resolved picture of the effect of thermally-activated vibrations +(Section 3.3), we relaxed a dimer structure using the B3LYP approximation for the exchange- + +6 +correlation functional and def2-TZVP basis set, and evaluated its normal modes and +frequencies. Using the harmonic approximation, we can relate the amplitude of these normal +modes with the thermal energy of a molecule. Thus, we distorted the dimer structure along +its lowest-frequency normal modes at a temperature of 300 K. In this manner, we generated +60 distortions of the dimer, that we then studied using TDDFT calculations using the ωPBE +functional. All these calculations were performed using the tools provided in the TURBOMOLE +package. +2.2. Convergence of G0W0+BSE calculations +We carefully tested that our GW+BSE results are converged. Due to the large size of a +BCL dimer, featuring more than 300 electrons, the calculation of the GW self-energy which +requires summation over virtual states is computationally demanding. We therefore used the +optimized virtual subspace method implemented in the MOLGW code, in which a reduced +virtual orbital subspace represented by a comparably small basis set is used to evaluate the +GW self-energy [73]. +Figure 2. Convergence of GW (a) HOMO-LUMO gap and (c) energy of the first excited +state of BCL a monomer as a function of the number of basis functions. Blue data points +correspond to calculations in which the same basis set is used for the occupied orbitals and the +virtual subspace. Red points correspond to calculations using the optimized virtual subspace +method. Lines are fits to these data points. Convergence of the HOMO-LUMO gap and energy +of the first excited state is shown in panel (b) and (d) for the B850 dimer, respectively. Here, +green corresponds to using the same basis set for the occupied orbitals and the virtual subspace +and pink to calculations using the optimized virtual subspace method. + +(c) ++ +SCF basis = GW basis +1.65 +SCF basis = GW basis +SCF basis = Def2-TZVP +SCF basis = Def2-TZVP +4.20 +1.60 +4.15 +.55 +4.10 +1.50 +4.05 +Monomer +Monomer +1.45 +4.00 +120014001600 +800 +1000 1200 1400 1600 +800 +1000 +N +(d) +N +b +basis +basis +SCF basis = GW basis +.65 +4.00 +SCF basis = GW basis + excitation (eV) +SCF basis = Def2-TZVP +SCF basis = Def2-TZVP +3.95 +1.60 +3.90 +1.55 +3.85 +1.50 +3.80 +>1.45 +3.75 +3.70 +Dimer +1.40 +Dimer +3.65 +1500 +2000 +2500 +3000 +3500 +1500 +2500 +3500 +2000 +3000 +N +N +basis +basis7 +We start by testing the convergence of the HOMO-LUMO gap, and the Qy and Qx +excitations of a BCL a monomer with respect to basis set size without the optimized virtual +subspace method (Table S1). In agreement with our previous results [62], we find that the +def2-TZVP basis set deviates by less than 10 meV from the considerably larger aug-cc-pVTZ +basis. We proceeded by calculating the convergence of the Qy and Qx excitations of the BCL +monomer as a function of the number of virtual orbitals Nvirt included in the evaluation of +the GW self-energy using the def2-TZVP basis (Figure S1). We find that for Nvirt = 500 +both excitations are converged to within 80 meV from the limit of infinite Nvirt. Based on +these findings we continued by evaluating the effect of using a smaller basis set for the +virtual subspace [73]. +The results for the HOMO-LUMO gap and the Qy excitation are +plotted in Figure 2a and c, and show that the optimized virtual subspace method leads to +an underestimation of the HOMO-LUMO gap and the Qy excitation energy as compared to +the conventional method in which the same basis set is used for all orbitals. We find that using +the aug-ccpVDZ basis for the optimized virtual subspace in conjunction with Nvirt = 500 leads +to a fortuitous error cancellation and results in a HOMO-LUMO gap and Qy and Qx excitation +energies that are within less than 50 meV of the results obtained with the conventional method +and Nvirt → ∞ (Figure S2). +For the dimer, we therefore chose Nvirt = 1000 and the same strategy for determining the +optimized virtual subspace. We find very similar results for the convergence of the HOMO- +LUMO gap and the first bright coupled Qy excitation shown in Figure 2b and d. All GW+BSE +results reported in this paper are therefore based on calculations using the def2-TZVP basis +set for the occupied orbitals and the aug-ccpVDZ basis for the optimized virtual subspace. +2.3. Construction of the Model Systems +We constructed our model systems from the x-ray crystallographic structures of the purple +bacteria Rhodobacter sphaeroides (structure ID 1M3X in the Protein Data Base) [76] and +Rhodoblastus acidophilus (structure ID 1NKZ) [77]. +In all structures, we replaced the +phytyl tail with hydrogen. Hydrogen atoms not resolved in the experimental crystal structure +were added using AVOGADRO and their positions were optimized while keeping the rest of +the structure fixed. These geometry optimizations were performed using TURBOMOLE and +the B3LYP exchange-correlation functional. The reaction center dimers PA – PB and PA – +BA (Figures 1a and b) were constructed using structure 1M3X while the B800 and B850 +ring dimers (Figure 1c) were extracted from 1NKZ. +These molecules correspond to ID +numbers BCL307 and BCL309 for the B800, and BCL302 and BCL303 for the B850 ring. +We additionally constructed an artificial dimer consisting of two exactly equivalent BCL a +molecules (using molecule PA) that we initially oriented in the same way as the special pair +dimer PA – PB by aligning their transition dipole moments (as calculated with TDDFT) with +those of PA and PB, respectively. We are providing all relevant structure files necessary to +reproduce the results of this article in the SM. + +8 +3. Discussion and Results +3.1. Charge-Transfer Excitations in RC and LHII Dimers +We start by comparing the excitation spectrum of the four dimeric systems shown in Figure 1a- +c using TDDFT and GW+BSE. The energies and oscillator strengths of the first 15 excitations +of each system can be found in Table S3 and S4. +The spectra are shown in Figure 3a +and b, respectively, and allow for several observations. +First, we find that TDDFT and +GW+BSE predict qualitatively very similar spectra. The most striking difference appears for +the B800 dimer, for which the coupled Qy excitations calculated with TDDFT are ∼0.3 eV +higher in energy than with GW+BSE while all other excitations are at similar energies. This +observation is consistent with our results for single BCL a molecules for which TDDFT with +optimally-tuned ωPBE consistently overestimates the Qy excitation energy by ∼0.3 eV [62] +and therefore leads to an underestimation of the Qy – Qx energy difference as compared +to experiment. +Interestingly, this overestimation as compared to GW+BSE, while still +present, is less pronounced for the other three dimers and seems to decrease with increasing +intermolecular coupling. +Figure 3. Excitation spectrum of B800, A-branch, B850, and SP dimers using (a) TDDFT +with ωPBE and (b) the G0W0@ωPBE+BSE approach. Arrows mark dark excitations without +(D) and with (CT) charge-transfer character. The shaded areas are calculated by folding the +excitation energies with Gaussian functions with a width of 0.08 eV as a guide to the eye. +Second, we find several dark excitations for all four systems, predicted at very similar +energies with TDDFT and the GW+BSE approach. We analyze the charge-transfer character + +(b) +(a) +TD-OPBE +G.W.@oPBE/BSE +0.8 +0.8 +Str +B800 +Str +B800 +lator +Oscillator +0.6 +0.6 +Oscil +0.4 +0.4 +D.D +2.3 +0 +0.2 +0.2 +0 +0 +0.8 +0.8 +Str +Str +A-branch +A-branch + 0.6 +lat +0.4 +0.4 +Osci +0.2 +0.2 +0 +0.8 +B850 +B850 +0.6 +Oscill +0.4 +CT +CT +0.2 +0.2 +0 +0.8 +Str + 0.8 +SP +SP +S +≥0.6 +lato +OsC +0.2 +0.2 +1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 +1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 +Excitation energy (eV) +Excitation energy (eV)9 +of these excitations by calculating their difference densities and integrating over subsystem +difference densities as described in Section 2.1. The energy and character of these dark +excitations considerably differs for our four dimers. For the B800 dimer, we find three dark +excitations, E5, E6, and E7, ∼0.7 eV above the coupled Qx excitations which are almost +degenerate. The difference densities (Figure S3 and Table 1) do not indicate any charge- +transfer character for these excitations - their charge distribution is primarily localized on +only one BCL in each excitation, and looks similar to those of the monomeric system. Charge- +transfer excitations can be found at around 3.0 eV, consistent with the large distance of 20 Å +between the B800 molecules, measured as the distance between their centers of masses. +dimer +molecule label +charge distribution +E3 +E4 +E5 +E6 +E7 +B800 +B307 +0 +0 +0 +0 +0 +B309 +0 +0 +0 +0 +0 +A-branch +PA +0 +0 +-0.97 +0 +0 +BA +0 +0 +0.97 +0 +0 +B850 +B302 +0 +-0.78 +-0.11 +0.91 +0 +B303 +0 +0.78 +0.11 +-0.91 +0 +SP +PA +-0.69 +0 +0 +0.83 +-0.76 +PB +0.69 +0 +0 +-0.83 +0.76 +Table 1. Difference density integrated over subsystem volumes. The first two excitations, +i.e., E1 and E2, are not included since their difference densities integrate to zero in all studied +systems. +The molecules PA and BA of the A-branch dimer are ∼13 Å apart, leading to stronger +intermolecular coupling and the appearance of a charge-transfer state in the energy range +considered here. Figure 3 shows that for this system the coupled Qy and Qx excitations +are split and the first dark excitation is ∼0.3 eV higher in energy than the second coupled +Qx excitation. Contrary to the B800 dimer, this dark excitation has clear charge-transfer +character (Table 1) and corresponds to P+ +A B− +A . The character of the two following dark states +is unchanged as compared to B800 apart from a redshift. +In the B850 dimer with ∼ 11Å distance, the stronger intermolecular coupling leads to +a further redshift of the dark excitations. We find that a dark state mixes with the coupled +Qx excitations leading to charge-transfer character in E4 and E5. Another charge-transfer +excitation in which charge is moved in the other direction is found ∼0.3 eV higher in energy. +The excitation spectrum of the special pair dimer SP is yet different. Due to the strong +intermolecular coupling of the two molecules which are only 9 Å apart, three charge-transfer +excitations appear at relatively low energies. +The first one is lower in energy than the +first coupled Qx excitation and corresponds to P+ +A P− +B , whereas the other two are above the +coupled Qx excitations and correspond to P− +A P+ +B and P+ +A P− +B , respectively. Note that due to + +10 +the overestimation of the coupled Qy excitations by TDDFT, GW+BSE predicts the energy +gap between the coupled Qy excitations and CT1 to be twice as large as TDDFT. Nonetheless, +since the qualitative features of all four excitation spectra and the charge-transfer character +of all excitations is similar, we use TDDFT for all further calculations and report GW+BSE +results in the SM. +3.2. Charge-Transfer Excitations in Artificial Dimer +The dimeric systems extracted from the RC and LHII crystal structures discussed in +Section 3.1, differ in their distance, relative orientation, and the structural details of the two +molecular subunits comprising the dimer. To disentangle these effects, we therefore proceeded +by performing TDDFT calculations for an artificial dimeric system constructed as discussed +in Section 2. The structural parameters that define the distance and relative orientations of +this dimer are shown in Figure 4. We measure the distance between the molecules r as the +distance between their centers of masses R1 and R2, i.e., r = |r| = |R1 − R2|. Their relative +orientation is defined by three angles α, β, and γ. The first angle, α, is a rotation around the +normal vector of the plane spanned by the Qy and Qx transition dipole moments of a single +molecule, i.e., it is approximately perpendicular to the porphyrin-ring plane. The second +rotation axis, associated with β, corresponds to r = R1 −R2. The third rotation, γ, is around +the axis given by the cross product of r and the normal vector of the Qy – Qx plane. For our +further discussion, we also distinguish between the four functional groups FG1, FG2, FG3, +and FG4, highlighted in Figure 4. +Figure 4. Structure of artificial dimer based on two identical PA molecules. We highlight four +functional groups FG1 (in green), FG2 (in red), FG3 (in pink), and FG4 (in orange). Hydrogen +atoms are omitted for clarity. +We start by investigating the effect of changing the distance r between the molecules PA1 +and PA2, fixing the relative orientation of the molecules such that it corresponds to the one +found in the special pair dimer SP. Figure 5a shows the excitation spectra of dimers separated +by 9, 11, and 13 Å, corresponding to the center-of-mass difference found in the special pair +SP, the B850 dimer, and the A-branch dimer of Section 3.1, respectively. Note that distances +smaller than 9 Å are not possible for the artificial dimer due to overlap between the FG3 +functional groups. Decreasing the center-of-mass difference leads to a redshift and splitting of +the coupled Qy excitations accompanied by a redistribution of oscillator strength between the + +P +P +A2 +FG +PaintX lite11 +two excitations, in accordance with expectations from Kasha’s exciton theory [78]. The effect +on the coupled Qx excitations cannot be discussed without also considering the higher-energy +charge-transfer excitations. The latter are redshifted when going from 13 Å to 11 Å, and mix +with the coupled Qx excitations at 9 Å, similar to the situation in the special pair dimer SP. +The corresponding charge distributions based on subsystem integrals of difference densities +are shown in Table 2 and demonstrate that for the system at r = 9 Å , all excitations in the +energy-range of the coupled Qx excitations and the higher energy dark states exhibit charge- +transfer character. We classify E4, which is in the energy range of the coupled Qx excitations +and corresponds to transfer of half an electron from PA1 to PA2 as a partial charge-transfer state +(PCT) in Figure 5a. Our results are qualitatively similar when using the GW+BSE approach, +as shown in Figure S5 and consistent with our discussion in Section 3.1. +Figure 5. +(a) Absorption spectra of artificial dimer with r = 9 Å (blue), r = 11 Å (red), +and r = 13 Å (green). Arrows mark excitations with charge-transfer character. The shaded +areas are calculated by folding the excitation energies with Gaussian functions with a width of +0.08 eV as a guide to the eye. (b) The excitation energy of the first two charge-transfer (CT1 +and CT2) excitations and the first four dark states (D1-D4) as a function of r. The color scale +represents the charge-transfer character of each excitation based on the absolute value of the +integrated subsystem difference densities. (c) ∆R (see main text) as a function of the rotation +angle α (top), β (middle), and γ (bottom). Blue lines are periodic fits and serve as a guide to +the eye. The color scale corresponds to the change in energy ∆E of CT1 as compared to the +unrotated reference structure. +These trends are even more apparent in Figure 5b, where we plot the energy of all +dark excitations as a function of distance and indicate their charge-transfer character in color. +In the energy range considered here, there are four dark excitations without charge-transfer +character which are essentially independent of distance and are only redshifted and acquire +substantial charge-transfer character at relatively small r. +The two charge-transfer states +exhibit a significant distance dependence and are red-shifted by almost 1 eV with decreasing +r but lose some of their charge-transfer character at the smallest distance where they start +mixing with the coupled Qx excitations. +For investigating the effect of the relative orientation of the two molecules, we fixed +the intermolecular distance at 13 Å. Shorter distances were not possible due to overlap of +functional groups for some orientations. Since rotations around the angles α, β, and γ do not +commute, we treat them separately from each other, i.e., we first consider rotations around + +(a) +(b) +(c) +0.9 +1.0 +2 +0.1 +α +■ CT, +●CT. +^D1 +←D2 +13 A +1 +0.9 +0.0 +0.8 +3.0 +11A +0 +0.8 +-1 +-0.1 +0.7 +９A +一 +-2 +Excitation energy (eV) +2.8 +-0.2 +0.7 +-3 +2 +0.1 +0.6 +2.6 +0.5 +0 +0.5 +△R +-1 +0.4 +-2 +2.4 +0.4 +-0.2 +-3 +0.3 +0.3 +0.1 +V +2.2 +0.2 +0.2 +0.0 +0 +PCT +CT +-1 +-0.1 +0.1 +0.1 +2.0 +-2 +-0.2 +-3 +0.0 +0.0 +1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 +8 +10 +12 +14 +18 +20 +22 +0 +50 +100 +150 + 200 +250 +¥300 +350 +16 +Angle +Excitation energy (eV) +r (A)12 +r (Å) +molecule +charge distribution +E4 +E5 +E6 +E7 +E8 +9 +PA1 +-0.48 +0.21 +0.28 +-0.62 +-0.63 +PA2 +0.48 +-0.21 +-0.28 +0.62 +0.63 +11 +PA1 +0 +-0.96 +0.96 +0 +0 +PA2 +0 +0.96 +-0.96 +0 +0 +13 +PA1 +0 +0 +0 +-0.99 +0.99 +PA2 +0 +0 +0 +0.99 +-0.99 +Table 2. +Charge distribution on each molecule in the artificial dimer upon excitation as +calculated by integration over subsystem difference densities. The first two excitations, i.e., +E1 and E2, are not included since their subsystem difference densities integrate to zero. +α for fixed β and γ, then rotations around β for fixed α and γ, and finally rotations around +γ for fixed α and β. For each structure, we determine the smallest intermolecular distance +between every two individual atoms in PA1 and PA2, R. The difference between R in the +reference (unrotated) structure from each rotated structure, ∆R = Rre f −Rrot, as a function of +rotation angle, is shown in Figure 5c. Since charge-transfer excitations CT1 and CT2 follow +similar trends, we only show the change in energy of CT1 upon rotation in Figure 5c. Negative +(positive) values of ∆ECT1 = ECT1 +ref −ECT1 +rot correspond to a redshift (blue-shift) of the excitation +energy. +Rotations around α and β correspond to orientations with smaller R than in the reference +structure. Consequently, we observe increased intermolecular coupling and hence a redshift +of the charge-transfer state by up to ∼0.2 eV. For the structure for which we observe the largest +effect (corresponding to a β rotation of 120 degrees), it is primarily the relative orientation and +distance of carbon chains determining the intermolecular coupling (Figure S7a). For many of +the other structures that show pronounced redshifts, we find that the functional groups of +the two BCLs highlighted in Figure 4 are in close spatial proximity (see Figure S7b for an +example). In contrast, the rotation around the angle γ results primarily in structures with +positive ∆R and a blueshift of the charge-transfer excitation by up to ∼0.1 eV. We note that in +the majority of structures rotated around γ, the functional groups FG1, FG2, and FG4 are +far apart from the second BCL. However, for some structures, overlap between FG2 and +the second BCL molecule led to unrealistic structures that were excluded from Figure 5c. +Overall, the γ rotation primarily leads to geometries with weaker intermolecular coupling and +an overall blueshift in energy of the charge-transfer excitation. +3.3. Vibrational Renormalization of Charge-Transfer Excitations +Excitations of different spatial localization and character are known to be affected in different +ways by molecular vibrations [79]. Our goal here is to provide a mode-resolved picture +of excitation energy renormalization in a BCL dimer due to thermally-activated vibrations, + +13 +following earlier work by Hele et al. [80]. For this purpose we started from the crystal +structure of the special pair dimer SP and performed a full geometry optimization using +the def2-TZVP basis set and B3LYP exchange-correlation functional. In the absence of the +protein environment and other co-factors, no external force fixes PA and PB in the parallel +configuration they have in vivo. +Consequently, the relaxed structure differs considerably +from SP, and is more akin to the A-branch dimer. Since our aim is to provide a qualitative +picture, we proceed with this structure which is dynamically stable, i.e., without imaginary +normal modes. We note, however, that the excitation spectrum of the relaxed dimer, shown +in Figure 6a, differs from the spectra discussed so far. In particular, the spectrum displays a +charge-transfer state CT1 at ∼1.6 eV (see also Table S8). This state mixes with the coupled +Qy excitations and corresponds to the transfer of 0.78 of an electron from PA to PB (see Table +S9). A second charge-transfer state CT2 mixes with the coupled Qx excitations, while the +third one, CT3, is energetically well-separated from the Q-band excitations at ∼2.7 eV. +Figure 6. (a) Absorption spectrum of relaxed dimer. Arrows mark the first three charge- +transfer excitations, (b) Excitation energy renormalization ∆E as a function of normal mode +frequency for CT1, CT2, and CT3. Negative (positive) values of ∆E correspond to a redshift +(blueshift), (c) Visualization of the first two normal modes which correspond to intermolecular +rotations (see main text). +We calculate the vibrational normal modes of the relaxed dimer using the same basis +set and exchange-correlation functional but with a very fine grid for the quadrature of the +exchange-correlation energy. We then distort the structure along the 60 first vibrational normal +modes with a distortion amplitude corresponding to a temperature of 300 K. The excitation +spectrum of each distorted structure is then calculated with TDDFT as before, i.e., with +ωPBE with ω = 0.171 a−1 +0 . We define the excitation energy renormalization of excitation +n as ∆En = En +ref − En +dis. Here we focus on how molecular vibrations affect charge-transfer +excitations, but note that ∆E for the coupled Qy and Qx excitations can also be substantial as +shown in Figure S8. +The excitation energy renormalization of the charge-transfer excitations CT1, CT2, and +CT3 is shown in Figure 6b. High-frequency modes correspond to intramolecular vibrations +such as C-C and C-H stretch modes, which are not thermally activated and only have a small +effect on the energy of the three charge-transfer states. In contrast, low-frequency modes +correspond to intermolecular vibrations that change the orbital overlap between neighboring +molecules and thus have a more substantial impact. In particular, we find that the two lowest- + +(a) +(b) +(c) +CT +0.5 +0.10 +CT +(2) +0.05 +O +30-0000 +-0.05 +0.10 +CT +-0.15 +0.1 +0 (1) +-0.20 +80 +1.6 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +20 +40 +60 +100 +140 +0 +160 +Excitation energy (eV) +0 (cm-l)14 +frequency modes lead to substantial changes of all three charge-transfer states. Both modes +correspond to a rotational motion of the porphyrin planes of the BCL molecules with respect to +each other as indicated in Figure 6c. The first modes leads to a redshift of all three excitations +which is with ∼0.2 eV most pronounced for CT1, the second one leads to a smaller blueshift +of CT1 and CT3 and a slight redshift of CT2. These results qualitatively agree with our results +in Section 3.2, suggesting that thermally-activated vibrational modes can significantly affect +the energy of charge-transfer excitations affecting their charge-transfer character and mixing +with other delocalized and localized excitations of the system. +4. Summary and Conclusions +In summary, we have presented a systematic first-principles study of charge-transfer +excitations in BCL dimers. Our model systems are inspired by molecular aggregates found +in the LHII complex and RC of purple bacteria and cover a wide range of intermolecular +coupling strengths, and consequently, excited-state structures. Charge-transfer excitations +can be found in a wide range of energies, primarily depending on intermolecular distance +and orientation. BCL molecules have a complex three-dimensional structure with several +functional groups, a long phytyl tail, and other carbon chains protruding out of the porphyrin +plane. In vivo, i.e., within the evolutionary-optimized protein networks of the photosynthetic +apparatus, the protein environment determines the distance, orientation, and structural +details of these aggregates. +Furthermore, the protein environment indirectly affects the +excited state structure and dynamics of BCL aggregates through dielectric screening and +electrostatic effects [75,81,82,82–89]. Therefore our results can not directly be used to infer +charge-transfer mechanisms in photosynthetic systems Nonetheless, they provide an intuitive +understanding and design rules for tailoring charge-transfer excitations in BCLs and similar +photoactive molecules. Furthermore, they explicitly confirm the importance of charge-transfer +excitations for a correct description of the Q-band excitations of BCL aggregates [40]. We +hope that our results inspire future calculations of the excited-state structure and dynamics of +pigment-protein complexes and chromophore aggregates based on model Hamiltonians, that +include charge-transfer excitations. +Furthermore, we have compared our results based on TDDFT with the optimally- +tuned ωPBE functional to calculations using the GW+BSE approach. +While charge- +transfer excitations appear at very similar energies with both approaches, coupled Qy +excitations are systematically overestimated by TDDFT as compared to the GW+BSE +approach. +Previous studies suggest that Qy excitation energies from GW+BSE are in +better agreement with wavefunction-based methods and experiment than TDDFT with ωPBE +[61, 62]. However, accurate benchmarks for larger molecular aggregates are missing and +we therefore do not think that a clear recommendation for using GW+BSE instead of +TDDFT is warranted. Nonetheless, with advances in code implementation [90–92] and in the +combination of GW+BSE with discrete and polarizable continuum models [93,94] and other +QM/MM methods [95], GW+BSE calculations of large molecular aggregates are becoming +computationally feasible, demonstrated in a recent study by Förster et al. [61]. Further study + +15 +of the accuracy and predictive power of TDDFT, with exchange-correlation functionals that +capture the nonlocal nature of charge-transfer excitations for such aggregates is necessary. +Supplementary Material +Additional convergence data, excitation energies, difference densities and transition densities +not shown in the main text, and structure files. +Acknowledgements +This work was supported by the Bavarian State Ministry of Science and the Arts through the +Elite Network Bavaria (ENB) and through computational resources provided by the Bavarian +Polymer Institute (BPI). +References +[1] Wahadoszamen M, Margalit I, Ara A M, van Grondelle R and Noy D 2014 Nat. Comm. 5 5287 URL +https://www.nature.com/articles/ncomms6287 +[2] Zoppi L and Baldridge K K 2018 Int. J. Quant. Chem. 118 e25413 URL https://onlinelibrary. +wiley.com/doi/abs/10.1002/qua.25413 +[3] Muntwiler M, Yang Q, Tisdale W A and Zhu X Y 2008 Phys. Rev. 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Comp. 14 6253–6268 + diff --git a/8NE1T4oBgHgl3EQfngRF/content/tmp_files/load_file.txt b/8NE1T4oBgHgl3EQfngRF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b142182af3a97c5d94763123131e2038c751ff0 --- /dev/null +++ b/8NE1T4oBgHgl3EQfngRF/content/tmp_files/load_file.txt @@ -0,0 +1,959 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf,len=958 +page_content='Mapping Charge-Transfer Excitations in Bacteriochlorophyll Dimers from First Principles Zohreh Hashemi1, Matthias Knodt1, Mario R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Marques1, Linn Leppert1,2 1)Institute of Physics, University of Bayreuth, Bayreuth 95440, Germany, 2)MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands E-mail: l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='leppert@utwente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='nl Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Photoinduced charge-transfer excitations are key to understand the primary processes of natural photosynthesis and for designing photovoltaic and photocatalytic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In this paper, we use Bacteriochlorophyll dimers extracted from the light harvesting apparatus and reaction center of a photosynthetic purple bacterium as model systems to study such excitations using first-principles numerical simulation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We distinguish four different regimes of intermolecular coupling, ranging from very weakly coupled to strongly coupled, and identify the factors that determine the energy and character of charge-transfer excitations in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We also construct an artificial dimer to systematically study the effects of intermolecular distance and orientation on charge-transfer excitations, as well as the impact of molecular vibrations on these excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Our results provide design rules for tailoring charge- transfer excitations in Bacteriochloropylls and related photoactive molecules, and highlight the importance of including charge-transfer excitations in accurate models of the excited-state structure and dynamics of Bacteriochlorophyll aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='03309v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='chem-ph] 9 Jan 2023 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Introduction Photoinduced charge-transfer excitations are of central importance to the primary processes of natural photosynthesis and for photovoltaic and photocatalytic applications [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In organic semiconductors, charge-transfer excitations are believed to be important intermediates between excited states localized on donor molecules and charge-separated electron-hole states on acceptor and donor units, respectively, even though the exact mechanism of charge- separation is debated [3–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In photosynthesis, the efficient conversion of solar energy into chemical energy is achieved by structurally complex aggregates of Bacteriochlorophylls (BCL), Chlorophylls, and other pigment molecules embedded in transmembrane proteins that modulate their structure and function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' These pigment-protein complexes form light- harvesting complexes and reaction centers that are responsible for photon absorption, excitation-energy transfer, and charge-separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Their main operating principles are well- understood due to a wealth of crystallographic and spectroscopic studies complemented by numerical modelling using semi-empirical and first-principles approaches [13–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Crystal structure of BCL aggregates in the reaction center (RC) and light-harvesting II (LHII) complex of the purple bacteria Rhodobacter sphaeroides and Rhodoblastus acidophilus, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Dimers of BCLs are highlighted in color using (a) pink for the special pair PA – PB, (b) orange for the A branch dimer PA – BA, (c) red for a dimer from the B800 and blue for a dimer from the B850 ring of the LHII complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Hydrogen atoms are omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In purple bacteria, charge separation occurs in reaction centers (RCs) comprising a hexameric aggregate of four BCLs and two Bacteriopheophytins, tightly surrounded by several protein chains [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The primary four BCL molecules of this reaction center are shown in Figure 1a, highlighting the so-called special pair (SP), a strongly-coupled dimer of BCLs called PA – PB in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Charge separation in the bacterial RC is initiated by a series of energy- and charge-transfer excitations that involve the SP and proceed along the A branch, the photoactive of the two pseudo-symmetric branches the RC consists of [19,26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In Figure 1b, we have highlighted the A-branch dimer PA – BA that has been speculated to be involved in the primary charge-separation step, although this assignment is debated in the literature [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Excitation energy reaches the RC through a cascade of excitation-energy c B B A B850 B B A B8003 transfer processes that are initiated in the light harvesting II (LHII) complex, consisting of two rings of BCL molecules dubbed B850 and B800, respectively, and shown in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Neighboring BCLs in the B800 ring are only weakly coupled and excitation-energy transfer is well-described by Förster dipole-dipole coupling [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In the B850 ring, neighboring BCL molecules are closer and intermediate between the weakly coupled B800 and the strongly coupled special pair BCLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The excited states that are believed to be responsible for excitation energy transfer in and between the light-harvesting complexes and the RC, are commonly thought of as Frenkel-like excitons that are spatially relatively localized on one or two BCL molecules [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Semi-empirical models based on Frenkel-excitons Hamiltonians have played an important role in modelling the excitation-energy and charge-transfer dynamics in large photosynthetic pigment-protein complexes [35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' However, for a reliable and predictive representation of the electronic coupling between adjacent pigments, charge-transfer excitations need to be included in these model Hamiltonians [36, 38–40], calling for accurate first-principles calculations of such excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For computationally efficient first-principles methods such as time-dependent density functional theory (TDDFT), charge-transfer excitations were long considered a major challenge due to their inherently nonlocal nature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=', the spatial separation of the occupied and virtual orbitals contributing to these excitations [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' TDDFT with optimally-tuned range-separated hybrid functionals is a viable solution to this problem, and has been used to predict excited states of molecular systems and solids with great success [42–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In these exchange-correlation functionals, the presence of long-range exact exchange leads to asymptotically correct potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Additionally, a parameter controlling the range-separation of exact and semilocal exchange can be used to tune the energies of the highest occupied and the lowest unoccupied orbitals to correspond to the negative of the ionization potentials and the electron affinity, respectively, within the conceptual framework of generalized Kohn-Sham [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Both conditions are crucial for accurately capturing charge-transfer excitations within linear-response TDDFT [50] and have been extended to solvated molecular systems [46, 51] and extended solids [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' An alternative approach for calculating charge-transfer excitations of molecules and solids is the GW+Bethe-Salpeter Equation (GW+BSE) approach [52,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' While this method was initially primarily applied to solids, recent years have witnessed a multitude of studies that have demonstrated the accuracy and predictive power of the GW+BSE method for small molecules [54–56] and larger molecular complexes [57–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In particular, we [62] and others [61] benchmarked the accuracy of the GW+BSE approach against experiment and wavefunction-based methods and found excellent agreement for the Qy and Qx excitations of a range of BCL and Chlorophyll molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We showed that both eigenvalue self- consistent GW calculations and one-shot G0W0 calculations where the zeroth-order single- particle Green’s function G0 and screened Coulomb interaction W0 were constructed from a DFT eigensystem obtained with an optimally-tuned range-separated hybrid functional lead to the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' TDDFT with an optimally-tuned hybrid-functional performed slightly worse and tended to overestimate the energy of the Qy excitations, in agreement with previous 4 studies [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In this article, we report a systematic first-principles study of charge-transfer excitations in BCL dimers - the smallest structural units in which excitations with intermolecular charge- transfer character can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' These BCL dimers, extracted from the LHII complex and RC of purple bacteria, constitute our model systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Our goal is to elucidate the factors that determine the energy and character of these excitations, in particular their mixing with the coupled Qy and Qx excitations of the dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We treat these dimers as representative of four different regimes of intermolecular coupling resulting in distinct charge-transfer properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The B800 dimer is very weakly coupled with Qy and Qx excitations resembling those of the monomeric units and high-energy charge-transfer excitations due to vanishing orbital overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The A-branch dimer is more strongly coupled and exhibits one charge-transfer excitation corresponding to electron transfer from PA to BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We use the notation P+ A B− A to indicate the direction of charge-transfer in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' This charge-transfer excitation is ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 eV higher in energy than the coupled Qx excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The B850 dimer is even more strongly coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The lowest-energy charge-transfer excitation mixes with the coupled Qx excitations and another charge-transfer state appears at higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Finally, the special pair SP is the most strongly coupled case with three charge-transfer excitations mixing with the coupled Qx excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Additionally, we construct an artificial BCL dimer and systematically study the effects of intermolecular distance and orientation on charge-transfer excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We also estimate the effect of molecular vibrations on charge-transfer excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We do this by calculating the vibrational normal modes of a dimeric system and determining the renormalization of excitation energies for structures distorted along normal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' This allows us to identify vibrational modes with pronounced effects on charge-transfer excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Finally, we comment on differences and similarities between TDDFT with an optimally-tuned range separated hybrid functional and the GW+BSE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Computational Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' First-Principles Methods and Computational Details For all calculations reported in this article, we used TDDFT as implemented in TURBOMOLE version 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='5 [63] and the GW+BSE approach as implemented in MOLGW version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Briefly, in the linear-response formulation of both methods the excitation energies Ωn can be obtained by solving the matrix eigenvalue equation CZ = Ω2 nZ, where C is Cijσ,klτ = (εiσ −εjσ)2δi jδ jlδστ +2�εiσ −εjσ √εkτ −εlτKi jσ,klτ (1) and the indices i,k refer to occupied, j,l to virtual orbitals and σ,τ to spin-indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Differences between TDDFT and the GW+BSE approach enter Equation 1 in two distinct ways: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Through the differences between virtual and occupied orbital energies εiσ − εjσ which are obtained from a (generalized) Kohn-Sham calculation in TDDFT and from the GW approach in GW+BSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Through the kernel matrix element Ki jσ,klτ, which depends on the exchange-correlation kernel fxc,σ - the functional derivative of the exchange-correlation 5 potential - in TDDFT, and on the screened Coulomb interaction W, typically evaluated in the random phase approximation and at zero frequency, in the BSE approach [65–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Here we use the optimally-tuned range-separated hybrid functional ωPBE for our TDDFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We use a range-separation parameter ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='171 a−1 0 , which we determined previously for a single BCL a molecule [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The optimal-tuning procedure follows the recipe by Stein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' and ensures that the HOMO eigenvalue corresponds to the ionization potential and the LUMO eigenvalue corresponds to the electron affinity of the molecule [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We do not perform a new tuning procedure for the dimers for general reasons: Using the same ω for each dimer allows us to compare the electronic and excited state structure of these systems on the same footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Furthermore, optimal tuning of conjugated systems of increasing size leads to artificially low values of ω and, thus, a dominance of semilocal exchange at long range, which deteriorates the description of charge-transfer excitations [46,70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For our GW+BSE calculations we use a "one-shot" G0W0 approach in which we construct the zeroth-order single-particle Green’s function G0 and the screened Coulomb interaction W0 from DFT eigenvalues and eigenfunctions calculated using the same ωPBE as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' This approach leads to excellent agreement with experimental excitation energies and reference values from wavefunction-based methods for a range of BCL and Chlorophyll molecules [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Range-separated hybrid functionals have been shown to lead to accurate charge-transfer excitations for larger molecular complexes as well [57,71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In all calculations we used a def2-TZVP basis set, and the frozen core and resolution-of-the-identity approximations (with the DeMon auxiliary basis set [72]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We did not apply the Tamm- Dancoff approximation in any of the results reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In our G0W0 calculations, we used the optimized virtual subspace method by Bruneval with an aug-cc-pVDZ basis set for the reduced virtual orbital subspace [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' With these settings, our excitation energies are converged to within 40 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Further details on our convergence tests can be found in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 and in the Supplemental Material (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For evaluating the character of the excited states, we calculated their transition densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Since the transition density vanishes for charge-transfer states, we calculated the difference density ∆ni = ni − n0 between the excited (ni) and the ground-state density (n0) for every excitation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The excited-state density ni is calculated as the diagonal part of the excited state density matrix γii(r,r′) = N � Ψi(r,r2,r3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=',rn)Ψi(r′,r2,r3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=',rn)dr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='drn, where N is the number of electrons and Ψi is the generalized Kohn-Sham excited-state wavefunction, that consists of a sum of Slater determinants of generalized Kohn-Sham orbitals with coefficients obtained from TDDFT [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' To quantify the magnitude of charge transfer we integrated over subsystem difference densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For this purpose, we subdivided the volume containing the difference densities of the dimer into subsystem volumes, each containing one pigment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Our aim is to assign each grid point of the difference-density grid to its closest pigment molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For achieving this, we used the distances between grid points and each molecule’s atomic coordinates (including hydrogen atoms), as previously done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Finally, to obtain a mode-resolved picture of the effect of thermally-activated vibrations (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='3), we relaxed a dimer structure using the B3LYP approximation for the exchange- 6 correlation functional and def2-TZVP basis set, and evaluated its normal modes and frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Using the harmonic approximation, we can relate the amplitude of these normal modes with the thermal energy of a molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Thus, we distorted the dimer structure along its lowest-frequency normal modes at a temperature of 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In this manner, we generated 60 distortions of the dimer, that we then studied using TDDFT calculations using the ωPBE functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' All these calculations were performed using the tools provided in the TURBOMOLE package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Convergence of G0W0+BSE calculations We carefully tested that our GW+BSE results are converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Due to the large size of a BCL dimer, featuring more than 300 electrons, the calculation of the GW self-energy which requires summation over virtual states is computationally demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We therefore used the optimized virtual subspace method implemented in the MOLGW code, in which a reduced virtual orbital subspace represented by a comparably small basis set is used to evaluate the GW self-energy [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Convergence of GW (a) HOMO-LUMO gap and (c) energy of the first excited state of BCL a monomer as a function of the number of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Blue data points correspond to calculations in which the same basis set is used for the occupied orbitals and the virtual subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Red points correspond to calculations using the optimized virtual subspace method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Lines are fits to these data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Convergence of the HOMO-LUMO gap and energy of the first excited state is shown in panel (b) and (d) for the B850 dimer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Here, green corresponds to using the same basis set for the occupied orbitals and the virtual subspace and pink to calculations using the optimized virtual subspace method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' (c) + SCF basis = GW basis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='65 SCF basis = GW basis SCF basis = Def2-TZVP SCF basis = Def2-TZVP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='05 Monomer Monomer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='00 120014001600 800 1000 1200 1400 1600 800 1000 N (d) N b basis basis SCF basis = GW basis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='00 SCF basis = GW basis excitation (eV) SCF basis = Def2-TZVP SCF basis = Def2-TZVP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='80 >1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='70 Dimer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='40 Dimer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='65 1500 2000 2500 3000 3500 1500 2500 3500 2000 3000 N N basis basis7 We start by testing the convergence of the HOMO-LUMO gap, and the Qy and Qx excitations of a BCL a monomer with respect to basis set size without the optimized virtual subspace method (Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In agreement with our previous results [62], we find that the def2-TZVP basis set deviates by less than 10 meV from the considerably larger aug-cc-pVTZ basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We proceeded by calculating the convergence of the Qy and Qx excitations of the BCL monomer as a function of the number of virtual orbitals Nvirt included in the evaluation of the GW self-energy using the def2-TZVP basis (Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We find that for Nvirt = 500 both excitations are converged to within 80 meV from the limit of infinite Nvirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Based on these findings we continued by evaluating the effect of using a smaller basis set for the virtual subspace [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The results for the HOMO-LUMO gap and the Qy excitation are plotted in Figure 2a and c, and show that the optimized virtual subspace method leads to an underestimation of the HOMO-LUMO gap and the Qy excitation energy as compared to the conventional method in which the same basis set is used for all orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We find that using the aug-ccpVDZ basis for the optimized virtual subspace in conjunction with Nvirt = 500 leads to a fortuitous error cancellation and results in a HOMO-LUMO gap and Qy and Qx excitation energies that are within less than 50 meV of the results obtained with the conventional method and Nvirt → ∞ (Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For the dimer, we therefore chose Nvirt = 1000 and the same strategy for determining the optimized virtual subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We find very similar results for the convergence of the HOMO- LUMO gap and the first bright coupled Qy excitation shown in Figure 2b and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' All GW+BSE results reported in this paper are therefore based on calculations using the def2-TZVP basis set for the occupied orbitals and the aug-ccpVDZ basis for the optimized virtual subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Construction of the Model Systems We constructed our model systems from the x-ray crystallographic structures of the purple bacteria Rhodobacter sphaeroides (structure ID 1M3X in the Protein Data Base) [76] and Rhodoblastus acidophilus (structure ID 1NKZ) [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In all structures, we replaced the phytyl tail with hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Hydrogen atoms not resolved in the experimental crystal structure were added using AVOGADRO and their positions were optimized while keeping the rest of the structure fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' These geometry optimizations were performed using TURBOMOLE and the B3LYP exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The reaction center dimers PA – PB and PA – BA (Figures 1a and b) were constructed using structure 1M3X while the B800 and B850 ring dimers (Figure 1c) were extracted from 1NKZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' These molecules correspond to ID numbers BCL307 and BCL309 for the B800, and BCL302 and BCL303 for the B850 ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We additionally constructed an artificial dimer consisting of two exactly equivalent BCL a molecules (using molecule PA) that we initially oriented in the same way as the special pair dimer PA – PB by aligning their transition dipole moments (as calculated with TDDFT) with those of PA and PB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We are providing all relevant structure files necessary to reproduce the results of this article in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Discussion and Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Charge-Transfer Excitations in RC and LHII Dimers We start by comparing the excitation spectrum of the four dimeric systems shown in Figure 1a- c using TDDFT and GW+BSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The energies and oscillator strengths of the first 15 excitations of each system can be found in Table S3 and S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The spectra are shown in Figure 3a and b, respectively, and allow for several observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' First, we find that TDDFT and GW+BSE predict qualitatively very similar spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The most striking difference appears for the B800 dimer, for which the coupled Qy excitations calculated with TDDFT are ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='3 eV higher in energy than with GW+BSE while all other excitations are at similar energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' This observation is consistent with our results for single BCL a molecules for which TDDFT with optimally-tuned ωPBE consistently overestimates the Qy excitation energy by ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='3 eV [62] and therefore leads to an underestimation of the Qy – Qx energy difference as compared to experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Interestingly, this overestimation as compared to GW+BSE, while still present, is less pronounced for the other three dimers and seems to decrease with increasing intermolecular coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Excitation spectrum of B800, A-branch, B850, and SP dimers using (a) TDDFT with ωPBE and (b) the G0W0@ωPBE+BSE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Arrows mark dark excitations without (D) and with (CT) charge-transfer character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The shaded areas are calculated by folding the excitation energies with Gaussian functions with a width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='08 eV as a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Second, we find several dark excitations for all four systems, predicted at very similar energies with TDDFT and the GW+BSE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We analyze the charge-transfer character (b) (a) TD-OPBE G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' @oPBE/BSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 Str B800 Str B800 lator Oscillator 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 Oscil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='D 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 Str Str A-branch A-branch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 lat 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 Osci 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 B850 B850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 Oscill 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 CT CT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 Str 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 SP SP S ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 lato OsC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 Excitation energy (eV) Excitation energy (eV)9 of these excitations by calculating their difference densities and integrating over subsystem difference densities as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The energy and character of these dark excitations considerably differs for our four dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For the B800 dimer, we find three dark excitations, E5, E6, and E7, ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='7 eV above the coupled Qx excitations which are almost degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The difference densities (Figure S3 and Table 1) do not indicate any charge- transfer character for these excitations - their charge distribution is primarily localized on only one BCL in each excitation, and looks similar to those of the monomeric system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Charge- transfer excitations can be found at around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 eV, consistent with the large distance of 20 Å between the B800 molecules, measured as the distance between their centers of masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' dimer molecule label charge distribution E3 E4 E5 E6 E7 B800 B307 0 0 0 0 0 B309 0 0 0 0 0 A-branch PA 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='97 0 0 BA 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='97 0 0 B850 B302 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='91 0 B303 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='91 0 SP PA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='69 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='76 PB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='69 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='76 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Difference density integrated over subsystem volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The first two excitations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=', E1 and E2, are not included since their difference densities integrate to zero in all studied systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The molecules PA and BA of the A-branch dimer are ∼13 Å apart, leading to stronger intermolecular coupling and the appearance of a charge-transfer state in the energy range considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Figure 3 shows that for this system the coupled Qy and Qx excitations are split and the first dark excitation is ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='3 eV higher in energy than the second coupled Qx excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Contrary to the B800 dimer, this dark excitation has clear charge-transfer character (Table 1) and corresponds to P+ A B− A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The character of the two following dark states is unchanged as compared to B800 apart from a redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In the B850 dimer with ∼ 11Å distance, the stronger intermolecular coupling leads to a further redshift of the dark excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We find that a dark state mixes with the coupled Qx excitations leading to charge-transfer character in E4 and E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Another charge-transfer excitation in which charge is moved in the other direction is found ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='3 eV higher in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The excitation spectrum of the special pair dimer SP is yet different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Due to the strong intermolecular coupling of the two molecules which are only 9 Å apart, three charge-transfer excitations appear at relatively low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The first one is lower in energy than the first coupled Qx excitation and corresponds to P+ A P− B , whereas the other two are above the coupled Qx excitations and correspond to P− A P+ B and P+ A P− B , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Note that due to 10 the overestimation of the coupled Qy excitations by TDDFT, GW+BSE predicts the energy gap between the coupled Qy excitations and CT1 to be twice as large as TDDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Nonetheless, since the qualitative features of all four excitation spectra and the charge-transfer character of all excitations is similar, we use TDDFT for all further calculations and report GW+BSE results in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Charge-Transfer Excitations in Artificial Dimer The dimeric systems extracted from the RC and LHII crystal structures discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1, differ in their distance, relative orientation, and the structural details of the two molecular subunits comprising the dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' To disentangle these effects, we therefore proceeded by performing TDDFT calculations for an artificial dimeric system constructed as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The structural parameters that define the distance and relative orientations of this dimer are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We measure the distance between the molecules r as the distance between their centers of masses R1 and R2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=', r = |r| = |R1 − R2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Their relative orientation is defined by three angles α, β, and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The first angle, α, is a rotation around the normal vector of the plane spanned by the Qy and Qx transition dipole moments of a single molecule, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=', it is approximately perpendicular to the porphyrin-ring plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The second rotation axis, associated with β, corresponds to r = R1 −R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The third rotation, γ, is around the axis given by the cross product of r and the normal vector of the Qy – Qx plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For our further discussion, we also distinguish between the four functional groups FG1, FG2, FG3, and FG4, highlighted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Structure of artificial dimer based on two identical PA molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We highlight four functional groups FG1 (in green), FG2 (in red), FG3 (in pink), and FG4 (in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Hydrogen atoms are omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We start by investigating the effect of changing the distance r between the molecules PA1 and PA2, fixing the relative orientation of the molecules such that it corresponds to the one found in the special pair dimer SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Figure 5a shows the excitation spectra of dimers separated by 9, 11, and 13 Å, corresponding to the center-of-mass difference found in the special pair SP, the B850 dimer, and the A-branch dimer of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Note that distances smaller than 9 Å are not possible for the artificial dimer due to overlap between the FG3 functional groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Decreasing the center-of-mass difference leads to a redshift and splitting of the coupled Qy excitations accompanied by a redistribution of oscillator strength between the P P A2 FG PaintX lite11 two excitations, in accordance with expectations from Kasha’s exciton theory [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The effect on the coupled Qx excitations cannot be discussed without also considering the higher-energy charge-transfer excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The latter are redshifted when going from 13 Å to 11 Å, and mix with the coupled Qx excitations at 9 Å, similar to the situation in the special pair dimer SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The corresponding charge distributions based on subsystem integrals of difference densities are shown in Table 2 and demonstrate that for the system at r = 9 Å , all excitations in the energy-range of the coupled Qx excitations and the higher energy dark states exhibit charge- transfer character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We classify E4, which is in the energy range of the coupled Qx excitations and corresponds to transfer of half an electron from PA1 to PA2 as a partial charge-transfer state (PCT) in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Our results are qualitatively similar when using the GW+BSE approach, as shown in Figure S5 and consistent with our discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' (a) Absorption spectra of artificial dimer with r = 9 Å (blue), r = 11 Å (red), and r = 13 Å (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Arrows mark excitations with charge-transfer character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The shaded areas are calculated by folding the excitation energies with Gaussian functions with a width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='08 eV as a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' (b) The excitation energy of the first two charge-transfer (CT1 and CT2) excitations and the first four dark states (D1-D4) as a function of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The color scale represents the charge-transfer character of each excitation based on the absolute value of the integrated subsystem difference densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' (c) ∆R (see main text) as a function of the rotation angle α (top), β (middle), and γ (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Blue lines are periodic fits and serve as a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The color scale corresponds to the change in energy ∆E of CT1 as compared to the unrotated reference structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' These trends are even more apparent in Figure 5b, where we plot the energy of all dark excitations as a function of distance and indicate their charge-transfer character in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In the energy range considered here, there are four dark excitations without charge-transfer character which are essentially independent of distance and are only redshifted and acquire substantial charge-transfer character at relatively small r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The two charge-transfer states exhibit a significant distance dependence and are red-shifted by almost 1 eV with decreasing r but lose some of their charge-transfer character at the smallest distance where they start mixing with the coupled Qx excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For investigating the effect of the relative orientation of the two molecules, we fixed the intermolecular distance at 13 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Shorter distances were not possible due to overlap of functional groups for some orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Since rotations around the angles α, β, and γ do not commute, we treat them separately from each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=', we first consider rotations around (a) (b) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1 α ■ CT, CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' ^D1 ←D2 13 A 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 11A 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='7 ９A 一 2 Excitation energy (eV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='7 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='5 △R 1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 0 PCT CT 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='7 8 10 12 14 18 20 22 0 50 100 150 200 250 ¥300 350 16 Angle Excitation energy (eV) r (A)12 r (Å) molecule charge distribution E4 E5 E6 E7 E8 9 PA1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='63 PA2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='63 11 PA1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='96 0 0 PA2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='96 0 0 13 PA1 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='99 PA2 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='99 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Charge distribution on each molecule in the artificial dimer upon excitation as calculated by integration over subsystem difference densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The first two excitations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=', E1 and E2, are not included since their subsystem difference densities integrate to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' α for fixed β and γ, then rotations around β for fixed α and γ, and finally rotations around γ for fixed α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For each structure, we determine the smallest intermolecular distance between every two individual atoms in PA1 and PA2, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The difference between R in the reference (unrotated) structure from each rotated structure, ∆R = Rre f −Rrot, as a function of rotation angle, is shown in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Since charge-transfer excitations CT1 and CT2 follow similar trends, we only show the change in energy of CT1 upon rotation in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Negative (positive) values of ∆ECT1 = ECT1 ref −ECT1 rot correspond to a redshift (blue-shift) of the excitation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Rotations around α and β correspond to orientations with smaller R than in the reference structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Consequently, we observe increased intermolecular coupling and hence a redshift of the charge-transfer state by up to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For the structure for which we observe the largest effect (corresponding to a β rotation of 120 degrees), it is primarily the relative orientation and distance of carbon chains determining the intermolecular coupling (Figure S7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For many of the other structures that show pronounced redshifts, we find that the functional groups of the two BCLs highlighted in Figure 4 are in close spatial proximity (see Figure S7b for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In contrast, the rotation around the angle γ results primarily in structures with positive ∆R and a blueshift of the charge-transfer excitation by up to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We note that in the majority of structures rotated around γ, the functional groups FG1, FG2, and FG4 are far apart from the second BCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' However, for some structures, overlap between FG2 and the second BCL molecule led to unrealistic structures that were excluded from Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Overall, the γ rotation primarily leads to geometries with weaker intermolecular coupling and an overall blueshift in energy of the charge-transfer excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Vibrational Renormalization of Charge-Transfer Excitations Excitations of different spatial localization and character are known to be affected in different ways by molecular vibrations [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Our goal here is to provide a mode-resolved picture of excitation energy renormalization in a BCL dimer due to thermally-activated vibrations, 13 following earlier work by Hele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' For this purpose we started from the crystal structure of the special pair dimer SP and performed a full geometry optimization using the def2-TZVP basis set and B3LYP exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In the absence of the protein environment and other co-factors, no external force fixes PA and PB in the parallel configuration they have in vivo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Consequently, the relaxed structure differs considerably from SP, and is more akin to the A-branch dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Since our aim is to provide a qualitative picture, we proceed with this structure which is dynamically stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=', without imaginary normal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We note, however, that the excitation spectrum of the relaxed dimer, shown in Figure 6a, differs from the spectra discussed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In particular, the spectrum displays a charge-transfer state CT1 at ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 eV (see also Table S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' This state mixes with the coupled Qy excitations and corresponds to the transfer of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='78 of an electron from PA to PB (see Table S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' A second charge-transfer state CT2 mixes with the coupled Qx excitations, while the third one, CT3, is energetically well-separated from the Q-band excitations at ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='7 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' (a) Absorption spectrum of relaxed dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Arrows mark the first three charge- transfer excitations, (b) Excitation energy renormalization ∆E as a function of normal mode frequency for CT1, CT2, and CT3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Negative (positive) values of ∆E correspond to a redshift (blueshift), (c) Visualization of the first two normal modes which correspond to intermolecular rotations (see main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We calculate the vibrational normal modes of the relaxed dimer using the same basis set and exchange-correlation functional but with a very fine grid for the quadrature of the exchange-correlation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We then distort the structure along the 60 first vibrational normal modes with a distortion amplitude corresponding to a temperature of 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The excitation spectrum of each distorted structure is then calculated with TDDFT as before, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=', with ωPBE with ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='171 a−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We define the excitation energy renormalization of excitation n as ∆En = En ref − En dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Here we focus on how molecular vibrations affect charge-transfer excitations, but note that ∆E for the coupled Qy and Qx excitations can also be substantial as shown in Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The excitation energy renormalization of the charge-transfer excitations CT1, CT2, and CT3 is shown in Figure 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' High-frequency modes correspond to intramolecular vibrations such as C-C and C-H stretch modes, which are not thermally activated and only have a small effect on the energy of the three charge-transfer states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In contrast, low-frequency modes correspond to intermolecular vibrations that change the orbital overlap between neighboring molecules and thus have a more substantial impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In particular, we find that the two lowest- (a) (b) (c) CT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='10 CT (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='05 O 30-0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='10 CT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='1 0 (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='20 80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='8 20 40 60 100 140 0 160 Excitation energy (eV) 0 (cm-l)14 frequency modes lead to substantial changes of all three charge-transfer states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Both modes correspond to a rotational motion of the porphyrin planes of the BCL molecules with respect to each other as indicated in Figure 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' The first modes leads to a redshift of all three excitations which is with ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2 eV most pronounced for CT1, the second one leads to a smaller blueshift of CT1 and CT3 and a slight redshift of CT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' These results qualitatively agree with our results in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='2, suggesting that thermally-activated vibrational modes can significantly affect the energy of charge-transfer excitations affecting their charge-transfer character and mixing with other delocalized and localized excitations of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Summary and Conclusions In summary, we have presented a systematic first-principles study of charge-transfer excitations in BCL dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Our model systems are inspired by molecular aggregates found in the LHII complex and RC of purple bacteria and cover a wide range of intermolecular coupling strengths, and consequently, excited-state structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Charge-transfer excitations can be found in a wide range of energies, primarily depending on intermolecular distance and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' BCL molecules have a complex three-dimensional structure with several functional groups, a long phytyl tail, and other carbon chains protruding out of the porphyrin plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' In vivo, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=', within the evolutionary-optimized protein networks of the photosynthetic apparatus, the protein environment determines the distance, orientation, and structural details of these aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Furthermore, the protein environment indirectly affects the excited state structure and dynamics of BCL aggregates through dielectric screening and electrostatic effects [75,81,82,82–89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Therefore our results can not directly be used to infer charge-transfer mechanisms in photosynthetic systems Nonetheless, they provide an intuitive understanding and design rules for tailoring charge-transfer excitations in BCLs and similar photoactive molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Furthermore, they explicitly confirm the importance of charge-transfer excitations for a correct description of the Q-band excitations of BCL aggregates [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' We hope that our results inspire future calculations of the excited-state structure and dynamics of pigment-protein complexes and chromophore aggregates based on model Hamiltonians, that include charge-transfer excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Furthermore, we have compared our results based on TDDFT with the optimally- tuned ωPBE functional to calculations using the GW+BSE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' While charge- transfer excitations appear at very similar energies with both approaches, coupled Qy excitations are systematically overestimated by TDDFT as compared to the GW+BSE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Previous studies suggest that Qy excitation energies from GW+BSE are in better agreement with wavefunction-based methods and experiment than TDDFT with ωPBE [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' However, accurate benchmarks for larger molecular aggregates are missing and we therefore do not think that a clear recommendation for using GW+BSE instead of TDDFT is warranted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Nonetheless, with advances in code implementation [90–92] and in the combination of GW+BSE with discrete and polarizable continuum models [93,94] and other QM/MM methods [95], GW+BSE calculations of large molecular aggregates are becoming computationally feasible, demonstrated in a recent study by Förster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Further study 15 of the accuracy and predictive power of TDDFT, with exchange-correlation functionals that capture the nonlocal nature of charge-transfer excitations for such aggregates is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Supplementary Material Additional convergence data, excitation energies, difference densities and transition densities not shown in the main text, and structure files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} +page_content=' Acknowledgements This work was supported by the Bavarian State Ministry of Science and the Arts through the Elite Network Bavaria (ENB) and through computational resources provided by the Bavarian Polymer Institute (BPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfngRF/content/2301.03309v1.pdf'} 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diff --git a/DdFRT4oBgHgl3EQfADdX/vector_store/index.pkl b/DdFRT4oBgHgl3EQfADdX/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..e91f2ac206d720c4b8ee757f8e3d1b9be5db3075 --- /dev/null +++ b/DdFRT4oBgHgl3EQfADdX/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ffb044d3d6c18122f155de26f77c183073e03ab4d7dc0f226be70cc068768b29 +size 76730 diff --git a/DtAzT4oBgHgl3EQfT_x1/content/tmp_files/2301.01259v1.pdf.txt b/DtAzT4oBgHgl3EQfT_x1/content/tmp_files/2301.01259v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..90d73748a58ae8417f992ca3167eed1aa73a2a94 --- /dev/null +++ b/DtAzT4oBgHgl3EQfT_x1/content/tmp_files/2301.01259v1.pdf.txt @@ -0,0 +1,6087 @@ +Higher categorical symmetries and gauging +in two-dimensional spin systems +Clement Delcamp� and Apoorv Tiwari� +�Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium +�Department of Physics, KTH Royal Institute of Technology, Stockholm, 106 91 Sweden +clement.delcamp@ugent.be, apoorvt@kth.se +We present a framework to systematically investigate higher categorical symmetries in two-dimensional +spin systems. Though exotic, such generalised symmetries have been shown to naturally arise as dual +symmetries upon gauging invertible symmetries. Our framework relies on an approach to dualities +whereby dual quantum lattice models only differ in a choice of module 2-category over some input +fusion 2-category. Given an arbitrary two-dimensional spin system with an ordinary symmetry, we +explain how to perform the (twisted) gauging of any of its sub-symmetries. We then demonstrate +that the resulting model has a symmetry structure encoded into the Morita dual of the input fu- +sion 2-category with respect to the corresponding module 2-category. We exemplify this approach +by specialising to certain finite group generalisations of the transverse-field Ising model, for which we +explicitly define lattice symmetry operators organised into fusion 2-categories of higher representations +of higher groups. +arXiv:2301.01259v1 [hep-th] 3 Jan 2023 + +Contents +1 +Introduction +2 +2 +Motivation: transverse-field Ising model +6 +2.1 +Z2-symmetric Hamiltonian +6 +2.2 +Gauging the Z2 symmetry +7 +2.3 +Symmetry operators +9 +3 +Gauging and dual symmetries +14 +3.1 +G-symmetric Hamiltonians +14 +3.2 +Dual Hamiltonians +17 +3.3 +Duality operators +21 +3.4 +Duality as twisted gauging +25 +3.5 +Dual symmetries +27 +3.6 +Higher representation theory +30 +3.7 +Morita duals +36 +3.8 +Gauging the transverse-field G-Ising model +40 +4 +Example: doubled transverse-field Ising model +45 +4.1 +Symmetric Hamiltonian and gauging +45 +4.2 +2Rep(Z2 +2) symmetry: invertible surface operators +47 +4.3 +2Rep(Z2 +2) symmetry: non-invertible surface operators +49 +5 +Example: transverse-field S3-Ising model +52 +5.1 +Symmetric Hamiltonian +52 +5.2 +Gauging sub-symmetries +54 +5.3 +2Rep(S3) symmetry +56 +5.4 +2VecG symmetry +59 +5.5 +2Rep(G)-symmetry +60 +6 +Discussion +63 +6.1 +Further examples +63 +6.2 +Self-dual models +64 +6.3 +Symmetry-twisted boundary conditions +65 +A Two-dimensional Zp gauge theory +66 +∼ 1 +∼ + +SECTION 1 +Introduction +Global symmetries have been playing a pivotal role in our understanding of quantum systems. Gener- +ally speaking, the existence of a global symmetry in a quantum system helps organise the spectrum of +states and operators into representations of the symmetry. In addition, symmetry typically imposes +strong constraints on the kinds of phases a quantum system can or cannot realise. These ideas have +led for instance to Landau’s classification scheme of phases of matter, and to organising principles for +the particle content of the Standard model. Despite its long and illustrious history, symmetry and its +manifestations in quantum theory is very much an evolving story. +Conventionally, given a Hamiltonian model, symmetries are implemented by operators that act +on all of space—i.e., one-codimensional operators in spacetime —commute with the Hamiltonian, and +satisfy fusion rules representative of a group. In contrast, the modern perspective on global symmetries +in quantum systems identifies the topological invariance of a symmetry operator within correlation +functions as its defining property [GKSW15]. This perspective lends itself to numerous generalised +notions of symmetry that have collectively come to be known as global categorical symmetries [FMT22], +in reference to the mathematical objects encoding them. Notably, these include symmetry structures +whose topological operators may be supported on higher codimensional sub-manifolds and/or are not +invertible, so that they do not obey fusion rules encoded into a group. +Relaxing the requirement that operators are one-codimensional has led to the concept of higher- +form symmetry. +Specifically, a p-form symmetry is defined with respect to topological operators +with support on (p+1)-codimensional sub-manifolds and act by linking with p-dimensional operators +[KT13, HLS18, DT19]. These operators being invertible, fusion rules are still encoded into a group, +but whenever p > 0, the corresponding group is necessarily abelian. Furthermore, it is possible to +combine higher form symmetries of various degrees in a non-trivial way. The corresponding groups +combine into categorifications of the notion of group known as higher groups [BL03], yielding the +concept of higher group symmetries [KT13, CDI18, BCH18, DT18]. Relaxing the requirement that +operators are invertible has led to symmetries encoded into higher algebraic structures. For instance, +given a (1+1)d system, non-invertible symmetries are encoded into fusion 1-categories [ENO10], and +the corresponding operators typically cannot be written as tensor products of local operators. More +generally, given a (d+1)-dimensional system, it is possible to have non-invertible symmetry operators of +varying degrees, in which case the algebraic structure is expected to be a fusion d-category [KLW+20a, +BBSNT22a, BSNT22]—a notion that remains partly elusive [DR18, GJF19]. +As it turns out, though somewhat exotic, non-invertible symmetries are not rare in one-dimensional +quantum models and have long been studied in the context of rational Conformal Field Theories +(CFTs). There, topological operators go by the name of Verlinde lines [Ver88, PZ00, BG04] and exist +in any rational CFT defined by a diagonal modular invariant. In particular, the fusion ring formed by +the Verlinde lines corresponds to that of representations of the chiral vertex algebra, i.e., the underlying +algebra of the given CFT, and is generically not group-like. A well-studied example is that of the +diagonal Ising CFT, that hosts three Verlinde lines embodying the Ising fusion category. It includes +in particular a non-invertible line known as the Kramers-Wannier duality defect [OA96a, OA96b, +FFRS04, FFRS06]. +Guided in part by integrability, the sub-algebra of topological defects within +rational CFTs was formalised for instance in ref. [FRS02, FRS04, BM09, CLS+18]. Furthermore, it +was already appreciated in this context that topological defects indeed embody a kind of symmetry +structure within a quantum field theory (QFT), and thus it is sensible to consider notions of ’t Hooft +anomalies and gauging thereof [FFRS09, Tac17, CLS+18]. +∼ 2 +∼ + +Naturally, a prolific source of topological operators are topological quantum field theories (TQFTs) +themselves. In fact, by definition, the entire spectrum of operators in a TQFT is topological. In +particular, a large class of TQFTs host topological defects that obey non-invertible fusion rules. The +most well-studied examples are provided by line defects in (2+1)d TQFTs, either in the continuum +[Wit89, RT90] or in the discrete [TV92, BW93]. Topological surface operators associated with braided +auto-equivalences of the quantum invariant assigned by the theory to the circle have also been studied +in this context [Bom10, KK11, BBCW14, THF15], but they are typically invertible. An important +development was the remark that these surface operators in (2+1)d TQFTs could be constructed by +condensing a suitable sub-algebras of topological line operators [CRS17], suggesting a mechanism to +generate a broader family of defects. This process was later formalised as the condensation completion +of the category of line operators [DR18, GJF19]. The resulting (possibly) non-invertible condensation +defects turn out to be rather ubiquitous in (2+1)d TQFTs [RSS22]. +In spite of these various developments, examples of non-invertible symmetry operators in higher di- +mensions have remained limited until recently. In the past year, various constructions of quantum sys- +tems with non-invertible symmetries have appeared that employ different kinds of generalised gauging +procedures. Generally speaking, it is understood that given a theory with an invertible symmetry, +gauging one of its sub-symmetries typically yields a theory with a different symmetry structure. Con- +cretely, gauging a p-form symmetry in a (d+1)-dimensional theory yields a dual (gauged) model whose +symmetry category contains (d−p−1)-dimensional topological operators labelled by irreducible rep- +resentations of the corresponding group. Whenever the group is non-abelian, these operators are in +particular non-invertible [Dri89, DPR91, dWP95, BT17]. Moreover, gauging a (normal) sub-symmetry +yields a theory possessing higher-group symmetries [Tac17]. As it turns out, the symmetry structures +resulting from these gauging procedures are even richer. +An early construction [KOZ21] of non-invertible defects in (3+1)d involved starting from a QFT +with a 0-form and 1-form mixed anomaly and gauging the 1-form symmetry. It was shown that this +inevitably generates a non-invertible symmetry structure. Another class of examples were inspired +by generalising the construction of the Kramers-Wannier duality defect in the (1+1)d Ising CFT to +(3+1)d self-dual QFTs [CCH+21, CCH+22, LRS22]. Yet another notable development pertained to +the relation between gauging certain symmetry along sub-manifolds of spacetime and the condensation +defects mentioned above [RSS22, LRS22]. Most relevant to the present work were a series of papers +[Del21, BBSNT22a, BSNW22, BBFP22a, BBSNT22b, BSNT22, BBFP22b] that considered starting +from a (3+1)d theory with an invertible symmetry structure encoded into a higher-group and gauging +one of its sub-symmetries. These works go beyond previous constructions in their analysis of the +resulting symmetry structure in terms of so-called higher representations of higher groups [Elg04, +GK06, BBFW08]. Finally, these types of non-invertible symmetries have been further discussed in +the context of various typical quantum field theories such as free field theories [NRS22], pure gauge +theories [KZZ22, AGR22], quantum electrodynamics [Kar22], axion models [CLS22c] and within other +physical contexts [CO22, CLS22a, CLS22b, CHKO22, GEI22]. +Notwithstanding the obvious recent interest in non-invertible symmetries, concrete lattice realisa- +tions of the corresponding topological operators have been largely unexplored, with some exceptions +[KNY21]. But, the lattice setting being concrete and tractable, it offers a welcome complimentary ap- +proach to understanding the most subtle aspects of these generalised categorical symmetries. Besides, +it paves the way for exploring the implications of such symmetries on the phase diagram of familiar +many-body systems. Furthermore, via the corresponding graphical calculs, the lattice setting is much +closer related to the category theoretic framework underlying these symmetry structures. +∼ 3 +∼ + +Our paper aims at further bridging the gap between the abstract concept of a generalised categorical +symmetry, as encoded into a higher mathematical structure, and its concrete realisation on a quantum +theory. More specifically, we wish to address the question, what does it mean to have symmetry oper- +ators encoded into fusion 2-categories of higher categorical representations of higher groups? Guided +by the Morita theory of fusion 2-categories [Del21, D´ec22], we address this question by providing a +framework that accomplishes—amongst other things—two tasks: Given an arbitrary two-dimensional +spin system with an ordinary global symmetry, it allows for the systematic twisted gauging of one +of its sub-symmetries, and the systematic identification of the resulting dual symmetry structure by +constructing the corresponding topological lattice operators. +The framework we introduce in this manuscript is inspired by the study of dualities in one- +dimensional quantum lattice models carried out in ref. [LDOV21, LDV22]. Within our framework, +a duality class of dualities is specified by an algebra of operators that is generated by a set of (ab- +stract) local operators. A representative of a duality class is obtained by choosing a Hilbert space +and correspondingly explicit matrix representations for the local operators. Concretely, the algebras +of operators we consider take as input data a finite group G—or rather, a fusion 2-category 2VecG of +G-graded 2-vector spaces—as well as a set of complex coefficients, which amounts to selecting certain +linear combinations of local operators. These choices completely determine the physical properties +of the duality class of models as encoded into their shared spectrum. Choosing a matrix represen- +tation then amounts to picking a so-called (indecomposable) module 2-category over 2VecG, i.e. a +2-category with a G-action. We think of the module 2-category as providing the physical degrees of +freedom—which may satisfy kinematical constraints—on which the local operators act. It follows that +Hamiltonian models that only differ in a choice of module 2-category are dual to one another. +In the framework described above, a duality operator amounts to a map between two module +2-categories, which provide matrix representations of the same local operators. +For consistencies, +the action of this map is required to commute with the G-action resulting in the notion of module +2-functor. Similarly, a symmetry operator amounts to a module 2-endofunctor between a module 2- +category and itself. More specifically, a module 2-endofunctor furnishes a topological surface operator +that commutes with the Hamiltonian. There is also a notion of map between module 2-functors that +are compatible with the G-action, namely module natural 2-transformations, which furnish topological +lines at the interfaces of (possibly distinct) topological surfaces. These data can be organised into a +2-category. Crucially, given an indecomposable module 2-category M, the composition of module 2- +endofunctors endows this 2-category with a fusion structure. The resulting fusion 2-category (2VecG)⋆ +M +is referred to the Morita dual of 2VecG with respect to M. This is the symmetry structure of the +model obtained by choosing the Hilbert space associated with the module 2-category M. Notice that +we can make this statement without referring to a specific duality class of models. As emphasised in +(1+1)d in ref. [LDOV21], this is because dualities are only sensitive to symmetry structures. Note that +it is always possible to choose 2VecG as a module 2-category over itself, in which case the symmetry +fusion 2-category of the resulting model is again 2VecG. In other words, it is a model with an ordinary +(0-form) G-symmetry. We can then show that choosing an alternative 2VecG-module 2-category has +the interpretation of performing a twisted gauging of one of its sub-symmetries. +One merit of our approach is our ability to provide lattice operators accompanying these ab- +stract statements, allowing to explicitly perform a twisted gauging in an arbitrary G-symmetric two- +dimensional spin system and prove that the resulting model does have the expected symmetry structure +by constructing the corresponding topological lattice operators. This ability extensively relies on the +tensor network study of topological phases of matter where such symmetry operators first appeared +in the form of matrix product operators in (1+1)d [cWB+14, LFH+20] and projected entangled pair +∼ 4 +∼ + +operators in (2+1)d [Del21]. In addition to providing a systematic recipe for generating new dual +models, this framework explicitly provides lattice operators embodying symmetry structures related +to higher representations of groups and categorifications thereof. Furthermore, we are also able to +construct duality lattice operators performing the transmutation of the local symmetric operators. +We can offer a different perspective on our approach to dualities: It is understood that a three- +dimensional TQFT as provided by the Reshetikhin-Turaev construction [RT91] possesses a state-sum +description if and only if it admits a non-trivial gapped boundary [FT20]. +These theories are of +the Turaev-Viro-Barrett-Westbury type, whose input data are spherical fusion 1-categories [TV92, +BW93]. More specifically, given a choice of gapped boundary condition, which can be encoded into a +module category over the input spherical fusion category, a state-sum can be obtained following the +construction outlined for instance in ref. [BGK16]. Distinct gapped boundary conditions yield distinct +state-sums. The corresponding state spaces are then spanned by topological tensor network states that +were defined in ref. [LFH+20]. In the same vein, we can construct a family of state-sums of the same +four-dimensional topological G gauge theory indexed by module 2-categories over 2VecG encoding +various choices of gapped boundary conditions. The corresponding state spaces are then spanned +by the topological tensor network states defined in ref. [Del21]. Importantly, it is possible to define +distinct state sums of the same theory in different regions of spacetime. The operators intertwining +these distinct lattice realisations then precisely correspond to the duality operators transmuting local +symmetric operators of a given Hamiltonian into local symmetric operators of one of its duals, as +considered in this manuscript. +We illustrate our approach with finite group generalisations of the transverse-field Ising model. +For an arbitrary finite group, we consider the gauging of the whole invertible symmetry, revealing a +dual symmetry structure in terms of 2-representations of the group. Supposing that the input group +is a semi-direct product, we also consider the gauging of its two constitutive sub-symmetries in detail, +revealing on the lattice dual symmetry structures in terms of 2-group-graded 2-vector spaces and 2- +representations of 2-groups. Further specialising to the Klein four-group and the symmetric group +of degree 3, we provide even more explicit expressions for the corresponding topological surfaces and +topological lines in terms of spin operators, allowing us to confirm on the lattice their fusion and +composition rules. +Organisation of the paper +We begin in sec. 2 with an in-depth analysis of the symmetry structure resulting from gauging the +global symmetry of the two-dimensional transverse-field Ising model. We emphasise in particular the +appearance of non-invertible surface operators. Guided by this example, we present in sec. 3 a general +framework to gauge invertible sub-symmetries of arbitrary two-dimensional quantum lattice models +and construct the dual symmetry operators as encoded into the corresponding Morita dual fusion +2-category. A few specific scenarios are discussed in detail. Finally, we exemplify our approach in +sec. 4 and 5 by specialising to finite group generalisations of the transverse-field Ising model for the +Klein group and the symmetric group of degree 3, respectively. +∼ 5 +∼ + +SECTION 2 +Motivation: transverse-field Ising model +We set the stage by exploring the higher categorical symmetries that emerge from gauging the Z2 +symmetry of the two-dimensional transverse-field Ising model. +2.1 +Z2-symmetric Hamiltonian +Let Σ be a closed oriented two-dimensional surface endowed with a (fixed) triangulation Σ△ whose +vertices, edges and plaquettes are denoted by v, e and p, respectively. Given an edge e ≡ (v1v2) oriented +from v1 to v2, we denote by s(e) := v1 and t(e) := v2 its source and target vertices, respectively. We +consider a variant of the well-known (2+1)d transverse-field Ising model. As in the usual model, qubit +degrees of freedom are assigned to vertices v ⊂ Σ△. We identify such an assignment with a choice +of 0-cochain m ∈ C0(Σ△, Z2) so the microscopic Hilbert space is provided by the tensor product + +v C[Z2] ≃  +v C2, where Z2 = ⟨r | r2 = 1⟩. Moreover, we denote by |m⟩ the state in the microscopic +Hilbert space associated with 0-cochain m. Throughout this manuscript, we write basis elements of +C[Z2] as |0⟩ and |1⟩, which are identified with the ‘up’ and ‘down’ state, respectively. Qubit degrees +of freedom are governed by the Hamiltonian +H = −J +ÿ +e +σz +s(e)σz +t(e) − Jκ +ÿ +v +σx +v − J˜κ +ÿ +v +σx +v +ź +(v v1v2) +exp +�iπ +4 (1 − σz +v1σz +v2) +� +, +(2.1) +where σx,z +v +is the usual shorthand for id b · · · b id b σx,z +v +b id b · · · b id, with the tensor product being +over all the vertices of the triangulation, and σx,z +v +are Pauli operators. +The first term in the Hamiltonian describes a ferromagnetic interaction between qubits. +The +second term is the usual paramagnetic term. The third term is a topologically twisted variant of the +paramagnetic term that includes phase factors associated with triangles (v v1v2) containing the vertex +v [LG12]. One can readily check that the model has a global (0-form) Z2 symmetry implemented by +surface operators1 acting on all of Σ△ +O0 = +ź +v +idv +and +O1 = +ź +v +σ1 +v , +(2.2) +with Z2 fusion rules +O1 d O1 = O0 = O0 d O0, +O1 d O0 = O1 = O0 d O1 . +(2.3) +Correspondingly, in the three extreme limits 1 ≫ κ, ˜κ, κ ≫ 1, ˜κ and ˜κ ≫ 1, κ, one obtains fixed-point +Hamiltonians with ferromagnetic, paramagnetic and symmetry-protected topological (SPT) ground +states, respectively. This Hamiltonian does not have any non-trivial 1-form symmetry as topological +lines on either surface operator must be the identity line. Furthermore, it is not possible for the surface +operator O1 to be open, i.e. to have support on a sub-region of Σ△. In other words, it is not possible +to define a (topological) line interface between O0 and O1. We describe below how, upon gauging of +the Z2 0-form global symmetry, one inevitably lands on a dual model with more a intricate symmetry +structure. +1Throughout this manuscript, we refer to operators that act on extended two-dimensional regions of Σ as topological +surface operators. These could act on all of Σ or on a sub-region. +∼ 6 +∼ + +2.2 +Gauging the Z2 symmetry +Although this was not immediately appreciated when the construction first appeared [Kog79, Sav80], it +is by now understood that gauging a 0-form Z2 symmetry yields a two-dimensional dual model hosting +Z2 topological Wilson lines labelled by representations of the group. In modern terminology, this is +the statement that the gauged model has a 1-form Z∨ +2 symmetry, with Z∨ +2 the Pontrjagin dual of Z2 +[GKSW15]. However, it was recently pointed out that this is only part of the story [Del21, BSNW22, +BBFP22a]. Indeed, the 1-form Z∨ +2 symmetry is only a component of the symmetry structure of the +gauged model in the sense that it does not encapsulate all possible topological operators. +In order to grasp the above statements, let us explicitly gauge the global Z2 symmetry in model +(2.1). To do so, we begin by assigning additional qubit degrees of freedom to edges e ⊂ Σ△. We +identify such an assignment with a choice of 1-cochain g ∈ C1(Σ△, Z2) so the model is now defined +on the extended microscopic Hilbert space provided by the tensor product  +e C[Z2]  +v C[Z2]. Let +us now promote generator O1 of the global Z2 symmetry to a local gauge transformation by defining +Gauß operators +Gv := σx +v +ź +e⊃v +σx +e . +(2.4) +Since Gauß operators obey the multiplication rule in Z2 and [Gv1, Gv2] = 0, for any v1, v2 ⊂ Σ△, they +are the generators of a Z2 gauge symmetry. Concretely, consider a basis state |g, m⟩ in the extended +microscopic Hilbert space. By definition, we have +σz +v |g, m⟩ = (−1)m[v]|g, m⟩ , +σz +e |g, m⟩ = (−1)g[e]|g, m⟩ , +(2.5) +where m[v] and g[e] denote the restrictions of m and g to v and e, respectively. One can now define a +general Gauß operator indexed by a 0-cochain x ∈ C0(Σ△, Z2) which acts as +G(x) := +ź +v +Gx[v] +v +: |g, m⟩ �→ |g + dx, m + x⟩ . +(2.6) +The gauge symmetry is imposed kinematically so that we only consider physical states in the +1 +eigenspace of G(x) for any x ∈ C0(Σ△, Z2). We then require the gauged Hamiltonian to commute with +Gauß operator G(x). This can be accomplished by minimally coupling Hamiltonian (2.1) with the edge +degrees of freedom: +Hg. = −J +ÿ +e +σz +s(e)σz +e σz +t(e) − Jκ +ÿ +v +σx +v − J˜κ +ÿ +v +σx +v +ź +(v v1v2) +exp +�iπ +4 (1 − σz +v1σz +(v1v2)σz +v2) +� ++ . . . , +(2.7) +where ‘. . . ’ refers to other gauge invariant terms that can potentially be added in the process of +gauging. A minimal example of such a term would be a product of σz +e operators around closed loops +in Σ△. For the sake of simplicity, we neglect these terms in what follows. We can readily confirm that +[Hg., G(x)] = 0 for any x ∈ C0(Σ△, Z2). +At this point, it is crucial to notice that the microscopic Hilbert space splits into super-selection +sectors labelled by eigenvalues of the operators ś +e⊂(v1v2v3) σz +e associated with every triangle (v1v2v3) ⊂ +Σ△. As customary, we shall restrict to a single super-selection sector, namely that given by +ź +e⊂(v1v2v3) +σz +e +!= id , +∀ (v1v2v3) ⊂ Σ△ . +(2.8) +∼ 7 +∼ + +These conditions are also imposed kinematically enforcing g to define a Z2 gauge field so that dg = 0. +Finally, notice that we have +σx +v +���� +Gv=id += +ź +e⊃v +σx +e +���� +Gv=id +. +(2.9) +Upon enforcing this operator equality on the physical Hilbert space, the model becomes classical +in the vertex degrees of freedom so they can be readily gauged away. Concretely, this operation is +implemented by a unitary operator performing the basis rotation |g, m⟩ �→ |g + dm, m⟩. Doing so +delivers the dual Hamiltonian +H∨ = −J +ÿ +e +σz +e − Jκ +ÿ +v +ź +e⊃v +σx +e − J˜κ +ÿ +v +ź +e⊃v +σx +e +ź +(v v1v2) +exp +�iπ +4 (1 − σz +(v1v2)) +� +, +(2.10) +which acts on the physical Hilbert space H∨ spanned by states |g⟩, where g ∈ Z1(Σ△, Z2). Assuming +for concreteness that Σ△ is the Poincar´e dual of the honeycomb lattice, let us explicitly write the +action of the various operators appearing in eq. (2.10). Firstly, +σz +e = +ÿ +g +(−1)g[e]|g⟩⟨g| , +(2.11) +which measures the Z2 gauge field along the edge e. Secondly, +ź +e⊃v +σx +e = +ÿ +g +|g + dxv⟩⟨g| ≡ +σx +σx +σx +σx +σx +σx +, +with xv[v1] = +� +1 if v = v1 +0 otherwise +, +(2.12) +which implements a Z2 gauge transformation at vertex v. Thirdly, denoting by +v the hexagonal +sub-complex centred around v and S := i +1 +2 (1−σz), +ź +e⊃v +σx +e +ź +(v v1v2) +exp +�iπ +4 (1 − σz +(v1v2)) +� +≡ +ÿ +g +exp +� +iπBock(g)[ +v] +� +|g + dxv⟩⟨g| ≡ +σx +σx +σx +σx +σx +σx +S +S +S +S +S +S +, +(2.13) +which implements a Z2 gauge transformation twisted by a sign depending on the number of ‘up’ states +along ∂ +v. In the above expression, Bock denotes the Bockstein homomorphism, a map of cohomology +classes +Bock : H1(Σ△, Z2) → H2(Σ△, Z2) , +(2.14) +induced from the short exact sequence +1 → Z2 → Z4 → Z2 → 1 . +(2.15) +Similar to the original model (2.1), the gauged model (2.7) also has three gapped phases. The case +1 ≫ κ, ˜κ corresponds to the confined phase, where the gauge fluctuations are energetically suppressed +[FS78, FS79]. Meanwhile, the κ ≫ 1, ˜κ and ˜κ ≫ 1, κ cases correspond to two topologically distinct +deconfined phases. More precisely these are the two topological Z2 gauge phases whose renormalisation +group fixed points are provided by the toric code and double semion model, respectively [DW90, Kit97, +LW04]. +∼ 8 +∼ + +2.3 +Symmetry operators +Let us now study the topological operators leaving the model H∨ invariant. We distinguish two surface +operators. These are the trivial operator or identity Utriv. and the non-trivial operator UZ2 defined as +follows:2 +UZ2[Σ△] := +ÿ +g∈Z1(Σ△,Z2) +Z2d(g)[Σ△] |g⟩⟨g| +(2.18) +≡ +1 +2#(Σ△) +ÿ +g∈Z1(Σ△,Z2) +b∈C1(Σ△,Z2) +n∈C0(Σ△,Z2) +exp +� +iπ +� +Σ△ +b⌣(dn + g) +� +|g⟩⟨g| , +where Z2d(g)[Σ△] is the partition function of a two-dimensional pure Z2 gauge theory coupled to +background Z2 gauge field g and #(Σ△) := |Σ0 +△| + |Σ2 +△| where |Σj +△| is the number of j-simplices in +the triangulation Σ△ (see app. A for details). Henceforth, when there is no scope for confusion, we +shall often omit specifying which sets the various cochains belong to for conciseness. The operator +UZ2[Σ△] commutes with the first term in the Hamiltonian as it acts diagonally in the σz +e basis. It also +commutes with the second and third terms in (2.10) by virtue of +Z2d(g)[Σ△] = Z2d(g + dx)[Σ△] +and +Bock(g + dxv)[ +v] = Bock(g)[ +v] . +(2.19) +Interestingly, this operator has non-invertible fusion rules [RSS22]: +(UZ2 d UZ2)[Σ△] = +1 +22#(Σ△) +ÿ +g,b,n +g′,b′,n′ +(−1) +� +Σ△b⌣(dn+g)+b′⌣(dn′+g′)|g⟩⟨g|g′⟩⟨g′| += +1 +22#(Σ△) +ÿ +g,n +b′,b+,n+ +(−1) +� +Σ△b+⌣(dn+g)+dn+⌣b′ +|g⟩⟨g| += +� +1 +2#(Σ△) +ÿ +b′,n+ +(−1) +� +Σ△b′⌣dn+� +1 +2#(Σ△) +ÿ +g,n,b+ +(−1) +� +Σ△b+⌣(dn+g)|g⟩⟨g| += Z2d[Σ△] · UZ2[Σ△] , +(2.20) +where the partition function Z2d[Σ△] of the pure Z2 gauge theory on Σ△ explicitly reads Z2d[Σ△] = +2b1(Σ)−b0(Σ), where bj is the jth Betti number of Σ. +Let us now attempt to rewrite the action of such a topological surface in terms of spin operators. +To do so, it is instructive to first sum over b in (2.18). Doing so delivers (see app. A) +UZ2[Σ△] = +1 +2χ(Σ) +ÿ +g,n +δdn,g |g⟩⟨g| , +(2.21) +2Here d is the simplicial or lattice codifferential operator d : Cn(Σ△, Z2) → Cn+1(Σ△, Z2) such that for q ∈ +Cn(Σ△, Z2) +dq[v1 . . . vn+1] = +n+1 +ÿ +j=1 +(−1)jq[v1 . . . ˆvj . . . vn+1] , +(2.16) +where (v1 . . . ˆvj . . . vn+1) denotes the n-simplex with the vertex vj omitted. Further ⌣ denotes the cup product ⌣: +Cn(Σ△, Z2) × Cm(Σ△, Z2) → Cn+m(Σ△, Z2) such that +q ⌣ p[v1 · · · vn+m] = q[v1 · · · vn] · p[vn · · · vn+m] , +(2.17) +where p ∈ Cm(Σ△, Z2). Note that these notions can be readily generalised to other finite abelian groups (see e.g. app. A +of ref. [BCH18].) +∼ 9 +∼ + +where χ(Σ) is the Euler characteristic of Σ, dn[v1v2] = n[v1] + n[v2], and δdn,g = δdn+g,0 is a Z2 +Dirac delta function that imposes dn + g = 0 mod 2. This operator can equivalently be expressed +by first introducing ‘virtual’ qubit degrees of freedom at vertices so as to temporarily enlarge the +physical Hilbert space from H∨ to H∨ b Hvirt. with Hvirt. =  +v C[Z2]. +Given |n⟩ ∈ Hvirt. with +n ∈ C0(Σ△, Z2), we thus require an operator that projects onto the constraint subspace of states |g, n⟩ +satisfying n[v1] = g[v1v2] + n[v2] at every edge (v1v2) ⊂ Σ△, before performing a partial trace over +Hvirt.. In symbols, +UZ2[Σ△] = +1 +2χ(Σ) trHvirt. +� ź +e +1 +2(id + σz +s(e)σz +e σz +t(e)) +� +. +(2.22) +Next, we ask, what are the line operators that commute with H∨? In addition to the identity line, a +line operator with support on any 1-cycle ℓ ∈ Z1(Σ△, Z2) labelled by the non-trivial character in Z∨ +2 +may be defined as +ź +e⊂ℓ +σz +e = +ÿ +g +ź +e⊂ℓ +(−1)g[e]|g⟩⟨g| . +(2.23) +One can readily check that these line operators commute with H∨. Moreover, they are topological by +virtue of the kinematical constraints (2.8), so that the sign ś +e⊂ℓ(−1)g[e] only depends on the homology +class of ℓ and is 1 whenever ℓ is a contractible cycle. More generally, any network of such lines can be +assigned a cohomology class in H1(Σ△, Z2). Then the sign obtained by such a network of lines can be +equivalently expressed via a representative cocycle f in H1(Σ△, Z2) as +Utriv.(f) = +ÿ +g +(−1) +� +Σ△f⌣g|g⟩⟨g| . +(2.24) +Consider for instance the following configuration: +σz +σz +σz +σz +σz +σz +σz +, +depicting a local patch of the triangular lattice Σ△ with a topological line operator (2.23) wrapping +along one of the non-contractible cycles. The blue lines represent the only edges where the represen- +tative 1-cocycle f evaluates to the non-trivial group element in Z2. Then for any choice of basis state +|g⟩ labelled by g ∈ Z1(Σ△, Z2), +� +Σ△ +f⌣g = +ź +e⊂ℓ +g[e] mod 2 . +(2.25) +For reference, the figure above also depicts a configuration g which is non-trivial on the red lines and +trivial elsewhere. The expressions on the left hand and right hand side of (2.25) evaluate to −1 for this +choice of g as the σz operators only cross a single red line, and similarly a single plaquette (coloured +in light grey) contributes to the cup product. +∼ 10 +∼ + +Summing over lines in (2.24) is equivalent to summing over f ∈ H1(Σ△, Z2):3 +1 +|H0(Σ△, Z2)| +ÿ +f +Utriv.(f) = +1 +|H0(Σ△, Z2)| +ÿ +g,f +(−1) +� +Σ△f⌣g|g⟩⟨g| += +1 +2#(Σ) +ÿ +g,b,n +(−1) +� +Σ△b⌣(dn+g)|g⟩⟨g| = UZ2[Σ△] , +(2.26) +where, in going to second line, we have introduced a Lagrange multiplier field n, which when summed +over imposes the cocycle condition on b ∈ C1(Σ△, Z2), recovering the first line. Interestingly, per- +forming such a sum yields the surface operator UZ2 defined in eq. (2.18). As we shall comment later +on, this is no mere coincidence. When gauging an abelian 0-form symmetry, one obtains a dual model +with topological line operators labelled by elements in the Pontrjagin dual, and more generally by +representations of the group when it is non-abelian. +Additionally, one obtains topological surface +operators, all of which can be understood by inserting networks (or condensing) suitable sub-algebras +of topological lines. Such surface topological defects have been under scrutiny lately under the name +of condensation defects [KW14, EN17, GJF19, BBSNT22a, RSS22, CCH+22]. It follows immediately +from the definition (2.24) that composition of such (networks of) lines within a surface operator Utriv. +are given by +Utriv.(f1 ◦ f2) = Utriv.(f1 + f2) . +(2.27) +Going back to definition (2.23), this is the statement that these line operators fuse like characters in +Z∨ +2 . Similarly, fusion rules of surface operators Utriv. with networks of lines inserted are given by +Utriv.(f1) d Utriv.(f2) = Utriv.(f1 + f2) . +(2.28) +As suggested by our notation, we shall think of topological lines Utriv.(f) as living on the trivial surface +operator Utriv.. +Next, we consider the operator UZ2[Σ△] defined with a collection of lines inserted. Going back to +definition (2.18) and given any 1-cycle ℓ ∈ Z1(Σ△, Z2), such a line operator acts with the Pauli σx +operator on the virtual qubits, which are traced over, at the vertices v ⊂ ℓ. More generally, any +network of such lines is found to be associated with a Z2-valued 1-cycle on the dual lattice Σ∨ +△, whose +Poincar´e dual is a 1-cocycle f ∈ Z1(Σ△, Z2) as before. The operator UZ2(f)[Σ△] with a network of +such lines inserted has the form +UZ2(f)[Σ△] = +1 +2χ(Σ) trHvirt. +� ź +e +1 +2(id + (−1)f[e]σz +s(e)σz +e σz +t(e)) +� += +1 +2#(Σ△) +ÿ +g,b,n +(−1) +� +Σ△b⌣(dn+f+g)|g⟩⟨g| . +(2.29) +3The choice of normalisation |H0(Σ△, Z2)|−1 is inherited from a convention in defining the partition function of +(d+1)-dimensional finite group gauge theories, namely that the theory assigns a one-dimensional Hilbert space to a +d-sphere for d > 1. +Note that in eq. (2.26), we sum over cohomology classes, rather than cocycles, therefore the +normalisation is |H0(Σ△, Z2)| instead of |C0(Σ△, Z2)|. +∼ 11 +∼ + +Consider for instance the following configuration: +σx +σx +σx +σx +σx +σx +σx +σx +, +depicting such an operator, where as before blue lines represent the only edges where the corresponding +1-cocycle f evaluates to the non-trivial element in Z2. +It readily follows from the definition that +composition rules of networks of lines within a surface operator UZ2[Σ△] are given by +UZ2(f1 ◦ f2)[Σ△] = UZ2(f1 + f2)[Σ△] . +(2.30) +Similarly, fusion rules of surface operators UZ2 with networks of lines inserted are given by +� +UZ2(f1) d UZ2(f2) +� +[Σ△] = UZ2(f1 + f2)[Σ△] . +(2.31) +Finally, we would like to consider the possibility of defining a surface operator UZ2 with support on +a sub-region of Σ△. This requires the existence of a topological line at the junction of topological +surfaces UZ2 and Utriv.. Such a line does exist and is simply obtained by restricting the definition +(2.18) to an open sub-complex Ξ△ ⊆ Σ△, i.e. +UZ2[Ξ△] = +ÿ +g∈Z1(Σ△,Z2) +Z2d(g)[Ξ△] |g⟩⟨g| , +(2.32) +with Dirichlet boundary conditions, i.e., b[∂Ξ△] = 0 imposed. We shall think of this operator as +describing a line operator from a topological surface UZ2 to Utriv.. +Conversely, we shall think of +the operator UZ2[Σ△\Ξ△] as describing a line operator from Utriv. to UZ2. Let us now consider the +composition of the former line operator with the latter. Specifically, we consider a setup where Σ is a +two-torus or a cylinder endowed with a triangulation Σ△, and Ξ△ is an annular strip of width a single +lattice spacing wrapping a non-contractible cycle: +Ξ△ +. +∼ 12 +∼ + +Then the composition of the lines is given by +UZ2[Ξ△] = +ÿ +g +Z2d[Ξ△] |g⟩⟨g| = +ÿ +g,f +(−1) +� +Ξ△f⌣g|g⟩⟨g| = id + +ź +e⊂ℓ +σz +e , +(2.33) +where ℓ refers here to the non-contractible cycle wrapped by Ξ△. In the second equality, the sum +is over f ∈ H1(Ξ△, ∂Ξ△, Z2) ∼= Z2, i.e., the relative cohomology group with Dirichlet boundary +conditions imposed. Note also that the normalisation was implicitly modified from 1/|H0(Ξ△, Z2)| +to 1/|H0(Ξ△, ∂Ξ△, Z2)| = 1. The non-trivial class in H1(Ξ△, ∂Ξ△, Z2) corresponds to an f-defect +wrapping the non-contractible cycle in Ξ△. For this choice of f, +� +∂Ξ△ f ⌣ g evaluates to ś +e⊂ℓ g[e] +mod 2. As expected, this results in a line operator living on Utriv., which is labelled by the regular +representation of Z2. The fusion rule of topological lines in (2.33) is closely related to the fusion rules +of Kramers-Wannier duality defects in the (1+1)d transverse-field Ising model [CCH+21, CCH+22]. +Now let us compute the composition of the topological line between Utriv. and UZ2 by considering a +thin annular strip of single lattice spacing width containing the identity operator Utriv., while the rest +of the lattice Σ△\Ξ△ containing UZ2. Let us specialise to the case where Σ is a two-torus such that +Σ△\Ξ△ is path-connected. Let us denote the left and right boundaries of Ξ△ as ∂LΞ△ and ∂RΞ△, +respectively. Then, the composition of lines is given by the operator +UZ2[Σ△\Ξ△] = +1 +2#(Σ△\Ξ△) +ÿ +g,b,n +(−1) +� +Σ△\Ξ△b⌣(dn+g)|g⟩⟨g| = +1 +2χ(Σ△\Ξ△) +ÿ +g,n +δΣ△\Ξ△ +dn,g +, +(2.34) +where in the first expression b ∈ C1(Σ△\Ξ△, Z2) with the Dirichlet condition b[∂LΞ△] = b[∂RΞ△] = 0 +imposed. Meanwhile n ∈ C0(Σ△, Z2)4 has no constraints imposed a priori. In the final expression, we +sum over b, which imposes the cocycle condition dn = g everywhere except within Ξ△, denoted by +δΣ△\Ξ△ +dn,g +. Besides, note that the Euler characteristic χ(Σ△\Ξ△) = 0, since Σ\Ξ is a cylinder. Now pick +a preferred edge in Ξ△. Naturally dn = g + s, where s is valued in Z2, on this edge. It follows from +conditions dg = 0 and dn = g on ∂Ξ△ that fixing s on any chosen edge in Ξ△ pins the configuration +to the same value of s for all other edges in Ξ△. Consider for instance the following configuration: +∂LΞ△ +∂RΞ△ +. +As before, the 1-cocycle g is non-trivial only on the red edges. The condition dn = g is satisfied +everywhere in Σ△\Ξ△, i.e., everywhere apart from the central region in grey. Then we consider n +to be fixed to a certain configuration on ∂LΞ△. Fixing the configuration of n on a single vertex (for +instance the one highlighted in green) on ∂RΞ△, pins the configuration on all other vertices on ∂RΞ△. +4n is a 0-cochain on Σ△ since C0(Σ△, Z) = C0(Σ△\Ξ△, Z). +∼ 13 +∼ + +The two choices at this vertex correspond to either the presence or absence of a line in VecZ2 traversing +Ξ△. Therefore, defining a Z2 cocycle f that evaluates to the non-trivial element in Z2 on every edge +in Ξ△ (indicated in blue in the above diagram) and to the identity element elsewhere, we obtain +UZ2[Σ△\Ξ△] = UZ2[Σ△] + UZ2(f)[Σ△] , +(2.35) +which amounts to a line operator living on UZ2[Σ△]. This concludes our analysis of the symmetry +structure of the gauged transverse-field Ising model. +We showed in this section that starting from a two-dimensional lattice model with arguably the +simplest kind of symmetry, namely a 0-form Z2 symmetry, gauging the symmetry results in a model +with non-invertible surface operators. It turns out that the surface and line operators, together with +their statistics, are organised into an algebraic structure referred to as the fusion 2-category of 2- +representation of Z2. In the next section, we present a framework allowing for the systematic gauging +of arbitrary invertible symmetries and analysis of the resulting symmetry structures in terms of higher +representations of groups, and categorifications thereof. +SECTION 3 +Gauging and dual symmetries +Motivated by the analysis of the (2+1)d transverse-field Ising model carried out above, we introduce in +this section a systematic approach to gauging invertible symmetries in (2+1)d quantum lattice models +and studying the resulting higher categorical symmetries. +3.1 +G-symmetric Hamiltonians +Throughout this manuscript, our starting point is always a two-dimensional quantum lattice model +with a global 0-form G symmetry, where G is a finite (possibly non-abelian) group. Concretely, it means +that the Hamiltonian commutes with topological operators supported on the whole two-dimensional +space, which are labelled by group elements of G, in such a way that the fusion of symmetry operators +is governed by the multiplication rule of the group. By definition of a group, these symmetry operators +are in particular invertible. +The modern approach to global symmetries in quantum field theories in terms of collections of +topological defects invites us to organise symmetry operators and their properties into higher cate- +gories. More specifically, given a (2+1)d quantum theory we expect symmetries to correspond to +fusion 2-categories in the sense of Douglas and Reutter [DR18], where objects label topological sur- +face operators and (1-)morphisms label topological line operators at the junctions of surface operators. +In this context, a G-symmetric Hamiltonian commutes with surface operators that form the so-called +fusion 2-category of G-graded 2-vector spaces. Let us present this fusion 2-category in some detail.5 +First of all, let us define a 2-vector space as a C-linear, finite, semisimple category. We can then +consider the 2-category 2Vec of 2-vector spaces, linear functors and natural transformations. It is +a prototypical example of fusion 2-category, where the monoidal structure is given by the Deligne +5In the vein of sec. 3.6, we shall think of 2VecG as a categorification of the fusion (1-)category VecG of G-graded +vector spaces, the same way we can think of VecG as a categorification of C[G], whereby the ring C is promoted to the +fusion category Vec. +∼ 14 +∼ + +tensor product. +Note that 2Vec has a unique equivalence class of simple objects, which is repre- +sented by the category Vec of complex vector spaces. Let us now consider the 2-groupoid6 [G, •, •] +with object-set G, no non-trivial 1-morphisms and no non-trivial 2-morphisms. Consider the cate- +gory 2Fun([G, •, •], 2Vec) of pseudofunctors, pseudonatural transformations and modifications between +[G, •, •] and 2Vec. +By definition, an object V in 2Fun([G, •, •], 2Vec) assigns to every g ∈ G a 2- +vector space Vg in 2Vec, and thus amounts to a G-graded 2-vector space of the form V = Ð +g∈G Vg. +Pseudonatural transformations in 2Fun([G, •, •], 2Vec) then correspond to grading preserving linear +functors, and modifications to natural transformations. The convolution product of pseudofunctors +[G, •, •] → 2Vec endows 2Fun([G, •, •], 2Vec) with the structure of a fusion 2-category according to +(V d W)g := +ð +x∈G +Vx b Vx−1g +(3.1) +with unit 1 satisfying 1g = δg,1G Vec. Henceforth, we denote by 2VecG this fusion 2-category. There +are |G|-many simple objects in 2VecG provided by the ‘one-dimensional’ 2-vector spaces Vecg, for every +g ∈ G, such that Vecg1 d Vecg2 ∼= Vecg1g2 and Hom2VecG(Vecg1, Vecg2) ∼= δg1,g2 Vec. At the end of the +day, it follows that simple objects can be safely identified with the corresponding group elements in +G, but the higher categorical perspective will be crucial in the following. +Let us now construct local operators that explicitly commute with symmetry operators labelled +by simple objects in 2VecG in the spirit of ref. [LDOV21] using the tools introduced in ref. [Del21]. +Let Σ be a closed oriented two-dimensional surface endowed with a triangulation Σ△. Although our +construction applies to arbitrary triangulations Σ△, let us assume for concreteness that Σ△ is isotopic +to the Poincar´e dual of a honeycomb lattice. We further assume that Σ△ has a total ordering of its +0-simplices (vertices), referred to as a choice of branching structure, such that the branching structure +in the neighbourhood of every vertex v ≡ (3) ⊂ Σ△ is of the form +0 +1 +4 +6 +5 +2 +3 +, +ˆu +ˆv +ˆ +w +. +(3.2) +Notice that a choice of branching structure induces an orientation of each 1-simplex (edge), which +is always chosen to be from the lowest ordered vertex to the higher ordered one. Let m denote an +assignment of group elements in G to vertices of Σ△. By a slight abuse of notation, we notate via +C0(Σ△, G) the collection of such assignments, which corresponds to a G-valued 0-cochain when G is +abelian. We define the microscopic Hilbert space of the system to be  +v C[G] and denote by |m⟩ the +assignment m regarded as an element of the microscopic Hilbert space. The restriction of |m⟩ to a given +vertex v ⊂ Σ△ is denoted by |m[v]⟩ ∈ C[G], and more generally, we notate via |m[Ξ△]⟩ :=  +v⊂Ξ△ |m[v]⟩ +the state associated with the restriction of m to a sub-complex Ξ△ ⊆ Σ△. +We are interested in G-symmetric local operators acting on the Hilbert space  +v C[G]. Given +a vertex v ⊂ Σ△, we notate via +v ⊆ Σ△ the hexagonal sub-complex centred around v. Let us +now consider the pinched interval cobordism +v×p.I ≡ +v ×p. [0, 1] defined as +v×I/ ∼, where the +6A 2-groupoid is a 2-category in which every morphism is an equivalence, in the same spirit of a 1-groupoid being a +(small) category in which every morhism is invertible. +∼ 15 +∼ + +equivalence relation ∼ is such that (x, i) ∼ (x, i′) for all (x, i), (x, i′) ∈ ∂ +v×I. Graphically, +I +v := +v ×p. I ≡ +0 +1 +4 +6 +5 +2 +3′ +3 +, +(3.3) +where the branching structure induced from that of eq. (3.2) is such that 2 < 3′ < 3. Notice that, by +definition, we have ∂ +I +v = +v×{0} ∪∂ +v +v×{1}. Henceforth, we employ the shorthand notations +0 +v ≡ +v×{0} and +1 +v ≡ +v×{1}. Given an assignment m ∈ C0( +I +v, G), we can define an operator +acting on a local neighbourhood of the vertex v as +��m[ +1 +v] +�� +m[ +0 +v] +��. Notice that this operator acts as +the identity operator at every vertex but v where it acts as +��m[v′] +�� +m[v] +��. More general operators can +then be constructed by considering linear combinations of the form +hv,n := +ÿ +m∈C0( +I +v,G) +hv,n +� +{m[v1]m[v2]−1}(v1v2)⊂ +I +v +� ��m[ +1 +v] +�� +m[ +0 +v] +�� , +(3.4) +where the coefficients hv,n are valued in U(1) and the oriented edges (v1v2) are always such that +v1 < v2. Note that, in practice, we typically consider models for which the coefficients hv,n are only +non-vanishing for specific choices of assignments m ∈ C0( +I +v, G). Any combinations of such operators +can finally be used to define a local Hamiltonian H ≡ ř +v hv := ř +v +ř +n hv,n. +By construction, any Hamiltonian thus defined is G-symmetric, whereby the symmetry is generated +by operators ś +v⊂Σ△ Rx +v with Rx +v : |m[v]⟩ �→ |m[v]x−1⟩ for any x ∈ G. This simply follows from a +redefinition of the variable m ∈ C0( +I +v, G) in the summation, together with the fact that the coefficients +hv,n only depend on {m[v1]m[v2]−1}(v1v2)⊂ +I +v and are therefore manifestly symmetric. Identifying every +m[v] ∈ G with the corresponding simple object Vecm[v] in 2VecG, one can equivalently state that any +Hamiltonian thus defined is 2VecG-symmetric. It is a straightforward exercise—which we carry out +below—to show that the transverse-field Ising model is of this form, and more generally, we can +argue that every local G-symmetric Hamiltonian can be written in terms of combinations of local +operators of the form (3.4). Note that Hamiltonians defined in this section only account for nearest or +next-nearest neighbours interactions. However, we can readily combine such local operators—which +geometrically amounts to concatenating complexes of the form (3.3)—so as to define local operators +simultaneously acting on a larger number of sites, thereby generating the whole algebra of G-symmetric +Hamiltonians on  +v C[G]. That being said, most familiar and physically relevant quantum systems, +e.g. Heisenberg-like models, are already included within the present formalism. +In prevision for the following section, let us slightly reformulate the previous construction. We +noticed above that the G symmetry of local operators (3.4) is guaranteed in particular by the fact that +the unitary coefficients hv,n only depend on the assignment m through group elements m[v1]m[v2]−1 for +every edge (v1v2) ⊂ +I +v. This leads us to contemplate the following alternative description: Consider +an assignment g of group elements in G to every edge of +I +v such that g[v1v2]g[v2v3] = g[v1v3] for +every 2-simplex (v1v2v3) ⊂ +I +v, which we shall think about as a flat gauge field on +I +v. By slight +∼ 16 +∼ + +abuse of notation, we notate via Z1( +I +v, G) the collection of such assignments. Let m ∈ C0( +I +v, G) be +an assignment as before but with the additional constraint that m[v1] = g[v1v2]m[v2] for every edge +(v1v2) ⊂ +I +v. In other words, we require dm = g, where dm[v1v2] = m[v1]m[v2]−1. We can now rewrite +the previous operators as follows: +hv,n = +ÿ +g∈Z1( +I +v,G) +hv,n(g) +ÿ +m∈C0( +I +v,G) +dm=g +��m[ +1 +v] +�� +m[ +0 +v] +�� . +(3.5) +Notice that given g, the condition dm = g does not fully constrain m. In the following section, we +consider generalizations of these operators yielding dual Hamiltonian models. +Back to the transverse-field Ising model +Let us illustrate our construction by recasting the transverse-field Ising model in terms of local op- +erators (3.5). We also discuss a finite group generalisation of this model in sec. 3.8. Consider the +Hamiltonian H = ř +v⊂Σ△ +ř4 +n=1 hv,n. For any vertex v ⊂ Σ△ and gauge field g ∈ Z1( +I +v, G), the +defining U(1)-coefficients hv,n(g) are chosen to be +hv,1(g) := −Jδg[v′v],0(−1)g[v v+ˆu] , +hv,2(g) := −Jδg[v′v],0(−1)g[v v+ˆv] , +hv,3(g) := −Jδg[v′v],0(−1)g[v v+ ˆ +w] , +hv,4(g) := −Jκ δg[v′v],1 , +(3.6) +where the branching structure of +I +v is that given in eq. (3.3). We can readily confirm that hv,4 acts as +−Jκσx +v , whereas local operators hv,n=1,2,3 act as −Jσz +v σz +v+ˆu, −Jσz +v σz +v+ˆv and −Jσz +v σz +v+ ˆ +w, respectively. +Putting everything together, we recover Hamiltonian (2.1) for ˜κ = 0. By construction, this model +has a 2VecZ2 symmetry such that the two simple objects Vec0 and Vec1 in 2VecZ2, where 0, 1 ∈ Z2, +are identified with the surface operators O0 and O1 defined in sec. 2, respectively. The fact that +the deformation class of models generated by (3.6) does not host any non-trivial line operators then +follows from Hom2VecZ2 (Vecg1, Vecg2) ∼= δg1,g2 Vec. +3.2 +Dual Hamiltonians +Given a Hamiltonian H = ř +v +ř +n hv,n with local operators hv,n as defined in eq. (3.5), we shall now +construct dual models. +In sec. 3.4, we shall relate these various dual models to twisted gauging +of the G symmetry or sub-symmetries thereof. +Our strategy goes as follows: Any finite group G +gives rise to an (abstract) algebra of local operators, in such a way that products of local operators +only make use of the multiplication in G. A duality class of models is then determined by choosing +certain linear combinations of local operators in the algebra. This choice is made through the set +of coefficients {hv,n(g)}g over g ∈ Z1( +I +v, G) in our context. This means that the group G together +with the collection hv,n of coefficients fully determine the physical characteristics of the duality class +of models as encoded into their common spectrum. Notice that we have not yet specified explicit +matrix/lattice representations of these local operators on a chosen Hilbert space. +As a matter of +fact, picking a representative of a duality class of models precisely corresponds to choosing such a +matrix representation. +Loosely speaking, this boils down to identifying a collection of degrees of +freedom providing a particular physical realisation of the properties encoded into the spectrum. In +other words, maintaining the same linear combination of symmetric operators, while choosing another +matrix realisation, yields a dual model. We explain below how such choices are made, thereby defining +duality classes of (2+1)d Hamiltonian models. +∼ 17 +∼ + +We begin our construction by noticing that picking a gauge field g ∈ Z1( +I +v, G) amounts to +assigning a simple object g[v1v2] ≡ Vecg[v1v2] in 2VecG to every edge (v1v2) ⊂ +I +v such that Vecg[v1v2] d +Vecg[v2v3] ∼= Vecg[v1v3] for every 2-simplex (v1v2v3) ⊂ +I +v. +In this context, picking an assignment +m ∈ C0( +I +v, G) such that dm = g amounts to assigning simple objects m[v1] ≡ Vecm[v1] in 2VecG such +that Vecg[v1v2] d Vecm[v2] ∼= Vecm[v1] for every edge (v1v2) ⊂ +I +v. We think of this latter assignment +as making a choice of degrees of freedom, and thus a choice of microscopic Hilbert space. As will +become clear in the following, this choice amounts to considering 2VecG as a module 2-category over +itself, inviting us to replace 2VecG by another module 2-category. In that spirit, let us first review the +notion of module 2-category over 2VecG as considered in ref. [D´e21, Del21]. +Succinctly, a module 2-category over 2VecG is a 2-category with a G-action. +More precisely, we +define a (left) 2VecG-module 2-category as a quadruple (M, ▷, α▷, π▷) consisting of a (C-linear finite +semisimple) 2-category M, a binary action 2-functor ▷ : 2VecG × M → M and an adjoint natural +2-equivalence α▷ : (− d −) ▷ − +∼ +−→ − ▷ (− ▷ −) satisfying a ‘pentagon axiom’ up to an invertible +modification π▷ whose components π▷ +Vecg1,Vecg2,Vecg3,M are defined via +(Vecg1 d (Vecg2 d Vecg3) ▷ M +Vecg1 ▷ ((Vecg2 d Vecg3) ▷ M) +((Vecg1 d Vecg2) d Vecg3) ▷ M +Vecg1 ▷ (Vecg2 ▷ (Vecg3 ▷ M)) +(Vecg1 d Vecg2) ▷ (Vecg3 ▷ M) +α▷ +Vecg1g2 ,Vecg3 ,M +α▷ +Vecg1 ,Vecg2 ,Vecg3 ▷M +1Vecg1 ▷α▷ +Vecg2 ,Vecg3 ,M +1Vecg1g2g3 ▷1M +α▷ +Vecg1 ,Vecg2g3 ,M +π▷ +Vecg1 ,Vecg2 ,Vecg3 ,M +, +(3.7) +for every g1, g2, g3 ∈ G and M ∈ M. The invertible modification π▷, which shall be referred to as the +left module pentagonator, is required to satisfy an ‘associahedron axiom’. For convenience, we shall +spell out this axiom employing an alternative to commutative diagrams in terms of string diagrams, +whereby regions represent objects, strings 1-morphisms, and blobs 2-morphisms. In practice, we shall +omit labelling regions but the corresponding objects can be recovered from the string labels. On these +diagrams, compositions of 1-morphisms is read from left to right, whereas the (vertical) composition +of 2-morphisms is read from top to bottom. For instance, the left module pentagonator π▷ can be +equivalently defined via the string diagram: +π▷ +11 +α▷ +1α▷ +α▷ +α▷ +, +(3.8) +where we omitted the d and ▷ symbols. The associahedron axiom satisfied by π▷ can then be conve- +∼ 18 +∼ + +niently expressed as the following equality of string diagrams: +11 +11 +α▷ +(11)α▷ +π1 +π▷ +π▷ +(11)1 +11 +(11)1 +α▷ +1α▷ 1(1α▷) +α▷ +α▷ +α▷ += +1(11) +1α▷ +1α▷ +α▷ +1(11) +1π▷ +π▷ +π▷ +(11)1 +11 +(11)1 +α▷ +1α▷ 1(1α▷) +α▷ +α▷ +α▷ +. +(3.9) +We are particularly interested in indecomposable 2VecG-module 2-categories constructed as follows +[Del21]:7 Given a subgroup A ⊆ G and a normalised 3-cocycle λ in H3(A, U(1)), let M(A, λ) be a +2-category with object-set the set G/A of left cosets. A left 2VecG-module structure can be defined +on M(A, λ) via Vecg ▷ M := (gr(M))A for any g ∈ G and M ∈ G/A, where r : G/A → G assigns to +every left coset its representative in G. Notice that in general we have gr(M) ̸= r(Vecg ▷ M) and we +denote by ag,M the group element in A satisfying +gr(M) = r(Vecg ▷ M)ag,M . +(3.10) +Associativity of the multiplication in G imposes +ag1g2,M = ag1,Vecg2▷M ag2,M , +∀ g1, g2 ∈ G and M ∈ G/A . +(3.11) +Endowing the abelian group Hom(G/A, U(1)) with a left G-module structure, we consider the 3-cochain +π▷ ∈ C3(G, Hom(G/A, U(1))) defined as +π▷(g1, g2, g3)(M) := λ(ag1,Vecg2g3▷M , ag2,Vecg2▷M , ag3,M) , +∀ g1, g2, g3 ∈ G and M ∈ G/A . +(3.12) +In virtue of the 3-cocycle condition dλ = 1 and eq. (3.11), we have +π▷(g2, g3, g4)(M) π▷(g1, g2g3, g4)(M) π▷(g1, g2, g3)(Vecg4 ▷ M) += π▷(g1g2, g3, g4)(M) π▷(g1, g2, g3g4)(M) , +∀ g1, g2, g3, g4 ∈ G and M ∈ G/A , +(3.13) +so that π▷ is a Hom(G/A, U(1))-valued 3-cocycle of G. Defining the invertible modification π▷ with +components +π▷ +Vecg1,Vecg2,Vecg3,M := π▷(g1, g2, g3)(M) · 1(g1g2g3r(M))A , +∀ g1, g2, g3 ∈ G and M ∈ G/A , +(3.14) +7Note that this construction does not give all 2VecG-module 2-categories. Physically, it only gives those module +2-categories that correspond to either spontaneously breaking the global symmetry down to a subgroup and/or pasting +a symmetry-protected topological phase labelled by a 3-cocycle of the preserved subgroup. Notably, we do not discuss +those module 2-categories that correspond to coupling to a inherently two-dimensional non-anomalous topological order. +It is expected that these topological orders would be contained in the completion of the 3-category of 2VecG-module +2-categories [BSNT22]. +∼ 19 +∼ + +we finally obtain that the quadruple M(A, λ) ≡ (M(A, λ), ▷, 1, π▷) does define a left 2VecG-module +2-category. For any group G, we can always choose the subgroup A to be either the trivial subgroup +{1G} or the whole group G, and the 3-cocycle to be trivial. The corresponding module 2-categories +are M({1g}, 1) ∼= 2VecG and M(G, 1) ∼= 2Vec with action 2-functors given by the monoidal product +in 2VecG and the forgetful functor 2VecG → 2Vec, respectively. +Let us now put these module 2-categories to use in order to construct dual Hamiltonians. Given a +vertex v ⊂ Σ△, a gauge field g ∈ Z1( +I +v, G) and a 2VecG-module 2-category M ≡ M(A, λ) as defined +above, let us consider an assignment m of simple objects m[v1] ∈ M to every vertex v1 ⊂ +I +v such that +m[v1] +!= Vecg[v1v2] ▷ m[v2] , +∀ (v1v2) ⊂ +I +v . +(3.15) +We notate via C0 +g( +I +v, M) the set of assignments m fulfilling conditions (3.15). Given such a pair +(g, m) ∈ Z1( +I +v, G) × C0 +g( +I +v, M), let us introduce the following phase factor: +π▷ +v (g, m) := +ź +(v1v2v3v4)⊂ +I +v +π▷(g[v1v2], g[v2v3], g[v3v4])(m[v4])ϵ(v1v2v3v4) , +(3.16) +where π▷ is the Hom(G/A, U(1))-valued 3-cocycle of G defined in eq. (3.12), and ϵ(v1v2v3v4) = ±1 +depends on the orientation of the 3-simplex (v1v2v3v4). Borrowing the notations of sec. 3.1, we finally +define new local operators as follows: +hM +v,n = +ÿ +g∈Z1( +I +v,G) +hv,n(g) +ÿ +m∈C0g( +I +v,M) +π▷ +v (g, m) +��(g, m)[ +1 +v] +�� +(g, m)[ +0 +v] +�� . +(3.17) +Notice immediately that choosing M to be 2VecG itself, we recover local operators hv,n defined in +eq. (3.5). We are now ready to state one of the main results of this manuscript: Hamiltonian models +that only differ in a choice of 2VecG-module 2-category are dual to one another via definition (3.17) +of the local operators. In other words, Hamiltonians HM = ř +v +ř +n hM +v,n and HM′ = ř +v +ř +n hM′ +v,n for any +two indecomposable 2VecG-module 2-categories M ≡ M(A, λ) and M′ ≡ M(A′, λ′) are dual to one +another. +As motivated above, duality between HM and HM′ follows from the fact 2VecG-module 2-categories +M and M′ merely encode distinct matrix realisations of the same local operators. More concretely, +regardless of the choice of 2VecG-module 2-category M, the set of local operators hM +v,n generate the +same algebra of operators characteristic of the duality class of models. In the same vein as the lower- +dimensional study carried out in ref. [LDOV21], we can readily confirm that products of local operators +indeed only involve the group G and coefficients hv,n. Concretely, computing products of local opera- +tors (3.17) involve algebraic manipulations that geometrically translate into three-dimensional Pachner +moves, which are encoded into the associahedron axiom given in eq. (3.9). The associahedron axiom +dictates that products of operators only depend on the monoidal structure of 2VecG via the monoidal +pentagonator π, which happens to be trivial, and is a fortiori independent of M. +An alternative +justification consists in showing that the Hamiltonian HM with M ≡ M(A, λ) is the result of the +λ-twisted gauging of the A sub-symmetry of H. This will be the purpose of sec. 3.4. +Importantly, dualities as considered in this manuscript systematically map symmetric local op- +erators to dual symmetric local operators—this almost tautologically follows from our definition of a +duality as a change of matrix realisation of the local operator encoded into a choice of 2VecG-module +2-category—whereas non-symmetric local operators are mapped to dual non-local non-symmetric op- +erators. These various mappings are realised via (typically non-local) lattice duality operators that +∼ 20 +∼ + +transmute in particular local operators into one another. Below, we explicitly construct these lattice +operators. +Back to the transverse-field Ising model +We explained in the previous section how to recast the transverse-field Ising model within our frame- +work. The input fusion 2-category being 2VecZ2, we distinguish three choices of module 2-categories, +namely 2VecZ2 itself, 2Vec and 2Vecλ, respectively, where λ corresponds to the non-identity element +in H3(Z2, U(1)) ≃ Z2. Given the coefficients (3.6), it readily follows from definition (3.17) of the +local operators that h2Vec +v,4 +acts as −Jκ ś +e⊃v σx +e , whereas local operators h2Vec +v,n=1,2,3 acts as −Jσz +(v v+ˆu), +−Jσz +(v v+ˆv) and −Jσz +(v v+ ˆ +w), respectively. Putting everything together, we recover the Hamiltonian +(2.10) with ˜κ = 0 resulting from gauging the Z2 symmetry of (2.1). Choosing instead the 2VecZ2- +module 2-category 2Vecλ amounts to the λ-twisted gauging of the Z2 symmetry and results in Hamil- +tonian (2.10) with κ = 0. +3.3 +Duality operators +We are interested in dualities between Hamiltonians HM and HM′ whose local operators hM +v,n and hM′ +v,n +only differ in the choice of 2VecG-module 2-category. Therefore, a duality operator should have the +interpretation of a map between the module 2-categories that is compatible with the action of 2VecG. +More concretely, given a pair of (left) 2VecG-module 2-categories (M, ▷, α▷, π▷) and (M′, ·▷, α·▷, π·▷), +we define a 2VecG-module 2-functor between them as a triple (F, ω, Ω) consisting of a 2-functor F : +M → M′ and an adjoint natural 2-equivalence ω : F(− ▷ −) +∼ +−→ − ·▷ F(−), with components ωVecg,M +for g ∈ G and M ∈ M, satisfying a ‘pentagon axiom’ up to an invertible modification Ω whose +components ΩVecg1,Vecg2,M are defined via +F(Vecg1 ▷ (Vecg2 ▷ M)) +Vecg1 ·▷ F(Vecg2 ▷ M) +F((Vecg1 d Vecg2) ▷ M) +Vecg1 ·▷(Vecg2 ·▷ F(M)) +(Vecg1 d Vecg2) ·▷ F(M) +ωVecg1g2 ,M +α·▷ +Vecg1 ,Vecg2 ,F (M) +1Vecg1 ·▷ ωVecg2 ,M +F (α▷ +Vecg1 ,Vecg2 ,M) +ωVecg1 ,Vecg2 ▷M +ΩVecg1 ,Vecg2 ,M +(3.18) +for every g1, g2 ∈ G and M ∈ M. As before, we shall prefer the equivalent definition in terms of the +string diagram +Ω +F (α▷) +ω +1ω +ω +α·▷ +. +(3.19) +∼ 21 +∼ + +This invertible modification Ω is required to satisfy an ‘associahedron axiom’ encoded into the following +equality of string diagrams: +F (α▷) +F (α▷) +ω +(11)ω +F (π▷) +Ω +Ω +F (11) F (α▷) F (1α▷) ω +1ω +1(1ω) +ω +α·▷ +α·▷ += +1ω +1α·▷ +α·▷ +11 +1Ω +Ω +π·▷ +F (11) F (α▷) F (1α▷) ω +1ω +1(1ω) +ω +α·▷ +α·▷ +. +(3.20) +Let us now use the data of a module 2-functor M → M′ to construct lattice operators that transmute +local operators hM +v,n into hM′ +v,n . Since module 2-functors can be composed—and by extension so do +the corresponding dualitiy operators—we can focus without loss of generality on 2VecG-module 2- +functors between 2VecG itself and M′ ≡ M(A′, λ′). +Every such module 2-functor is of the form +(− ▷ M ′, 1, π▷ +−,−,−,M ′), with M ′ ∈ M′, in which case the associahedron axiom (3.20) boils down to +(3.9). In the spirit of our definition of the local operators, let us consider the complex8 +v×I ≡ +2 +4 +0 +1 +5 +6 +2′ +4′ +0′ +1′ +5′ +6′ +3 +3′ +, +(3.21) +centred around v ≡ (3). The branching structure agrees with that of eq. (3.3), and in particular we have +v′ +1 < v1 for every v1 ⊂ +v. Given a simple object M ′ ∈ M′ interpreted as a 2VecG-module 2-functor +2VecG → M′, we consider the following assignment of degrees of freedom: First, let g ∈ Z1( +0 +v, G) +and g′ ∈ Z1( +1 +v, G) such that g[v1v2] = g′[v′ +1v′ +2] for every v1, v2 ⊂ +v. We then consider an assignment +m of simple objects m[v1] ∈ 2VecG to every vertex v1 ⊂ +0 +v and an assignment m′ of simple objects +m′[v′ +1] ∈ M′ to every vertex v′ +1 ⊂ +1 +v such that m[v1] = Vecg[v1v2] d m[v2] for every (v1v2) ∈ +0 +v, +m′[v′ +1] = Vecg′[v′ +1v′ +2] ▷ m′[v′ +2] for every (v′ +1v′ +2) ∈ +1 +v, and m[v1] ▷ M ′ = m′[v′ +1] for every (v′ +1v1) ⊂ +v×I. +As before, we notate via C0 +g( +0 +v, G) the collection of assignments m fulfilling the conditions spelt +out above. Notice that assignment m ∈ C0 +g( +0 +v, G) together with M ′ ∈ M′ uniquely specifies an +assignment m′ via the constraints m[v1] ▷ M ′ = m′[v′ +1] for every (v′ +1v1) ⊂ +v ×I, and we denote by +m ▷ M ′ this assignemnt. +8Alternative operators more suited to the more general case of an arbitrary fusion 2-category can be found in +ref. [Del21]. +∼ 22 +∼ + +Given assignments (g, m) ∈ Z1( +0 +v, G) × C0 +g( +0 +v, G), let us introduce the following phase factor: +π▷ +v (g, m, M ′) := +ź +(v1v2v3)⊂ +0 +v +π▷(g[v1v2], g[v2v3], m[v3])(M ′)ϵ(v1v2v3) , +(3.22) +where π▷ is the Hom(G/A′, U(1))-valued 3-cocycle of G defined previously, and ϵ(v1v2v3) = ±1 depends +on the orientation of the 2-simplex (v1v2v3). We finally define the duality operator labelled by M ′ ∈ M′ +acting at vertex v ⊂ Σ△ as follows: +dM ′ +v += +ÿ +g∈Z1( +0 +v ,G) +m∈C0 +g( +0 +v ,G) +π▷ +v (g, m, M ′) |g, m ▷ M ′⟩⟨g, m| . +(3.23) +What is the action of these operators? +On the one hand, the operator turns degrees of freedom +provided by simple objects in 2VecG—thought as a module 2-category over itself—into simple objects +in M′ via the module 2-functor − ▷ M ′. On the other hand, it acts by scalar multiplication by the +phase factors π▷ +v (g, m, M ′). It follows that acting with dM ′ +v +on hv,n yields hM′ +v,n , i.e., +dM ′ +v +◦ hv,n = hM′ +v,n ◦ dM ′ +v +. +(3.24) +The only non-trivial aspect to confirm is the compatibility of the various phase factors. It is convenient +to do so using the geometrical interpretations of the operators. +Geometrically, the commutation +relation (3.24) can be represented as follows: += +, +(3.25) +where we think of the complex +I +v on the l.h.s. as supporting the local operator hv,n and that on +the r.h.s. as supporting hM′ +v,n . Recall that the phase factors entering the definition of hv,n are trivial, +whereas those entering the definition of hM′ +v,n are given by π▷. Similarly, the phase factors entering +the definition of dM ′ +v +evaluates to π▷. It follows from the associahedron axiom satisfied by the module +pentagonator π▷—or rather the cocycle condition of the 3-cocycle it evaluates to—as well as the +fact that every pair of neighbouring 3-simplices share a 2-simplex, that the commutation relation is +∼ 23 +∼ + +satisfied. Indeed, eq. (3.20) graphically translates as +v1 +v4 +v2 +v3 +v′ +2 +v′ +4 +v′ +1 +v′ +3 +v′ +3 +v′ +1 += +v′ +2 +v′ +3 +v′ +1 +v′ +4 +v1 +v3 +v2 +v4 +v4 +v1 +, +(3.26) +where vertices carrying the same label are identified. Applying the assignment rules of degrees of +freedom presented above, we find that the phase factors associated with the 3-simplices on the l.h.s. +and r.h.s. are 1 and π▷(g[v1v2], g[v2v3], g[v3v4])(m′[v′ +4]), respectively. Similarly, we associate to the +prisms on the l.h.s. the phase factors π▷(g[v1v2], g[v2v3], m[v3])(M ′) and π▷(g[v1v3], g[v3v4], m[v4])(M ′), +whereas we associate to the prisms on the r.h.s. the phase factors π▷(g[v2v3], g[v3v4], m[v4])(M ′) and +π▷(g[v1v2], g[v2v4], m[v4])(M ′). Choosing g[v1v2] ≡ g1, g[v2v3] ≡ g2, g[v3v4] ≡ g3 and m[v4] ≡ g4 ∈ G, +it follows from the various assignment rules—e.g. m′[v′ +4] = m[v4] ▷ M ′ = Vecg4 ▷ M ′—that eq. (3.26) +exactly encodes eq. (3.13). Applying eq. (3.26) for every 3-simplex in +I +v, we find that all the phase +factors of the form π▷(g[v1v3], g[v3v4], m[v4])(M ′) and π▷(g[v2v3], g[v3v4], m[v4])(M ′) cancel two-by-two +resulting in eq. (3.25). More generally, given local operators hM +v,n and hM′ +v,n , the analogue of eq. (3.26) +will be guaranteed by the associahedron axiom (3.20) fulfilled by the 2VecG-module structure of the +2-functor M → M′ (see sec. 3.5 for the case of module 2-endofunctors). +So we have found duality operators performing the transmutation of local operators hv,n into hM′ +v,n . +In order to perform this operation to the whole Hamiltonian, it suffices to extend the definition of our +duality operator to the whole Σ△ following exactly the same construction: +dM ′ = +ÿ +g∈Z1(Σ0 +△,G) +m∈C0 +g(Σ0 +△,G) +� +ź +(v1v2v3)⊂Σ0 +△ +π▷(g[v1v2], g[v2v3], m[v3])(M ′)ϵ(v1v2v3) +� +|g, m ▷ M ′⟩⟨g, m| . +(3.27) +Let us conclude with a couple of important remarks: Firstly, duality operators are oblivious to the +details of the definition of the local operators. +In particular they act in the same way regardless +of the coefficients hv,n, and as such are valid for infinitely many lattice models. +This is because +duality operators only care about matrix/lattice realisations of a given symmetry and not specific +choices of algebra of local operators. +In other words, a given duality operator will systematically +transmute every symmetric operator with respect to a given lattice realisation of a symmetry into a +symmetric operator with respect to another realisation, regardless of the precise definition of these +operators. These symmetries will be analysed in detail in sec. 3.5. Secondly, the knowledge of such +a duality operator is not sufficient to rigorously write down an isometry mapping the corresponding +Hamiltonians to one another. As detailed in ref. [LDV22] for the lower-dimensional setting, defining +∼ 24 +∼ + +such an isometry would require analysing all the topological sectors of the models. We comment on +this aspect in sec. 6 but a detailed analysis will be carried out elsewhere. +Back to the transverse-field Ising model +We established in the previous section how, given the coefficients (3.6), choosing the 2VecZ2-module +2-categories 2VecZ2, 2Vec or 2Vecλ yields the transverse-field Ising model, its Z2-gauged dual or its +λ-twisted Z2-gauged dual. The results obtained above now allow us to construct the lattice operators +performing the transmutations of the corresponding local symmetric operators. Succinctly, there is a +unique 2VecZ2-module functor from 2VecZ2 to 2Vec, namely the forgetful functor, identified with the +unique simple object Vec ∈ 2Vec. The corresponding duality operator acts as dVec : |g, m⟩ �→ |g, m▷Vec⟩ +for any (g, m) ∈ Z1(Σ△, Z2) × C0 +g(Σ△, VecZ2). But m ▷ Vec ∼= Vec and g is fully constrained by m +according to m[v1] = Vecg[v1v2] d m[v2] for any (v1v2) ⊂ Σ△. It follows that the duality operator +effectively acts as dVec : |m⟩ �→ |dm⟩ in the notation of (3.5). It readily follows that dVec : σz +s(e)σz +t(e) �→ σz +e +and dVec : σx +v �→ ś +e⊃v σx +e , as expected. The treatment of the duality 2VecZ2 → 2Vecλ follows the same +steps. Note that these duality operators were already obtained in [Del21] exploiting the graphical +calculus of monoidal 2-categories. +3.4 +Duality as twisted gauging +Let us now clarify in which sense the dual Hamiltonians described above are the results of applying +some (twisted) gauging to the original G-symmetric Hamiltonian. We begin by providing an alternative +expression for local operators (3.17). By definition of our notations, local operators hM +v,n act on degrees +of freedom located at vertices and edges labelled by simple objects in M(A, λ) and 2VecG, respectively. +However, these degrees of freedom must satisfy (3.15), which we shall think of as kinematical con- +straints. Resolving these kinematical constraints allow us to consider a smaller effective microscopic +Hilbert space. Consider for instance the 2VecG-module 2-category M({1G}, 1) ∼= 2VecG. A choice m +of assignments of objects in M to every vertex v1 ⊂ +I +v fully constraints g ∈ Z1( +I +v, G) in virtue of +eq. (3.15), so that we should consider the effective Hilbert space  +v C[G], at which point the oper- +ators (3.17) boil down to (3.5) as expected. More generally, given M(A, λ) and a pair (m[v1], m[v2]) +of simple objects in M(A, λ), there are exactly |A|-many distinct group elements g ∈ G such that +m[v1] ∼= Vecg ▷ m[v2].9 Consequently, local operators (3.17) effectively act on a microscopic Hilbert +space constituted of degrees of freedom at vertices labelled by simple objects in M(A, λ) and degrees +of freedom at edges labelled by group elements in A—or rather simple objects in 2VecA. Given a pair +(g, m) ∈ Z1( +I +v, G) × C0 +g( +I +v, M), we denote by ag,m the assignment of group elements ag,m[v1v2] to +every edge (v1v2) ⊂ +I, where +ag,m[v1v2] := r(m[v1])−1g[v1v2] r(m[v2]) ≡ ag[v1v2],m[v2] , +(3.28) +where we used in the last identification the notation introduced in eq. (3.10) when defining M(A, λ). +Note that in virtue of eq. (3.15), we have ag,m[v1v2]ag,m[v2v3] = ag,m[v1v3] for every (v1v2v3) ⊂ +I. +Recalling the definition of the module pentagonator π▷, we introduce +π▷ +v (ag,m) := +ź +(v1v2v3v4)⊂ +I +v +λ(ag,m[v1v2], ag,m[v2v3], ag,m[v3v4])ϵ(v1v2v3v4) . +(3.29) +9Given a pair (m[v1], m[v2]) of simple objects in M(A, λ), there must exist g ∈ G such that Vecg ▷ m[v2] ∼ += m[v1]. Let +g′ ∈ G such that m[v1] ∼ += Vecg′g ▷ m[v2] ∼ += Vecg′ ▷ m[v1]. This requires Vecg′ to be in the stabiliser of m[v1], which in +turn requires g′ ∈ r(m[v1])Ar(m[v1])−1. Our statement finally follows from |r(m[v1])Ar(m[v1])−1| = |A|. +∼ 25 +∼ + +Putting everything together, we find that local operators (3.17) act on the effective Hilbert space as +hM(A,λ) +v,n +eff. += +ÿ +g∈Z1( +I +v,G) +hv,n(g) +ÿ +m∈C0g( +I +v,M) +π▷ +v (ag,m) +��(ag,m, m) +� +1 +v +��� +(ag,m, m) +� +0 +v +��� . +(3.30) +In practice, this is the expression we shall employ when discussing explicit models. +Let now employ this expression to clarify why hM(A,λ) +v,n +is the result of a λ-twisted gauging of the +A sub-symmetry of the G-symmetric Hamiltonian defined in terms of local operators (3.5). Consider +the (untwisted) gauging of the whole G symmetry. Typically, this operation goes as follows: The +macroscopic Hilbert space is enlarged by the introduction of a G-gauge field and a (local) Gauß +constraint is imposed at every vertex in such a way that they commute with one another. We then +require the Hamiltonian to commute with such Gauß constraints, which is accomplished by minimally +coupling the Hamiltonian with the gauge field. Finally, the Gauß constraints are imposed kinematically +allowing for the initial (matter) degrees of freedom to be gauged away. Within our framework, these +operations are simply accomplished by considering the 2VecG-module 2-category M ≡ M(G, 1) ∼= +2Vec. In particular, it follows form the definition that we have ag,m = g. +More generally, let us consider the gauging of the A sub-symmetry of a G-symmetric Hamiltonian. +As above, we begin by introducing an A-gauge field a ∈ Z1(Σ△, A) and impose the following Gauß +constraints at every vertex: +Gv := +1 +|A| +ÿ +x∈A +� ź +e→v +Rx +e +� +Rx +v +� ź +e←v +Lx +e +� != id , +(3.31) +where +Rx +e : |a[e]⟩ �→ |a[e]x−1⟩ , +Lx +e : |a[e]⟩ �→ |xa[e]⟩ , +(3.32) +so that the physical Hilbert space does not have a tensor product structure anymore. In order to +kinematically enforce these Gauß constraints, it is convenient to disentangle degrees of freedom by +applying the following unitary: +U := +ź +v +� ź +e→v +cRv,e +ź +e←v +cLv,e +� +, +(3.33) +where we introduced the controlled group actions +cRv,e : |g1⟩v b |g2⟩e �→ |g1⟩v b |g2g−1 +1 ⟩e , +cLv,e : |g1⟩v b |g2⟩e �→ |g1⟩v b |g1g2⟩e . +(3.34) +In particular, we have +UGvU† = +1 +|A| +ÿ +x∈A +Rx +v +!= id , +(3.35) +so that imposing the Gauß constraints amounts to considering an effective microscopic Hilbert space +whereby degrees of freedom at vertices are labelled by elements in G/A, or rather simple objects m[v] +in M(A, 1). Notice finally that +cL−1 +v,(vv1) (Lx +v b Lx +(vv1)) cLv,(vv1) : |m[v], a[vv1]⟩ �→ |Cx ▷ m[v], ax,m[v]a[vv1]⟩ ≡ |Cx ▷ m[v], a[v′v1]⟩ , +cR−1 +v,(v1v) (Lx +v b Rx +(v1v)) cRv,(v1v) : |m[v], a[v1v]⟩ �→ |Cx ▷ m[v], a[v1v]ax−1,Vecx▷m[v]⟩ ≡ |Cx ▷ m[v], a[v1v′]⟩ , +where we identified x ≡ g[v′v], at which point we recover the image of the operator Lx +v under the +duality map, as encoded into local operators of the form (3.30) with M ≡ M(A, 1). +Given this +∼ 26 +∼ + +understanding of choosing the 2VecG-module 2-category M(A, 1) as gauging the A sub-symmetry, we +interpret choosing M(A, λ) with λ a non-trivial 3-cocycle in H3(A, U(1)) as performing a λ-twisted +gauging of the A sub-symmetry. More details on this gauging perspective are provided in sec. 3.8 for +the case of the finite group generalisation of the transverse-field Ising model. +3.5 +Dual symmetries +We commented earlier that dualities considered in this manuscript map local symmetric operators to +dual local symmetric operators. However, we have not yet revealed what the symmetry of a given +Hamiltonian HM is. Notice that we are still not choosing any specific Hamiltonian, it is enough to +know that it is defined in terms of local operators of the form (3.17). +Recall that we introduced in sec. 3.3 the notion of 2VecG-module 2-functors and explained how +these provide duality operators between local operators that only differ in a choice of 2VecG-module +category. Given a Hamiltonian HM = ř +v +ř +n hM +v,n, a module 2-functor from M to itself should thus +correspond to a symmetry operator of the model. Indeed, we shall demonstrate that 2VecG-module +2-endofunctors of M label surface symmetry operators. Furthermore, these surface operators can host +topological line operators. More generally, surface operators are not necessarily closed, in which case +topological lines living at the junctions of distinct topological surfaces are required so the Hamiltonian +is left invariant. Mathematically, these topological lines are captured by the notion of module natural +2-transformation between module 2-functors: Given a pair of left 2VecG-module 2-functors (F, ω, Ω) +and ( ˜F, ˜ω, ˜Ω), we define a 2VecG-module natural 2-transformation between them as a tuple (θ, Θ) +consisting of a natural 2-transformation θ : F ⇒ ˜F satisfying a coherence axiom up to an invertible +modification Θ with components ΘVecg,M defined according to the string diagram +Θ +ω +1θ +θ +ω′ +. +(3.36) +This invertible modification is required to satisfy a coherence axiom encoded into the following equality +of string diagrams: +ω +1θ +Ω +Θ +F (α▷) +ω +1ω +1(1θ) +θ +˜ω +α·▷ += +1θ +1˜ω +˜ω +˜ +F (α▷) +1Θ +Θ +˜Ω +F (α▷) +ω +1ω +1(1θ) +θ +˜ω +α·▷ +. +(3.37) +∼ 27 +∼ + +Furthermore, given a pair of 2VecG-module natural 2-transformations (θ, Θ) and (˜θ, ˜Θ), we can define +a 2VecG-module modification from (θ, Θ) to (˜θ, ˜Θ) as a modification ϑ : θ ⇛ ˜θ such that +˜ΘVecg,M ◦ (11Vecg ·▷ ϑM) = ϑVecg▷M ◦ ΘVecg,M +(3.38) +for all g ∈ G and M ∈ M. +Given a pair (M, M′) of 2VecG-module 2-categories, we shall refer to 2Fun2VecG(M, M′) as the +2-category whose objects are 2VecG-module 2-functors M → M′, 1-morphisms are 2VecG-module 2- +natural transformations, and 2-morphisms are 2VecG-module modifications. In the present context, we +are specifically interested in 2-categories of the form (2VecG)⋆ +M(A,λ) := 2Fun2VecG(M(A, λ), M(A, λ)) +that shall be referred to as ‘Morita duals’ of 2VecG with respect to M(A, λ). Crucially, these inherit +a fusion structure from the composition of module 2-functors. We shall now demonstrate that for any +2VecG-module category M ≡ M(A, λ), the Hamiltonian HM is left invariant by topological operators +organised into the Morita dual 2-category (2VecG)⋆ +M. +Let us begin by constructing topological surface operators labelled by simple objects in the fusion +2-category (2VecG)⋆ +M. Given the complex +v×I depicted in eq. (3.21) and a simple object (F, ω, Ω) +in (2VecG)⋆ +M, we consider the following assignment of degrees of freedom: First, we assign as before +the same gauge field g ∈ Z1( +v, G) to +0 +v and +1 +v. We then consider an assignment m of simple +objects m[v1], m[v′ +1] ∈ M to every vertex v1 ⊂ +0 +v and v′ +1 ⊂ +1 +v such that m[v1] = Vecg[v1v2] ▷ m[v2] for +every (v1v2) ∈ +0 +v, m[v′ +1] = Vecg[v′ +1v′ +2] ▷ m[v′ +2] for every (v′ +1v′ +2) ∈ +1 +v. We notate via C0 +g( +v×I, M) the +collection of assignments m fulfilling these conditions. Every edge of the form (v′ +1v1) ⊂ +v×I is further +allocated a simple 1-morphism f[v′ +1v1] in the (possibly terminal) hom-category HomM(F(m[v1]), m[v′ +1]). +Given any prism (v1v2v3)×I ⊂ +v ×I, every plaquette (v1v2)×I ≡ (v′ +1v′ +2v1v2) is labelled by a basis +vector f[v′ +1v′ +2v1v2] in the vector space V ϵ(v′ +1v′ +2v1v2)� +(g, m, f)[v′ +1v′ +2v1v2] +� +given by +V +� +(g, m, f)[v′ +1v′ +2v1v2] +� +:= HomM +� +1m[v′ +1] ◦ (Vecg[v1v2] ▷ f[v′ +2v2]) ◦ ωVecg[v1v2],m[v2] , f[v′ +1v1] ◦ F(1m[v1]) +� +, +V −� +(g, m, f)[v′ +1v′ +2v1v2] +� +:= HomM +� +f[v′ +1v1] ◦ F(1m[v1]) , 1m[v′ +1] ◦ (Vecg[v1v2] ▷ f[v′ +2v2]) ◦ ωVecg[v1v2],m[v2] +� +, +(3.39) +where ϵ(v′ +1v′ +2v1v2) = ±1 depends on the orientation of (v1v2) relative to that of (v1v2v3). For conve- +nience, we summarise these various notations below: +m[v1] +m[v3] +F (m[v1])⊃m[v′ +1] +m[v′ +3]⊂F (m[v3]) +m[v′ +2]⊂F (m[v2]) +m[v2] +g[v1v2] +g[v1v2] +g[v2v3] +g[v2v3] +g[v1v3] +g[v1v3] +f[v′ +1v1] +f[v′ +3v3] +f[v′ +2v2] +f[v′ +1v′ +2v1v2] +f[v′ +2v′ +3v2v3] +f[v′ +1v′ +3v1v3] +. +(3.40) +Given assignments (g, m, f) described above, we finally associate to the prism (v1v2v3)×I the symbol +Ωϵ(v1v2v3)� +(g, m, f)[(v1v2v3)×I] +� +defined as the matrix entry corresponding to the vectors f[v′ +1v′ +2v1v2], +∼ 28 +∼ + +f[v′ +2v′ +3v2v3] and f[v′ +1v′ +3v1v3] of the map +V +� +(g, m, f)[v′ +1v′ +2v1v2] +� +b V +� +(g, m, f)[v′ +2v′ +3v2v3] +� ∼ +−→ V +� +(g, m, f)[v′ +1v′ +3v1v3] +� +(3.41) +that is determined by the component ΩVecg[v1v2],Vecg[v2v3],m[v3] of the invertible modification Ω. +By +convention, we set the symbols of this map to vanish whenever assignment f of 1-morphisms and basis +vectors in M is such that one of hom-categories associated with edges of the form (v′ +1v1) is the terminal +category or one of the vector spaces associated with plaquettes of the form (v1v2)×I is the zero vector +space. +Applying the rules described above, we find that the associahedron axiom (3.20) yields an equation +in terms of the symbols defined above that graphically translates as eq. (3.26). Summing over all +possible labels associated with simplices shared by several prisms +v×I that fulfil all the rules described +above, as well as tracing over basis vectors associated with plaquettes shared by two prisms via the +canonical pairing +V −� +(g, m, f)[v′ +1v′ +2v1v2] +� +b V +� +(g, m, f)[v′ +1v′ +2v1v2] +� +→ C , +(3.42) +yields an operator that commutes with hM +v,n. In order to construct the surface operator commuting +with the whole Hamiltonian HM, it suffices to extend the previous construction to the whole Σ△, +thereby summing over all simple objects and simple 1-morphisms, and tracing over all basis basis +vectors associated with plaquettes: +ÿ +g∈Z1( +0 +v ,G) +m∈C0 +g( +v×I,M) +f +� +ź +(v1v2v3)⊂Σ△ +Ωϵ(v1v2v3)� +(g, m, f)[(v1v2v3)×I] +����(g, m) +� +Σ1 +△ +��� +(g, m) +� +Σ0 +△ +��� . +(3.43) +It follows from the construction that fusion of surface operators is provided by the composition of the +corresponding module 2-endofunctors. +Given the above, let us now consider line operators at the junction of two surface operators. In many +ways, the derivation is merely a lower-dimensional analogue of that above. Given a pair of simple +objects F ≡ (F, ω, Ω) and ˜F ≡ ( ˜F, ˜ω, ˜Ω) in (2VecG)⋆ +M, let (θ, Θ) be a simple 1-morphism in (2VecG)⋆ +M +between them. As previously, we shall define the line operators by means of a labelled complex. Given +a cube (˜v1˜v2v1v2)×I, we consider an assignment of degrees of freedom that resembles that of the prisms: +We assign the same group variable g[v1v2] to edges (v1v2), (v′ +1v′ +2), (˜v1˜v2) and (˜v′ +1˜v′ +2), as well as simple +objects m[v1], m[v′ +1], m[˜v1], m[˜v′ +1] ∈ M to vertices of the cube such that m[˜v1] = m[v1], m[˜v′ +1] = m[v′ +1], +m[v1] = Vecg[v1v2]▷m[v2], m[v′ +1] = Vecg[v1v2]▷m[v′ +2]. We further allocate to the corresponding edges sim- +ple 1-morphisms f[v′ +1v1] and ˜f[˜v′ +1˜v1] in the (possibly terminal) hom-categories HomM(F(m[v1]), m[v′ +1]) +and ∈ HomM( ˜F(m[˜v1]), m[˜v′ +1]), respectively. Plaquettes (v′ +1v′ +2v1v2) and (˜v′ +1˜v′ +2˜v1˜v2) are labelled by basis +vectors f[v′ +1v′ +2v1v2] and ˜f[˜v′ +1˜v′ +2˜v1˜v2] in (possibly zero) vector spaces V ϵ(v′ +1v′ +2v1v2)� +(g, m, f)[v′ +1v′ +2v1v2] +� +and +V ϵ(˜v′ +1˜v′ +2˜v1˜v2)� +(g, m,˜f)[˜v′ +1˜v′ +2˜v1˜v2] +� +as defined in eq. (3.39), respectively, whereas plaquettes (˜v′ +1˜v1v′ +1v1) are +labelled by basis vectors s[˜v′ +1˜v1v′ +1v1] in vector spaces V ϵ(˜v′ +1˜v1v′ +1v1)� +(m, f,˜f, s)[˜v′ +1˜v1v′ +1v1] +� +given by +V +� +(m, f,˜f, s)[˜v′ +1˜v1v′ +1v1] +� +:= HomM +�˜f[˜v′ +1˜v1] ◦ θm[v1] , f[v′ +1v1] +� +, +V −� +(m, f,˜f, s)[˜v′ +1˜v1v′ +1v1] +� +:= HomM +� +f[v′ +1v1] , ˜f[˜v′ +1˜v1] ◦ θm[v1] +� +. +(3.44) +∼ 29 +∼ + +As before, let us summarise our notations via the following diagram: +m[v1] +m[v2] +m[v1] +m[v2] +m[v1] +m[v2]⊂F (m[v2]) +m[v1] +m[v2]⊂ ˜ +F (m[v2]) +g[v1v2] +g[v1v2] +g[v1v2] +g[v1v2] +f[v′ +1v1] +˜f[˜v′ +2˜v2] +˜f[˜v′ +1˜v1] +f[v′ +2v2] +˜f[˜v′ +1˜v′ +2˜v1˜v2] +s[˜v′ +1˜v1˜v′ +1˜v1] +s[˜v′ +2˜v2˜v′ +2˜v2] +f[v′ +1v′ +2 v1v2] +. +(3.45) +We finally associate to the cube (˜v1˜v2v1v2)×I the symbol Θϵ(˜v1˜v2v1v2)� +(g, m, f,˜f, s)[(v1v2v3)×I] +� +corre- +sponding to the vectors f[v′ +1v′ +2v1v2], ˜f[˜v′ +1˜v′ +2˜v1˜v2], s(˜v′ +1˜v1v′ +1v1) and s(˜v′ +2˜v2v′ +2v2) of the map +V +� +(g, m, f)[v′ +1v′ +2v1v2] +� +b V +� +(m, f,˜f, s)[˜v′ +1˜v1v′ +1v1] +� +∼ +−→ V +� +(m, f,˜f, s)[˜v′ +2˜v2v′ +2v2] +� +b V +� +(g, m,˜f)[˜v′ +1˜v′ +2˜v1˜v2] +� +(3.46) +that is determined by the component ΘVecg[v1v2],m[v2] of the invertible modification Θ. By convention, +we set matrix entries of this map to vanish whenever one of the hom-categories associated with edges of +the form (v′ +1v1) or (˜v′ +1˜v1) is the terminal category, or one of the hom-spaces associated with plaquettes +(˜v′ +1˜v1v′ +1v1) is the zero vector space. Applying all the rules introduced above to the following complexes +v′ +2 +v′ +1 +v′ +3 +˜v′ +1 +˜v′ +3 +v2 +˜v1 +˜v3 +v1 +v3 += +˜v′ +2 +v′ +2 +v′ +2 +v′ +1 +v′ +3 +v1 +v3 +˜v1 +˜v3 +˜v2 +v2 +v2 +˜v′ +1 +˜v′ +3 +(3.47) +yields an equation in terms of symbols of ΩVecg[v1v2],Vecg[v2v3],m[v3], ΘVecg[v1v3],m[v3] on the l.h.s. +and +˜ΩVecg[v1v2],Vecg[v2v3],m[v3], ΘVecg[v1v2],Vecg[v2v3]▷m[v3], ΘVecg[v2v3],m[v3] on the r.h.s., where as before we trace +over basis vectors associated with plaquettes shared by complexes. This equation is guaranteed by the +coherence axiom (3.37). Concretely, this equation means that a simple object in Hom(2VecG)⋆ +M(F, ˜F) +defines a topological line operator at the interface of two surface operators labelled by simple objects +(F, ω, Ω) and ( ˜F, ˜ω, ˜Ω) in (2VecG)⋆ +M. Vertical composition of 2VecG-module natural 2-transformations +in (2VecG)⋆ +M finally provides the fusion of topological lines. +3.6 +Higher representation theory +We demonstrated above that symmetry operators of Hamiltonians HM with M ≡ M(A, λ) form the +fusion 2-category (2VecG)⋆ +M. Concretely, this means that starting from a G-symmetric Hamiltonian +that has been rewritten in terms of local operators (3.4), simply replacing the implicit choice of 2VecG- +module 2-category 2VecG by any other indecomposable 2VecG-module 2-category M(A, λ) yields a +∼ 30 +∼ + +dual model with a fusion 2-categorical (2VecG)⋆ +M symmetry in virtue of the demonstration above. +Within this context, identifying the symmetry of a dual model thus boils down to computing the +Morita dual fusion 2-category (2VecG)⋆ +M of 2VecG with respect to M. Interestingly—and to some +extent this is the purpose of the following sections—knowing from the general demonstration that a +given model possesses a (2VecG)⋆ +M symmetry does not mean it is easily verifiable in terms of explicit +lattice operators.10 We shall explicitly compute Morita duals in sec. 3.7 but, in order to understand +the resulting fusion 2-categories, we first need to discuss higher representation theory. +We set the stage with a review of the category theoretic viewpoint on (ordinary) representation +theory. Given a finite group G, we denote by [G, •] the 1-groupoid with object-set G and no non- +trivial morphisms. Let us consider the category Fun([G, •], Vec) of functors from [G, •] to the category +Vec. By definition, an object V in Fun([G, •], Vec) assigns to every g ∈ G a vector space Vg in Vec, +and thus amounts to a G-graded vector space of the form V = À +g∈G Vg. Natural transformations +in Fun([G, •], Vec) then correspond to grading preserving linear maps. The convolution product of +functors [G, •] → Vec, which descends from the multiplication rule of G, further endows Fun([G, •], Vec) +with the structure of a fusion (1-)category according to +(V d W)g := +à +x∈G +Vx b Wx−1g , +(3.48) +with unit 1 satisfying 1g = δg,1G C. Henceforth, we denote this fusion category by VecG. There are +|G|-many simple objects in VecG provided by the one-dimensional vector spaces Cg, for every g ∈ G, +such that Cg dCh ≃ Cgh and HomVecG(Cg, Ch) ≃ δg,h C. Notice that VecG can be equivalently defined +as the fusion category Mod(CG) of modules over the algebra CG of functions on G. Fusion category +VecG is merely the lower categorical analogue of the fusion 2-category 2VecG we have been considering. +More specifically, we shall think of 2VecG as a categorification of VecG. +Another way to treat a finite group G as a 1-category is to consider the delooping of G defined +as the 1-groupoid [•, G] with a single object • and Hom[•,G](•, •) = G such that the composition +of morphisms is given by the multiplication rule of G. +As before, we can consider the category +Fun([•, G], Vec) of functors [•, G] → Vec. By definition, an object ρ in Fun([•, G], Vec) assigns to the +unique object • a vector space V := ρ(•), and to every morphism g : • → • a linear map ρ(g) : V → V +fulfilling ρ(g) ◦ ρ(h) = ρ(gh) for every g, h ∈ G. In other words, ρ is a representation of G. It follows +that natural transformations in Fun([•, G], Vec) correspond to intertwiners. The symmetric monoidal +structure of Vec endows Fun([•, G], Vec) with the structure of a fusion 1-category according to +(ρ d ϱ)(•) := ρ(•) b ϱ(•) ≡ V b W , +(ρ d ϱ)(g) := ρ(g) b ϱ(g) ∈ End(V b W) , +(3.49) +where the tensor products on the r.h.s. are that in Vec. Henceforth, we denote by Rep(G) this fusion +category. Note that the simple objects are provided by the irreducible representations of G. Given +the equivalence between representations of G and modules over the group algebra C[G], we also have +Rep(G) ∼= Mod(C[G]). +In the following, we shall find that Morita duals of 2VecG are often related to ‘higher’ notions of group +representation obtained by following the ethos categorification. +We remarked above that a group +representation is equivalent to a module over the group algebra. Let us adopt this viewpoint and +10Already in (1+1)d systems, it is easy to construct models with Rep(G)-symmetries for instance, which are very +tedious to confirm without a systematic approach analogous to the one employed in this manuscript [LDV22]. +∼ 31 +∼ + +categorify it. Recall that C[G] is the associative algebra whose elements are given by formal linear +combinations of group elements over C and multiplication rule descends from that of the group. Loosely +speaking, categorifying the notion of group algebra requires in particular to loosen the associativity +condition so that it only holds up to isomorphisms [BD98]. +But this would be inconsistent with +having coefficients valued in C. One solution is to consider instead coefficients valued in Vec, where +we should think of Vec as being a categorification of C. The result is the fusion category VecG, whose +definition was reviewed above, thought as a group ‘2-algebra’. This incites us to consider a notion +of ‘2-representation’ of a group G as a module category over VecG. We shall refine this notion of +2-representation later, but let us accept it for the moment and proceed. +The notion of VecG-module category is defined in close analogy with that of 2VecG-module 2- +category reviewed in sec. 3.1. Concretely, a (left) VecG-module category can be defined as a triple +(N, ▷, α▷) consisting of a (C-linear finite semisimple) category N, a binary action functor ▷ : VecG × +N → N and a natural isomorphism α▷ : (− d −) ▷ − +∼ +−→ − ▷ (− ▷ −) referred to as the left module +associator, which is required to satisfy a ‘pentagon axiom’ akin to (3.7).11 Indecomposable module +categories over VecG are obtained following the same recipe as in the 2VecG case: Given a subgroup +B ⊆ G and a normalised 2-cocycle ψ in H2(B, U(1)), let N(B, ψ) be a category with object-set the set +G/B of left cosets. A left VecG-module structure can be defined on N(B, ψ) via Cg▷N := (gr(N))B for +any g ∈ G and N ∈ G/B, where r : G/B → G assigns to every left coset its representative element in G. +As before, we notate via bg,N the group element in B satisfying gr(N) = r(Cg ▷ N)bg,N. Associativity +of the multiplication in G imposes bg1g2,N = bg1,Cg2▷N bg2,N, for every g1, g2 ∈ G and N ∈ G/B, in +exact analogy with eq. (3.11). Thinking of the abelian group Hom(G/B, U(1)) as a left G-module, let +us consider the 2-cochain α▷ ∈ C2(G, Hom(G/A, U(1))) defined as α▷(g1, g2)(N) := ψ(bg1,Cg2▷N, bg2,N) +for any g1, g2 ∈ G and N ∈ G/B. In virtue of the cocycle condition dψ = 1 and the equation above, +we have +α▷(g2, g3)(N) α▷(g1, g2g3)(N) = α▷(g1g2, g3)(N) α▷(g1, g2)(Cg3 ▷ N) , +(3.50) +for every g1, g2, g3 ∈ G and N ∈ G/B, so that α▷ is a Hom(G/B, U(1))-valued 2-cocycle of G. +Defining the natural isomorphism α▷ with components α▷ +Cg1,Cg2,N := α▷(g1, g2)(N) · 1(g1g2r(N))B : +(Cg1 d Cg2) ▷ N +∼ +−→ Cg1 ▷ (Cg2 ▷ N) for every g1, g2 ∈ G and N ∈ G/B, we find that the triple +(N(B, ψ), ▷, α▷) does define a left VecG-module category. It is a result of Ostrik that all indecomposable +module categories over VecG are of this form [Ost01, Ost02]. +If VecG-module categories admit an interpretation as 2-representations of the group G, then VecG- +module functors should be understood as 1-intertwiners between them. As previously, the notion of +VecG-module functor is defined in immediate analogy with that of 2VecG-module 2-functor. Concretely, +given a pair of (left) VecG-module categories (N, ▷, α▷) and (N ′, ·▷, α·▷), we define a module functor +between them as a pair (F, ω) consisting of a functor F : N → N ′ and a natural isomorphism +ω : F(− ▷ −) +∼ +−→ − ·▷ F(−), which is required to satisfy a pentagon axiom involving both α▷ and α·▷ +akin to (3.18). There is also a notion of map between module functors, which within our context shall +be interpreted as ‘2-intertwiners’. More precisely, given a pair of left VecG-module functors (F, ω) +and ( ˜F, ˜ω), we define a module natural transformation between them as a natural transformation +θ : F ⇒ ˜F satisfying (1Cg ·▷ θM) ◦ ωCg,M = ˜ωCg,M ◦ θCg▷M for all g ∈ G and M ∈ M. +It follows from the definitions that, given a pair (N, N ′) of VecG-module categories, 1- and 2- +intertwiners form a category that we denote by FunVecG(N, N ′). Special attention is paid to Morita +dual fusion categories (VecG)⋆ +N (B,ψ) := FunVecG(N(B, ψ), N(B, ψ)) of VecG with respect to N(B, ψ) +11Since we shall encounter both VecG-module categories and 2VecG-module 2-categories at the same time in the +following, we notate them via N and M, respectively, to facilitate the distinction. +∼ 32 +∼ + +[M¨u03, EGNO16]. +In particular, these categories inherit a fusion structure from the composition +of module functors [ENO08], so that considering module endofunctors of indecomposable module +categories is a way to construct new fusion categories. +Treating VecG as a module category over +itself, we have for instance (VecG)⋆ +VecG ∼= VecG. Since the composition of module functors is a well- +defined operation, we can further consider the 2-category Mod(VecG) consisting of (left) VecG-module +categories and hom-categories of VecG-module functors. This is another example of a fusion 2-category, +still in the sense of Douglas and Reutter [DR18], where the fusion structure is obtained by defining a +VecG-module structure on N b N ′ via Cg ▷ (N b N ′) := (Cg ▷ N) b (Cg ▷ N ′) for every g ∈ G, N ∈ N +and N ′ ∈ N ′ [Gre10]. Explicit formulae for the fusion of indecomposable VecG-module categories +N(B, ψ) can be found in ref. [ENO10] for the abelian case and in ref. [Gre10] for the non-abelian one. +• We are now ready to refine the notion of 2-representation alluded to above. Firstly, we require +a categorification of the notion of vector space, a natural candidate being the notion of 2-vector +space introduced in sec. 3.1. Secondly, let [•, G, •] be the 2-groupoid with unique simple object •, 1- +morphisms labelled by group elements in G, and no non-trivial 2-morphisms. Mimicking the definition +of Rep(G), we would like to consider the 2-category 2Rep(G) := 2Fun([•, G, •], 2Vec) of pseudofunctors, +pseudonatural transformations and modifications between [•, G, •] and 2Vec. Unpacking the definition, +one finds that an object ρ in 2Rep(G) is a map +ρ : [•, G, •] → 2Vec +: +• +�→ ρ(•) =: V +: • +g−→ • �→ V +ρ(g) +−−→ V ∈ End(V) +(3.51) +assigning to the unique object • a 2-vector space V := ρ(•) and to every 1-morphism g : • → • a linear +functor ρ(g) : V → V, in sush a way that composition of 1-morphisms is only preserved up to natural +2-isomorphisms, i.e. ρ further assigns to every pair of 1-morphisms labelled by g1, g2 ∈ G a natural +2-isomorphism +ρg1,g2 : ρ(g1) ◦ ρ(g2) +∼ +=⇒ ρ(g1g2) , +(3.52) +which is required to fulfill12 +ρg1,g2g3 · [1ρ(g1) ◦ ρg2,g3] = ρg1g2,g3 · [ρg2,g3 ◦ 1ρ(g3)] , +(3.53) +for any g1, g2, g3 ∈ G. Introducing the notations ▷ : VecG × V → V, whereby Cg ▷ M := ρ(g)(M) for +every M ∈ V, and α▷ +Cg1,Cg2,M := (ρg1,g2)M, it follows from the 2-cocycle condition (3.53) that +[1Cg1 ▷ α▷ +Cg2,Cg3,M] ◦ α▷ +Cg1,Cg2dCg3,M ◦ 1Cg1g2g3▷M = α▷ +Cg1,Cg2,Cg3▷M ◦ α▷ +Cg1dCg2,Cg3,M +(3.54) +holds for any g1, g2, g3 ∈ G and M ∈ V. Consequently, the triple (V, ▷, α▷) thus constructed defines +a left VecG-module category. Similarly, we can readily check that pseudonatural transformations and +modifications in 2Rep(G) corresponds to VecG-module functors and VecG-module natural transfor- +mations, respectively. Putting everything together, we have an equivalence 2Rep(G) ∼= Mod(VecG), +thereby justifying referring to VecG-module categories as 2-representations of the group [GK06, Bar09]. +The symmetric monoidal structure of 2Vec endows 2Rep(G) with the expected fusion structure ac- +cording to +(ρ d ϱ)(•) := ρ(•) b ρ(•) ≡ V b ˜V , +(ρ d ϱ)(g) := ρ(g) b ϱ(g) ∈ End(V b ˜V) . +(3.55) +12As customary, we notate via ◦ and · the horizontal and vertical compositions of 2-morphisms, respectively. +∼ 33 +∼ + +The main reason to define 2-representations of G as pseudofunctors [•, G, •] → 2Vec is that it is readily +generalisable to other scenarios relevant to our study. We present two such scenarios below. +• Let G be a 2-group with homotopy groups Q and L in degree one and two, respectively [BL03]. +Succinctly, a 2-group is a monoidal groupoid such that every object has a weak inverse. Concretely, +the 2-group G has object-set Q, hom-sets HomG(q, q) = L with composition rule13 +(q +l1 +−→ q) ◦ (q +l2 +−→ q) = (q +l1+l2 +−−−→ q) , +(3.56) +and monoidal structure +(q1 +l1 +−→ q1) d (q2 +l2 +−→ q2) = (q1q2 +l1+φq1(l2) +−−−−−−−→ q1q2) , +(3.57) +for any q1, q1 ∈ Q and l1, l2 ∈ L, where φ− : Q → Aut(L). As for a group, we distinguish two ways +to treat a 2-group as a 2-groupoid, namely [Q, L, •] and [•, Q, L]. Let us focus for now on the former. +We are interested in pseudofunctors between [Q, L, •] and 2Vec. Unpacking the definition, one finds +that such a pseudofunctor is a map +ρ : [Q, L, •] → 2Vec +: +q +�→ ρ(q) =: Vq +: q +l−→ q �→ Vq +ρ(l) +−−→ Vq ∈ End(Vq) +, +(3.58) +assigning to every group element q ∈ Q a 2-vector space Vq := ρ(q) and to every 1-morphism l : q → q +a linear functor ρ(l) : Vq → Vq in such a way that composition of the 1-morphisms is only preserved +up to natural 2-isomorphisms, i.e. ρ further assigns to every pair of 1-morphisms labelled by l1, l2 ∈ L +a natural 2-isomorphism +ρl1,l2 : ρ(l1) ◦ ρ(l2) +∼ +=⇒ ρ(l1 + l2), +(3.59) +which is required to fulfil eq. (3.53). In close analogy with the constructions presented so far, we deduce +that ρ amounts to a Q-graded 2-vector space V := Ð +q∈Q Vq such that every homogeneous component +Vq has the structure of a (left) VecL-module category, or alternatively of a 2-representation of L. +Pseudonatural transformations between pseudofunctors [Q, L, •] → 2Vec provide the corresponding +1-morphisms, which amount to Q-grading preserving VecL-module functors. More concretely, every +group element q ∈ Q together with an indecomposable VecL-module category V furnishes a simple +object Vq in 2Fun([Q, L, •], 2Vec). The hom-category between two such simple objects Vq1 and ˜Vq2 +is then provided by δq1,q2 FunVecQ(Vq1, ˜Vq2), i.e., it is terminal unless q1 = q2. Finally, the convolu- +tion product of pseudofunctors [Q, L, •] → 2Vec endows 2Fun([Q, L, •], 2Vec) with a fusion structure, +whereby Vq1 d ˜Vq2 is the L-graded 2-vector space with homogeneous components +(Vq1 d ˜Vq2)q = +ð +x∈Q +(Vq1)x b (˜Vq2)x−1q = δq,q1q2 Vq1 b ˜Vq2 , +(3.60) +equipped with a VecL-module structure defined by +Cl ▷ (N b N ′) := (Cl ▷ N) b (Cφq−1 +1 +(l) ▷ N ′) , +(3.61) +for any N ∈ Vq1, N ′ ∈ V′ +q2 and l ∈ L. Henceforth, we denote by 2VecG this fusion 2-category and refer +to it as the 2-category of G-graded 2-vector spaces [DR18]. For our applications, the monoidal product +13Notice that we write the product rule in L as an addition to emphasise the fact that it is an abelian group. +∼ 34 +∼ + +of simple objects will not play a crucial role. We shall rather be interested in the monoidal product of +simple 1-morphisms, which is of the same form as (3.61). Consider for instance the monoidal product +of the identity 1-endomorphism 1Vq of a 1-simple object Vq with any simple 1-endomorphism of Vec1Q. +By virtue of (VecL)⋆ +Vec ∼= Rep(L), which establishes the Morita equivalence between VecL and Rep(L) +[EGNO16], a simple 1-endomorphism of the simple object Vec1Q in 2VecG is labelled by a character +ρ(−) of L. It follows in particular from eq. (3.61) that 1Vq d ρ(−) = ρ(φq−1(−)) d 1Vq. +• In the notation of the above paragraph, let us finally consider the 2-category 2Fun([•, Q, L], 2Vec), +where we recall that [•, Q, L] is the 2-groupoid with single object • and hom-category Hom[•,Q,L](•, •) = +G such that horizontal and vertical compositions are provided by the monoidal product and the com- +position in G, respectively. This 2-category was investigated in detail in ref. [Elg04, BM04, BBFW08], +or more recently in ref. [BBFP22a]. +As such, we shall keep our exposition brief and merely re- +view the salient features of this 2-category following the description in terms of module categories +and module functors proposed in ref. [Del22]. Unpacking the definition we find that an object in +2Fun([•, Q, L], 2Vec) is a map +ρ : +[•, Q, L] +→ 2Vec +: +• +�→ ρ(•) =: V +: +• +q−→ • +�→ V +ρ(q) +−−→ V ∈ End(V) +: • +• +q +q +l +�→ V +V +ρ(q) +ρ(q) +ρ(l) +∈ EndEnd(V)(ρ(q)) +, +(3.62) +assigning to the unique simple object • a 2-vector space V, to every morphism q : • → • a linear +functor ρ(q) : V → V, and to every 2-morphism l : q ⇒ q a natural transformation. +Moreover, +vertical and horizontal compositions of 2-morphisms are strictly preserved, whereas the composition +of 1-morphisms is only preserved up to natural 2-isomorphisms, i.e. ρ further assigns to every pair of +1-morphisms labelled by q1, q2 ∈ Q a natural 2-isomorphism +ρq1,q2 : ρ(q1) ◦ ρ(q2) +∼ +=⇒ ρ(q1q2) , +(3.63) +which is required to fulfil eq. (3.53). Given two objects ρ and ϱ, a 1-morphism θ : ρ → ϱ between +them is a pseudonatural transformation that assigns to • an object θ• in Fun(ρ(•), ϱ(•)) and to every +morphism q : • → • a natural 2-isomorphism defined by +θq : θ• ◦ ρ(q) +∼ +=⇒ ϱ(q) ◦ θ• +(3.64) +such that θ1Q = 1θ•. Compatibility with the composition of 1-morphisms in G requires the following +coherence relation to be satisfied: +[ϱq1,q2 ◦ 1θ•] · [1ϱ(q1) ◦ θq2] · [θq1 ◦ 1ϱ(q2)] = θq1q2 · [1θ• ◦ ρq1,q2] , +(3.65) +for every q1, q2 ∈ Q, whereas naturality stipulates that +θq · [1θ• ◦ ρ(l)] = [ϱ(l) ◦ 1θ•] · θq , +(3.66) +for every 2-morphism l : q ⇒ q. +In the same vein as the previous derivations, we find that a 2- +vector space V together with endofunctors ρ(q) : V → V and natural 2-isomorphisms ρq1,q2 satisfying +∼ 35 +∼ + +(3.53) amounts to a (left) VecQ-module category. As a natural transformation, ρ(l) : ρ(q) ⇒ ρ(q) +assigns to every simple object N ∈ V an endomorphism ρ(q)(N) → ρ(q)(N), which, together with +ρ(l1) · ρ(l2) = ρ(l1 · l2), implies that ρ further assigns to every simple object N ∈ V a representation +ρ(−)N : L → EndV(ρ(q)(N)). Crucially, ρ(l1) ◦ ρ(l2) = ρ(l1 ◦ l2) requires the following condition: +ρ(φq(−))N = ρ(−)ρ(q)(N) , +∀ q ∈ Q and N ∈ V . +(3.67) +Given the above, a 1-morphism ρ → ϱ in 2Fun([•, Q, L], 2Vec) amounts to a VecQ-module functor +(θ•, (θ−)−) between the corresponding VecQ-module categories, which, in virtue of the naturality +condition (3.66), must satisfy the additional requirement +(θq)N ◦ θ•(ρ(l)N) = ϱ(l)θ•(N) ◦ (θq)N , +(3.68) +for every 2-morphism l : q ⇒ q and N ∈ V. Finally, the symmetric monoidal structure of 2Vec endows +2Fun([•, Q, L], 2Vec) with a fusion structure. Henceforth, we denote by 2Rep(G) this fusion 2-category +and refer to it as the 2-category of 2-representations of the 2-group G. +3.7 +Morita duals +Guided by the derivations above, we shall now compute Morita duals of 2VecG with respect to various +choices of 2VecG-module 2-categories. In the context of this manuscript, this will establish that given a +generic two-dimensional G-symmetric Hamiltonian, the models obtained by gauging the G symmetry, +or sub-symmetries thereof, are left invariant by symmetry operators organised into fusion 2-categories +of some higher representations. Combined with the results of sec. 3.5, this provides an answer to the +question, what does it mean for a lattice model to commute with symmetry operators labelled by +higher representations? +• Choosing G as a subgroup of itself and the trivial cocycle in H3(G, U(1)) yields the module 2- +category 2Vec via the forgetful functor 2VecG → 2Vec. The Morita dual (2VecG)⋆ +Vec was found in +in ref. [Del21, D´ec22] to be equivalent as a monoidal 2-category to 2Rep(G), whose definition was +given in sec. 3.6. Let us briefly review this derivation for completeness, we encourage the reader to +consult ref. [Del21] for detail. By definition, an object in 2Fun2VecG(2Vec, 2Vec) consists of a 2-functor +F : 2Vec → 2Vec, which is fully determined by a 2-vector space V := F(2Vec), an adjoint natural +2-equivalence ω prescribed by +ωg : Vecg ▷ F(Vec) +∼ +−→ F(Vecg ▷ Vec) ∈ Fun(V, V) , +∀ g ∈ G , +(3.69) +as well as an invertible modification Ω defined as per eq. (3.19) with components +Ωg1,g2 ∈ HomFun(V,V)(ωg1 ◦ ωg2, ωg1g2) +∀ g1, g2 ∈ G . +(3.70) +Isomorphisms ωg provide an action 2-functor ▷ : VecG × V → V via Cg ▷ M := ωg(M) for any M ∈ V, +whereas maps Ωg1,g2 yields natural isomorphisms α▷ +Cg1,Cg2,M := (Ωg1,g2)M. It follows from the asso- +ciahedron axiom (3.20) that the triple (V, ▷, α▷) defines a left VecG-module category. Given a pair +(F, ω, Ω) and ( ˜F, ˜ω, ˜Ω) of 2VecG-module 2-endofunctors of 2Vec, a 2VecG-module natural transforma- +tion between them is given by a choice of natural transformation θ : F ⇒ ˜F specified by a choice of +functor ˆF ∈ Fun(V, ˜V) between the corresponding VecG-module categories. The invertible modification +Θ defined as per eq. (3.36) is prescribed by a collection of natural transformations +Θg ∈ HomFun(V,˜V)( ˆF ◦ ωg, ˜ωg ◦ ˆF) , +∀ g ∈ G , +(3.71) +∼ 36 +∼ + +endowing ˆF with a VecG-module structure ˆωCg,M := (Θg)M, so that 1-morphisms are given by VecG- +module functors ( ˆF, ˆω) between the corresponding VecG-module categories, as expected. Similarly, we +can show that 2VecG-module natural 2-transformations are identified with VecG-module natural trans- +formations. Finally, it follows from the composition of 2VecG-module functors (F, ω, Ω) and ( ˜F, ˜ω, ˜Ω) +that the monoidal structure is obtained by defining a VecG-module structure on (F ◦ ˜F)(Vec) ≡ V b ˜V +via ωg b ˜ωg : V b ˜V → V b ˜V. Putting everything together, this shows the monoidal equivalence +(2VecG)⋆ +2Vec ∼= Mod(VecG) ∼= 2Rep(G). +In the context of our work, this computation together with the results of sec. 3.5 shows that gauging +the G symmetry of (2+1)d quantum theory results in a theory with a 2Rep(G) symmetry. This result +appeared in ref. [Del21] and was recovered in ref. [BBFP22a, BSNW22] in terms of separable algebras +in fusion 2-categories [D´ec22]. Concretely, this means that a theory with a gauged G symmetry host +non-trivial topological surface operators labelled by indecomposable VecG-module categories as well +as topological line operators labelled by VecG-module functors. Our construction further teaches us +how these operators explicitly act on a lattice model. We have already seen an example in sec. 2, +and we shall see further examples below, but in general the surface operator associated with the +indecomposable left VecG-module category (N ≡ N(B, ψ), ▷, α▷) is proportional to14 +ÿ +g∈Z1(Σ△,G) +n∈C0 +g(Σ△,N ) +� +ź +(v1v2v3)⊂Σ△ +α▷(g[v1v2], g[v2v3])(n[v1])ϵ(v1v2v3) +� +|g⟩⟨g| , +(3.72) +such that α▷ +Cg1,Cg2,N ≡ α▷(g1, g2)(N)·1(g1g2r(N))B and C0 +g(Σ△, N) refers to the collection of assignments +n of simple objects in N at every vertex of Σ△ such that n[v1] = Cg[v1v2] ▷ n[v2] for every (v1v2) ⊂ Σ△. +Let us emphasise that the conditions n[v1] = Cg[v1v2] ▷ n[v2] are with respect to the module structure +of the VecG-module 1-category N. The same operators appeared in ref. [Del21] in the context of the +(3+1)d gauge models of topological phases of matter. In the case where B = {1G}, it is convenient +to think of assignments n ∈ C0 +g(Σ△, N) as (virtual) matter fields n ∈ C0(Σ△, G) fulfilling dn = g, as +we did in sec. 2. Note finally that we recover through Hom2Rep(G)(Vec, Vec) = FunVecG(Vec, Vec) ∼= +Rep(G) that Wilson lines generate a 1-form symmetry for the G-gauged theory (see sec. 3.8 for more +details). More generally, given a surface operator labelled by a VecG-module category N(B, ψ), we +can insert topological lines on it that organise into the Morita dual fusion 1-category (VecG)⋆ +N (B,ψ) = +FunVecG(N(B, ψ), N(B, ψ)). Combining the construction of sec. 3.5 and the computations above, these +line operators can be implemented on the lattice in terms of matrix product operators whose building +blocks evaluate to the module structure of the functors in (VecG)⋆ +N (B,ψ) [LFH+20, LDOV21, LDV22]. +We commented above that (VecG)⋆ +Vec ∼= Rep(G) signifies that the fusion 1-categories VecG and +Rep(G) are Morita equivalent, which implies in particular the equivalence Mod(VecG) ∼= Mod(Rep(G)) +[EGNO16]. Interestingly, the fusion 2-category Mod(Rep(G)) can be thought of as the idempotent +completion of the delooping of the braided fusion 1-category Rep(G), which encodes the line operators +of the trivial surface operator. Physically, this idempotent completion amounts to including surface +operators obtained by condensing suitable algebras of line operators in Rep(G), and as such these +surface operators are often referred to as condensation defects. In this context, the surface operators +in 2Rep(G) are labelled by algebra objects in Rep(G). By definition, such an algebra object in Rep(G) +is a G-algebra, i.e. an associative unital algebra equipped with a G-action by algebra automorphisms. +We know from the Morita equivalence between Rep(G) and VecG that Morita classes of indecomposable +14In the notations of sec. 3.5, we are using the fact that simple morphisms f[v′ +1v1] are given by simple objects in +hom-categories that are equivalent N. +∼ 37 +∼ + +algebra objects in Rep(G) are labelled by pairs (B, ψ). Davydov then provided in ref. [Dav09] a recipe +to explicitly construct the corresponding G-algebras. We already showed in eq. (2.26) for the case of +G = Z2 how to construct the non-trivial surface operator from this condensation perspective, and we +shall provide additional comments along these lines in sec. 5. +Note finally that, more generally, picking a representative of a non-trivial cohomology class +in H3(G, U(1)) yields a 2VecG-module 2-category M(G, λ) ≡ 2Vecλ that only differ from 2Vec +in the choice of module pentagonator π▷, which is such that π▷ +Vecg1,Vecg2,Vecg3,C = λ(g1, g2, g3) for +any g1, g2, g3 ∈ G. +Importantly, it follows immediately from the definitions that we still have +(2VecG)⋆ +2Vecλ ∼= 2Rep(G). +• The subsequent examples require the group G to be isomorphic to a semi-direct product Q ⋉φ L +with L abelian, in which the multiplication is given by +(q1, l1)(q1, l2) = (q1q2, l1 + φq1(l2)) , +(3.73) +for any q1, q2 ∈ Q and l1, l2 ∈ L, where we are using the notation of sec. 3.6. Introducing projection +maps ϖQ : G → Q and ϖL : G → L, every group element in G admits a decomposition of the form +g ≡ (ϖQ(g), ϖL(g)) ∈ Q ⋉φ L. Consider the 2VecG-module 2-category M(L, 1) ∼= 2VecQ.15 We find +that the Morita dual (2VecG)⋆ +2VecQ is equivalent as a monoidal 2-category to the fusion 2-category +2VecG := 2Fun([Q, L, •], 2Vec) of G-graded 2-vector spaces defined in the previous section. We sketch +below the main steps of this derivation. +Any 2VecG-module 2-endofunctor of 2VecQ is in particular an object in (2VecQ)⋆ +2VecQ ∼= 2VecQ so +that an object Vq1 ∈ 2VecQ determines a 2VecG-module endofunctor of 2VecQ of the form F(−) = +− d Vq1. The 2VecG-module structure on F is then prescribed by a collection of functors +ωg,q ∈ Hom2VecQ +� +Vecg▷F(Vecq), F(Vecg▷Vecq) +� += Hom2VecQ(VecϖQ(g)qdVq1, VecϖQ(g)qdVq1) (3.74) +satisfying a coherence relation up to an invertible modification Ω defined as per eq. (3.19) with com- +ponents +Ωg1,g2,q ∈ Hom2VecQ +� +ωg1,ϖQ(g2)q ◦ ωg2,q , ωg1g2,q +� +∀ g1, g2 ∈ G and q ∈ G . +(3.75) +As before, 1-morphisms between such objects correspond to 2VecG-module natural transformations +between the corresponding 2-endofunctors of 2VecQ. Before analysing more carefully this 2-category +(2VecG)⋆ +2VecQ, let us immediately consider its fusion structure. Recall that the monoidal product of +objects in (2VecG)⋆ +2VecQ is provided by the composition of the corresponding module 2-endofunctors. +Let (F, ω, Ω) and ( ˜F, ˜ω, ˜Ω) be two 2VecG-module 2-endofunctors of 2VecQ such that F(−) = − d Vq1 +and ˜F(−) = − d ˜Vq2, respectively. Composition yields a new module 2-endofunctor of 2VecQ given +by ( ˜F ◦ F)(−) = − d (Vq1 d ˜Vq2), whose 2VecG-module structure is provided by +(˜ω ◦ ω)g,q : Vecg ▷ ( ˜F ◦ F)(Vecq) +˜ωg,qq1 +−−−−→ ˜F(Vecg ▷ F(Vecq)) +˜ +F (ωg,q) +−−−−−→ ( ˜F ◦ F)(Vecg ▷ Vecq) += ˜F(ωg,q) ◦ ˜ωg,qq1 ∈ Hom2VecQ(VecϖQ(g)q d (Vq1 d ˜Vq2), VecϖQ(g)q d (Vq1 d ˜Vq2)) . +(3.76) +15By definition, L ⊆ G is a normal subgroup so that G/L isomorphic to Q with the isomorphism being provided by +the composition of the natural embedding L → G and the natural projection G → G/L. It follows that we can identify +M(L, 1) with 2VecQ with the 2VecG-module structure being provided by Vecg ▷ Vecq := VecϖQ(g)q, for all g ∈ G and +q ∈ Q. +∼ 38 +∼ + +Let us now consider the monoidal equivalence given by16 +(Vq1, ωg,q, Ωg1,g2,q) �→ (Vq1, ωl := ω(1Q,l),1Q, Ωl1,l2 := Ω(1Q,l1),(1Q,l2),1Q) +(Vq1, ωl, Ωl1,l2) �→ (Vq1, ωg,q := ωag,q, Ωg1,g2,q := Ωag1,ϖQ(g2)q,ag2,q) , +(3.77) +where ag,q ∈ L was defined in eq. (3.10). Invoking this equivalence, we find that the Q-graded 2- +vector space Vq1 has the structure of a VecL-module category with the module action and the module +associator being provided by the collection of maps ωl and Ωl1,l2, respectively. Furthermore, the fusion +structure is now obtained by endowing Vq1 d ˜Vq2 with the VecL-module structure +(˜ω ◦ ω)l = (˜ω ◦ ω)(1Q,l),1Q = ω(1Q,l),1Q d ˜ω(1Q,l),q1 = ωl d ˜ωφq−1 +1 +(l) , +(3.78) +where we used the fact that a(1Q,l),q1 = φq−1 +1 (l), which agrees with eq. (3.61). We can now check +that 1-morphisms in (2VecG)⋆ +2VecQ amounts to Q-grading preserving VecL-module functors. Putting +everything together, this motivates the equivalence (2VecG)⋆ +2VecQ ∼= 2VecG, where G is the 2-group +defined in sec. 3.6. +Together with previous results, this computation shows that gauging the L sub-symmetry of a +(Q ⋉φ L)-symmetric (2+1)d quantum theory results in a theory with a 2VecG symmetry. Although it +is a little bit tedious to explicitly write down the lattice realisations of the corresponding topological +surfaces and lines in general—but these can be obtained from the construction in sec. 3.5—we shall +consider specific examples in sec. 5. +• Still assuming G ≃ Q⋉φ L, let us now consider the 2VecG-module 2-category M(Q, 1) ∼= 2VecG/Q.17 +Even though G/Q is not isomorphic to L as a group, we have |G/Q| = |L| and thus we label simple +objects in M(Q, 1) by group elements in L. We thus write the 2VecG-structure of M(Q, 1) as Vecg ▷ +Vecl := VecϖL(g)+φϖQ(g)(l) for every g ∈ G and l ∈ L. Analogously to the previous computation, +objects in (2VecG)⋆ +2VecG/Q are functors F(−) = − d V with V = Ð +l1∈L Vl1 ∈ 2VecG/Q equipped with +a 2VecG-module structure provided by +ωg,l ∈ Hom2VecG/Q(Vecg ▷ (Vecl d V), VecϖL(g)+φϖQ(l) d V) = Hom2VecG/Q(V, V) +(3.79) +satisfying a coherence relation to up an invertible modification Ω defined as per eq. (3.19) with com- +ponents +Ωg1,g2,l ∈ Hom2VecG/Q(ωg1,ϖL(g1)+φϖQ(g1)(l) ◦ ωg2,l, ωg!g2,l) , +∀ g1, g2 ∈ G and l ∈ L . +(3.80) +Still in the same vein as the previous computation, let us consider the equivalence provided by +(V, ωg,l, Ωg1,g2,l) �→ (V, ωq := ω(q,0L),0L, Ωq1,q2 := Ω(q1,0L),(q2,0L),0L) +(V, ωq, Ωq1,q2) �→ (V, ωg,l := ωag,l, Ωg1,g2,l := Ωag1,ϖL(g1)+φϖQ(g1)(l),ag2,l) , +(3.81) +where ag,l ∈ Q was defined in eq. (3.10). In particular, invoking this equivalence, we find that the +L-graded 2-vector space V has the structure of a VecQ-module category with the module action ▷ +and the module associator α▷ being provided by the collections of maps ωq and Ωq1,q2, respectively. +16This equivalence essentially follows the isomorphism Hn(G, Hom(G/L, U(1))) ≃ Hn(L, U(1)) provided by Shapiro’s +lemma. +17Note the slight abuse of notation. Since Q is not a normal subgroup, the quotient G/Q is not equipped with a group +structure and thus the 2-category 2VecG/Q is not monoidal. +∼ 39 +∼ + +Moreover, going back to eq. (3.79), we find that ωq = À +l1∈L ωg|Vl1 where ωq|Vl1 : Vφq(l1) → Vl1. +Associating to every object N ∈ V a group element l1(N) in such a way that N ∈ Vl1(N), we must have +Cq ▷ N := ωq|Vl1 (N) ∈ Vl1 for every N ∈ Vφq(l1) and thus l1(Cq ▷ N) = φq−1(l1(N)). Finally, invoking +the isomorphism L ≃ L∨ = Hom(L, U(1)) and defining a Q-action on L∨ via q ▷ ρ(−) = ρ(φq−1(−)), +we can equivalently state that an object in (2VecG)⋆ +2VecQ/G corresponds to a VecQ-module category V +such that for every object N ∈ V, we assign a character ρ(−)N such that ρ(φq(−))N = ρ(−)Cq▷N, for +every q ∈ Q. This is precisely the defining condition given in eq. (3.67). Analysing 1- and 2-morphisms +under the same scope, it is then fairly immediate to obtain (2VecG)⋆ +2VecG/Q ∼= 2Rep(G), where G is the +2-group defined in sec. 3.6. +We summarise in the diagram below the Morita equivalences evoked in this section for a finite group +G ≃ Q ⋉φ L: +2VecG +2Rep(G) +2VecG +2Rep(G) +2Vec +2Vec +2VecQ +2Rep(Q) +2VecG/Q +2Rep(G/Q) +with +2VecG := 2Fun([G, •, •], 2Vec) , +2Rep(G) := 2Fun([•, G, •], 2Vec) , +2VecG := 2Fun([Q, L, •], 2Vec) , +2Rep(G) := 2Fun([•, Q, L], 2Vec) , +(3.82) +where fusion 2-categories connected by a double arrow are Morita equivalent with respect to the module +2-category labelling the arrow. Note that the equivalence (2VecG)⋆ +2Vec ∼= 2Rep(G) holds for arbitrary +G as demonstrated at the beginning of this section. Although we have not explicitly constructed all +the Morita equivalences displayed above, we included them in this diagram for completeness.18 We +leave to future work a more systematic and general treatment of such Morita equivalences. +3.8 +Gauging the transverse-field G-Ising model +Let us now illustrate some of the concepts presented in this section with a series of examples. Starting +from a finite group generalisation of the transverse-field Ising model, we shall construct various dual +models obtained by gauging sub-symmetries. Within our approach, this amounts to writing the initial +model in terms of local operators (3.17), and then simply replacing the initial module 2-category +by another one. +By virtue of our construction, we already know that the resulting Hamiltonians +will commute with symmetry operators encoded into Morita duals with respect to the corresponding +module 2-categories. For now, the focus will be on deriving the various dual models using effective local +operators (3.30) obtained after resolving kinematical constraints of the form (3.15). In the following +sections, we shall choose specific groups and analyse in detail the dual symmetries by translating the +symmetry operators defined in sec. 3.5 into explicit spin operators. +Given a finite group G, let M ≡ M(A, λ) be an indecomposable 2VecG-module 2-category. We +are interested in Hamiltonians of the form +HM = +ÿ +v⊂Σ△ +4ÿ +n=1 +hM +v,n . +(3.83) +18Equivalence (2VecG)⋆ +2Vec ∼ += 2Rep(G) was established in ref. [D´ec22] +∼ 40 +∼ + +For any vertex v ⊂ Σ△ and gauge field g ∈ Z1( +I +v, G), the defining U(1)-coefficients hv,n(g) are chosen +to be +hv,1(g) := −Jδg[v′v],1δg[v v+ˆu],1 , +hv,2(g) := −Jδg[v′v],1δg[v v+ˆv],1 , +hv,3(g) := −Jδg[v′v],1δg[v v+ ˆ +w],1 , +hv,4(g) := − Jκ +|G| , +(3.84) +where the branching structure of +I +v is that given in eq. (3.3). +This is all the data required to +define a Morita class of Hamiltonian models. Specific matrix realisations of this Hamiltonian, i.e. +representatives of the Morita class, are obtained by choosing specific 2VecG-module 2-categories M. +We consider below four different choices. +• Let us begin with the choice M(A = {1G}, 1) ∼= 2VecG, i.e. the module 2-category 2VecG over +itself. Recall that local operators (3.30) act on the effective Hilbert space obtained after resolving +the kinematical constraints (3.15). Since the kinematical constraints are such that degrees of freedom +assigned to edges are fully determined by those assigned to vertices, we are left with a tensor product +Hilbert space of the form  +v C[G] ∋ |m⟩, where m ∈ C0(Σ△, G). In the notation of sec. 3.4, this is +the statement that the assignment ag,m is such that ag,m[v1v2] = 1G for every edge (v1v2) ⊂ +I +v. It +immediately follows from the definition of the effective local operators and the choice of coefficients +(3.84) that h2VecG +v,4 +acts as +h2VecG +v,4 += −Jκ +|G| +ÿ +x∈G +Lx +v , +(3.85) +where we recall that Lx +v : |m[v]⟩ �→ |xm[v]⟩. Similarly, we find that local operators h2VecG +v,n=1,2,3 act as +−JΠ1G +v,v+ˆu, −JΠ1G +v,v+ˆv and −JΠ1G +v,v+ ˆ +w, respectively, where the vectors (ˆu, ˆv, ˆw) were introduced in (3.2) +and Π1G +v1,v2 := ř +m δm[v1]−1m[v2],1G|m⟩⟨m|. Putting everything together, we obtain +H2VecG = −J +ÿ +e +Π1G +s(e),t(e) − Jκ +|G| +ÿ +v +ÿ +x∈G +Lx +v , +(3.86) +which we recognise as the finite group generalisation of the transverse-field Ising model. We can readily +check that this Hamiltonian possesses a G symmetry, which is consistent with (2VecG)⋆ +2VecG ∼= 2VecG. +Let us now construct dual models resulting from gauging sub-symmetries. +• Within our framework, gauging the whole G symmetry—or rather 2VecG symmetry—amounts to +choosing the 2VecG-module 2-category M(G, 1) ∼= Vec. Clearly, with this choice, there are no degrees +of freedom left at vertices so the effective microscopic Hilbert space is spanned by states |g⟩ ∈  +e C[G], +where g ∈ Z1(Σ△, G). The condition dg = 1G imposes kinematical constraints g[v1v2]g[v2v3] = g[v1v3] +for every triangle (v1v2v3) ⊂ Σ△ so that g defines a G-gauge field. In the notation of sec. 3.4, we have +ag,m = g. Going back to the definition of the local operators (3.30), it readily follows from the flatness +condition of g that +h2Vec +v,4 += − Jκ +|G| +ÿ +x∈G +� ź +e→v +Rx +e +�� ź +e←v +Lx +e +� +≡ − Jκ +|G| +ÿ +x∈G +Ax +v ≡ −JκAv , +(3.87) +where e → v and e ← v refer to edges e ⊂ Σ△ such that t(e) = v and s(e) = v, respectively. Similarly, +we find that the local operators h2VecG +v,n=1,2,3 read −JΠ1G +(v v+ˆu), −JΠ1G +(v v+ˆv) and −JΠ1G +(v v+ ˆ +w), respectively, +where Π1G +e += ř +g δg[e],1G|g⟩⟨g|. Putting everything together, we obtain +H2Vec = −J +ÿ +e +Π1G +e +− Jκ +ÿ +v +Av , +(3.88) +∼ 41 +∼ + +which we recognise as the pure Ising G-gauge theory. We know from the general construction that this +model has a (2VecG)⋆ +2Vec ∼= 2Rep(G) symmetry and we provided in eq. (3.72) a formula for constructing +the corresponding surface and operators. In the following section, we shall study these symmetry +operators on the lattice in more detail for specific choices of input group G, but let us make a few +general comments in the meantime. Straightforward examples of surface operators are those associated +with simple objects of the form Vecψ ∈ 2Rep(G), where Vecψ is the VecG-module category that only +differs from Vec in the choice of module associator α▷, which is such that α▷ +Cg1,Cg2,C = ψ(g1, g2) for every +g1, g2 ∈ G. Going back to the general definition provided in sec. 3.5 we find these surface operators +act diagonally by multiplication by the evaluation of the 3-cocycle characterising the corresponding +module associator. In symbols, these are proportional to +ÿ +g∈Z1(Σ△,G) +� +ź +(v1v2v3) +ψ(g[v1v2], g[v2v3])ϵ(v1v2v3)� +|g⟩⟨g| . +(3.89) +In particular, the commutation relation with operators Av introduced in eq. (3.88) follows from the +2-cocycle condition dψ = 1. Topological lines living on such a surface operator were shown in sec. 3.5 +to be labelled by 1-endomorphisms of Vecψ in 2Rep(G), which correspond by definition to VecG- +module endofunctors of Vecψ in FunVecG(Vecψ, Vecψ) ∼= Rep(G). Therefore, these amount to ordinary +Wilson lines labelled by representations of G. Explicitly, the closed Wilson line operator labelled by +ρ ∈ Rep(G) with support the closed path ℓ reads +ÿ +g∈Z1(Σ△,G) +tr +� ź +e⊂ℓ +ρ(g[e]) +� +|g⟩⟨g| . +(3.90) +The fact that these commute with the Hamiltonian eq. (3.86), and in particular with operators Av, +follows from the gauge invariance of Wilson loop operators. These generate the 1-form Rep(G) sym- +metry of the model. We could also consider an open version of the surface operator (3.89) labelled by +Vecψ, together with topological lines living at the interface of this surface operator and the trivial one +labelled by Vec ∈ 2Rep(G). Mimicking the derivation of (VecG)⋆ +Vec ∼= Rep(G) [EGNO16], we immedi- +ately find that such topological lines are labelled by simple objects in FunVecG(Vec, Vecψ) ∼= Repψ(G), +i.e. projective representations of the group G with Schur’s multiplier ψ. Explicit examples of such +symmetry operators are presented in the next section. +Note finally that instead of considering the 2VecG-module 2-category 2Vec, we could have consid- +ered instead M(G, λ) ∼= 2Vecλ where λ is a non-trivial 3-cocycle in H3(G, U(1)). Recall from sec. 3.7 +that 2Vecλ only differs from 2Vec in the choice of module pentagonator π▷. Physically, this amounts +to the λ-twisted gauging of the G symmetry. In the case of Z2, and given the only non-trivial 3-cocycle +in H3(Z2, U(1)) ≃ Z2, the resulting model would precisely correspond to that considered in sec. 2. We +already commented on the fact that (2VecG)⋆ +2Vecλ ∼= 2Rep(G) so the resulting twisted G-gauge theory +would have the symmetry operators as H2Vec. +• In order to proceed with the next two cases, we further assume that the group G is a semi-direct +product of the form G ≃ Q ⋉φ L with L abelian. +Recall that we write group elements as g ≡ +(ϖQ(g), ϖL(g)) ∈ Q ⋉φ L and the multiplication is given by (3.73). Let us now consider the gauging +of the L sub-symmetry, which amounts to choosing the 2VecG-module 2-category M(A = L, 1) ∼= +2VecQ. +We know from sec. 3.4 that the effective microscopic Hilbert space is spanned by states +|l, m⟩ ∈  +e C[L]  +v C[Q], where l ∈ Z1(Σ△, L). Let us now work out how the local operators hM(L,1) +v,n +∼ 42 +∼ + +act on this Hilbert space. Going back to the definition, we have +hM(L,1) +v,4 += − Jκ +|G| +ÿ +g∈Z1( +I +v,G) +m∈C0 +g( +I +v,M) +��(ag,m, m) +� +1 +v +��� +(ag,m, m) +� +0 +v +��� . +(3.91) +We shall first consider the operator associated with a fixed pair (g, m) ∈ Z1( +I +v, G) × C0 +g( +I +v, M). Up +to the scalar prefactor −Jκ +|G| , this operator acts on the state |m[v]⟩ as +|m[v]⟩ �→ |m[v′]⟩ = |Vecg[v′v] ▷ m[v]⟩ = |ϖQ(g[v′v]) m[v]⟩ , +(3.92) +while it acts on a state |l[vv1]⟩ ≡ |ag,m[vv1]⟩ = |ag[vv1],m[v1]⟩ as +|l[vv1]⟩ �→ |ag[v′v]g[vv1],m[v1]⟩ = |l[vv1] + ag,m[v′v]⟩ , +(3.93) +where we made use of (the abelian version of) eq. (3.10). +Similarly, it acts on a state |l[v1v]⟩ ≡ +|ag,m[v1v]⟩ = |ag[v1v],m[v]⟩ as +|l[v1v]⟩ �→ |ag[v1v]g[v′v]−1,g[v′v]▷m[v]⟩ = |l[v1v] + ag,m[vv′]⟩ . +(3.94) +Invoking eq. (3.28), we have ag,m[v1v2] = φm[v1]−1� +ϖL(g[v1v2]) +� +so that +ag,m[v′v] = φm[v′]−1� +ϖL(g[v′v]) +� +, +ag,m[vv′] = −φm[v′]−1� +ϖL(g[v′v]) +� +, +(3.95) +where we used the fact that g[vv′] = g[v′v]−1 ≡ +� +ϖQ(g[v′v])−1, −φϖQ(g[v′v])−1� +ϖL(g[v′v]) +�� +. These +expressions in turn allow us to rewrite the actions (3.93) and (3.94) more explicitly. Keeping in mind +that m[v′] = ϖQ(g[v′v]) m[v], we obtain the following expression for the local operator hM(L,1) +v,4 +: +hM(L,1) +v,4 += − Jκ +|G| +ÿ +x∈G +� ź +e→v +φRx +e,v +�� ź +e←v +φLx +e,v +� +LϖQ(x) +v +≡ − Jκ +|G| +ÿ +x∈G +φAx +v ≡ −JκφAv , +(3.96) +where +φLx +e,v : |l[e]⟩ �→ |l[e] + φm[v]−1(ϖL(x))⟩ , +φRx +e,v : |l[e]⟩ �→ |l[e] − φm[v]−1(ϖL(x))⟩ . +(3.97) +Moreover, it immediately follows from the definitions that that local operators hM(L,1) +v,n=1,2,3 act as +−JΠ1Q,0L +(v v+ˆu), −JΠ1Q,0L +(v v+ˆv) and −JΠ1Q,0L +(v v+ ˆ +w), respectively, where +Π1Q,0L +(v1v2) := +ÿ +m,l +δm[v1]−1m[v2],1Q δl[v1v2],0L |l, m⟩⟨l, m| . +(3.98) +Putting everything together, we obtain19 +HM(L,1) = −J +ÿ +e +Π1Q,0L +e +− Jκ +ÿ +v +φAv . +(3.99) +19An alternative Hamiltonian resulting from the gauging of a normal subgroup sub-symmetry of H2VecG is often found +in the literature, see e.g. ref. [WBV17, TJV21, TVV22]. The Hamiltonian found in these references is related to ours +via unitary transformation |φm(l), m⟩⟨l, m|, where φm(l)[e] = φm[s(e)](l[e]). The point of this additional unitary is for the +remaining 0-form 2VecQ to be on-site so it can be subsequently gauged following the canonical approach. However, the +resulting model then possesses a twisted 1-form Rep(L) symmetry. +∼ 43 +∼ + +We know from the general construction of sec. 3.5 that this Hamiltonian must commute with symmetry +operators encoded into the Morita dual (2VecG)⋆ +M(L,1), which was shown in sec. 3.7 to be equivalent to +the fusion 2-category 2VecG of G-graded 2-vector spaces. In particular, the model possesses a 1-form +Rep(L) symmetry, which is acted upon by a 0-form 2VecQ symmetry that is not on-site. Instead +of explaining the lattice implementations of these symmetry operators in the general case, we shall +provide explicit parametrisations in terms of spin operators in sec. 5 for the case of the symmetric +group S3 of degree 3. +• Finally, we consider gauging the Q sub-symmetry, which amounts to choosing the 2VecG-module +2-category M(A = Q, 1) ∼= 2VecG/Q. The effective microscopic Hilbert space is spanned by states +|q, m⟩ ∈  +e C[Q]  +v C[L] where q ∈ Z1(Σ△, Q).20 Mimicking the previous derivation, given a fixed +pair (g, m) ∈ Z1( +I +v, G) × C0 +g( +I +v, M) and up to the scalar prefactor −Jκ +|G| , we have an operator that +acts on the state |m[v]⟩ as +|m[v]⟩ �→ |m[v′]⟩ = |Vecg[v′v] ▷ m[v]⟩ = +��ϖL(g[v′v]) + φϖQ(g[v′v])(m[v]) +� +, +(3.100) +while it acts on states |q[vv1]⟩ ≡ |ag,m[vv1]⟩ and |q[v1v] ≡ ||ag,m[vv1]⟩ as +|q[vv1]⟩ �→ +��ϖQ(g[v′v])q[vv1] +� +and +|q[v1v]⟩ �→ +��q[v1v]ϖQ(g[v′v])−1� +, +(3.101) +respectively. In the latter equations, we used the fact ag,m[v1v2] = ag[v1v2],m[v2] = ϖQ(g[v1v2]). We +thus obtain the following expression for the local operators hM(Q,1) +v,4 +: +hM(Q,1) +v,4 += − Jκ +|G| +ÿ +x∈G +� ź +e→v +RϖQ(x) +e +� +φLx +v +� ź +e←v +LϖQ(x) +e +� +≡ − Jκ +|G| +ÿ +x∈G +φ�Ax +v ≡ −Jκφ�Av +(3.102) +where +φLx +v : |m[v]⟩ �→ +��ϖL(x) + φϖQ(x)(m[v]) +� +. +(3.103) +Similarly, we find that local operators hM(Q,1) +v,n=1,2,3 acts as −JΠ0L,1Q +(v v+ˆu), −JΠ0L,1Q +(v v+ˆv) and −JΠ0L,1Q +(v v+ ˆ +w), re- +spectively, where +Π0L,1Q +(v1v2) := +ÿ +m,q +δm[v1],m[v2] δq[v1v2],1Q |q, m⟩⟨q, m| . +(3.104) +Putting everything together, we obtain +HM(Q,1) = −J +ÿ +e +Π0L,1Q +e +− Jκ +ÿ +v +φ�Av . +(3.105) +We know from the general construction of sec. 3.5 that this Hamiltonian must commute with symmetry +operators encoded into the Morita dual (2VecG)⋆ +M(Q,1), which was shown in sec. 3.7 to be equivalent +to the fusion 2-category 2Rep(G) of 2-representations of the 2-group G. As for the previous example, +we shall refrain from describing the lattice implementations of the corresponding topological surfaces +and topological lines in the general case, and shall rather focus in sec. 5 on the specific case of the +symmetric group S3. +20Recall that even though G/Q is not isomorphic to L as a group, we can still identify objects M in M(Q, 1) with +group elements in l ∈ L such that M = lQ. +∼ 44 +∼ + +Back to the transverse-field Ising model +We conclude this section by specialising once more to the case of the transverse-field (Z2-)Ising model. +Let us focus on Hamiltonian (2.10) obtained by choosing the 2VecZ2-module category 2Vec. We es- +tablished that by construction this model has a 2Rep(Z2) symmetry. There are two simple objects +in 2Rep(Z2) provided by the two indecomposable VecZ2-module categories, namely Vec and VecZ2. +The corresponding surface operators were notated via Utriv. and UZ2 in sec. 2, respectively. It follows +from the alternative definition provided in eq. (2.22) and the preceding paragraph that UZ2 indeed +corresponds to surface operator (3.72) for N = VecZ2. Line operators living on the surface operator +Utriv. are now identified with simple 1-morphisms in Hom2Rep(Z2)(Vec, Vec) ∼= Rep(Z2). Line oper- +ators labelled by the non-trivial representation of Z2 as defined in eq. (3.90) readily correspond to +(2.23). Similarly, we recover line operators on the surface operator UZ2 as simple 1-morphisms in +Hom2Rep(Z2)(VecZ2, VecZ2) ∼= VecZ2. What about line operators at the junctions of surface operators +UZ2 and Utriv.? We established in sec. 2 that such lines are unique up to isomorphisms. Within +the framework of this section, these correspond to the unique simple objects in the hom-categories +Hom2Rep(Z2)(VecZ2, Vec) = FunVecZ2 (VecZ2, Vec) ∼= Vec and FunVecZ2 (Vec, VecZ2) ∼= Vec. Moreover, +composition of VecZ2-module functors FunVecZ2 (VecZ2, Vec) × FunVecZ2 (Vec, VecZ2) → Rep(Z2) informs +us that composing the corresponding line operators yields a line operator living on Utriv. labelled by +the regular representation in Rep(Z2), which is compatible with eq. (2.33). Similarly, composition of +VecZ2-module functors FunVecZ2 (Vec, VecZ2) × FunVecZ2 (VecZ2, Vec) → VecZ2 informs us that compos- +ing the corresponding line operators yields a line operator living on UZ2 labelled by the object C0 ‘C1 +in VecZ2, which is compatible with eq. (2.35). Finally, the monoidal structure of 2Rep(Z2) is such that +VecZ2 d VecZ2 ∼= VecZ2 ‘ VecZ2, which amounts to (2.20) when Σ is the two-torus. +SECTION 4 +Example: doubled transverse-field Ising model +In this section, we study in detail the symmetry structure of the model obtained by gauging the Z2 +2 +symmetry of the doubled transverse-field Ising model. +4.1 +Symmetric Hamiltonian and gauging +The starting point is the doubled transverse-field Ising model on a triangulation Σ△ of a closed oriented +surface Σ. Pairs of qubit degrees of freedom are assigned to vertices v ⊂ Σ△. We identify such an +assignment with a choice of 0-cochain m ∈ C0(Σ△, Z2 +2) so the microscopic Hilbert space is provided by +the tensor product  +v C[Z2 +2] ≃ C4, on which two sets of Pauli operators denoted as σµ,I +v +with I = 1, 2 +act. The doubled transverse-field Ising model is then defined via the Hamiltonian +H +2VecZ2 +2 = − +2ÿ +I=1 +� +JI,1 +ÿ +e +σz,I +s(e)σz,I +t(e) + JI,2 +ÿ +v +σx,I +v +� +. +(4.1) +The model has a (0-form) global Z2 +2 symmetry implemented by surface operators +Og = +ź +v +(σx,1 +v +)g1(σx,2 +v +)g2 , +(4.2) +for every g ≡ (g1, g2) ∈ Z2 +2. Fusion rules of these surface operators are dictated by the multiplication +rule in Z2 +2. +∼ 45 +∼ + +In the language of the previous section, the symmetry structure of this model is encapsulated in +the fusion 2-category 2VecZ2 +2, whose four simple objects correspond to the surface operators O(0,0), +O(0,1), O(1,0) and O(1,1), respectively. Moreover, recall that for any simple object Vecg in 2VecZ2 +2, +its endo-category is equivalent to Vec, whose unique simple object corresponds to the identity line +operator living on Og. Finally, since there are no 1-morphisms in 2VecZ2 +2 between distinct simple +objects, there are no topological lines between distinct surface operators. +Note that for conciseness we only consider a minimal Z2 +2-symmetric transverse-field Ising model, +which realises the symmetric paramagnetic and symmetry-broken gapped phases. In particular, this +Hamiltonian does not realise any SPT phase.21 However, as was explained in the previous section, +details of the Hamiltonian are irrelevant to the ensuing analysis of the gauging procedure and the +symmetry structure of the gauged model, so that the following derivations hold for any model with +the same Z2 +2 symmetry. +Given the input fusion 2-category 2VecZ2 +2, Hamiltonian (4.1) is implicitly defined with respect +to the module 2-category 2VecZ2 +2 over itself. In this case, gauging the Z2 +2 symmetry simply amounts +to choosing instead the 2VecZ2 +2-module 2-category 2Vec. That being said, since this Hamiltonian is +merely a doubled version of that considered in sec. 2, we can immediately infer from the procedure +outlined there the resulting dual Hamiltonian: +H2Vec = − +2ÿ +I=1 +� +JI,1 +ÿ +e +σz,I +e ++ JI,2 +ÿ +v +ź +e⊃v +σx,I +e +� +, +(4.4) +which acts on the physical Hilbert space spanned by states |g⟩, where g ≡ (g1, g2) ∈ Z1(Σ△, Z2 +2). +Recall that we chose the basis such that +σz,I +e +|g⟩ = (−1)gI[e]|g⟩ . +(4.5) +In this basis, the first term in the Hamiltonian (4.4) acts diagonally, while an arbitrary combination +of operators ś +e⊃v σx,I +e +indexed by a 0-cochain x ≡ (x1, x2) ∈ C0(Σ△, Z2 +2) acts as +Ax := +ź +v⊂Σ△ +ź +e⊃v +2 +â +I=1 +(σx,I +e +)xI[v] = +ÿ +g +|g + dx⟩⟨g| . +(4.6) +We know from the results of sec. 3.2 that Hamiltonian (4.4) must commute with various surface and +line operators that are organised into the Morita dual fusion 2-category (2VecZ2 +2)⋆ +Vec ∼= 2Rep(Z2 +2). +Recall from sec. 3.6 that simple objects in 2Rep(Z2 +2) are provided by indecomposable VecZ2 +2-module +categories N(B, ψ), which are conveniently labelled by tuples (B, ψ) consisting of a subgroup B ⊆ Z2 +2 +and a 2-cocycle ψ in H2(B, U(1)). Therefore, we count six simple objects in 2Rep(Z2 +2) labelled by +the tuples (Z2 +2, 1), (Z2 +2, ψ), (Z(1) +2 , 1), (Z(2) +2 , 1), (Z(diag.) +2 +, 1) and (Z1, 1), respectively, where ψ refers here +to a normalised representative of the non-trivial cohomology class in H2(Z2 +2, U(1)) ≃ Z2. Each such +21The classification of SPT phases with 0-form global Z2 +2 symmetry is given by the cohomology group H3(Z2 +2, U(1)) ≃ +Z3 +2. Therefore, there are eight distinct topological phases which can be labelled by p ≡ (p1, p2, p3) ∈ Z3 +2. The corre- +sponding fixed-point Hamiltonians, which we denote as Hp, have the form +H(1,0,0) = − +ÿ +v +σx,1 +v +ź +(v v1v2) +exp +� iπ +4 (1 − σz,1 +v1 σz,1 +v2 ) +� +, +H(0,1,0) = − +ÿ +v +σx,2 +v +ź +(v v1v2) +exp +� iπ +4 (1 − σz,2 +v1 σz,2 +v2 ) +� +, +H(0,0,1) = − +ÿ +v +σx,1 +v +ź +(v v1v2) +exp +� iπ +4 (1 − σz,1 +v1 σz,1 +v2 ) +� +− +ÿ +v +σx,2 +v +ź +(v v1v2) +exp +� iπ +4 (1 − σz,2 +v1 σz,2 +v2 ) +� +. +(4.3) +∼ 46 +∼ + +simple object provides a surface operator commuting with (4.4). Furthermore, there are various line +operators within each surface operator as well as at interfaces between surface operators associated +with distinct simple objects in 2Rep(Z2 +2).22 The remainder of this section is dedicated to explicitly +constructing these various operators. +4.2 +2Rep(Z2 +2) symmetry: invertible surface operators +We begin our detailed analysis of the symmetry structure of Hamiltonian (4.4) by enumerating the +invertible surface operators. Firstly, there is of course the identity operator23 +Utriv. = +ź +e +ide, +(4.7) +which corresponds to the identity object N(Z2 +2, 1) ∼= Vec in 2Rep(Z2 +2). As explained in sec. 3.8, line +operators living on this trivial operator form the hom-category +Hom2Rep(Z2 +2)(Vec, Vec) ∼= FunVecZ2 +2 (Vec, Vec) ∼= Rep(Z2 +2) . +(4.8) +We provided in eq. (3.90) a general formula for such line operators but we can make it more explicit by +specialising to G = Z2 +2. Given a 1-cycle ℓ on Σ△ and an irreducible representation (ρ1, ρ2) ∈ Rep(Z2 +2), +we may define such a line operator as +ÿ +g +� ź +e⊂ℓ +ź +I +ρI(gI[e]) +� +|g⟩⟨g| , +(4.9) +which readily commutes with (4.4). +For instance, choosing both ρ1 and ρ2 to be the non-trivial +irreducible representation of Z2, the operator above can be equivalently defined as ś +e⊂ℓ σz,1 +e +σz,2 +e +. +More generally, an operator corresponding to a network of lines in Rep(Z2 +2) can be defined as +Utriv.(f) = +ÿ +g +(−1) +� +Σ△(f1⌣g1+f2⌣g2)|g⟩⟨g| , +(4.10) +where f = (f1, f2) ∈ Z1(Σ△, Z2 +2). It follows directly from the definition that the composition such lines +satisfies +Utriv.(f1 ◦ f2) = Utriv.(f1 + f2) , +(4.11) +which does amount to the monoidal structure in Rep(Z2 +2). Similarly, the fusion of lines read +Utriv.(f1) d Utriv.(f2) = Utriv.(f1 + f2) . +(4.12) +as predicted by the monoidal structure in 2Rep(Z2 +2). The second and final invertible surface corresponds +to the simple object N(Z2 +2, ψ) ∼= Vecψ in 2Rep(Z2 +2), which as a VecZ2 +2-module category only differs +from Vec in the choice of module associator. We provided in eq. (3.89) a general expression for the +corresponding type of surface operator. Writing ψ(g, g′) := (−1)g1g′ +2, we find it is a non-trivial operator +that acts diagonally on basis states as +Uψ[Σ△] = +ÿ +g +(−1) +� +Σ△g1⌣g2|g⟩⟨g| . +(4.13) +22Note that two objects that have a non-trivial 1-morphism between them are said to belong to the same Schur +component. Physically, this means that there is a condensation process relating both objects [GJF19, DR18, Reu22]. +23Notice that, in a way that is reminiscent of the construction of the corresponding module categories, we notate the +surface operator associated with the tuple (B, ψ) via UG/B,ψ. +∼ 47 +∼ + +This operator can be shown to commute with the Hamiltonian. On the one hand, it is diagonal in the +chosen computational basis, thus it clearly commutes with the first term in (4.4). On the other hand, +the SPT being non-anomalous is gauge invariant, and thus commutes with the second term. The +operator can also be defined with non-trivial line insertions which are also labelled by f ∈ Z1(Σ△, Z2 +2) +as +Uψ(f)[Σ△] = +ÿ +g +(−1) +� +Σ△g1⌣g2+f1⌣g1+f2⌣g2|g⟩⟨g| , +(4.14) +which satisfy composition rules analogous to (4.11). +It follows that such lines also encoded into +Rep(Z2 +2). As evoked in sec. 3.8, this is explained by the fact that FunVecZ2 +2 (Vecψ, Vecψ) ∼= Rep(Z2 +2). +The operators Uψ[Σ△] and Utriv. satisfy Z2 fusion rules of the form +� +Uψ d Uψ� +[Σ△] = Utriv. , +� +Uψ d Utriv.� +[Σ△] = Uψ[Σ△] = +� +Uψ d Utriv.� +[Σ△] , +(4.15) +which readily follows from eq. (4.13). +Similarly, composition rules for surface operators with line +operators inserted take the form +� +Uψ(f1) d Uψ(f2) +� +[Σ△] = Utriv.(f1 + f2)[Σ△] , +� +Uψ(f1) d Utriv.(f2) +� +[Σ△] = Uψ(f1 + f2)[Σ△] , +� +Utriv.(f1) d Uψ(f2) +� +[Σ△] = Uψ(f1 + f2)[Σ△] . +(4.16) +Interestingly, one may also define the operator Uψ on an open sub-complex Ξ△ ⊆ Σ△. Equivalently, +this is the statement that there is a topological line operator between the operators Utriv. and Uψ. +Naively, the operator Uψ[Ξ△] simply defined by restricting definition (4.13) to the open sub-complex +Ξ△ does not commute with the Hamiltonian (4.4) since +AxUψ[Ξ△] = +ÿ +g +exp +� +iπ +� +Ξ△ +g1 ⌣g2 +� +|g + dx⟩⟨g| , +Uψ[Ξ△]Ax = +ÿ +g +exp +� +iπ +� +Ξ△ +g1 ⌣g2 + iπ +� +∂Ξ△ +ζ(g, x) +� +|g + dx⟩⟨g| , +(4.17) +where dζ(g, x) = (g1+dx)⌣(g2+dx)−g1 ⌣ g2. Such a lack of commutation is remedied by appending +a line operator on the boundary ∂Ξ△, which has the form +Uψ|triv.[∂Ξ△] = +ÿ +g +Zanom.(g)[∂Ξ△]|g⟩⟨g| , +(4.18) +where the amplitude Zanom.(g)[∂Ξ△] can be understood as the partition function of a quantum me- +chanical (i.e., (0 + 1)-dimensional) system with an anomalous Z2 +2 symmetry encoded into the (unique) +irreducible projective representation with Schur multiplier ψ. Concretely, this operator can be con- +structed by considering auxiliary Z2 +2-valued vertex degrees of freedom p, q on ∂Ξ△ such that +Zanom.(g)[∂Ξ△] = +ÿ +b,n +exp +� +iπ +� +∂Ξ△ +� +δI,JbI ⌣(dnJ + gJ) + dn1 ⌣dn2 +�� +, +(4.19) +which has the required property +Zanom.(g + dx)[∂Ξ△] = exp +� +iπ +� +∂Ξ△ +ζ(g, x) +� +Zanom.(g)[∂Ξ△] . +(4.20) +∼ 48 +∼ + +It follows that the combined operator +Uψ→triv.[Ξ△] := Uψ[Ξ△] · Uψ|triv.[∂Ξ△] , +(4.21) +commutes with the Hamiltonian and is therefore a symmetry operator. Graphically, we depict such a +configuration as +(4.22) +where the gray area depicts Ξ△ and the blue coloured edges the support of the line operator. From a +category theoretic standpoint, recall that line operators at the interface of Uψ and Utriv. are organised +into the fusion category FunVecZ2 +2 (Vec, Vecψ) ∼= Repψ(Z2 +2) of ψ-projective representations of Z2 +2, which +is compatible with the above construction. Similarly, we define an operator Utriv.→ψ[Ξ△]. +Finally, the composition of the topological line Uψ|triv.[∂Ξ△] follows from the tensor product of +representations. Since the line Uψ|triv.[∂Ξ△] carries a ψ-projective representation of Z2 +2, composing it +with itself must yield an object labelled by the trivial representation in Rep(Z2 +2). We can compose this +line between the trivial surface operators Utriv. and the non-trivial operator Uψ in two ways so as to +recover the trivial representation line either within the trivial surface or within the non-trivial surface +operator, i.e. +Uψ→triv.[Ξ△] ◦ Utriv.→ψ[Σ△/Ξ△] = Uψ[Σ△] , +Utriv.→ψ[Ξ△] ◦ Uψ→triv.[Σ△/Ξ△] = Utriv. , +(4.23) +which is mathematically encoded into the composition of the corresponding VecZ2 +2-module functors. +This concludes our analysis of the invertible surface operators. +4.3 +2Rep(Z2 +2) symmetry: non-invertible surface operators +We continue our analysis of the symmetry structure of Hamiltonian (4.4) with the study of surface +operators that have non-invertible fusion rules. In terms of simple objects in 2Rep(Z2 +2), these are the +ones labelled by the tuples (Z(1) +2 , 1), (Z(2) +2 , 1), (Zdiag. +2 +, 1) and (Z1, 1). Let us focus for now on the first +three surface operators. Mimicking the definition of the non-invertible surface operator described in +sec. 2, one finds: +UZ(1) +2 [Σ△] = +1 +2#(Σ△) +ÿ +g,n,b +(−1) +� +Σ△b⌣(dn+g1)|g⟩⟨g| = +1 +2χ(Σ△) +ÿ +g,n +δdn,g1|g⟩⟨g| , +UZ(2) +2 [Σ△] = +1 +2#(Σ△) +ÿ +g,n,b +(−1) +� +Σ△b⌣(dn+g2)|g⟩⟨g| = +1 +2χ(Σ△) +ÿ +g,n +δdn,g2|g⟩⟨g| , +(4.24) +UZ(diag.) +2 +[Σ△] = +1 +2#(Σ△) +ÿ +g,n,b +(−1) +� +Σ△b⌣(dn+g1+g2)|g⟩⟨g| = +1 +2χ(Σ△) +ÿ +g,n +δdn,g1+g2|g⟩⟨g| , +where the summation variables are n ∈ C0(Σ△, Z2) and b ∈ C1(Σ△, Z2). +As in sec. 2, we have +defined #(Σ△) = 2|Σ0 +△|+|Σ2 +△| and χ(Σ△) is the Euler characteristic of Σ△. Summing over b, imposes +∼ 49 +∼ + +a constraint that pins dn on each edge of the triangulation to be g1, g2 and g1 + g2 for the three +operators UZ(1) +2 , UZ(2) +2 +and UZ(diag.) +2 +, respectively. These operators should be thought of as an explicit +version of the general operator (3.72). +There, n refers to an assignment of simple objects in the +VecZ2 +2-module categories N(Z(1) +2 , 1), N(Z(2) +2 , 1) and N(Z(diag.) +2 +), which are all equivalent to VecZ2 as +categories, satisfying n[v1] = Cg[v1v2] ▷ n[v2] for every edge (v1v2) ⊂ Σ△. In particular, the VecZ2 +2- +module structure on N(Z(diag.) +2 +, 1) is given by Cg ▷N := (g1 +g2)+N mod 2, for any g ≡ (g1, g2) ∈ Z2 +2 +and N ∈ VecZ2. This amounts to the condition dn = g1 + g2 in the equation above. +Summing over n instead of b in eq. (4.24) imposes db = 0. +Summing over equivalence classes of +1-cocycles, i.e., f ∈ H1(Σ△, Z2), one finds +UZ(1) +2 [Σ△] = +1 +|H0(Σ△, Z2)| +ÿ +g,f +(−1) +� +Σ△f⌣g1|g⟩⟨g| , +UZ(2) +2 [Σ△] = +1 +|H0(Σ△, Z2)| +ÿ +g,f +(−1) +� +Σ△f⌣g2|g⟩⟨g| , +UZ(diag.) +2 +[Σ△] = +1 +|H0(Σ△, Z2)| +ÿ +g,f +(−1) +� +Σ△f⌣(g1+g2)|g⟩⟨g| . +(4.25) +From these expressions, it is clear that these operators are condensation defects of the Rep(Z2 +2) lines +described previously. It turns out that this alternative form of the operators is particularly convenient +to demonstrate the commutativity of these operators with the Hamiltonian. +It follows from the +operators being diagonal in the chosen basis and invariance under the action of Ax. +Still in analogy with the non-invertible surface operator considered in sec. 2, the surfaces (4.24) +can be defined with topological lines inserted. Recall that these must be encoded into Morita duals +of VecZ2 +2 with respect to the corresponding module categories. For instance, lines living on UZ(1) +2 +form +the fusion 1-category +FunVecZ2 +2 (VecZ(1) +2 , VecZ(1) +2 ) ∼= VecZ(1) +2 +b Rep(Z(2) +2 ) . +(4.26) +The corresponding surface operator with a network of lines inserted can be constructed by choosing +1-cocycles ˜f1, f2 ∈ Z1(Σ△, Z2) as +UZ(1) +2 (˜f1, f2)[Σ△] = +1 +2#(Σ△) +ÿ +g,n,b +(−1) +� +Σ△b⌣(dn+g1+˜f1)+f2⌣g2|g⟩⟨g| += +1 +2χ(Σ△) +ÿ +g,n +δdn,g1+˜f1(−1) +� +Σ△ f2∪g2|g⟩⟨g| , +(4.27) +where ˜f1, which twists the cocycle condition on n, corresponds to the VecZ(1) +2 +lines, whereas f2 cor- +responds to the usual Wilson lines. By analogy, the topological lines living on the surface operators +UZ(2) +2 [Σ△] and UZ(diag.) +2 +[Σ△] form the fusion 1-categories VecZ(2) +2 bRep(Z(1) +2 ) and VecZ(1) +2 bRep(Z(diag.) +2 +), +respectively, where a networks of lines within these two operators take the form +UZ(2) +2 (f1,˜f2)[Σ△] = +1 +2#(Σ△) +ÿ +g,n,b +(−1) +� +Σ△f1⌣g1+b⌣(dn+g2+˜f2)|g⟩⟨g| , +UZ(diag.) +2 +(f,˜f)[Σ△] = +1 +2#(Σ△) +ÿ +g,n,b +(−1) +� +Σ△b⌣(dn+(g1+g2)+˜f)+f⌣(g1+g2)|g⟩⟨g| . +(4.28) +∼ 50 +∼ + +Let us now describe the final symmetry surface operator in the fusion 2-category 2Rep(Z2 +2). It is that +labelled by the VecZ2 +2-module category N(Z1, 1) ∼= VecZ2 +2: +UZ2 +2[Σ△] = +1 +22#(Σ△) +2 +ź +I=1 +ÿ +g +ÿ +nI,bI +(−1) +� +Σ△δI,JbI⌣(dnJ+gJ)|g⟩⟨g| = +1 +2χ(Σ△) +2 +ź +I=1 +ÿ +nI +δdnI,gI|g⟩⟨g| . +(4.29) +Similarly, UZ2 +2[Σ△] can be defined with line operator insertions: +UZ2 +2(˜f1,˜f2)[Σ△] = +1 +22#(Σ△) +ź +I +ÿ +g,nI,bI +(−1) +� +Σ△δI,JbI⌣(dnJ+gJ+˜fJ)|g⟩⟨g| . +(4.30) +The commutation of UZ2 +2(˜f1,˜f2)[Σ△] with the Hamiltonian can be demonstrated as before. +Having described all the surface operators associated with simple objects in 2Rep(Z2 +2), let now compute +their fusion rules. For instance, we have +(UZ(1) +2 +d UZ(1) +2 )[Σ△] = +1 +22#(Σ△) +ÿ +g,b,n +g′,b′,n′ +(−1) +� +Σ△b⌣(dn+g1)+b′⌣(dn′+g′ +1)|g⟩⟨g|g′⟩⟨g′| += +� +1 +2#(Σ△) +ÿ +b′,n+ +(−1) +� +Σ△b′⌣dn+� +1 +2#(Σ△) +ÿ +g,b+,n +(−1) +� +Σ△b+⌣(dn+g1)|g⟩⟨g| += Z2d[Σ△] · UZ(1) +2 [Σ△] , +(4.31) +where in the second line, we have defined b+ = b + b′ and n+ = n + n′. The pre-factor Z2d[Σ△] in +the fusion rule outcome is the partition function of the pure two-dimensional Z2 gauge theory on Σ△ +as in sec. 2 and app. A. In the case where Σ is a two-torus, we recover exactly the fusion structure of +2Rep(Z2 +2) according to which +VecZ(1) +2 +d VecZ(1) +2 +∼= VecZ(1) +2 +‘ VecZ(1) +2 +. +(4.32) +The remaining fusion rules can be computed analogously: +� +UZ(I) +2 +d UZ(J) +2 � +[Σ△] = +� +Z2d[Σ△] · UZ(I) +2 [Σ△] +if I = J , +UZ2 +2[Σ△] +otherwise , +(4.33) +where I, J ∈ {1, 2, diag.}. Finally, the fusion rules between the symmetry operator UZ2 +2[Σ△] and the +three other non-invertible surfaces are given by +� +UZ2 +2 d UZI +2� +[Σ△] = +� +UZJ +2 d UZ2 +2� +[Σ△] = Z2d[Σ] × UZ2 +2[Σ△] . +(4.34) +Finally, fusing UZ2 +2 with itself yields +� +UZ2 +2 d UZ2 +2� +[Σ△] = (Z2d[Σ△])2 · UZ2 +2[Σ△] , +(4.35) +where the coefficient (Z2d[Σ△])2 amounts to the partition function of the pure two-dimensional Z2 +2 +topological gauge theory on Σ△. +In order to conclude our analysis of the symmetry structure of (4.4) as encoded into 2Rep(Z2 +2), we are +left to consider topological lines between distinct surfaces as well as the corresponding composition +∼ 51 +∼ + +rules. It largely mimics the case presented in sec. 2. A topological surface operator can be defined on +a triangulation of the form Σ△ = (Σ△\Ξ△) ⊔∂Ξ△ Ξ△, which locally looks like UZ(I) +2 +and UZ(J) +2 +in the +regions Σ△\Ξ△ and Ξ△, respectively. Such an operator has the form +UZ(I) +2 +,Z(J) +2 [Σ△\Ξ△, Ξ△] = +1 +2#(Σ△) +ÿ +g,n,b +˜n,˜b +(−1) +� +Σ△\Ξ△b⌣(dn+gI)+ +� +Ξ△ +˜b⌣(d˜n+gJ)|g⟩⟨g| , +(4.36) +where I, J ∈ {1, 2, diag.} and gdiag. := g1 + g2. +Moreover, we imposed in the previous equation +Dirichlet boundary conditions b[∂Ξ△] = ˜b[∂Ξ△] = 0 along the interface. Similarly, one may define +a topological surface operator that interpolates between UZ2 +2 and UZ(I) +2 +by defining UZ2 +2 on Σ△\Ξ△, +UZ(I) +2 +on Ξ△, and imposing suitable Dirichlet boundary conditions along the interface ∂Ξ△. +Let us now compute the composition rules between topological interfaces separating regions with +locally distinct symmetry operators. To do so, we consider a setup closely resembling that of sec. 2. +Let Ξ△ be a thin annular strip of single lattice spacing width supporting a surface operator UZ(J) +2 , +while the rest of the lattice Σ△\Ξ△ supports UZ(I) +2 . We denote the left and right boundaries of Ξ△ by +∂LΞ△ and ∂RΞ△, respectively. The corresponding composition of lines is then given by the operator +à +f +UZ(I) +2 (f)[Σ△] , +(4.37) +where the sum is over the four simple topological lines of UZ(I) +2 [Σ△] traversing the (relative) homology +cycle of Ξ△ with Dirichlet conditions b[∂LΞ△] = b[∂RΞ△] = 0 imposed. +These fusion rules are +reminiscent of the Z2 +2 Tambara-Yamagami fusion category. +SECTION 5 +Example: transverse-field S3-Ising model +In this section, we consider the higher-categorical symmetry structures of the models obtained by gaug- +ing various sub-symmetries of the transverse-field S3-Ising model. We shall focus on features specific +to dealing with a non-abelian group. +5.1 +Symmetric Hamiltonian +For our final series of examples, we consider a transverse-field Ising model with a non-abelian symmetry +group, namely the symmetric group S3 of degree 3. In sec. 3.8, we explained how to perform the gauging +of various sub-symmetries of this model for an arbitrary group. Moreover, we elucidated there the +symmetry structures of the resulting models in terms of fusion 2-categories of higher representations of +groups, and categorifications thereof. Although the construction of sec. 3.5 provides a general recipe +to realise on the lattice operators commuting with dual Hamiltonians, it remains somewhat formal. +The goal of this section is to describe these symmetry operators more explicitly in the spirit of sec. 4. +Let us begin by reviewing the group structure of S3. The permutation group S3 is the group with +presentation +S3 = ⟨r, s | r2 = s3 = (rs)2 = 1⟩ . +(5.1) +Both the cyclic groups Z2 = ⟨r | r2 = 1⟩ and Z3 = ⟨s | s3 = 1⟩ are subgroups of S3. Due to the action +of Z2 on Z3 given by φ− : Z2 → Aut(Z3) such that φr(s) = s2, we have an isomorphism S3 ≃ Z2⋉φZ3. +Therefore, it is a group of the form considered in sec. 3.8. +∼ 52 +∼ + +The microscopic Hilbert space of the transverse-field S3-model is given by the tensor product + +v C[S3] ∋ |m⟩, where m is an assignment of group elements in S3 to every vertex of Σ△. +Due +to the semi-direct product structure of S3, the local Hilbert space can be rather spanned by states +|m[v]⟩ ≡ |ϖZ2(m[v]), ϖZ3(m[v])⟩, that is a pair of qubit and qutrit degrees of freedom at every vertex. +Given the above, it is useful to define the following operators +σx = +�0 1 +1 0 +� +, +σz = +�1 0 +0 −1 +� +, +Σx = +� +� +0 0 1 +1 0 0 +0 1 0 +� +� , +Σz = +� +� +1 0 0 +0 ω 0 +0 0 ω2 +� +� , +Γ = +� +� +1 0 0 +0 0 1 +0 1 0 +� +� . +(5.2) +The σ matrices satisfy the Pauli algebra σxσz = −σzσx as before, while the Σ matrices represent the +Z3 clock and shift operators, which satisfy ΣzΣx = ωΣxΣz and (Σx)3 = (Σz)3 = id with ω = exp( 2πi +3 ). +Additionally, the operator Γ implements the action of Z2 on Z3 by automorphisms. This operator +satisfies the relations ΓΣxΓ = Σx† and ΓΣzΓ = Σz†. We work in the basis such that +(id b Σz +v)|m[v]⟩ = ωϖZ3(m[v])|m[v]⟩ , +(σz +v b id)|m[v]⟩ = (−1)ϖZ2(m[v])|m[v]⟩ , +(5.3) +effectively identifying group elements in Z2 with {0, 1} and those in Z3 with {0, 1, 2}. The Lg +v operators +which act by left multiplication (see e.g. below eq. (3.85)) have the following explicit form in this basis: +Lr +v : |m[v]⟩ �→ (σx b Γ)v|m[v]⟩ = |r · m[v]⟩ , +Ls +v : |m[v]⟩ �→ (id b Σx)v|m[v]⟩ = |s · m[v]⟩ . +(5.4) +The qubit and qutrit degrees of freedom are subject to the 2VecS3-symmetric Hamiltonian whose +expression we reproduce below: +H2VecG = −J +ÿ +e +Π +1S3 +s(e),t(e) − Jκ +6 +ÿ +v +ÿ +g∈G +Lg +v . +(5.5) +Both terms appearing in this Hamiltonian can be rewritten more explicitly in terms of the matrices +introduced above. On the one hand, we have +Π +1S3 +s(e),t(e) = 1 +6 +� +id + σz +s(e)σz +t(e) +� +b +� +id + Σz +s(e)(Σz +t(e))† + (Σz +s(e))†Σz +t(e) +� +, +(5.6) +which implicitly makes use of the fact that 1 + ω + ¯ω = 0. Such a term is analogous to the ferro- +magnetic term in the Z2-symmetric transverse field Ising model. It energetically favors homogenous +configurations in the computational basis. In the limit κ → 0, the ground state spontaneously breaks +the S3 global symmetry with m[v] = m0, for all v, and there are |S3| ground states associated with +different choices of m0 ∈ S3. On the other hand, the term proportional to κ is a combination of +operators Lg +v which act on the basis by left multiplication. The linear combination of Lg +v operators +has the following explicit form: +1 +6 +ÿ +g∈S3 +Lg +v = 1 +6 +ÿ +g∈S3 +(σx b Γ)ϖZ2(g)(id b Σx)ϖZ3(g) , +(5.7) +which acts as a projector onto the one-dimensional subspace of C[S3] (at the vertex v) transforming in +the trivial representation of S3. This term is analogous to the paramagnetic term in the Z2-symmetric +transverse field Ising model and favours a unique ground state that preserves the full S3 symmetry. +∼ 53 +∼ + +One can readily check that this model has a global 0-form S3 symmetry implemented by surface +operators +Og = +ź +v +Rg +v , +(5.8) +where Rg +v acts on the basis state at vertex v by right multiplication. These operators satisfy the fusion +rules provided by the multiplication rules in S3, i.e., Og1 d Og2 = Og1g2. In order to rewrite operators +Rg +v more explicitly, it is convenient to introduce the following controlled gate: +cΣx : C2 b C3 → C2 b C3 +: +|q, l⟩ +�→ +� +id b (Σx)1+q� +|q, l⟩ , +(5.9) +where the qubit plays the role of the control. These controlled gates satisfy in particular the following +commutation relation +cΣx(σx b Γ) = (σx b Γ)cΣx , +(5.10) +and adaptations thereof. The explicit action of right multiplication operators Rg +v now reads: +Rr +v : |m[v]⟩ �→ (σx b id)v|m[v]⟩ = |m[v] · r⟩ , +Rs +v : |m[v]⟩ �→ (cΣx)† +v|m[v]⟩ = |m[v] · s2⟩ . +(5.11) +Given this action, commutation of Og with the Hamiltonian is ensured by the various commutation +relations listed above. +5.2 +Gauging sub-symmetries +Given a finite group isomorphic to a semi-direct product, we explained in sec. 3.8 how to obtain three +dual Hamiltonians. In the present context, these are obtained by gauging the S3, Z3 and Z2 sub- +symmetries of the S3-symmetric Hamiltonian, respectively. In the language of sec. 3.8, these amount +to choosing the 2VecS3-module 2-categories M(S3, 1) ∼= 2Vec, M(Z3, 1) ∼= 2VecZ2 and M(Z2, 1) ∼= +2VecS3/Z2, respectively. In preparation for the analysis of the symmetry structures, we provide below +more explicit expressions for the terms appearing in the definitions of these Hamiltonians. +• Let us first consider gauging the full S3 symmetry. The resulting Hamiltonian was found in eq. (3.88) +to be of the form +H2Vec = −J +ÿ +e +Π +1S3 +e +− Jκ +ÿ +v +Av . +(5.12) +The edge term explicitly reads +Π +1S3 +e += 1 +6 +� +id + σz +e +� +b +� +id + Σz +e + (Σz +e)†� +, +(5.13) +whereas the contribution of the generators of S3 to the vertex term Av = 1 +6 +ř +g∈S3 Ag +v can be depicted +as +Ar +v ≡ +σx +σx +σxΓ +σxΓ +σxΓ +σx +, +As +v ≡ +cΣx†cΣx† +Σx +Σx +Σx +cΣx† +, +(5.14) +where we omitted b symbols for convenience. The operators Ag +v for the remaining g ∈ S3 can be +obtained by suitably composing Ar +v and As +v. The model (5.12) describes an S3 lattice gauge theory. +∼ 54 +∼ + +The first term suppresses the S3 fluctuations. In the limit κ → 0, the model is in the confined phase, +which is the dual analogue of the symmetry-broken phase in the pre-gauged model. Instead, the second +term is responsible for gauge fluctuations. In the κ → ∞ limit, the model is in the deconfined phase +whose renormalisation group fixed point is provided by the Hamiltonian realisation of S3 Dijkgraaf- +Witten theory with trivial cohomological twist [DW90, dWP95]. +• In the same vein, let us now consider gauging the Z3 sub-symmetry. The resulting Hamiltonian was +found in eq. (3.99) to be of the form +H2VecZ2 = −J +ÿ +e +Π +0Z2,0Z3 +e +− Jκ +ÿ +v +φAv . +(5.15) +The edge term explicitly reads +Π +0Z2,0Z3 +e += 1 +6 +� +id + σz +s(e)σz +t(e) +� +b +� +id + Σz +e + (Σz +e)†� +. +(5.16) +In order to rewrite the vertex term more explicitly, we require the controlled gates introduced pre- +viously. We shall apply here these gates between a qutrit assigned to an edge and a qubit assigned +to a vertex. It is convenient to graphically depict such controlled gates by means of a dotted line +connecting a control qubit identified by ‘c’ and a target qutrit. The contribution of the generators of +S3 to the vertex term φAv = 1 +6 +ř +g∈S3 +φAg +v can now be depicted as +φAr +v = σx +v ≡ +σx +, +φAs +v = +ź +e←v +cΣx +e +ź +e→v +(cΣx +e )† ≡ +Σx† Σx† +Σx +Σx +Σx +Σx† +c +, +(5.17) +where all the gates on the r.h.s. are controlled by the qubit living at the vertex v the operator acts on. +The model (5.15) describes a Z3 lattice gauge theory coupled to a Z2-Ising matter model. In the limit +κ → 0, the Z3 gauge sector is in the confined phase while the Z2 matter sector is in the ferromagnetic +phase. Conversely, in the κ → ∞ limit, the gauge sector is in the deconfined phase while the matter +sector is in the paramagnetic phase. In this limit the model describes up to a unitary (see footnote +19) a Z3 topological gauge theory enriched by a global Z2 symmetry [WBV17, TJV21, TVV22]. +• Finally, let us consider gauging the Z2 sub-symmetry. +The resulting Hamiltonian was found in +eq. (3.105) to be of the form +H2VecS3/Z2 = −J +ÿ +e +Π +0Z3,0Z2 +e +− Jκ +ÿ +v +φ�Av . +(5.18) +The edge term explicitly reads +Π +0Z3,0Z2 +e += 1 +6 +� +id + σz +e +� +b +� +id + Σz +s(e)(Σz +t(e))† + (Σz +s(e))†Σz +t(e) +� +, +(5.19) +whereas the contribution of the generators of S3 to the vertex term φ�Av = 1 +6 +ř +g∈S3 +φ�Ag +v can be depicted +as +φ�As +v = +Σx +, +φ�Ar +v = +σx +σx +σx +σx +σx +σx +Γ +. +(5.20) +∼ 55 +∼ + +The two phases of this Hamiltonian can be interpreted in the same vein as for the other models. +Having described the models (5.12), (5.15) and (5.18), which are obtained by gauging the different +subgroups of S3, we detail below the symmetry structures corresponding to each of these (partially) +gauged models. +5.3 +2Rep(S3) symmetry +Let us study the symmetry structure of Hamiltonian (5.12) acting on a Hilbert space spanned by states +|g⟩ ∈  +e C[G], where g ∈ Z1(Σ△, G). Following the general discussions in sec. 3.2 and 3.8, we know +that Hamiltonian (5.12) must have a symmetry structure embodying the fusion 2-category 2Rep(S3) +of 2-representations of S3. In particular, this means that (5.12) hosts topological surfaces associated +with simple objects in 2Rep(S3). Recall from sec. 3.6 that simple objects in 2Rep(S3) are provided by +indecomposable VecS3-module categories N(B, ψ), which are conveniently labelled by tuples (B, ψ) +consisting of a subgroup B ⊆ S3 and a 2-cocycle ψ in H2(B, U(1)). Since H2(B, U(1)) is trivial for +any B ⊆ S3, we count four simple objects in 2Rep(S3) associated with each subgroup of S3, namely +N(S3, 1) ∼= Vec, N(Z3, 1) ∼= VecZ2, N(Z2, 1) ∼= VecS3/Z2 and N(Z1, 1) ∼= VecS3. We provided in (3.72) +a general formula to construct the corresponding surface operators, but let us unpack it further here +in the spirit of sec. 2. +Generally speaking, defining topological surfaces associated with simple objects in 2Rep(S3) re- +quires introducing virtual degrees of freedom n[v] at every vertex v ⊂ Σ△, whose configuration space +is given by the set of isomorphism classes of simple objects in the corresponding VecS3-module cat- +egory. Concretely, given the topological surface associated with N(B, 1), virtual degrees of freedom +are valued in the collection G/B of left cosets. These virtual degrees of freedom are then coupled to +the physical degrees of freedom via the conditions spelt out below (3.72) involving the VecS3-module +structure of N(B, 1), which we recall is given by the natural action of S3 on the left cosets. +Suppose for instance that the subgroup B is the whole group S3. There is a single left coset in +S3/S3 ≃ Z1, namely S3 itself, on which S3 acts invariantly. It follows that the resulting operator acts +as the identity. We denote it by Utriv. in accordance with sec. 2 and 4: +Utriv. = +ź +e +ide . +(5.21) +This topological surface, which is associated with the VecS3-module category Vec, is the only invertible +one for this model. +Let us now consider non-invertible topological surfaces. We first focus on that associated with +the VecS3-module category N(Z3, 1) ∼= VecZ2. By definition, simple objects in N(Z3, 1) are left cosets +in S3/Z3 ≃ +� +{1, s, s2} , {r, sr, s2r} +� +≃ Z2. The corresponding surface operator amounts to summing +over configurations n ∈ C0(Σ△, Z2) of virtual degrees of freedom, which are coupled to the physical +degrees of freedom by imposing conditions n[v1] = Cg[v1v2] ▷ n[v2] = ϖZ2(g[v1v2]) + n[v2] at every +edge (v1v2) ⊂ Σ△. These conditions can be enforced explicitly by introducing Lagrange multiplier +b ∈ C1(Σ△, Z2) as follows: +UZ2[Σ△] = +1 +2#(Σ△) +ÿ +g,n,b +exp +� +πi +� +Σ△ +b⌣ +� +dn − ϖZ2(g) +�� +|g⟩⟨g| , += +1 +2χ(Σ△) +ÿ +g,n +ź +(v1v2)⊂Σ△ +δCg[v1v2]▷n[v2],n[v1] |g⟩⟨g| , +(5.22) +∼ 56 +∼ + +where (dn)[v1v2] = n[v1] − n[v2]. +The topological surface associated with the VecS3-module category N(Z2, 1) ∼= VecS3/Z2 is con- +structed similarly. By definition, simple objects in N(Z2, 1) are provided by left cosets in S3/Z2 ≃ +� +{1, r}, {r, sr}, {s2, s2r} +� +. The non-normal subgroup Z2 ⊂ S3 acts trivially on S3/Z3, while the re- +maining elements permute the elements in S3/Z2. The corresponding surface operator amounts to sum- +ming over configurations n ∈ C0(Σ△, Z3) of virtual degrees of freedom, which are coupled to the physi- +cal degrees of freedom by imposing conditions n[v1] = Cg[v1v2]▷n[v2] = ϖZ3(g[v1v2])+φϖZ2(g[v1v2])(n[v2]) +at every edge (v1v2) ⊂ Σ△, where we are using that S3/Z2 ≃ Z3 as a set. These conditions can be +enforced explicitly by introducing Lagrange multiplier b ∈ C1(Σ△, Z3) as follows: +US3/Z2[Σ△] = +1 +3#(Σ△) +ÿ +g,n,b +exp +�2πi +3 +� +Σ△ +b⌣ +� +dgn − ϖZ3(g) +�� +|g⟩⟨g| , += +1 +3χ(Σ△) +ÿ +g,n +ź +(v1v2)⊂Σ△ +δCg[v1v2]▷n[v2],n[v1] |g⟩⟨g| , +(5.23) +where we have used the twisted differential +(dgn)[v1v2] := n[v1] − φϖZ2(g[v1v2])(n[v2]) . +(5.24) +Finally, the remaining topological surface associated with the VecS3-module category N(Z1, 1) ∼= VecS3 +can be simply expressed as +US3[Σ△] = +1 +|S3|χ(Σ△) +ÿ +g,n +ź +(v1v2)⊂Σ△ +δg[v1v2]n[v2],n[v1] |g⟩⟨g| . +(5.25) +Let us now briefly confirm that all these surface operators do commute with (5.12), and are thus +part of the symmetry structure of the model. Firstly, edge terms Π +1S3 +e +straightforwardly commute +with all the topological surfaces since all the operators are diagonal in the chosen computational +basis. Vertex operators Av perform averagings over the group of gauge transformations. An arbitrary +combination of gauge transformations indexed by an assignment x ∈ C0(Σ△, S3) acts as |g⟩ �→ |xg⟩, +where xg[e] = x[s(e)] · g[e] · x[t(e)]−1. Commutation with the surface operators is simply obtained by +absorbing the gauge transformation of g into a redefinition of n, which is summed over. +To summarise, we have thus far obtained symmetry surface operators associated with each simple of +object of 2Rep(S3). Let us now compute the corresponding fusion rules. We now by construction that +these must correspond to the monoidal structure of 2Rep(S3). Briefly, fusion rules follow from the fact +that acting consecutively with two surface operators amounts to taking a Cartesian product of the local +configuration space assigned to each vertex. This Cartesian product can be subsequently decomposed +into disjoint unions of isomorphism classes of S3-sets according to the Burnside ring multiplication +rule [Gre10]. Concretely, let N(B, 1) and N(B′, 1) be two indecomposable VecS3-module categories, +the fusion of the corresponding topological surfaces read +� +UG/B d UG/B′� +[Σ△] = +���� +B × B′ +G × G +���� +χ(Σ△) +ÿ +g,g′,n,n′ +ź +(v1v2)⊂Σ△ +δCg[v1v2]▷n[v2],n[v1] δCg′[v1v2]▷n′[v2],n′[v1] |g⟩⟨g|g′⟩⟨g′| += +���� +B × B′ +G × G +���� +χ(Σ△) ÿ +g,n,n′ +ź +(v1v2)⊂Σ△ +δCg[v1v2]▷(n,n′)[v2],(n,n′)[v1] |g⟩⟨g| , +(5.26) +∼ 57 +∼ + +which is a topological surface with virtual degrees of freedom valued in the Cartesian product S3/B × +S3/B′ that is acted upon diagonally by S3. Decomposing this Cartesian product into a disjoint union +of S3-sets then yields the decomposition of the topological surface into simple ones. For instance, when +taking the fusion of topological surfaces US3/Z2 d US3/Z2, one has +S3/Z2 × S3/Z2 ≃ +� +{1, r}, {s, sr}, {s2, s2r} +� +× +� +{1, r}, {s, sr}, {s2, s2r} +� +≃ +�� +{1, r}, {1, r} +� +, +� +{s, sr}, {s, sr} +� +, +� +{s2, s2r}, {s2, s2r} +�� +⊔ +�� +{1, r}, {s, sr} +� +, +� +{1, r}, {s2, s2r} +� +, +� +{s, sr}, {1, r} +� +, +� +{s, sr}, {s2, s2r} +� +, +� +{s2, s2r}, {1, r} +� +, +� +{s2, s2r}, {s, sr} +�� +, +(5.27) +which decomposes into a three dimensional (diagonal) orbit and a six-dimensional off-diagonal orbit +under S3. Then it follows from (5.26) that +� +US3/Z2 d US3/Z2� +[Σ△] = +1 +3χ(Σ△) · US3/Z2[Σ△] ‘ +�2 +3 +�χ(Σ△) +· US3[Σ△] . +(5.28) +Similarly, the fusion rules for all the remaining operators on a general (path-connected) Σ read +� +US3 d US3� +[Σ△] = 61−χ(Σ△) · US3[Σ△] , +� +US3/Z2 d US3� +[Σ△] = 31−χ(Σ△) · US3[Σ△] , +� +UZ2 d UZ2� +[Σ△] = 21−χ(Σ△) · UZ2[Σ△] , +� +US3/Z2 d UZ2� +[Σ△] = US3[Σ△] , +� +UZ2 d US3� +[Σ△] = 21−χ(Σ△) · US3[Σ△] . +(5.29) +Note that p1−χ(Σ△) is the partition function for the Zp gauge theory on a general path connected +manifold Σ with Σ△. Choosing Σ to be a two-torus, we recover the monoidal structure of 2Rep(S3) +[Gre10]. +Let us finally describe the topological lines living on the various surface operators constructed above. +We established in sec. 3.7 that give a surface operator associated with a VecS3-module category +N(B, 1), we can insert topological lines that form the Morita dual fusion 1-category (VecS3)⋆ +N (B,1) = +FunVecS3 (N(B, 1), N(B, 1)). Concretely, we have +(VecS3)⋆ +Vec ∼= Rep(S3) , +(VecS3)⋆ +VecZ2 ∼= VecS3 , +(VecS3)⋆ +VecS3/Z2 ∼= Rep(S3) , +(VecS3)⋆ +VecS3 ∼= VecS3 . +(5.30) +Generally speaking, the topological line associated with a simple object of such a Morita dual fusion +1-category can be realised on the lattice in terms of matrix product operators whose building blocks +evaluate to the matrix elements of the module structure of the corresponding VecS3-module functors +[LDOV21, Del21]. In particular, we have FunVecS3 (Vec, Vec) ∼= Rep(S3), which indicates topological +lines of the identity surface operator are labelled by irreducible representations of S3 and amount to +ordinary Wilson lines. Recall that Rep(S3) has three simple objects, namely the trivial 0, the sign 1 and +the standard representation 2. We provided in eq. (3.90) the corresponding lattice operators. These +implement a non-invertible 1-form symmetry of the model. Indeed, commutation with edge terms Π +1S3 +e +follows from the fact that both operators act diagonally on basis states |g⟩, while commutation with +vertex terms Av amounts to the gauge invariance of Wilson lines. Finally, it follows straightforwardly +from (3.90) that composition of lines amounts to the monoidal structure of Rep(S3) with 0 the unit +and 1 b 1 ≃ 0, 1 b 2 ≃ 2 and 2 b 2 ≃ 0 ‘ 1 ‘ 2. +∼ 58 +∼ + +Finally, we mentioned in sec. 3.5 that we could recover the topological surfaces considered above as +condensation defects obtained by condensing suitable algebras of topological lines in Rep(S3). Specif- +ically, topological surfaces in 2Rep(S3) can be identified with (separable) algebra objects in Rep(S3). +We further commented in sec. 3.5 that these algebra objects are given by S3-algebras, i.e., associative +unital algebras equipped with an S3-action. Since none of the subgroups of S3 has a non-trivial second +cohomology group, these admit a simple definition: Given a subgroup B ⊆ S3, the corresponding S3- +algebra is provided by the permutation representation C[G/B] with pointwise multiplication. Besides, +we have C[S3/S3] ≃ 0, C[S3/Z3] ≃ 0 ‘ 1, C[S3/Z2] ≃ 0 ‘ 2 and C[S3] ≃ 0 ‘ 1 ‘ 2. Concretely, this +means for instance that we can reconstruct the topological surface US3/Z2 by inserting a network of +topological lines labelled by 0 ‘ 2 ∈ Rep(S3). +5.4 +2VecG symmetry +We now turn to the symmetry structure of Hamiltonian (5.15) acting on a Hilbert space spanned by +states |l, m⟩ ∈  +e C[Z3]  +v C[Z2], where l ∈ Z1(Σ△, Z3) and m ∈ C0(Σ△, Z2). Following the general +discussions in sec. 3.2 and 3.8, we know that Hamiltonian (5.12) must have a symmetry structure +embodying the fusion 2-category 2VecG of 2-vector spaces graded by the 2-group G with homotopy +groups Z2 and Z3 in degree one and two, respectively, as defined in sec. 3.6. In particular, Hamiltonian +(5.15) must commute with topological surfaces labelled by Z2-graded vector spaces of the form Vq, +where q ∈ Z2 and Vq has the structure of a VecZ3-module category. We thus count four topological +surfaces identified with Vec0, Vec1, (VecZ3)0 and (VecZ3)1. Firstly, as always, there is the identity +operator +Utriv. = +ź +e +ide , +(5.31) +which is here identified with Vec0. Next, there is a non-invertible surface operator defined as +UZ3[Σ△] = +1 +3#(Σ△) +ÿ +l,m,n,b +exp +�2πi +3 +� +Σ△ +b⌣(dn − l) +� +|l, m⟩⟨l, m| , +(5.32) +where b ∈ Z1(Σ△, Z3) and n ∈ C0(Σ△, Z2). This is the surface operator identified with (VecZ2)0. +Summing over n, one obtains the presentation of this operator as a condensation defect of Rep(Z3) +lines. The third operator, which is identified with Vec1, implements the 0-form Z2 symmetry that +is left over from the initial S3 0-form symmetry after gauging of the Z3 sub-symmetry. +This Z2 +operator is somewhat unconventional owing to the fact that Z2 acted non-trivially on Z3 via the outer +automorphism φ in S3. Concretely, the 0-form Z2 symmetry operator is +O1 = +ź +v +σx +v +ź +e +Γe . +(5.33) +Commutation with Hamiltonian (5.15), and more specifically with the vertex terms, then follows from +eq. (5.10). This operator was obtained by applying the general recipe of sec. 3.5: Recall that 2VecG +arises as the Morita dual (2VecS3)⋆ +2VecZ2 of 2VecS3 with respect to 2VecZ2. In this context, the operator +O1 is associated with the module 2-endofunctor − d Vec1. As such, it acts on degrees of freedom at +vertices valued in the set of isomorphism classes of simple objects of 2VecZ2 by mulitplication by +the non-trivial element in Z2 (hence σx +v ), whereas the action on edge degrees of freedom simply +follows from the identification of the effective degrees of freedom according to the formula ag,m[v1v2] = +φm[v1]−1� +ϖL(g[v1v2]) +� +. The fourth and final operator denoted by UZ3,1[Σ△], which is identified with +the simple object (VecZ2)1, can be simply defined as the fusion (O1 d UZ2)[Σ△]. The fusion rules of +∼ 59 +∼ + +these various topological surfaces can be computed using the explicit formulas above following methods +employed for the other examples. We write below the non-trivial fusion rules: +� +UZ3 d UZ3� +[Σ△] = 31−χ(Σ△) · UZ3[Σ△] , +� +UZ3 d O1� +[Σ△] = UZ3,r[Σ△] , +� +UZ3 d UZ3,r� +[Σ△] = 31−χ(Σ△) · UZ3[Σ△] , +� +O1 d UZ3,r� +[Σ△] = UZ3[Σ△] , +� +UZ3,r d UZ3,r� +[Σ△] = 31−χ(Σ△) · UZ3[Σ△] , +Or d O1 = Utriv. , +(5.34) +which matches the monoidal structure of 2VecG as defined in sec. 3.6. +Let us now analyse the topological lines living on the four topological surfaces described above. We +know that these are labelled by simple objects in the endo-categories of 2VecG, and we explained in +sec. 3.6 that these amount to Z2-grading preserving VecZ3-module functors. In particular, this means +that the operator O1 must act on the whole space and cannot have support on an (open) sub-region +of Σ△, while the topological lines on Utriv. are labelled by simple objects (VecZ3)⋆ +Vec ∼= Rep(Z3). +The characterisation in terms of module functors also indicates that topological lines living on UZ3 +and UZ3,1 are labelled by simple objects in (VecZ3)⋆ +VecZ3 +∼= VecZ3 and can be constructed mimicking +previous constructions. +Let us focus on the topological (Wilson) lines of Utriv.. Given a 1-cycle ℓ on Σ△ and an irreducible +representation ρ ∈ Rep(Z3), we may define such a line operator as +ÿ +l,m +� ź +e⊂ℓ +ρ(l[e]) +� +|l, m⟩⟨l, m| . +(5.35) +For instance, given the non-trivial irreducible representation such that ρ(s) = ω, this operator can +be equivalently defined as ś +e⊂ℓ Σz +e. It is then straightforward to confirm that these commute with +(5.15). Furthermore, composition of these lines is provided by the monoidal structure of Rep(Z3). +Interestingly, the 0-form operator O1 acts non-trivially on these topological lines via the action of Z2 +on Z∨ +3 , mapping a topological line labelled by ρ to one labelled by the dual representation ρ⋆. Indeed, +due to the presence of the Γ matrices in eq. (5.33), we have for instance +O1� ź +e⊂ℓ +Σz +e +� += +� ź +e⊂ℓ +Σz +e +†� +O1 , +(5.36) +and vice versa. We explained this feature below eq. (3.61) in terms of the monoidal structure of 2VecG. +Similarly, the operator O1 acts non-trivially on the topological lines living on the topological surfaces +UZ3 and UZ3,1, which can for instance be traced back to the fact that these surfaces results from +condensing topological lines in Rep(Z3). This concludes our analysis of the symmetry structure of +Hamiltonian (5.15) as encoded into 2VecG. +5.5 +2Rep(G)-symmetry +We finally describe the symmetry structure of Hamiltonian (5.18), obtained by gauging the non- +normal Z2 subgroup of S3 in the transverse field S3-Ising model (5.5). This model acts on a Hilbert +space spanned by states |q, m⟩ ∈  +e C[Z2]  +v C[Z3], where q ∈ Z1(Σ△, Z2) and m ∈ C0(Σ△, Z3). +Following the general discussions in sec. 3.2 and 3.8, we know that Hamiltonian (5.12) must have a +symmetry structure embodying the fusion 2-category 2Rep(G) of 2-representations of the same 2-group +G considered above. Invoking the results of sec. 3.7, Hamiltonian (5.18) must commute in particular +with topological surfaces labelled by tuples (V, {l(N)}N∈V) consisting of an indecomposable VecZ2- +module category V and a collection {l(N)}N∈V of group elements in Z3 for each simple object in V such +∼ 60 +∼ + +that l(Cq ▷ N) = φq−1(l(N)) for every q ∈ Z2. Concretely, we count three simple topological surfaces +identified with tuples (Vec, {0}), (VecZ2, (0, 0)) and (VecZ2, (1, −1)).24 Note that by exchanging the +roles of 1 and 2 in Z3, we find a simple object (VecZ2, (−1, 1)) that is equivalent to (VecZ2, (1, −1)). +As usual, we begin with the identity operator +Utriv. = +ź +e +ide , +(5.37) +which is now identified with (Vec, {0}). Next, there is a non-invertible surface operator defined as +UZ2,0[Σ△] = +1 +2#(Σ△) +ÿ +q,m,b,n +� +iπ +� +Σ△ +b⌣(dn + q) +� +|q, m⟩⟨q, m| , +(5.38) +where n ∈ C0(Σ△, Z2) and b ∈ C1(Σ△, Z2). This is the surface operator identified with (VecZ2, (0, 0)), +which is an ordinary condensation defect as we described for the transverse field Z2-Ising model in +sec. 2. The third and fourth operators, which are identified with (VecZ2, (1, −1)) and (VecZ2, (−1, 1)), +respectively, implement the 0-form Z3 symmetry that is left over from the initial S3 0-form symmetry +after gauging the Z2 sub-symmetry. Conventionally, a Z3 0-form symmetry generator would take the +form ś +v Σx +v , but this operator clearly does not commute vertex terms depicted eq. (5.20) due to the +presence of the Γ matrix. Instead, one may define the following topological surface operators +UZ2,±1[Σ△] = +1 +2#(Σ△) +ÿ +q,m,b,n +exp +� +iπ +� +Σ△ +b⌣(dn + q) +� +|q, m ± φn(1)⟩⟨q, m| , +(5.39) +where φn(1)[v] := φn[v](1) for all v ⊂ Σ△, which is identified with (VecZ2, (±1, ∓1)). In contrast, +the conventional 0-form Z3 operators would be given by O±1 = ř +q,m |q, m ± 1⟩⟨q, m|. It follows that +UZ2,±1 locally acts like O±1 or O∓1 depending on the configuration n of virtual degrees of freedom. +More precisely, UZ2,1 is a Z2 condensation defect that additionally acts as Σx +v at a vertex v ⊂ Σ△ if +n[v] = 0 and Σx +v +† if n[v] = 1. Commutation of this surface operator with Hamiltonian (5.18) can be +demonstrated as follows: The only non-trivial commutation to check is that with vertex terms φ�Ar +v as +depicted in eq. (5.20). This operator acts on the computational basis as +φ�Ar +v : |q, m⟩ �→ |q + dxv, φxv(m)⟩ , +(5.40) +where xv ∈ C0(Σ△, Z2) is trivial everywhere except at the vertex v. Let us now separately evaluate +φ�Ar +v d UZ2,±1[Σ△] and UZ2,±1[Σ△] d φ�Ar +v. On the one hand, we have +UZ2,±1[Σ△] d φ�Ar +v = +1 +2#(Σ△) +ÿ +q,m,b,n +q′,m′ +(−1) +� +Σ△b⌣(dn+q)|q, m ± φn(1)⟩⟨q, m|q′ + dxv, φxv(m′)⟩⟨q′, m′| += +1 +2#(Σ△) +ÿ +q,m,b,n +(−1) +� +Σ△b⌣(dn+q+dxv)|q + dxv, φxv(m) ± φn(1)⟩⟨q, m| += +1 +2#(Σ△) +ÿ +q,m,b,n +(−1) +� +Σ△b⌣(dn+q)|q + dxv, φxv(m) ± φn−xv(1)⟩⟨q, m| , +(5.41) +24Recall that we write group elements in Z3 as {0, 1, 2} so that the mutliplication is given by the addition modulo 3. +∼ 61 +∼ + +where in the last line we performed the change of variable n �→ n − xv. On the other hand, we have +φ�Ar +v d UZ2,±1[Σ△] = +1 +2#(Σ△) +ÿ +q,m,b,n +q′,m′ +(−1) +� +Σ△b⌣(dn+q)|q′ + dxv, φxv(m′)⟩⟨q′, m′|q, m ± φn(1)⟩⟨q, m| += +1 +2#(Σ△) +ÿ +q,m,b,n +(−1) +� +Σ△b⌣(dn+q)|q + dxv, φxv(m) ± φxvφn(1)⟩⟨q, m| . +(5.42) +Since φn−xv(1) = φxvφn(1), (5.41) equals (5.42), establishing that UZ2,±1 is indeed a symmetry operator +of the model (5.18). +Guided by the presentation of 2Rep(G) provided in sec. 3.6, topological lines living on the surfaces +described above can be constructed mimicking previous examples. Let us rather focus on the fusion +rules of these surface operators. The surface operator UZ2,0 being identical to that encountered in +the study of the transverse-field Z2-Ising model, we already know that it satisfies fusion rules (2.20). +However, here we present an alternative derivation, which is more suitable to compute fusion rules of +the remaining operators: +� +UZ2,0 d UZ2,0� +[Σ△] = +1 +22#(Σ△) +ÿ +q1,2,m1,2 +n1,2,b1,2 +(−1) +� +Σ△δI,JbI⌣(dnJ+qJ)|q1, m1⟩⟨q1, m1|q2, m2⟩⟨q2, m2| += +1 +22#(Σ△) +ÿ +q,m +b1,2,n1,2 +(−1) +� +Σ△b1⌣(dn1+q)+b2⌣(dn2+q)|q, m⟩⟨q, m| += +1 +22χ(Σ△) +ÿ +q,m,n1,2 +δdn1,qδdn2,q|q, m⟩⟨q, m| . +(5.43) +At this point, notice that the configuration space of virtual degrees of freedom at each vertex is given +by the Cartesian product Z2 × Z2, which can be decomposed into the disjoint union of two Z2-sets, +namely Z(1) +2 +≡ {(0, 0), (1, 1)} and Z(2) +2 +≡ {(0, 1), (1, 0)}. Therefore, we can decompose the summation +over n1,2 into +ÿ +n1,2 += +ÿ +n∈C0(Σ△,Z(1) +2 +) +‘ +ÿ +n′∈C0(Σ△,Z(2) +2 +) +. +(5.44) +Using (5.44) in (5.43), we immediately obtain +� +UZ2,0 d UZ2,0� +[Σ△] = +1 +2χ(Σ△) · UZ2 +0 [Σ△] ‘ +1 +2χ(Σ△) · UZ2 +0 [Σ△] = 21−χ(Σ△) · UZ2 +0 [Σ△] . +(5.45) +Note that the prefactor 21−χ(Σ△) is precisely the partition function Z2d[Σ△] of the pure Z2 gauge +theory for a path-connected manifold Σ. Whenever Σ is a torus so that χ(Σ△) = 0, the fusion rules +reproduce the monoidal product of VecZ2-module categories, namely +VecZ2 d VecZ2 ∼= VecZ2 ‘ VecZ2 , +(5.46) +where the first copy of VecZ2 on the r.h.s. has simple objects C0 bC0 and C1 bC1, whereas the second +copy has simple objects C0 b C1 and C1 b C0. +Guided by the derivation above, we can compute the fusion of two surface operators UZ2,l1 and +UZ2,l2 with l1, l2 ∈ {0, ±1}: +∼ 62 +∼ + +� +UZ2,l1 d UZ2,l2� +[Σ△] = +1 +22χ(Σ△) +ÿ +q,m,n1,2 +δdn1,qδdn2,q|q, m + φn1(l1) + φn2(l2)⟩⟨q, m| += +1 +22χ(Σ△) +ÿ +q,m +n∈C0(Σ△,Z(1) +2 +) +δdn,q|q, m + φn1(l1) + φn2(l2)⟩⟨q, m| +‘ +1 +22χ(Σ△) +ÿ +q,m +n′∈C0(Σ△,Z(2) +2 +) +δdn′,q1|q, m + φn′ +1(l1) + φn′ +2(l2)⟩⟨q, m| . +(5.47) +Let us describe various choices of l1, l2 in some detail: First of all, choosing l1 = l2 = 0, we recover +the fusion rules (5.45). Similarly, choosing l2 = 0, we find +� +UZ2,l d UZ2,0� +[Σ△] = +1 +2χ(Σ△) · +� +UZ2,l[Σ△] ‘ UZ2,l[Σ△] +� +. +(5.48) +Let us now suppose that l1 = l2 = 1. Given n ∈ C0(Σ△, Z(1) +2 ) so that n1 = n2, we find that the +first operator appearing in the decomposition of the fusion product is a Z2 condensation defect that +additionally acts as Σx +v +† = (Σx +v )2 at a vertex v ⊂ Σ△ if n[v] = (0, 0) and Σx +v = (Σx +v +†)2 if n[v] = (1, 1). +Conversely, given n′ ∈ C0(Σ△, Z(2) +2 ) so that n′ +1 ̸= n′ +2, we find that the second operator appearing in +the decomposition is a plain Z2 condensation defect since it acts as Σx +v Σx +v +† = id at a vertex v ⊂ Σ△ if +n′[v] = (0, 1) and Σx +v +†Σx +v = id if n′[v] = (1, 0). Putting everything together, we find +� +UZ2,1 d UZ2,1 +1 +� +[Σ△] = +1 +2χ(Σ△) · +� +UZ2,−1[Σ△] ‘ UZ2,0[Σ△] +� +. +(5.49) +More generally, fusion rules of arbitrary topological surfaces read +� +UZ2,l1 d UZ2,l2 +1 +� +[Σ△] = +1 +2χ(Σ△) · +� +UZ2,l1+l2[Σ△] ‘ UZ2,l1−l2[Σ△] +� +. +(5.50) +Choosing Σ to be the two-torus, we recover the fusion rules provided by the monoidal structure of +2Rep(G). These can be immediately inferred from the treatment of eq. (5.46) [Del22]. +SECTION 6 +Discussion +We conclude with a discussion of extensions and generalisations of the results presented in this +manuscript. +6.1 +Further examples +The focus of this manuscript was on developing a general framework for gauging invertible symmetries +of two-dimensional quantum models and studying the resulting higher categorical symmetries. In order +to illustrate our constructions, we considered finite group generalisations of the transverse-field Ising +model. It will be very interesting to employ the present formalism in order to tackle more challenging +as well as physically more relevant models. Indeed, using the framework presented in this manuscript, +the entire parameter space of symmetric Hamiltonians generated by the local operators described in +sec. 3.1 can be investigated. +∼ 63 +∼ + +Symmetries strongly constrain various aspects of the low-energy or infra-red phase diagrams of +symmetric quantum systems. For instance, the kinds of phases realised, the spectrum of excitations +within each phase, universality classes of phase transitions and dualities acting on the parameter +space of symmetric models, can all be studied from the lens of the symmetry structure. Therefore it +is natural to study the phase diagrams of the quantum spin models introduced in this work from the +perspective of their higher categorical symmetries. +It is also possible to extend the current framework so as to consider more general models as +well as more general higher categorical symmetries. For instance, given a group G ≃ Q ⋉φ L, our +framework readily accommodates models with 2Vecπ +G-symmetry where π is a (non-trivial) 4-cocycle +in H4(G, U(1)) characterising the monoidal pentagonator of the fusion 2-category.25 Such a monoidal +pentagonator encapsulates an anomaly revealing an obstruction to gauging the whole symmetry. For +certain choices of 4-cocycle π, gauging the L-sub-symmetry would result in a model with a 2VecG- +symmetry, where G is a 2-group with a non-trivial Postnikov class [Tho20], thereby going beyond the +examples considered in the current manuscript. +More generally, the framework employed in this manuscript can be extended so as to accommo- +date arbitrary fusion 2-categories generalising further the one-dimensional framework presented in +[LDOV21]. For instance, given an input fusion 2-category and a choice of module 2-category over it, +local operators can be defined of the form (3.17) evaluating to matrix entries of the corresponding +module pentagonator. In particular, it would be interesting to consider models built from the data of +fusion 2-categories obtained by idempotent completions of deloopings of braided fusion 1-categories +[DR18, GJF19]. The corresponding module 2-categories were discussed in ref. [D´e21]. +6.2 +Self-dual models +A celebrated result in low-dimensional condensed matter physics is the exact localisation of the critical +point of the (1+1)d transverse-field Ising model by invoking its self-duality [KW41]. The relevant +duality in this case is the Kramers-Wannier duality, which, up to a local unitary, is obtained by +gauging its Z2-symmetry. As we reviewed in this manuscript, this phenomenon is specific to (1+1)d +since the (2+1)d transverse-field Ising model that possesses an ordinary Z2-symmetry is dual to a +model that possesses in particular a 1-form Z∨ +2 -symmetry. This begs the question, how to construct +(non-trivially) self-dual symmetric spin systems in two spatial dimensions? +It turns out that the mathematical framework developed in this manuscript allows us to rule +out many possibilities. Indeed, a necessary condition for self-duality is that the fusion 2-categories +of symmetry operators associated with a model and its dual are monoidally equivalent, in addition +to being Morita equivalent. +This is the case of the (1+1)d Ising model, where the initial VecZ2- +symmetry is monoidally equivalent to the dual Rep(Z2)-symmetry. In contrast, 2VecZ2 and 2Rep(Z2) +are clearly not monoidally equivalent. As a matter of fact, starting from a G-symmetric theory, any +duality involving the gauging of a non-trivial subgroup of G would result in a model whose symmetry +is not monoidally equivalent to 2VecG, making it impossible for the model to be self-dual. Recent +computations of the Brauer-Picard group of 2Rep(G), which informs us about auto-equivalences of +the algebraic structure encoding the super-selection sectors of a G-symmetric model, suggests that +the only candidate dualities may be of the form 2Vec → 2Vecλ, which only involve a change of +module pentagonator amounting to the pasting of a (2+1)d symmetry-protected topological phase +[KLW+20b, D´e22]. This special type of duality, and the possibility of defining self-dual models with +respect to it, will be investigated elsewhere. +25Within our framework, this is simply accomplished by choosing 2Vecπ +G as a module 2-category over itself when +defining the local operators (3.17), so that the module pentagonator coincides with the monoidal pentagonator. +∼ 64 +∼ + +6.3 +Symmetry-twisted boundary conditions +Throughout this manuscript, we have purposefully been somewhat vague regarding the role of bound- +ary conditions. For instance, given a two-dimensional surface with the topology of a torus, our results +would implicitly assume periodic boundary conditions. But our framework can be extended so as +to accommodate symmetry-twisted boundary conditions. In the case of a G-symmetric model, we +expect symmetry-twisted boundary conditions along the pair of non-contractible cycles of the torus +to be labelled by commuting group elements in G. Commutativity in G should be required so as to +preserve translation invariance of the model up to local unitary transformations. Importantly, the +original G-symmetry interacts with these boundary conditions in such a way that one is typically left +with a smaller symmetry in the presence of non-trivial symmetry twists. Concretely, given symmetry +twists (g1, g2) ∈ G2 such that g1g2 = g2g1, the leftover symmetry group is given by the stabiliser +subgroup of group elements x ∈ G satisfying (xg1x−1, xg2x−1) = (g1, g2). It follows in particular +that every pair of commuting twists (g1, g2) and (g′ +1, g′ +2) for which there exists x ∈ G such that +(g′ +1, g′ +2) = (xg1x−1, xg2x−1) possess the same stabiliser subgroup, thereby defining equivalence classes +of boundary conditions. Given such an equivalence class and one of its representatives, the resulting +Hamiltonian would decompose into symmetry charge sectors labelled by irreducible representations of +the stabiliser subgroup of the representative. +Interestingly, the same data labelling the super-selection sectors described above, i.e. symmetry- +twisted boundary conditions together with twisted symmetry charge sectors, appeared before in a +different context. Indeed, these correspond to the simple modules of tube algebras, which were first +considered in ref. [Del17] and generalised in ref. [BD19a, BD19b], that classify and characterise loop-like +excitations in (3+1)d Hamiltonian realisations of Dijkgraaf-Witten theory [DW90]. More specifically, +such a simple module labels a loop-like flux, to which a point-like charge may be attached, while being +threaded by an auxiliary string-like flux. A complimentary field-theoretic approach was developed in +ref. [CTR15, TCR16, CTNR17, TCSR17] to extract topological line and surface operators and the +braiding phases of (3+1)d topological finite group gauge theories from their gapless surface theories. +A similar interplay between super-selection sectors of symmetric models and topological excita- +tions of a higher-dimensional topological model exist in (1+1)d. Indeed, super-selection sectors of a +symmetric model are known to be labelled by simple objects in the monoidal centre of the symmetry +fusion 1-category [AFM20, KZ22, MMT22, LOST22, LDV22], and the interplay between dualities +and super-selection sectors was recently studied in detail for arbitrary one-dimensional quantum lat- +tice models in ref. [MMT22, LDV22]. But the monoidal center of a fusion 1-category encodes the +anyonic excitations of the corresponding Hamiltonian realisation of the Turaev-Viro-Barrett-Westbury +state-sum invariant [TV92, BW93, LW05]. Interestingly, the notion of monoidal centre of a fusion +2-category also exists [BN96, Cra98] and was computed explicitly in a few cases [KTZ19]. Perhaps +surprisingly, given a topological model with input datum a certain (spherical) fusion 2-category, the +excitation content encoded into its monoidal centre differs from that described by the tube algebras +mentioned above [BD20]. The precise relation between both algebraic structures together with the +corresponding physical implications were clarified in ref. [BD21]. Concretely, this means that in con- +trast to the one-dimensional scenario, super-selection sectors of a G-symmetric model on a torus are +not labelled by simple objects in the monoidal centre of 2VecG. In light of the results obtained in +ref. [BD20, BD21], we rather conjecture that the monoidal centre in the higher dimensional case would +encode super-selection sectors of a model defined on a cylinder hosting a combination of open and +closed boundary conditions. +∼ 65 +∼ + +Acknowledgements +CD would like to thank Thibault D´ecoppet for numerous discussions on Morita equivalence of fusion +2-categories, as well as Laurens Lootens, Frank Verstraete and Gerardo Ortiz for collaborations on the +lower-dimensional setting. AT would like to thank Lakshya Bhardwaj, Lea Bottini, Heidar Moradi, +Faroogh Moosavian and Sakura Sch¨afer Nameki for numerous discussions on related topics. This work +has received funding from the Research Foundation Flanders (FWO) through postdoctoral fellowship +No. 1228522N awarded to CD. AT is supported by the Swedish Research Council (VR) through grants +number 2019-04736 and 2020-00214. +APPENDIX A +Two-dimensional Zp gauge theory +In this appendix, we collect some basic facts about different presentations of the two-dimensional Zp +topological gauge theory, which appear in the definition of condensation defects. In particular, the +case p = 2 appears throughout the manuscript. Given a closed oriented surface Σ endowed with a +triangulation Σ△, the partition function of the theory reads26 +Z(p) +2d [Σ△] = +1 +p#(Σ△) +ÿ +b,n +exp +�2πi +p +� +Σ△ +b⌣dn +� +, +(A.1) +where #(Σ△) := |Σ0 +△| + |Σ2 +△| and |Σj +△| is the number of j-simplices in the triangulation Σ△. In the +above partition sum, b ∈ C1(Σ△, Zp) and n ∈ C0(Σ△, Zp). As in the main text, the symbols d and ⌣ +denote the simplicial codifferential and the cup product, respectively. +Let us begin by listing a couple of important identities that are used in several occurences in the +main text +ÿ +n +exp +�2πi +p +� +Σ△ +n⌣(db − q2) +� += p|Σ2 +△|δdb,q2 , +ÿ +b +exp +�2πi +p +� +Σ△ +b⌣(dn − q1) +� += p|Σ1 +△|δdn,q1 , +(A.2) +where qj ∈ Cj(Σ△, Zp). We can now evaluate the partition function by summing over n. First we +perform an integration by parts such that the codifferential acts on b. Then summing over n imposes +a Zp delta function on each 2-simplex, giving an overall factor of p|Σ2 +△|. Putting everything together, +one obtains +Z(p) +2d [Σ△] = +p|Σ2 +△| +p#(Σ△) +ÿ +b +δdb,0 = |Z1(Σ△, Zp)| +p|Σ0 +△| += |H1(Σ△, Zp)| × |B1(Σ△, Zp)| +p|Σ0 +△| += |H1(Σ△, Zp)| +|H0(Σ△, Zp)| = pb1(Σ)−b0(Σ) . +(A.3) +Notice that the sum over b gives a factor of |Z1(Σ△, Zp)| due to the Zp cocycle constraint. Moreover, +we used H1(Σ△, Zp) = Z1(Σ△, Zp)/B1(Σ△, Zp), where Z1(Σ△, Zp) and B1(Σ△, Zp) are the set of +1-cocycles and 1-coboundaries, respectively. Finally we employed the expression +B1(Σ△, Zp) ≃ C0(Σ△, Zp)/Z0(Σ△, Zp) ≃ C0(Σ△, Zp)/H0(Σ△, Zp) . +(A.4) +26In the main text, we denote the partition function of the two dimensional Z2 gauge theory Z2d[Σ△] := Z(2) +2d [Σ△]. +∼ 66 +∼ + +Notice that the final expression in eq. (A.3), which is a topological invariant, can equivalently be +expressed in terms of the 1st and 2nd Betti numbers b1,2(Σ) of Σ. +It is also instructive to compute (A.1) by summing over b instead of summing over n: +Z(p) +2d = +p|Σ1 +△| +p#(Σ△) +ÿ +n +δdn,0 = +1 +pχ(Σ) +ÿ +n +δdn,0 = |H0(Σ△, Zp)| +pχ(Σ) += pb1(Σ)−b2(Σ) = pb1(Σ)−b0(Σ) . +(A.5) +We first used eq. (A.2), as well as the expression +χ(Σ) := |Σ0 +△| − |Σ1 +△| + |Σ2 +△| = b0(Σ) − b1(Σ) + b2(Σ) , +(A.6) +defining the Euler characteristic χ(Σ) of the surface Σ. 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Phys. 121 +(1989) 351–399. +∼ 73 +∼ + diff --git a/DtAzT4oBgHgl3EQfT_x1/content/tmp_files/load_file.txt b/DtAzT4oBgHgl3EQfT_x1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..20da0b17debf7b4a0a4249111694341cb628e91e --- /dev/null +++ b/DtAzT4oBgHgl3EQfT_x1/content/tmp_files/load_file.txt @@ -0,0 +1,2807 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf,len=2806 +page_content='Higher categorical symmetries and gauging in two-dimensional spin systems Clement Delcamp� and Apoorv Tiwari� �Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium �Department of Physics, KTH Royal Institute of Technology, Stockholm, 106 91 Sweden clement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='delcamp@ugent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='be, apoorvt@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='se We present a framework to systematically investigate higher categorical symmetries in two-dimensional spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Though exotic, such generalised symmetries have been shown to naturally arise as dual symmetries upon gauging invertible symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Our framework relies on an approach to dualities whereby dual quantum lattice models only differ in a choice of module 2-category over some input fusion 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given an arbitrary two-dimensional spin system with an ordinary symmetry, we explain how to perform the (twisted) gauging of any of its sub-symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We then demonstrate that the resulting model has a symmetry structure encoded into the Morita dual of the input fu- sion 2-category with respect to the corresponding module 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We exemplify this approach by specialising to certain finite group generalisations of the transverse-field Ising model, for which we explicitly define lattice symmetry operators organised into fusion 2-categories of higher representations of higher groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='01259v1 [hep-th] 3 Jan 2023 Contents 1 Introduction 2 2 Motivation: transverse-field Ising model 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1 Z2-symmetric Hamiltonian 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 Gauging the Z2 symmetry 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3 Symmetry operators 9 3 Gauging and dual symmetries 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1 G-symmetric Hamiltonians 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 Dual Hamiltonians 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3 Duality operators 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4 Duality as twisted gauging 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 Dual symmetries 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6 Higher representation theory 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7 Morita duals 36 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8 Gauging the transverse-field G-Ising model 40 4 Example: doubled transverse-field Ising model 45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1 Symmetric Hamiltonian and gauging 45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 2Rep(Z2 2) symmetry: invertible surface operators 47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3 2Rep(Z2 2) symmetry: non-invertible surface operators 49 5 Example: transverse-field S3-Ising model 52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1 Symmetric Hamiltonian 52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 Gauging sub-symmetries 54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3 2Rep(S3) symmetry 56 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4 2VecG symmetry 59 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 2Rep(G)-symmetry 60 6 Discussion 63 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1 Further examples 63 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 Self-dual models 64 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3 Symmetry-twisted boundary conditions 65 A Two-dimensional Zp gauge theory 66 ∼ 1 ∼ SECTION 1 Introduction Global symmetries have been playing a pivotal role in our understanding of quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Gener- ally speaking, the existence of a global symmetry in a quantum system helps organise the spectrum of states and operators into representations of the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In addition, symmetry typically imposes strong constraints on the kinds of phases a quantum system can or cannot realise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These ideas have led for instance to Landau’s classification scheme of phases of matter, and to organising principles for the particle content of the Standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Despite its long and illustrious history, symmetry and its manifestations in quantum theory is very much an evolving story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Conventionally, given a Hamiltonian model, symmetries are implemented by operators that act on all of space—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', one-codimensional operators in spacetime —commute with the Hamiltonian, and satisfy fusion rules representative of a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In contrast, the modern perspective on global symmetries in quantum systems identifies the topological invariance of a symmetry operator within correlation functions as its defining property [GKSW15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This perspective lends itself to numerous generalised notions of symmetry that have collectively come to be known as global categorical symmetries [FMT22], in reference to the mathematical objects encoding them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notably, these include symmetry structures whose topological operators may be supported on higher codimensional sub-manifolds and/or are not invertible, so that they do not obey fusion rules encoded into a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Relaxing the requirement that operators are one-codimensional has led to the concept of higher- form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Specifically, a p-form symmetry is defined with respect to topological operators with support on (p+1)-codimensional sub-manifolds and act by linking with p-dimensional operators [KT13, HLS18, DT19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These operators being invertible, fusion rules are still encoded into a group, but whenever p > 0, the corresponding group is necessarily abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Furthermore, it is possible to combine higher form symmetries of various degrees in a non-trivial way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The corresponding groups combine into categorifications of the notion of group known as higher groups [BL03], yielding the concept of higher group symmetries [KT13, CDI18, BCH18, DT18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Relaxing the requirement that operators are invertible has led to symmetries encoded into higher algebraic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For instance, given a (1+1)d system, non-invertible symmetries are encoded into fusion 1-categories [ENO10], and the corresponding operators typically cannot be written as tensor products of local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More generally, given a (d+1)-dimensional system, it is possible to have non-invertible symmetry operators of varying degrees, in which case the algebraic structure is expected to be a fusion d-category [KLW+20a, BBSNT22a, BSNT22]—a notion that remains partly elusive [DR18, GJF19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As it turns out, though somewhat exotic, non-invertible symmetries are not rare in one-dimensional quantum models and have long been studied in the context of rational Conformal Field Theories (CFTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' There, topological operators go by the name of Verlinde lines [Ver88, PZ00, BG04] and exist in any rational CFT defined by a diagonal modular invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, the fusion ring formed by the Verlinde lines corresponds to that of representations of the chiral vertex algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', the underlying algebra of the given CFT, and is generically not group-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A well-studied example is that of the diagonal Ising CFT, that hosts three Verlinde lines embodying the Ising fusion category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It includes in particular a non-invertible line known as the Kramers-Wannier duality defect [OA96a, OA96b, FFRS04, FFRS06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Guided in part by integrability, the sub-algebra of topological defects within rational CFTs was formalised for instance in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [FRS02, FRS04, BM09, CLS+18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Furthermore, it was already appreciated in this context that topological defects indeed embody a kind of symmetry structure within a quantum field theory (QFT), and thus it is sensible to consider notions of ’t Hooft anomalies and gauging thereof [FFRS09, Tac17, CLS+18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 2 ∼ Naturally, a prolific source of topological operators are topological quantum field theories (TQFTs) themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In fact, by definition, the entire spectrum of operators in a TQFT is topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, a large class of TQFTs host topological defects that obey non-invertible fusion rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The most well-studied examples are provided by line defects in (2+1)d TQFTs, either in the continuum [Wit89, RT90] or in the discrete [TV92, BW93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Topological surface operators associated with braided auto-equivalences of the quantum invariant assigned by the theory to the circle have also been studied in this context [Bom10, KK11, BBCW14, THF15], but they are typically invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' An important development was the remark that these surface operators in (2+1)d TQFTs could be constructed by condensing a suitable sub-algebras of topological line operators [CRS17], suggesting a mechanism to generate a broader family of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This process was later formalised as the condensation completion of the category of line operators [DR18, GJF19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The resulting (possibly) non-invertible condensation defects turn out to be rather ubiquitous in (2+1)d TQFTs [RSS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In spite of these various developments, examples of non-invertible symmetry operators in higher di- mensions have remained limited until recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the past year, various constructions of quantum sys- tems with non-invertible symmetries have appeared that employ different kinds of generalised gauging procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Generally speaking, it is understood that given a theory with an invertible symmetry, gauging one of its sub-symmetries typically yields a theory with a different symmetry structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Con- cretely, gauging a p-form symmetry in a (d+1)-dimensional theory yields a dual (gauged) model whose symmetry category contains (d−p−1)-dimensional topological operators labelled by irreducible rep- resentations of the corresponding group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Whenever the group is non-abelian, these operators are in particular non-invertible [Dri89, DPR91, dWP95, BT17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Moreover, gauging a (normal) sub-symmetry yields a theory possessing higher-group symmetries [Tac17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As it turns out, the symmetry structures resulting from these gauging procedures are even richer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' An early construction [KOZ21] of non-invertible defects in (3+1)d involved starting from a QFT with a 0-form and 1-form mixed anomaly and gauging the 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It was shown that this inevitably generates a non-invertible symmetry structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Another class of examples were inspired by generalising the construction of the Kramers-Wannier duality defect in the (1+1)d Ising CFT to (3+1)d self-dual QFTs [CCH+21, CCH+22, LRS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Yet another notable development pertained to the relation between gauging certain symmetry along sub-manifolds of spacetime and the condensation defects mentioned above [RSS22, LRS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Most relevant to the present work were a series of papers [Del21, BBSNT22a, BSNW22, BBFP22a, BBSNT22b, BSNT22, BBFP22b] that considered starting from a (3+1)d theory with an invertible symmetry structure encoded into a higher-group and gauging one of its sub-symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These works go beyond previous constructions in their analysis of the resulting symmetry structure in terms of so-called higher representations of higher groups [Elg04, GK06, BBFW08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, these types of non-invertible symmetries have been further discussed in the context of various typical quantum field theories such as free field theories [NRS22], pure gauge theories [KZZ22, AGR22], quantum electrodynamics [Kar22], axion models [CLS22c] and within other physical contexts [CO22, CLS22a, CLS22b, CHKO22, GEI22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notwithstanding the obvious recent interest in non-invertible symmetries, concrete lattice realisa- tions of the corresponding topological operators have been largely unexplored, with some exceptions [KNY21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' But, the lattice setting being concrete and tractable, it offers a welcome complimentary ap- proach to understanding the most subtle aspects of these generalised categorical symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Besides, it paves the way for exploring the implications of such symmetries on the phase diagram of familiar many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Furthermore, via the corresponding graphical calculs, the lattice setting is much closer related to the category theoretic framework underlying these symmetry structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 3 ∼ Our paper aims at further bridging the gap between the abstract concept of a generalised categorical symmetry, as encoded into a higher mathematical structure, and its concrete realisation on a quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More specifically, we wish to address the question, what does it mean to have symmetry oper- ators encoded into fusion 2-categories of higher categorical representations of higher groups?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Guided by the Morita theory of fusion 2-categories [Del21, D´ec22], we address this question by providing a framework that accomplishes—amongst other things—two tasks: Given an arbitrary two-dimensional spin system with an ordinary global symmetry, it allows for the systematic twisted gauging of one of its sub-symmetries, and the systematic identification of the resulting dual symmetry structure by constructing the corresponding topological lattice operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The framework we introduce in this manuscript is inspired by the study of dualities in one- dimensional quantum lattice models carried out in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [LDOV21, LDV22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Within our framework, a duality class of dualities is specified by an algebra of operators that is generated by a set of (ab- stract) local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A representative of a duality class is obtained by choosing a Hilbert space and correspondingly explicit matrix representations for the local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, the algebras of operators we consider take as input data a finite group G—or rather, a fusion 2-category 2VecG of G-graded 2-vector spaces—as well as a set of complex coefficients, which amounts to selecting certain linear combinations of local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These choices completely determine the physical properties of the duality class of models as encoded into their shared spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Choosing a matrix represen- tation then amounts to picking a so-called (indecomposable) module 2-category over 2VecG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' a 2-category with a G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We think of the module 2-category as providing the physical degrees of freedom—which may satisfy kinematical constraints—on which the local operators act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows that Hamiltonian models that only differ in a choice of module 2-category are dual to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the framework described above, a duality operator amounts to a map between two module 2-categories, which provide matrix representations of the same local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For consistencies, the action of this map is required to commute with the G-action resulting in the notion of module 2-functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, a symmetry operator amounts to a module 2-endofunctor between a module 2- category and itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More specifically, a module 2-endofunctor furnishes a topological surface operator that commutes with the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' There is also a notion of map between module 2-functors that are compatible with the G-action, namely module natural 2-transformations, which furnish topological lines at the interfaces of (possibly distinct) topological surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These data can be organised into a 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Crucially, given an indecomposable module 2-category M, the composition of module 2- endofunctors endows this 2-category with a fusion structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The resulting fusion 2-category (2VecG)⋆ M is referred to the Morita dual of 2VecG with respect to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This is the symmetry structure of the model obtained by choosing the Hilbert space associated with the module 2-category M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notice that we can make this statement without referring to a specific duality class of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As emphasised in (1+1)d in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [LDOV21], this is because dualities are only sensitive to symmetry structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note that it is always possible to choose 2VecG as a module 2-category over itself, in which case the symmetry fusion 2-category of the resulting model is again 2VecG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In other words, it is a model with an ordinary (0-form) G-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We can then show that choosing an alternative 2VecG-module 2-category has the interpretation of performing a twisted gauging of one of its sub-symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' One merit of our approach is our ability to provide lattice operators accompanying these ab- stract statements, allowing to explicitly perform a twisted gauging in an arbitrary G-symmetric two- dimensional spin system and prove that the resulting model does have the expected symmetry structure by constructing the corresponding topological lattice operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This ability extensively relies on the tensor network study of topological phases of matter where such symmetry operators first appeared in the form of matrix product operators in (1+1)d [cWB+14, LFH+20] and projected entangled pair ∼ 4 ∼ operators in (2+1)d [Del21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In addition to providing a systematic recipe for generating new dual models, this framework explicitly provides lattice operators embodying symmetry structures related to higher representations of groups and categorifications thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Furthermore, we are also able to construct duality lattice operators performing the transmutation of the local symmetric operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We can offer a different perspective on our approach to dualities: It is understood that a three- dimensional TQFT as provided by the Reshetikhin-Turaev construction [RT91] possesses a state-sum description if and only if it admits a non-trivial gapped boundary [FT20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These theories are of the Turaev-Viro-Barrett-Westbury type, whose input data are spherical fusion 1-categories [TV92, BW93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More specifically, given a choice of gapped boundary condition, which can be encoded into a module category over the input spherical fusion category, a state-sum can be obtained following the construction outlined for instance in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [BGK16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Distinct gapped boundary conditions yield distinct state-sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The corresponding state spaces are then spanned by topological tensor network states that were defined in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [LFH+20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the same vein, we can construct a family of state-sums of the same four-dimensional topological G gauge theory indexed by module 2-categories over 2VecG encoding various choices of gapped boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The corresponding state spaces are then spanned by the topological tensor network states defined in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Del21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Importantly, it is possible to define distinct state sums of the same theory in different regions of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The operators intertwining these distinct lattice realisations then precisely correspond to the duality operators transmuting local symmetric operators of a given Hamiltonian into local symmetric operators of one of its duals, as considered in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We illustrate our approach with finite group generalisations of the transverse-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For an arbitrary finite group, we consider the gauging of the whole invertible symmetry, revealing a dual symmetry structure in terms of 2-representations of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Supposing that the input group is a semi-direct product, we also consider the gauging of its two constitutive sub-symmetries in detail, revealing on the lattice dual symmetry structures in terms of 2-group-graded 2-vector spaces and 2- representations of 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Further specialising to the Klein four-group and the symmetric group of degree 3, we provide even more explicit expressions for the corresponding topological surfaces and topological lines in terms of spin operators, allowing us to confirm on the lattice their fusion and composition rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Organisation of the paper We begin in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2 with an in-depth analysis of the symmetry structure resulting from gauging the global symmetry of the two-dimensional transverse-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We emphasise in particular the appearance of non-invertible surface operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Guided by this example, we present in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3 a general framework to gauge invertible sub-symmetries of arbitrary two-dimensional quantum lattice models and construct the dual symmetry operators as encoded into the corresponding Morita dual fusion 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A few specific scenarios are discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, we exemplify our approach in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 4 and 5 by specialising to finite group generalisations of the transverse-field Ising model for the Klein group and the symmetric group of degree 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 5 ∼ SECTION 2 Motivation: transverse-field Ising model We set the stage by exploring the higher categorical symmetries that emerge from gauging the Z2 symmetry of the two-dimensional transverse-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1 Z2-symmetric Hamiltonian Let Σ be a closed oriented two-dimensional surface endowed with a (fixed) triangulation Σ△ whose vertices, edges and plaquettes are denoted by v, e and p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given an edge e ≡ (v1v2) oriented from v1 to v2, we denote by s(e) := v1 and t(e) := v2 its source and target vertices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We consider a variant of the well-known (2+1)d transverse-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As in the usual model, qubit degrees of freedom are assigned to vertices v ⊂ Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We identify such an assignment with a choice of 0-cochain m ∈ C0(Σ△, Z2) so the microscopic Hilbert space is provided by the tensor product  v C[Z2] ≃  v C2, where Z2 = ⟨r | r2 = 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Moreover, we denote by |m⟩ the state in the microscopic Hilbert space associated with 0-cochain m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Throughout this manuscript, we write basis elements of C[Z2] as |0⟩ and |1⟩, which are identified with the ‘up’ and ‘down’ state, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Qubit degrees of freedom are governed by the Hamiltonian H = −J ÿ e σz s(e)σz t(e) − Jκ ÿ v σx v − J˜κ ÿ v σx v ź (v v1v2) exp �iπ 4 (1 − σz v1σz v2) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1) where σx,z v is the usual shorthand for id b · · · b id b σx,z v b id b · · · b id, with the tensor product being over all the vertices of the triangulation, and σx,z v are Pauli operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The first term in the Hamiltonian describes a ferromagnetic interaction between qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The second term is the usual paramagnetic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The third term is a topologically twisted variant of the paramagnetic term that includes phase factors associated with triangles (v v1v2) containing the vertex v [LG12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' One can readily check that the model has a global (0-form) Z2 symmetry implemented by surface operators1 acting on all of Σ△ O0 = ź v idv and O1 = ź v σ1 v , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2) with Z2 fusion rules O1 d O1 = O0 = O0 d O0, O1 d O0 = O1 = O0 d O1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3) Correspondingly, in the three extreme limits 1 ≫ κ, ˜κ, κ ≫ 1, ˜κ and ˜κ ≫ 1, κ, one obtains fixed-point Hamiltonians with ferromagnetic, paramagnetic and symmetry-protected topological (SPT) ground states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This Hamiltonian does not have any non-trivial 1-form symmetry as topological lines on either surface operator must be the identity line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Furthermore, it is not possible for the surface operator O1 to be open, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' to have support on a sub-region of Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In other words, it is not possible to define a (topological) line interface between O0 and O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We describe below how, upon gauging of the Z2 0-form global symmetry, one inevitably lands on a dual model with more a intricate symmetry structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 1Throughout this manuscript, we refer to operators that act on extended two-dimensional regions of Σ as topological surface operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These could act on all of Σ or on a sub-region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 6 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 Gauging the Z2 symmetry Although this was not immediately appreciated when the construction first appeared [Kog79, Sav80], it is by now understood that gauging a 0-form Z2 symmetry yields a two-dimensional dual model hosting Z2 topological Wilson lines labelled by representations of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In modern terminology, this is the statement that the gauged model has a 1-form Z∨ 2 symmetry, with Z∨ 2 the Pontrjagin dual of Z2 [GKSW15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' However, it was recently pointed out that this is only part of the story [Del21, BSNW22, BBFP22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Indeed, the 1-form Z∨ 2 symmetry is only a component of the symmetry structure of the gauged model in the sense that it does not encapsulate all possible topological operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In order to grasp the above statements, let us explicitly gauge the global Z2 symmetry in model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' To do so, we begin by assigning additional qubit degrees of freedom to edges e ⊂ Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We identify such an assignment with a choice of 1-cochain g ∈ C1(Σ△, Z2) so the model is now defined on the extended microscopic Hilbert space provided by the tensor product  e C[Z2]  v C[Z2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now promote generator O1 of the global Z2 symmetry to a local gauge transformation by defining Gauß operators Gv := σx v ź e⊃v σx e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) Since Gauß operators obey the multiplication rule in Z2 and [Gv1, Gv2] = 0, for any v1, v2 ⊂ Σ△, they are the generators of a Z2 gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, consider a basis state |g, m⟩ in the extended microscopic Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By definition, we have σz v |g, m⟩ = (−1)m[v]|g, m⟩ , σz e |g, m⟩ = (−1)g[e]|g, m⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5) where m[v] and g[e] denote the restrictions of m and g to v and e, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' One can now define a general Gauß operator indexed by a 0-cochain x ∈ C0(Σ△, Z2) which acts as G(x) := ź v Gx[v] v : |g, m⟩ �→ |g + dx, m + x⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6) The gauge symmetry is imposed kinematically so that we only consider physical states in the +1 eigenspace of G(x) for any x ∈ C0(Σ△, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We then require the gauged Hamiltonian to commute with Gauß operator G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This can be accomplished by minimally coupling Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1) with the edge degrees of freedom: Hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' = −J ÿ e σz s(e)σz e σz t(e) − Jκ ÿ v σx v − J˜κ ÿ v σx v ź (v v1v2) exp �iπ 4 (1 − σz v1σz (v1v2)σz v2) � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7) where ‘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ’ refers to other gauge invariant terms that can potentially be added in the process of gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A minimal example of such a term would be a product of σz e operators around closed loops in Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For the sake of simplicity, we neglect these terms in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We can readily confirm that [Hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', G(x)] = 0 for any x ∈ C0(Σ△, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' At this point, it is crucial to notice that the microscopic Hilbert space splits into super-selection sectors labelled by eigenvalues of the operators ś e⊂(v1v2v3) σz e associated with every triangle (v1v2v3) ⊂ Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As customary, we shall restrict to a single super-selection sector, namely that given by ź e⊂(v1v2v3) σz e !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='= id , ∀ (v1v2v3) ⊂ Σ△ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8) ∼ 7 ∼ These conditions are also imposed kinematically enforcing g to define a Z2 gauge field so that dg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, notice that we have σx v ���� Gv=id = ź e⊃v σx e ���� Gv=id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='9) Upon enforcing this operator equality on the physical Hilbert space, the model becomes classical in the vertex degrees of freedom so they can be readily gauged away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, this operation is implemented by a unitary operator performing the basis rotation |g, m⟩ �→ |g + dm, m⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Doing so delivers the dual Hamiltonian H∨ = −J ÿ e σz e − Jκ ÿ v ź e⊃v σx e − J˜κ ÿ v ź e⊃v σx e ź (v v1v2) exp �iπ 4 (1 − σz (v1v2)) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10) which acts on the physical Hilbert space H∨ spanned by states |g⟩, where g ∈ Z1(Σ△, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Assuming for concreteness that Σ△ is the Poincar´e dual of the honeycomb lattice, let us explicitly write the action of the various operators appearing in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Firstly, σz e = ÿ g (−1)g[e]|g⟩⟨g| , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='11) which measures the Z2 gauge field along the edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Secondly, ź e⊃v σx e = ÿ g |g + dxv⟩⟨g| ≡ σx σx σx σx σx σx , with xv[v1] = � 1 if v = v1 0 otherwise , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12) which implements a Z2 gauge transformation at vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Thirdly, denoting by v the hexagonal sub-complex centred around v and S := i 1 2 (1−σz), ź e⊃v σx e ź (v v1v2) exp �iπ 4 (1 − σz (v1v2)) � ≡ ÿ g exp � iπBock(g)[ v] � |g + dxv⟩⟨g| ≡ σx σx σx σx σx σx S S S S S S , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='13) which implements a Z2 gauge transformation twisted by a sign depending on the number of ‘up’ states along ∂ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the above expression, Bock denotes the Bockstein homomorphism, a map of cohomology classes Bock : H1(Σ△, Z2) → H2(Σ△, Z2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='14) induced from the short exact sequence 1 → Z2 → Z4 → Z2 → 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15) Similar to the original model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1), the gauged model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7) also has three gapped phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The case 1 ≫ κ, ˜κ corresponds to the confined phase, where the gauge fluctuations are energetically suppressed [FS78, FS79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Meanwhile, the κ ≫ 1, ˜κ and ˜κ ≫ 1, κ cases correspond to two topologically distinct deconfined phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More precisely these are the two topological Z2 gauge phases whose renormalisation group fixed points are provided by the toric code and double semion model, respectively [DW90, Kit97, LW04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 8 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3 Symmetry operators Let us now study the topological operators leaving the model H∨ invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We distinguish two surface operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These are the trivial operator or identity Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' and the non-trivial operator UZ2 defined as follows:2 UZ2[Σ△] := ÿ g∈Z1(Σ△,Z2) Z2d(g)[Σ△] |g⟩⟨g| (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18) ≡ 1 2#(Σ△) ÿ g∈Z1(Σ△,Z2) b∈C1(Σ△,Z2) n∈C0(Σ△,Z2) exp � iπ � Σ△ b⌣(dn + g) � |g⟩⟨g| , where Z2d(g)[Σ△] is the partition function of a two-dimensional pure Z2 gauge theory coupled to background Z2 gauge field g and #(Σ△) := |Σ0 △| + |Σ2 △| where |Σj △| is the number of j-simplices in the triangulation Σ△ (see app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Henceforth, when there is no scope for confusion, we shall often omit specifying which sets the various cochains belong to for conciseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The operator UZ2[Σ△] commutes with the first term in the Hamiltonian as it acts diagonally in the σz e basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It also commutes with the second and third terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10) by virtue of Z2d(g)[Σ△] = Z2d(g + dx)[Σ△] and Bock(g + dxv)[ v] = Bock(g)[ v] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='19) Interestingly, this operator has non-invertible fusion rules [RSS22]: (UZ2 d UZ2)[Σ△] = 1 22#(Σ△) ÿ g,b,n g′,b′,n′ (−1) � Σ△b⌣(dn+g)+b′⌣(dn′+g′)|g⟩⟨g|g′⟩⟨g′| = 1 22#(Σ△) ÿ g,n b′,b+,n+ (−1) � Σ△b+⌣(dn+g)+dn+⌣b′ |g⟩⟨g| = � 1 2#(Σ△) ÿ b′,n+ (−1) � Σ△b′⌣dn+� 1 2#(Σ△) ÿ g,n,b+ (−1) � Σ△b+⌣(dn+g)|g⟩⟨g| = Z2d[Σ△] · UZ2[Σ△] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20) where the partition function Z2d[Σ△] of the pure Z2 gauge theory on Σ△ explicitly reads Z2d[Σ△] = 2b1(Σ)−b0(Σ), where bj is the jth Betti number of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now attempt to rewrite the action of such a topological surface in terms of spin operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' To do so, it is instructive to first sum over b in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Doing so delivers (see app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A) UZ2[Σ△] = 1 2χ(Σ) ÿ g,n δdn,g |g⟩⟨g| , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='21) 2Here d is the simplicial or lattice codifferential operator d : Cn(Σ△, Z2) → Cn+1(Σ△, Z2) such that for q ∈ Cn(Σ△, Z2) dq[v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' vn+1] = n+1 ÿ j=1 (−1)jq[v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ˆvj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' vn+1] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='16) where (v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ˆvj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' vn+1) denotes the n-simplex with the vertex vj omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Further ⌣ denotes the cup product ⌣: Cn(Σ△, Z2) × Cm(Σ△, Z2) → Cn+m(Σ△, Z2) such that q ⌣ p[v1 · · · vn+m] = q[v1 · · · vn] · p[vn · · · vn+m] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17) where p ∈ Cm(Σ△, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note that these notions can be readily generalised to other finite abelian groups (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [BCH18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=') ∼ 9 ∼ where χ(Σ) is the Euler characteristic of Σ, dn[v1v2] = n[v1] + n[v2], and δdn,g = δdn+g,0 is a Z2 Dirac delta function that imposes dn + g = 0 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This operator can equivalently be expressed by first introducing ‘virtual’ qubit degrees of freedom at vertices so as to temporarily enlarge the physical Hilbert space from H∨ to H∨ b Hvirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' with Hvirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' =  v C[Z2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given |n⟩ ∈ Hvirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' with n ∈ C0(Σ△, Z2), we thus require an operator that projects onto the constraint subspace of states |g, n⟩ satisfying n[v1] = g[v1v2] + n[v2] at every edge (v1v2) ⊂ Σ△, before performing a partial trace over Hvirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='. In symbols, UZ2[Σ△] = 1 2χ(Σ) trHvirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' � ź e 1 2(id + σz s(e)σz e σz t(e)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='22) Next, we ask, what are the line operators that commute with H∨?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In addition to the identity line, a line operator with support on any 1-cycle ℓ ∈ Z1(Σ△, Z2) labelled by the non-trivial character in Z∨ 2 may be defined as ź e⊂ℓ σz e = ÿ g ź e⊂ℓ (−1)g[e]|g⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='23) One can readily check that these line operators commute with H∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Moreover, they are topological by virtue of the kinematical constraints (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8), so that the sign ś e⊂ℓ(−1)g[e] only depends on the homology class of ℓ and is 1 whenever ℓ is a contractible cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More generally, any network of such lines can be assigned a cohomology class in H1(Σ△, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Then the sign obtained by such a network of lines can be equivalently expressed via a representative cocycle f in H1(Σ△, Z2) as Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f) = ÿ g (−1) � Σ△f⌣g|g⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='24) Consider for instance the following configuration: σz σz σz σz σz σz σz , depicting a local patch of the triangular lattice Σ△ with a topological line operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='23) wrapping along one of the non-contractible cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The blue lines represent the only edges where the represen- tative 1-cocycle f evaluates to the non-trivial group element in Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Then for any choice of basis state |g⟩ labelled by g ∈ Z1(Σ△, Z2), � Σ△ f⌣g = ź e⊂ℓ g[e] mod 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='25) For reference, the figure above also depicts a configuration g which is non-trivial on the red lines and trivial elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The expressions on the left hand and right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='25) evaluate to −1 for this choice of g as the σz operators only cross a single red line, and similarly a single plaquette (coloured in light grey) contributes to the cup product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 10 ∼ Summing over lines in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='24) is equivalent to summing over f ∈ H1(Σ△, Z2):3 1 |H0(Σ△, Z2)| ÿ f Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f) = 1 |H0(Σ△, Z2)| ÿ g,f (−1) � Σ△f⌣g|g⟩⟨g| = 1 2#(Σ) ÿ g,b,n (−1) � Σ△b⌣(dn+g)|g⟩⟨g| = UZ2[Σ△] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='26) where, in going to second line, we have introduced a Lagrange multiplier field n, which when summed over imposes the cocycle condition on b ∈ C1(Σ△, Z2), recovering the first line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Interestingly, per- forming such a sum yields the surface operator UZ2 defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As we shall comment later on, this is no mere coincidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' When gauging an abelian 0-form symmetry, one obtains a dual model with topological line operators labelled by elements in the Pontrjagin dual, and more generally by representations of the group when it is non-abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Additionally, one obtains topological surface operators, all of which can be understood by inserting networks (or condensing) suitable sub-algebras of topological lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Such surface topological defects have been under scrutiny lately under the name of condensation defects [KW14, EN17, GJF19, BBSNT22a, RSS22, CCH+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows immediately from the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='24) that composition of such (networks of) lines within a surface operator Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' are given by Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f1 ◦ f2) = Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f1 + f2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='27) Going back to definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='23), this is the statement that these line operators fuse like characters in Z∨ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, fusion rules of surface operators Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' with networks of lines inserted are given by Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f1) d Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f2) = Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f1 + f2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='28) As suggested by our notation, we shall think of topological lines Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f) as living on the trivial surface operator Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='. Next, we consider the operator UZ2[Σ△] defined with a collection of lines inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Going back to definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18) and given any 1-cycle ℓ ∈ Z1(Σ△, Z2), such a line operator acts with the Pauli σx operator on the virtual qubits, which are traced over, at the vertices v ⊂ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More generally, any network of such lines is found to be associated with a Z2-valued 1-cycle on the dual lattice Σ∨ △, whose Poincar´e dual is a 1-cocycle f ∈ Z1(Σ△, Z2) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The operator UZ2(f)[Σ△] with a network of such lines inserted has the form UZ2(f)[Σ△] = 1 2χ(Σ) trHvirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' � ź e 1 2(id + (−1)f[e]σz s(e)σz e σz t(e)) � = 1 2#(Σ△) ÿ g,b,n (−1) � Σ△b⌣(dn+f+g)|g⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='29) 3The choice of normalisation |H0(Σ△, Z2)|−1 is inherited from a convention in defining the partition function of (d+1)-dimensional finite group gauge theories, namely that the theory assigns a one-dimensional Hilbert space to a d-sphere for d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note that in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='26), we sum over cohomology classes, rather than cocycles, therefore the normalisation is |H0(Σ△, Z2)| instead of |C0(Σ△, Z2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 11 ∼ Consider for instance the following configuration: σx σx σx σx σx σx σx σx , depicting such an operator, where as before blue lines represent the only edges where the corresponding 1-cocycle f evaluates to the non-trivial element in Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It readily follows from the definition that composition rules of networks of lines within a surface operator UZ2[Σ△] are given by UZ2(f1 ◦ f2)[Σ△] = UZ2(f1 + f2)[Σ△] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='30) Similarly, fusion rules of surface operators UZ2 with networks of lines inserted are given by � UZ2(f1) d UZ2(f2) � [Σ△] = UZ2(f1 + f2)[Σ△] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='31) Finally, we would like to consider the possibility of defining a surface operator UZ2 with support on a sub-region of Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This requires the existence of a topological line at the junction of topological surfaces UZ2 and Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='. Such a line does exist and is simply obtained by restricting the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18) to an open sub-complex Ξ△ ⊆ Σ△, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' UZ2[Ξ△] = ÿ g∈Z1(Σ△,Z2) Z2d(g)[Ξ△] |g⟩⟨g| , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='32) with Dirichlet boundary conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', b[∂Ξ△] = 0 imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We shall think of this operator as describing a line operator from a topological surface UZ2 to Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='. Conversely, we shall think of the operator UZ2[Σ△\\Ξ△] as describing a line operator from Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' to UZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now consider the composition of the former line operator with the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Specifically, we consider a setup where Σ is a two-torus or a cylinder endowed with a triangulation Σ△, and Ξ△ is an annular strip of width a single lattice spacing wrapping a non-contractible cycle: Ξ△ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 12 ∼ Then the composition of the lines is given by UZ2[Ξ△] = ÿ g Z2d[Ξ△] |g⟩⟨g| = ÿ g,f (−1) � Ξ△f⌣g|g⟩⟨g| = id + ź e⊂ℓ σz e , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='33) where ℓ refers here to the non-contractible cycle wrapped by Ξ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the second equality, the sum is over f ∈ H1(Ξ△, ∂Ξ△, Z2) ∼= Z2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', the relative cohomology group with Dirichlet boundary conditions imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note also that the normalisation was implicitly modified from 1/|H0(Ξ△, Z2)| to 1/|H0(Ξ△, ∂Ξ△, Z2)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The non-trivial class in H1(Ξ△, ∂Ξ△, Z2) corresponds to an f-defect wrapping the non-contractible cycle in Ξ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For this choice of f, � ∂Ξ△ f ⌣ g evaluates to ś e⊂ℓ g[e] mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As expected, this results in a line operator living on Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', which is labelled by the regular representation of Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The fusion rule of topological lines in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='33) is closely related to the fusion rules of Kramers-Wannier duality defects in the (1+1)d transverse-field Ising model [CCH+21, CCH+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Now let us compute the composition of the topological line between Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' and UZ2 by considering a thin annular strip of single lattice spacing width containing the identity operator Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', while the rest of the lattice Σ△\\Ξ△ containing UZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us specialise to the case where Σ is a two-torus such that Σ△\\Ξ△ is path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us denote the left and right boundaries of Ξ△ as ∂LΞ△ and ∂RΞ△, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Then, the composition of lines is given by the operator UZ2[Σ△\\Ξ△] = 1 2#(Σ△\\Ξ△) ÿ g,b,n (−1) � Σ△\\Ξ△b⌣(dn+g)|g⟩⟨g| = 1 2χ(Σ△\\Ξ△) ÿ g,n δΣ△\\Ξ△ dn,g , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='34) where in the first expression b ∈ C1(Σ△\\Ξ△, Z2) with the Dirichlet condition b[∂LΞ△] = b[∂RΞ△] = 0 imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Meanwhile n ∈ C0(Σ△, Z2)4 has no constraints imposed a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the final expression, we sum over b, which imposes the cocycle condition dn = g everywhere except within Ξ△, denoted by δΣ△\\Ξ△ dn,g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Besides, note that the Euler characteristic χ(Σ△\\Ξ△) = 0, since Σ\\Ξ is a cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Now pick a preferred edge in Ξ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Naturally dn = g + s, where s is valued in Z2, on this edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows from conditions dg = 0 and dn = g on ∂Ξ△ that fixing s on any chosen edge in Ξ△ pins the configuration to the same value of s for all other edges in Ξ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Consider for instance the following configuration: ∂LΞ△ ∂RΞ△ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As before, the 1-cocycle g is non-trivial only on the red edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The condition dn = g is satisfied everywhere in Σ△\\Ξ△, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', everywhere apart from the central region in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Then we consider n to be fixed to a certain configuration on ∂LΞ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Fixing the configuration of n on a single vertex (for instance the one highlighted in green) on ∂RΞ△, pins the configuration on all other vertices on ∂RΞ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 4n is a 0-cochain on Σ△ since C0(Σ△, Z) = C0(Σ△\\Ξ△, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 13 ∼ The two choices at this vertex correspond to either the presence or absence of a line in VecZ2 traversing Ξ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Therefore, defining a Z2 cocycle f that evaluates to the non-trivial element in Z2 on every edge in Ξ△ (indicated in blue in the above diagram) and to the identity element elsewhere, we obtain UZ2[Σ△\\Ξ△] = UZ2[Σ△] + UZ2(f)[Σ△] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='35) which amounts to a line operator living on UZ2[Σ△].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This concludes our analysis of the symmetry structure of the gauged transverse-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We showed in this section that starting from a two-dimensional lattice model with arguably the simplest kind of symmetry, namely a 0-form Z2 symmetry, gauging the symmetry results in a model with non-invertible surface operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It turns out that the surface and line operators, together with their statistics, are organised into an algebraic structure referred to as the fusion 2-category of 2- representation of Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the next section, we present a framework allowing for the systematic gauging of arbitrary invertible symmetries and analysis of the resulting symmetry structures in terms of higher representations of groups, and categorifications thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' SECTION 3 Gauging and dual symmetries Motivated by the analysis of the (2+1)d transverse-field Ising model carried out above, we introduce in this section a systematic approach to gauging invertible symmetries in (2+1)d quantum lattice models and studying the resulting higher categorical symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1 G-symmetric Hamiltonians Throughout this manuscript, our starting point is always a two-dimensional quantum lattice model with a global 0-form G symmetry, where G is a finite (possibly non-abelian) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, it means that the Hamiltonian commutes with topological operators supported on the whole two-dimensional space, which are labelled by group elements of G, in such a way that the fusion of symmetry operators is governed by the multiplication rule of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By definition of a group, these symmetry operators are in particular invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The modern approach to global symmetries in quantum field theories in terms of collections of topological defects invites us to organise symmetry operators and their properties into higher cate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More specifically, given a (2+1)d quantum theory we expect symmetries to correspond to fusion 2-categories in the sense of Douglas and Reutter [DR18], where objects label topological sur- face operators and (1-)morphisms label topological line operators at the junctions of surface operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In this context, a G-symmetric Hamiltonian commutes with surface operators that form the so-called fusion 2-category of G-graded 2-vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us present this fusion 2-category in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 First of all, let us define a 2-vector space as a C-linear, finite, semisimple category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We can then consider the 2-category 2Vec of 2-vector spaces, linear functors and natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It is a prototypical example of fusion 2-category, where the monoidal structure is given by the Deligne 5In the vein of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6, we shall think of 2VecG as a categorification of the fusion (1-)category VecG of G-graded vector spaces, the same way we can think of VecG as a categorification of C[G], whereby the ring C is promoted to the fusion category Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 14 ∼ tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note that 2Vec has a unique equivalence class of simple objects, which is repre- sented by the category Vec of complex vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now consider the 2-groupoid6 [G, •, •] with object-set G, no non-trivial 1-morphisms and no non-trivial 2-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Consider the cate- gory 2Fun([G, •, •], 2Vec) of pseudofunctors, pseudonatural transformations and modifications between [G, •, •] and 2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By definition, an object V in 2Fun([G, •, •], 2Vec) assigns to every g ∈ G a 2- vector space Vg in 2Vec, and thus amounts to a G-graded 2-vector space of the form V = Ð g∈G Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Pseudonatural transformations in 2Fun([G, •, •], 2Vec) then correspond to grading preserving linear functors, and modifications to natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The convolution product of pseudofunctors [G, •, •] → 2Vec endows 2Fun([G, •, •], 2Vec) with the structure of a fusion 2-category according to (V d W)g := ð x∈G Vx b Vx−1g (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1) with unit 1 satisfying 1g = δg,1G Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Henceforth, we denote by 2VecG this fusion 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' There are |G|-many simple objects in 2VecG provided by the ‘one-dimensional’ 2-vector spaces Vecg, for every g ∈ G, such that Vecg1 d Vecg2 ∼= Vecg1g2 and Hom2VecG(Vecg1, Vecg2) ∼= δg1,g2 Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' At the end of the day, it follows that simple objects can be safely identified with the corresponding group elements in G, but the higher categorical perspective will be crucial in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now construct local operators that explicitly commute with symmetry operators labelled by simple objects in 2VecG in the spirit of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [LDOV21] using the tools introduced in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Del21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let Σ be a closed oriented two-dimensional surface endowed with a triangulation Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Although our construction applies to arbitrary triangulations Σ△, let us assume for concreteness that Σ△ is isotopic to the Poincar´e dual of a honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We further assume that Σ△ has a total ordering of its 0-simplices (vertices), referred to as a choice of branching structure, such that the branching structure in the neighbourhood of every vertex v ≡ (3) ⊂ Σ△ is of the form 0 1 4 6 5 2 3 , ˆu ˆv ˆ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2) Notice that a choice of branching structure induces an orientation of each 1-simplex (edge), which is always chosen to be from the lowest ordered vertex to the higher ordered one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let m denote an assignment of group elements in G to vertices of Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By a slight abuse of notation, we notate via C0(Σ△, G) the collection of such assignments, which corresponds to a G-valued 0-cochain when G is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We define the microscopic Hilbert space of the system to be  v C[G] and denote by |m⟩ the assignment m regarded as an element of the microscopic Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The restriction of |m⟩ to a given vertex v ⊂ Σ△ is denoted by |m[v]⟩ ∈ C[G], and more generally, we notate via |m[Ξ△]⟩ :=  v⊂Ξ△ |m[v]⟩ the state associated with the restriction of m to a sub-complex Ξ△ ⊆ Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We are interested in G-symmetric local operators acting on the Hilbert space  v C[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a vertex v ⊂ Σ△, we notate via v ⊆ Σ△ the hexagonal sub-complex centred around v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now consider the pinched interval cobordism v×p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='I ≡ v ×p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [0, 1] defined as v×I/ ∼, where the 6A 2-groupoid is a 2-category in which every morphism is an equivalence, in the same spirit of a 1-groupoid being a (small) category in which every morhism is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 15 ∼ equivalence relation ∼ is such that (x, i) ∼ (x, i′) for all (x, i), (x, i′) ∈ ∂ v×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Graphically, I v := v ×p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' I ≡ 0 1 4 6 5 2 3′ 3 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3) where the branching structure induced from that of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2) is such that 2 < 3′ < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notice that, by definition, we have ∂ I v = v×{0} ∪∂ v v×{1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Henceforth, we employ the shorthand notations 0 v ≡ v×{0} and 1 v ≡ v×{1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given an assignment m ∈ C0( I v, G), we can define an operator acting on a local neighbourhood of the vertex v as ��m[ 1 v] �� m[ 0 v] ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notice that this operator acts as the identity operator at every vertex but v where it acts as ��m[v′] �� m[v] ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More general operators can then be constructed by considering linear combinations of the form hv,n := ÿ m∈C0( I v,G) hv,n � {m[v1]m[v2]−1}(v1v2)⊂ I v � ��m[ 1 v] �� m[ 0 v] �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) where the coefficients hv,n are valued in U(1) and the oriented edges (v1v2) are always such that v1 < v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note that, in practice, we typically consider models for which the coefficients hv,n are only non-vanishing for specific choices of assignments m ∈ C0( I v, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Any combinations of such operators can finally be used to define a local Hamiltonian H ≡ ř v hv := ř v ř n hv,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By construction, any Hamiltonian thus defined is G-symmetric, whereby the symmetry is generated by operators ś v⊂Σ△ Rx v with Rx v : |m[v]⟩ �→ |m[v]x−1⟩ for any x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This simply follows from a redefinition of the variable m ∈ C0( I v, G) in the summation, together with the fact that the coefficients hv,n only depend on {m[v1]m[v2]−1}(v1v2)⊂ I v and are therefore manifestly symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Identifying every m[v] ∈ G with the corresponding simple object Vecm[v] in 2VecG, one can equivalently state that any Hamiltonian thus defined is 2VecG-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It is a straightforward exercise—which we carry out below—to show that the transverse-field Ising model is of this form, and more generally, we can argue that every local G-symmetric Hamiltonian can be written in terms of combinations of local operators of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note that Hamiltonians defined in this section only account for nearest or next-nearest neighbours interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' However, we can readily combine such local operators—which geometrically amounts to concatenating complexes of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3)—so as to define local operators simultaneously acting on a larger number of sites, thereby generating the whole algebra of G-symmetric Hamiltonians on  v C[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' That being said, most familiar and physically relevant quantum systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Heisenberg-like models, are already included within the present formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In prevision for the following section, let us slightly reformulate the previous construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We noticed above that the G symmetry of local operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) is guaranteed in particular by the fact that the unitary coefficients hv,n only depend on the assignment m through group elements m[v1]m[v2]−1 for every edge (v1v2) ⊂ I v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This leads us to contemplate the following alternative description: Consider an assignment g of group elements in G to every edge of I v such that g[v1v2]g[v2v3] = g[v1v3] for every 2-simplex (v1v2v3) ⊂ I v, which we shall think about as a flat gauge field on I v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By slight ∼ 16 ∼ abuse of notation, we notate via Z1( I v, G) the collection of such assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let m ∈ C0( I v, G) be an assignment as before but with the additional constraint that m[v1] = g[v1v2]m[v2] for every edge (v1v2) ⊂ I v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In other words, we require dm = g, where dm[v1v2] = m[v1]m[v2]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We can now rewrite the previous operators as follows: hv,n = ÿ g∈Z1( I v,G) hv,n(g) ÿ m∈C0( I v,G) dm=g ��m[ 1 v] �� m[ 0 v] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5) Notice that given g, the condition dm = g does not fully constrain m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the following section, we consider generalizations of these operators yielding dual Hamiltonian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Back to the transverse-field Ising model Let us illustrate our construction by recasting the transverse-field Ising model in terms of local op- erators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We also discuss a finite group generalisation of this model in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Consider the Hamiltonian H = ř v⊂Σ△ ř4 n=1 hv,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For any vertex v ⊂ Σ△ and gauge field g ∈ Z1( I v, G), the defining U(1)-coefficients hv,n(g) are chosen to be hv,1(g) := −Jδg[v′v],0(−1)g[v v+ˆu] , hv,2(g) := −Jδg[v′v],0(−1)g[v v+ˆv] , hv,3(g) := −Jδg[v′v],0(−1)g[v v+ ˆ w] , hv,4(g) := −Jκ δg[v′v],1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6) where the branching structure of I v is that given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We can readily confirm that hv,4 acts as −Jκσx v , whereas local operators hv,n=1,2,3 act as −Jσz v σz v+ˆu, −Jσz v σz v+ˆv and −Jσz v σz v+ ˆ w, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Putting everything together, we recover Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1) for ˜κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By construction, this model has a 2VecZ2 symmetry such that the two simple objects Vec0 and Vec1 in 2VecZ2, where 0, 1 ∈ Z2, are identified with the surface operators O0 and O1 defined in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The fact that the deformation class of models generated by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6) does not host any non-trivial line operators then follows from Hom2VecZ2 (Vecg1, Vecg2) ∼= δg1,g2 Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 Dual Hamiltonians Given a Hamiltonian H = ř v ř n hv,n with local operators hv,n as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5), we shall now construct dual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4, we shall relate these various dual models to twisted gauging of the G symmetry or sub-symmetries thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Our strategy goes as follows: Any finite group G gives rise to an (abstract) algebra of local operators, in such a way that products of local operators only make use of the multiplication in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A duality class of models is then determined by choosing certain linear combinations of local operators in the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This choice is made through the set of coefficients {hv,n(g)}g over g ∈ Z1( I v, G) in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This means that the group G together with the collection hv,n of coefficients fully determine the physical characteristics of the duality class of models as encoded into their common spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notice that we have not yet specified explicit matrix/lattice representations of these local operators on a chosen Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As a matter of fact, picking a representative of a duality class of models precisely corresponds to choosing such a matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Loosely speaking, this boils down to identifying a collection of degrees of freedom providing a particular physical realisation of the properties encoded into the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In other words, maintaining the same linear combination of symmetric operators, while choosing another matrix realisation, yields a dual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We explain below how such choices are made, thereby defining duality classes of (2+1)d Hamiltonian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 17 ∼ We begin our construction by noticing that picking a gauge field g ∈ Z1( I v, G) amounts to assigning a simple object g[v1v2] ≡ Vecg[v1v2] in 2VecG to every edge (v1v2) ⊂ I v such that Vecg[v1v2] d Vecg[v2v3] ∼= Vecg[v1v3] for every 2-simplex (v1v2v3) ⊂ I v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In this context, picking an assignment m ∈ C0( I v, G) such that dm = g amounts to assigning simple objects m[v1] ≡ Vecm[v1] in 2VecG such that Vecg[v1v2] d Vecm[v2] ∼= Vecm[v1] for every edge (v1v2) ⊂ I v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We think of this latter assignment as making a choice of degrees of freedom, and thus a choice of microscopic Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As will become clear in the following, this choice amounts to considering 2VecG as a module 2-category over itself, inviting us to replace 2VecG by another module 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In that spirit, let us first review the notion of module 2-category over 2VecG as considered in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [D´e21, Del21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Succinctly, a module 2-category over 2VecG is a 2-category with a G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More precisely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' we define a (left) 2VecG-module 2-category as a quadruple (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ▷,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' α▷,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' π▷) consisting of a (C-linear finite semisimple) 2-category M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' a binary action 2-functor ▷ : 2VecG × M → M and an adjoint natural 2-equivalence α▷ : (− d −) ▷ − ∼ −→ − ▷ (− ▷ −) satisfying a ‘pentagon axiom’ up to an invertible modification π▷ whose components π▷ Vecg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='M are defined via (Vecg1 d (Vecg2 d Vecg3) ▷ M Vecg1 ▷ ((Vecg2 d Vecg3) ▷ M) ((Vecg1 d Vecg2) d Vecg3) ▷ M Vecg1 ▷ (Vecg2 ▷ (Vecg3 ▷ M)) (Vecg1 d Vecg2) ▷ (Vecg3 ▷ M) α▷ Vecg1g2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='M α▷ Vecg1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg3 ▷M 1Vecg1 ▷α▷ Vecg2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='M 1Vecg1g2g3 ▷1M α▷ Vecg1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg2g3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='M π▷ Vecg1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7) for every g1, g2, g3 ∈ G and M ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The invertible modification π▷, which shall be referred to as the left module pentagonator, is required to satisfy an ‘associahedron axiom’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For convenience, we shall spell out this axiom employing an alternative to commutative diagrams in terms of string diagrams, whereby regions represent objects, strings 1-morphisms, and blobs 2-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In practice, we shall omit labelling regions but the corresponding objects can be recovered from the string labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' On these diagrams, compositions of 1-morphisms is read from left to right, whereas the (vertical) composition of 2-morphisms is read from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For instance, the left module pentagonator π▷ can be equivalently defined via the string diagram: π▷ 11 α▷ 1α▷ α▷ α▷ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8) where we omitted the d and ▷ symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The associahedron axiom satisfied by π▷ can then be conve- ∼ 18 ∼ niently expressed as the following equality of string diagrams: 11 11 α▷ (11)α▷ π1 π▷ π▷ (11)1 11 (11)1 α▷ 1α▷ 1(1α▷) α▷ α▷ α▷ = 1(11) 1α▷ 1α▷ α▷ 1(11) 1π▷ π▷ π▷ (11)1 11 (11)1 α▷ 1α▷ 1(1α▷) α▷ α▷ α▷ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='9) We are particularly interested in indecomposable 2VecG-module 2-categories constructed as follows [Del21]:7 Given a subgroup A ⊆ G and a normalised 3-cocycle λ in H3(A, U(1)), let M(A, λ) be a 2-category with object-set the set G/A of left cosets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A left 2VecG-module structure can be defined on M(A, λ) via Vecg ▷ M := (gr(M))A for any g ∈ G and M ∈ G/A, where r : G/A → G assigns to every left coset its representative in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notice that in general we have gr(M) ̸= r(Vecg ▷ M) and we denote by ag,M the group element in A satisfying gr(M) = r(Vecg ▷ M)ag,M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10) Associativity of the multiplication in G imposes ag1g2,M = ag1,Vecg2▷M ag2,M , ∀ g1, g2 ∈ G and M ∈ G/A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='11) Endowing the abelian group Hom(G/A, U(1)) with a left G-module structure, we consider the 3-cochain π▷ ∈ C3(G, Hom(G/A, U(1))) defined as π▷(g1, g2, g3)(M) := λ(ag1,Vecg2g3▷M , ag2,Vecg2▷M , ag3,M) , ∀ g1, g2, g3 ∈ G and M ∈ G/A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12) In virtue of the 3-cocycle condition dλ = 1 and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='11), we have π▷(g2, g3, g4)(M) π▷(g1, g2g3, g4)(M) π▷(g1, g2, g3)(Vecg4 ▷ M) = π▷(g1g2, g3, g4)(M) π▷(g1, g2, g3g4)(M) , ∀ g1, g2, g3, g4 ∈ G and M ∈ G/A , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='13) so that π▷ is a Hom(G/A, U(1))-valued 3-cocycle of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Defining the invertible modification π▷ with components π▷ Vecg1,Vecg2,Vecg3,M := π▷(g1, g2, g3)(M) · 1(g1g2g3r(M))A , ∀ g1, g2, g3 ∈ G and M ∈ G/A , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='14) 7Note that this construction does not give all 2VecG-module 2-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Physically, it only gives those module 2-categories that correspond to either spontaneously breaking the global symmetry down to a subgroup and/or pasting a symmetry-protected topological phase labelled by a 3-cocycle of the preserved subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notably, we do not discuss those module 2-categories that correspond to coupling to a inherently two-dimensional non-anomalous topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It is expected that these topological orders would be contained in the completion of the 3-category of 2VecG-module 2-categories [BSNT22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 19 ∼ we finally obtain that the quadruple M(A, λ) ≡ (M(A, λ), ▷, 1, π▷) does define a left 2VecG-module 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For any group G, we can always choose the subgroup A to be either the trivial subgroup {1G} or the whole group G, and the 3-cocycle to be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The corresponding module 2-categories are M({1g}, 1) ∼= 2VecG and M(G, 1) ∼= 2Vec with action 2-functors given by the monoidal product in 2VecG and the forgetful functor 2VecG → 2Vec, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now put these module 2-categories to use in order to construct dual Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a vertex v ⊂ Σ△, a gauge field g ∈ Z1( I v, G) and a 2VecG-module 2-category M ≡ M(A, λ) as defined above, let us consider an assignment m of simple objects m[v1] ∈ M to every vertex v1 ⊂ I v such that m[v1] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='= Vecg[v1v2] ▷ m[v2] , ∀ (v1v2) ⊂ I v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15) We notate via C0 g( I v, M) the set of assignments m fulfilling conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given such a pair (g, m) ∈ Z1( I v, G) × C0 g( I v, M), let us introduce the following phase factor: π▷ v (g, m) := ź (v1v2v3v4)⊂ I v π▷(g[v1v2], g[v2v3], g[v3v4])(m[v4])ϵ(v1v2v3v4) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='16) where π▷ is the Hom(G/A, U(1))-valued 3-cocycle of G defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12), and ϵ(v1v2v3v4) = ±1 depends on the orientation of the 3-simplex (v1v2v3v4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Borrowing the notations of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1, we finally define new local operators as follows: hM v,n = ÿ g∈Z1( I v,G) hv,n(g) ÿ m∈C0g( I v,M) π▷ v (g, m) ��(g, m)[ 1 v] �� (g, m)[ 0 v] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17) Notice immediately that choosing M to be 2VecG itself, we recover local operators hv,n defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We are now ready to state one of the main results of this manuscript: Hamiltonian models that only differ in a choice of 2VecG-module 2-category are dual to one another via definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17) of the local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In other words, Hamiltonians HM = ř v ř n hM v,n and HM′ = ř v ř n hM′ v,n for any two indecomposable 2VecG-module 2-categories M ≡ M(A, λ) and M′ ≡ M(A′, λ′) are dual to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As motivated above, duality between HM and HM′ follows from the fact 2VecG-module 2-categories M and M′ merely encode distinct matrix realisations of the same local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More concretely, regardless of the choice of 2VecG-module 2-category M, the set of local operators hM v,n generate the same algebra of operators characteristic of the duality class of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the same vein as the lower- dimensional study carried out in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [LDOV21], we can readily confirm that products of local operators indeed only involve the group G and coefficients hv,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, computing products of local opera- tors (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17) involve algebraic manipulations that geometrically translate into three-dimensional Pachner moves, which are encoded into the associahedron axiom given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The associahedron axiom dictates that products of operators only depend on the monoidal structure of 2VecG via the monoidal pentagonator π, which happens to be trivial, and is a fortiori independent of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' An alternative justification consists in showing that the Hamiltonian HM with M ≡ M(A, λ) is the result of the λ-twisted gauging of the A sub-symmetry of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This will be the purpose of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Importantly, dualities as considered in this manuscript systematically map symmetric local op- erators to dual symmetric local operators—this almost tautologically follows from our definition of a duality as a change of matrix realisation of the local operator encoded into a choice of 2VecG-module 2-category—whereas non-symmetric local operators are mapped to dual non-local non-symmetric op- erators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These various mappings are realised via (typically non-local) lattice duality operators that ∼ 20 ∼ transmute in particular local operators into one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Below, we explicitly construct these lattice operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Back to the transverse-field Ising model We explained in the previous section how to recast the transverse-field Ising model within our frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The input fusion 2-category being 2VecZ2, we distinguish three choices of module 2-categories, namely 2VecZ2 itself, 2Vec and 2Vecλ, respectively, where λ corresponds to the non-identity element in H3(Z2, U(1)) ≃ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given the coefficients (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6), it readily follows from definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17) of the local operators that h2Vec v,4 acts as −Jκ ś e⊃v σx e , whereas local operators h2Vec v,n=1,2,3 acts as −Jσz (v v+ˆu), −Jσz (v v+ˆv) and −Jσz (v v+ ˆ w), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Putting everything together, we recover the Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10) with ˜κ = 0 resulting from gauging the Z2 symmetry of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Choosing instead the 2VecZ2- module 2-category 2Vecλ amounts to the λ-twisted gauging of the Z2 symmetry and results in Hamil- tonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10) with κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3 Duality operators We are interested in dualities between Hamiltonians HM and HM′ whose local operators hM v,n and hM′ v,n only differ in the choice of 2VecG-module 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Therefore, a duality operator should have the interpretation of a map between the module 2-categories that is compatible with the action of 2VecG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More concretely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' given a pair of (left) 2VecG-module 2-categories (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ▷,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' α▷,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' π▷) and (M′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ·▷,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' α·▷,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' π·▷),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' we define a 2VecG-module 2-functor between them as a triple (F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Ω) consisting of a 2-functor F : M → M′ and an adjoint natural 2-equivalence ω : F(− ▷ −) ∼ −→ − ·▷ F(−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' with components ωVecg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='M for g ∈ G and M ∈ M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' satisfying a ‘pentagon axiom’ up to an invertible modification Ω whose components ΩVecg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='M are defined via F(Vecg1 ▷ (Vecg2 ▷ M)) Vecg1 ·▷ F(Vecg2 ▷ M) F((Vecg1 d Vecg2) ▷ M) Vecg1 ·▷(Vecg2 ·▷ F(M)) (Vecg1 d Vecg2) ·▷ F(M) ωVecg1g2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='M α·▷ Vecg1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='F (M) 1Vecg1 ·▷ ωVecg2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='M F (α▷ Vecg1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='M) ωVecg1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg2 ▷M ΩVecg1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='Vecg2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='M (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18) for every g1, g2 ∈ G and M ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As before, we shall prefer the equivalent definition in terms of the string diagram Ω F (α▷) ω 1ω ω α·▷ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='19) ∼ 21 ∼ This invertible modification Ω is required to satisfy an ‘associahedron axiom’ encoded into the following equality of string diagrams: F (α▷) F (α▷) ω (11)ω F (π▷) Ω Ω F (11) F (α▷) F (1α▷) ω 1ω 1(1ω) ω α·▷ α·▷ = 1ω 1α·▷ α·▷ 11 1Ω Ω π·▷ F (11) F (α▷) F (1α▷) ω 1ω 1(1ω) ω α·▷ α·▷ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20) Let us now use the data of a module 2-functor M → M′ to construct lattice operators that transmute local operators hM v,n into hM′ v,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Since module 2-functors can be composed—and by extension so do the corresponding dualitiy operators—we can focus without loss of generality on 2VecG-module 2- functors between 2VecG itself and M′ ≡ M(A′, λ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Every such module 2-functor is of the form (− ▷ M ′, 1, π▷ −,−,−,M ′), with M ′ ∈ M′, in which case the associahedron axiom (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20) boils down to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the spirit of our definition of the local operators, let us consider the complex8 v×I ≡ 2 4 0 1 5 6 2′ 4′ 0′ 1′ 5′ 6′ 3 3′ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='21) centred around v ≡ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The branching structure agrees with that of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3), and in particular we have v′ 1 < v1 for every v1 ⊂ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a simple object M ′ ∈ M′ interpreted as a 2VecG-module 2-functor 2VecG → M′, we consider the following assignment of degrees of freedom: First, let g ∈ Z1( 0 v, G) and g′ ∈ Z1( 1 v, G) such that g[v1v2] = g′[v′ 1v′ 2] for every v1, v2 ⊂ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We then consider an assignment m of simple objects m[v1] ∈ 2VecG to every vertex v1 ⊂ 0 v and an assignment m′ of simple objects m′[v′ 1] ∈ M′ to every vertex v′ 1 ⊂ 1 v such that m[v1] = Vecg[v1v2] d m[v2] for every (v1v2) ∈ 0 v, m′[v′ 1] = Vecg′[v′ 1v′ 2] ▷ m′[v′ 2] for every (v′ 1v′ 2) ∈ 1 v, and m[v1] ▷ M ′ = m′[v′ 1] for every (v′ 1v1) ⊂ v×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As before, we notate via C0 g( 0 v, G) the collection of assignments m fulfilling the conditions spelt out above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notice that assignment m ∈ C0 g( 0 v, G) together with M ′ ∈ M′ uniquely specifies an assignment m′ via the constraints m[v1] ▷ M ′ = m′[v′ 1] for every (v′ 1v1) ⊂ v ×I, and we denote by m ▷ M ′ this assignemnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 8Alternative operators more suited to the more general case of an arbitrary fusion 2-category can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Del21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 22 ∼ Given assignments (g, m) ∈ Z1( 0 v, G) × C0 g( 0 v, G), let us introduce the following phase factor: π▷ v (g, m, M ′) := ź (v1v2v3)⊂ 0 v π▷(g[v1v2], g[v2v3], m[v3])(M ′)ϵ(v1v2v3) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='22) where π▷ is the Hom(G/A′, U(1))-valued 3-cocycle of G defined previously, and ϵ(v1v2v3) = ±1 depends on the orientation of the 2-simplex (v1v2v3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We finally define the duality operator labelled by M ′ ∈ M′ acting at vertex v ⊂ Σ△ as follows: dM ′ v = ÿ g∈Z1( 0 v ,G) m∈C0 g( 0 v ,G) π▷ v (g, m, M ′) |g, m ▷ M ′⟩⟨g, m| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='23) What is the action of these operators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' On the one hand, the operator turns degrees of freedom provided by simple objects in 2VecG—thought as a module 2-category over itself—into simple objects in M′ via the module 2-functor − ▷ M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' On the other hand, it acts by scalar multiplication by the phase factors π▷ v (g, m, M ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows that acting with dM ′ v on hv,n yields hM′ v,n , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', dM ′ v hv,n = hM′ v,n ◦ dM ′ v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='24) The only non-trivial aspect to confirm is the compatibility of the various phase factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It is convenient to do so using the geometrical interpretations of the operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Geometrically, the commutation relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='24) can be represented as follows: = , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='25) where we think of the complex I v on the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' as supporting the local operator hv,n and that on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' as supporting hM′ v,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall that the phase factors entering the definition of hv,n are trivial, whereas those entering the definition of hM′ v,n are given by π▷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, the phase factors entering the definition of dM ′ v evaluates to π▷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows from the associahedron axiom satisfied by the module pentagonator π▷—or rather the cocycle condition of the 3-cocycle it evaluates to—as well as the fact that every pair of neighbouring 3-simplices share a 2-simplex, that the commutation relation is ∼ 23 ∼ satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Indeed, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20) graphically translates as v1 v4 v2 v3 v′ 2 v′ 4 v′ 1 v′ 3 v′ 3 v′ 1 = v′ 2 v′ 3 v′ 1 v′ 4 v1 v3 v2 v4 v4 v1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='26) where vertices carrying the same label are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Applying the assignment rules of degrees of freedom presented above, we find that the phase factors associated with the 3-simplices on the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' are 1 and π▷(g[v1v2], g[v2v3], g[v3v4])(m′[v′ 4]), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, we associate to the prisms on the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' the phase factors π▷(g[v1v2], g[v2v3], m[v3])(M ′) and π▷(g[v1v3], g[v3v4], m[v4])(M ′), whereas we associate to the prisms on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' the phase factors π▷(g[v2v3], g[v3v4], m[v4])(M ′) and π▷(g[v1v2], g[v2v4], m[v4])(M ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Choosing g[v1v2] ≡ g1, g[v2v3] ≡ g2, g[v3v4] ≡ g3 and m[v4] ≡ g4 ∈ G, it follows from the various assignment rules—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' m′[v′ 4] = m[v4] ▷ M ′ = Vecg4 ▷ M ′—that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='26) exactly encodes eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Applying eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='26) for every 3-simplex in I v, we find that all the phase factors of the form π▷(g[v1v3], g[v3v4], m[v4])(M ′) and π▷(g[v2v3], g[v3v4], m[v4])(M ′) cancel two-by-two resulting in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More generally, given local operators hM v,n and hM′ v,n , the analogue of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='26) will be guaranteed by the associahedron axiom (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20) fulfilled by the 2VecG-module structure of the 2-functor M → M′ (see sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 for the case of module 2-endofunctors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' So we have found duality operators performing the transmutation of local operators hv,n into hM′ v,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In order to perform this operation to the whole Hamiltonian, it suffices to extend the definition of our duality operator to the whole Σ△ following exactly the same construction: dM ′ = ÿ g∈Z1(Σ0 △,G) m∈C0 g(Σ0 △,G) � ź (v1v2v3)⊂Σ0 △ π▷(g[v1v2], g[v2v3], m[v3])(M ′)ϵ(v1v2v3) � |g, m ▷ M ′⟩⟨g, m| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='27) Let us conclude with a couple of important remarks: Firstly, duality operators are oblivious to the details of the definition of the local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular they act in the same way regardless of the coefficients hv,n, and as such are valid for infinitely many lattice models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This is because duality operators only care about matrix/lattice realisations of a given symmetry and not specific choices of algebra of local operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In other words, a given duality operator will systematically transmute every symmetric operator with respect to a given lattice realisation of a symmetry into a symmetric operator with respect to another realisation, regardless of the precise definition of these operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These symmetries will be analysed in detail in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Secondly, the knowledge of such a duality operator is not sufficient to rigorously write down an isometry mapping the corresponding Hamiltonians to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As detailed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [LDV22] for the lower-dimensional setting, defining ∼ 24 ∼ such an isometry would require analysing all the topological sectors of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We comment on this aspect in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 6 but a detailed analysis will be carried out elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Back to the transverse-field Ising model We established in the previous section how, given the coefficients (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6), choosing the 2VecZ2-module 2-categories 2VecZ2, 2Vec or 2Vecλ yields the transverse-field Ising model, its Z2-gauged dual or its λ-twisted Z2-gauged dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The results obtained above now allow us to construct the lattice operators performing the transmutations of the corresponding local symmetric operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Succinctly, there is a unique 2VecZ2-module functor from 2VecZ2 to 2Vec, namely the forgetful functor, identified with the unique simple object Vec ∈ 2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The corresponding duality operator acts as dVec : |g, m⟩ �→ |g, m▷Vec⟩ for any (g, m) ∈ Z1(Σ△, Z2) × C0 g(Σ△, VecZ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' But m ▷ Vec ∼= Vec and g is fully constrained by m according to m[v1] = Vecg[v1v2] d m[v2] for any (v1v2) ⊂ Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows that the duality operator effectively acts as dVec : |m⟩ �→ |dm⟩ in the notation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It readily follows that dVec : σz s(e)σz t(e) �→ σz e and dVec : σx v �→ ś e⊃v σx e , as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The treatment of the duality 2VecZ2 → 2Vecλ follows the same steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note that these duality operators were already obtained in [Del21] exploiting the graphical calculus of monoidal 2-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4 Duality as twisted gauging Let us now clarify in which sense the dual Hamiltonians described above are the results of applying some (twisted) gauging to the original G-symmetric Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We begin by providing an alternative expression for local operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By definition of our notations, local operators hM v,n act on degrees of freedom located at vertices and edges labelled by simple objects in M(A, λ) and 2VecG, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' However, these degrees of freedom must satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15), which we shall think of as kinematical con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Resolving these kinematical constraints allow us to consider a smaller effective microscopic Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Consider for instance the 2VecG-module 2-category M({1G}, 1) ∼= 2VecG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A choice m of assignments of objects in M to every vertex v1 ⊂ I v fully constraints g ∈ Z1( I v, G) in virtue of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15), so that we should consider the effective Hilbert space  v C[G], at which point the oper- ators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17) boil down to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5) as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More generally, given M(A, λ) and a pair (m[v1], m[v2]) of simple objects in M(A, λ), there are exactly |A|-many distinct group elements g ∈ G such that m[v1] ∼= Vecg ▷ m[v2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='9 Consequently, local operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17) effectively act on a microscopic Hilbert space constituted of degrees of freedom at vertices labelled by simple objects in M(A, λ) and degrees of freedom at edges labelled by group elements in A—or rather simple objects in 2VecA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a pair (g, m) ∈ Z1( I v, G) × C0 g( I v, M), we denote by ag,m the assignment of group elements ag,m[v1v2] to every edge (v1v2) ⊂ I, where ag,m[v1v2] := r(m[v1])−1g[v1v2] r(m[v2]) ≡ ag[v1v2],m[v2] , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='28) where we used in the last identification the notation introduced in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10) when defining M(A, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note that in virtue of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15), we have ag,m[v1v2]ag,m[v2v3] = ag,m[v1v3] for every (v1v2v3) ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recalling the definition of the module pentagonator π▷, we introduce π▷ v (ag,m) := ź (v1v2v3v4)⊂ I v λ(ag,m[v1v2], ag,m[v2v3], ag,m[v3v4])ϵ(v1v2v3v4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='29) 9Given a pair (m[v1], m[v2]) of simple objects in M(A, λ), there must exist g ∈ G such that Vecg ▷ m[v2] ∼ = m[v1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let g′ ∈ G such that m[v1] ∼ = Vecg′g ▷ m[v2] ∼ = Vecg′ ▷ m[v1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This requires Vecg′ to be in the stabiliser of m[v1], which in turn requires g′ ∈ r(m[v1])Ar(m[v1])−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Our statement finally follows from |r(m[v1])Ar(m[v1])−1| = |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 25 ∼ Putting everything together, we find that local operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17) act on the effective Hilbert space as hM(A,λ) v,n eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' = ÿ g∈Z1( I v,G) hv,n(g) ÿ m∈C0g( I v,M) π▷ v (ag,m) ��(ag,m, m) � 1 v ��� (ag,m, m) � 0 v ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='30) In practice, this is the expression we shall employ when discussing explicit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let now employ this expression to clarify why hM(A,λ) v,n is the result of a λ-twisted gauging of the A sub-symmetry of the G-symmetric Hamiltonian defined in terms of local operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Consider the (untwisted) gauging of the whole G symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Typically, this operation goes as follows: The macroscopic Hilbert space is enlarged by the introduction of a G-gauge field and a (local) Gauß constraint is imposed at every vertex in such a way that they commute with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We then require the Hamiltonian to commute with such Gauß constraints, which is accomplished by minimally coupling the Hamiltonian with the gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, the Gauß constraints are imposed kinematically allowing for the initial (matter) degrees of freedom to be gauged away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Within our framework, these operations are simply accomplished by considering the 2VecG-module 2-category M ≡ M(G, 1) ∼= 2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, it follows form the definition that we have ag,m = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More generally, let us consider the gauging of the A sub-symmetry of a G-symmetric Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As above, we begin by introducing an A-gauge field a ∈ Z1(Σ△, A) and impose the following Gauß constraints at every vertex: Gv := 1 |A| ÿ x∈A � ź e→v Rx e � Rx v � ź e←v Lx e � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='= id , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='31) where Rx e : |a[e]⟩ �→ |a[e]x−1⟩ , Lx e : |a[e]⟩ �→ |xa[e]⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='32) so that the physical Hilbert space does not have a tensor product structure anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In order to kinematically enforce these Gauß constraints, it is convenient to disentangle degrees of freedom by applying the following unitary: U := ź v � ź e→v cRv,e ź e←v cLv,e � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='33) where we introduced the controlled group actions cRv,e : |g1⟩v b |g2⟩e �→ |g1⟩v b |g2g−1 1 ⟩e , cLv,e : |g1⟩v b |g2⟩e �→ |g1⟩v b |g1g2⟩e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='34) In particular, we have UGvU† = 1 |A| ÿ x∈A Rx v !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='= id , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='35) so that imposing the Gauß constraints amounts to considering an effective microscopic Hilbert space whereby degrees of freedom at vertices are labelled by elements in G/A, or rather simple objects m[v] in M(A, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notice finally that cL−1 v,(vv1) (Lx v b Lx (vv1)) cLv,(vv1) : |m[v], a[vv1]⟩ �→ |Cx ▷ m[v], ax,m[v]a[vv1]⟩ ≡ |Cx ▷ m[v], a[v′v1]⟩ , cR−1 v,(v1v) (Lx v b Rx (v1v)) cRv,(v1v) : |m[v], a[v1v]⟩ �→ |Cx ▷ m[v], a[v1v]ax−1,Vecx▷m[v]⟩ ≡ |Cx ▷ m[v], a[v1v′]⟩ , where we identified x ≡ g[v′v], at which point we recover the image of the operator Lx v under the duality map, as encoded into local operators of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='30) with M ≡ M(A, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given this ∼ 26 ∼ understanding of choosing the 2VecG-module 2-category M(A, 1) as gauging the A sub-symmetry, we interpret choosing M(A, λ) with λ a non-trivial 3-cocycle in H3(A, U(1)) as performing a λ-twisted gauging of the A sub-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More details on this gauging perspective are provided in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8 for the case of the finite group generalisation of the transverse-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 Dual symmetries We commented earlier that dualities considered in this manuscript map local symmetric operators to dual local symmetric operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' However, we have not yet revealed what the symmetry of a given Hamiltonian HM is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notice that we are still not choosing any specific Hamiltonian, it is enough to know that it is defined in terms of local operators of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall that we introduced in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3 the notion of 2VecG-module 2-functors and explained how these provide duality operators between local operators that only differ in a choice of 2VecG-module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a Hamiltonian HM = ř v ř n hM v,n, a module 2-functor from M to itself should thus correspond to a symmetry operator of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Indeed, we shall demonstrate that 2VecG-module 2-endofunctors of M label surface symmetry operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Furthermore, these surface operators can host topological line operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More generally, surface operators are not necessarily closed, in which case topological lines living at the junctions of distinct topological surfaces are required so the Hamiltonian is left invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Mathematically, these topological lines are captured by the notion of module natural 2-transformation between module 2-functors: Given a pair of left 2VecG-module 2-functors (F, ω, Ω) and ( ˜F, ˜ω, ˜Ω), we define a 2VecG-module natural 2-transformation between them as a tuple (θ, Θ) consisting of a natural 2-transformation θ : F ⇒ ˜F satisfying a coherence axiom up to an invertible modification Θ with components ΘVecg,M defined according to the string diagram Θ ω 1θ θ ω′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='36) This invertible modification is required to satisfy a coherence axiom encoded into the following equality of string diagrams: ω 1θ Ω Θ F (α▷) ω 1ω 1(1θ) θ ˜ω α·▷ = 1θ 1˜ω ˜ω ˜ F (α▷) 1Θ Θ ˜Ω F (α▷) ω 1ω 1(1θ) θ ˜ω α·▷ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='37) ∼ 27 ∼ Furthermore, given a pair of 2VecG-module natural 2-transformations (θ, Θ) and (˜θ, ˜Θ), we can define a 2VecG-module modification from (θ, Θ) to (˜θ, ˜Θ) as a modification ϑ : θ ⇛ ˜θ such that ˜ΘVecg,M ◦ (11Vecg ·▷ ϑM) = ϑVecg▷M ◦ ΘVecg,M (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='38) for all g ∈ G and M ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a pair (M, M′) of 2VecG-module 2-categories, we shall refer to 2Fun2VecG(M, M′) as the 2-category whose objects are 2VecG-module 2-functors M → M′, 1-morphisms are 2VecG-module 2- natural transformations, and 2-morphisms are 2VecG-module modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the present context, we are specifically interested in 2-categories of the form (2VecG)⋆ M(A,λ) := 2Fun2VecG(M(A, λ), M(A, λ)) that shall be referred to as ‘Morita duals’ of 2VecG with respect to M(A, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Crucially, these inherit a fusion structure from the composition of module 2-functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We shall now demonstrate that for any 2VecG-module category M ≡ M(A, λ), the Hamiltonian HM is left invariant by topological operators organised into the Morita dual 2-category (2VecG)⋆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us begin by constructing topological surface operators labelled by simple objects in the fusion 2-category (2VecG)⋆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given the complex v×I depicted in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='21) and a simple object (F, ω, Ω) in (2VecG)⋆ M, we consider the following assignment of degrees of freedom: First, we assign as before the same gauge field g ∈ Z1( v, G) to 0 v and 1 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We then consider an assignment m of simple objects m[v1], m[v′ 1] ∈ M to every vertex v1 ⊂ 0 v and v′ 1 ⊂ 1 v such that m[v1] = Vecg[v1v2] ▷ m[v2] for every (v1v2) ∈ 0 v, m[v′ 1] = Vecg[v′ 1v′ 2] ▷ m[v′ 2] for every (v′ 1v′ 2) ∈ 1 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We notate via C0 g( v×I, M) the collection of assignments m fulfilling these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Every edge of the form (v′ 1v1) ⊂ v×I is further allocated a simple 1-morphism f[v′ 1v1] in the (possibly terminal) hom-category HomM(F(m[v1]), m[v′ 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given any prism (v1v2v3)×I ⊂ v ×I, every plaquette (v1v2)×I ≡ (v′ 1v′ 2v1v2) is labelled by a basis vector f[v′ 1v′ 2v1v2] in the vector space V ϵ(v′ 1v′ 2v1v2)� (g, m, f)[v′ 1v′ 2v1v2] � given by V +� (g, m, f)[v′ 1v′ 2v1v2] � := HomM � 1m[v′ 1] ◦ (Vecg[v1v2] ▷ f[v′ 2v2]) ◦ ωVecg[v1v2],m[v2] , f[v′ 1v1] ◦ F(1m[v1]) � , V −� (g, m, f)[v′ 1v′ 2v1v2] � := HomM � f[v′ 1v1] ◦ F(1m[v1]) , 1m[v′ 1] ◦ (Vecg[v1v2] ▷ f[v′ 2v2]) ◦ ωVecg[v1v2],m[v2] � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='39) where ϵ(v′ 1v′ 2v1v2) = ±1 depends on the orientation of (v1v2) relative to that of (v1v2v3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For conve- nience, we summarise these various notations below: m[v1] m[v3] F (m[v1])⊃m[v′ 1] m[v′ 3]⊂F (m[v3]) m[v′ 2]⊂F (m[v2]) m[v2] g[v1v2] g[v1v2] g[v2v3] g[v2v3] g[v1v3] g[v1v3] f[v′ 1v1] f[v′ 3v3] f[v′ 2v2] f[v′ 1v′ 2v1v2] f[v′ 2v′ 3v2v3] f[v′ 1v′ 3v1v3] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='40) Given assignments (g, m, f) described above, we finally associate to the prism (v1v2v3)×I the symbol Ωϵ(v1v2v3)� (g, m, f)[(v1v2v3)×I] � defined as the matrix entry corresponding to the vectors f[v′ 1v′ 2v1v2], ∼ 28 ∼ f[v′ 2v′ 3v2v3] and f[v′ 1v′ 3v1v3] of the map V +� (g, m, f)[v′ 1v′ 2v1v2] � b V +� (g, m, f)[v′ 2v′ 3v2v3] � ∼ −→ V +� (g, m, f)[v′ 1v′ 3v1v3] � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='41) that is determined by the component ΩVecg[v1v2],Vecg[v2v3],m[v3] of the invertible modification Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By convention, we set the symbols of this map to vanish whenever assignment f of 1-morphisms and basis vectors in M is such that one of hom-categories associated with edges of the form (v′ 1v1) is the terminal category or one of the vector spaces associated with plaquettes of the form (v1v2)×I is the zero vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Applying the rules described above, we find that the associahedron axiom (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20) yields an equation in terms of the symbols defined above that graphically translates as eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Summing over all possible labels associated with simplices shared by several prisms v×I that fulfil all the rules described above, as well as tracing over basis vectors associated with plaquettes shared by two prisms via the canonical pairing V −� (g, m, f)[v′ 1v′ 2v1v2] � b V +� (g, m, f)[v′ 1v′ 2v1v2] � → C , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='42) yields an operator that commutes with hM v,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In order to construct the surface operator commuting with the whole Hamiltonian HM, it suffices to extend the previous construction to the whole Σ△, thereby summing over all simple objects and simple 1-morphisms, and tracing over all basis basis vectors associated with plaquettes: ÿ g∈Z1( 0 v ,G) m∈C0 g( v×I,M) f � ź (v1v2v3)⊂Σ△ Ωϵ(v1v2v3)� (g, m, f)[(v1v2v3)×I] ����(g, m) � Σ1 △ ��� (g, m) � Σ0 △ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='43) It follows from the construction that fusion of surface operators is provided by the composition of the corresponding module 2-endofunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given the above, let us now consider line operators at the junction of two surface operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In many ways, the derivation is merely a lower-dimensional analogue of that above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a pair of simple objects F ≡ (F, ω, Ω) and ˜F ≡ ( ˜F, ˜ω, ˜Ω) in (2VecG)⋆ M, let (θ, Θ) be a simple 1-morphism in (2VecG)⋆ M between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As previously, we shall define the line operators by means of a labelled complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a cube (˜v1˜v2v1v2)×I, we consider an assignment of degrees of freedom that resembles that of the prisms: We assign the same group variable g[v1v2] to edges (v1v2), (v′ 1v′ 2), (˜v1˜v2) and (˜v′ 1˜v′ 2), as well as simple objects m[v1], m[v′ 1], m[˜v1], m[˜v′ 1] ∈ M to vertices of the cube such that m[˜v1] = m[v1], m[˜v′ 1] = m[v′ 1], m[v1] = Vecg[v1v2]▷m[v2], m[v′ 1] = Vecg[v1v2]▷m[v′ 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We further allocate to the corresponding edges sim- ple 1-morphisms f[v′ 1v1] and ˜f[˜v′ 1˜v1] in the (possibly terminal) hom-categories HomM(F(m[v1]), m[v′ 1]) and ∈ HomM( ˜F(m[˜v1]), m[˜v′ 1]), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Plaquettes (v′ 1v′ 2v1v2) and (˜v′ 1˜v′ 2˜v1˜v2) are labelled by basis vectors f[v′ 1v′ 2v1v2] and ˜f[˜v′ 1˜v′ 2˜v1˜v2] in (possibly zero) vector spaces V ϵ(v′ 1v′ 2v1v2)� (g, m, f)[v′ 1v′ 2v1v2] � and V ϵ(˜v′ 1˜v′ 2˜v1˜v2)� (g, m,˜f)[˜v′ 1˜v′ 2˜v1˜v2] � as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='39), respectively, whereas plaquettes (˜v′ 1˜v1v′ 1v1) are labelled by basis vectors s[˜v′ 1˜v1v′ 1v1] in vector spaces V ϵ(˜v′ 1˜v1v′ 1v1)� (m, f,˜f, s)[˜v′ 1˜v1v′ 1v1] � given by V +� (m, f,˜f, s)[˜v′ 1˜v1v′ 1v1] � := HomM �˜f[˜v′ 1˜v1] ◦ θm[v1] , f[v′ 1v1] � , V −� (m, f,˜f, s)[˜v′ 1˜v1v′ 1v1] � := HomM � f[v′ 1v1] , ˜f[˜v′ 1˜v1] ◦ θm[v1] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='44) ∼ 29 ∼ As before, let us summarise our notations via the following diagram: m[v1] m[v2] m[v1] m[v2] m[v1] m[v2]⊂F (m[v2]) m[v1] m[v2]⊂ ˜ F (m[v2]) g[v1v2] g[v1v2] g[v1v2] g[v1v2] f[v′ 1v1] ˜f[˜v′ 2˜v2] ˜f[˜v′ 1˜v1] f[v′ 2v2] ˜f[˜v′ 1˜v′ 2˜v1˜v2] s[˜v′ 1˜v1˜v′ 1˜v1] s[˜v′ 2˜v2˜v′ 2˜v2] f[v′ 1v′ 2 v1v2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='45) We finally associate to the cube (˜v1˜v2v1v2)×I the symbol Θϵ(˜v1˜v2v1v2)� (g, m, f,˜f, s)[(v1v2v3)×I] � corre- sponding to the vectors f[v′ 1v′ 2v1v2], ˜f[˜v′ 1˜v′ 2˜v1˜v2], s(˜v′ 1˜v1v′ 1v1) and s(˜v′ 2˜v2v′ 2v2) of the map V +� (g, m, f)[v′ 1v′ 2v1v2] � b V +� (m, f,˜f, s)[˜v′ 1˜v1v′ 1v1] � ∼ −→ V +� (m, f,˜f, s)[˜v′ 2˜v2v′ 2v2] � b V +� (g, m,˜f)[˜v′ 1˜v′ 2˜v1˜v2] � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='46) that is determined by the component ΘVecg[v1v2],m[v2] of the invertible modification Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By convention, we set matrix entries of this map to vanish whenever one of the hom-categories associated with edges of the form (v′ 1v1) or (˜v′ 1˜v1) is the terminal category, or one of the hom-spaces associated with plaquettes (˜v′ 1˜v1v′ 1v1) is the zero vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Applying all the rules introduced above to the following complexes v′ 2 v′ 1 v′ 3 ˜v′ 1 ˜v′ 3 v2 ˜v1 ˜v3 v1 v3 = ˜v′ 2 v′ 2 v′ 2 v′ 1 v′ 3 v1 v3 ˜v1 ˜v3 ˜v2 v2 v2 ˜v′ 1 ˜v′ 3 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='47) yields an equation in terms of symbols of ΩVecg[v1v2],Vecg[v2v3],m[v3], ΘVecg[v1v3],m[v3] on the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' and ˜ΩVecg[v1v2],Vecg[v2v3],m[v3], ΘVecg[v1v2],Vecg[v2v3]▷m[v3], ΘVecg[v2v3],m[v3] on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', where as before we trace over basis vectors associated with plaquettes shared by complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This equation is guaranteed by the coherence axiom (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, this equation means that a simple object in Hom(2VecG)⋆ M(F, ˜F) defines a topological line operator at the interface of two surface operators labelled by simple objects (F, ω, Ω) and ( ˜F, ˜ω, ˜Ω) in (2VecG)⋆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Vertical composition of 2VecG-module natural 2-transformations in (2VecG)⋆ M finally provides the fusion of topological lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6 Higher representation theory We demonstrated above that symmetry operators of Hamiltonians HM with M ≡ M(A, λ) form the fusion 2-category (2VecG)⋆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, this means that starting from a G-symmetric Hamiltonian that has been rewritten in terms of local operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4), simply replacing the implicit choice of 2VecG- module 2-category 2VecG by any other indecomposable 2VecG-module 2-category M(A, λ) yields a ∼ 30 ∼ dual model with a fusion 2-categorical (2VecG)⋆ M symmetry in virtue of the demonstration above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Within this context, identifying the symmetry of a dual model thus boils down to computing the Morita dual fusion 2-category (2VecG)⋆ M of 2VecG with respect to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Interestingly—and to some extent this is the purpose of the following sections—knowing from the general demonstration that a given model possesses a (2VecG)⋆ M symmetry does not mean it is easily verifiable in terms of explicit lattice operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10 We shall explicitly compute Morita duals in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7 but, in order to understand the resulting fusion 2-categories, we first need to discuss higher representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We set the stage with a review of the category theoretic viewpoint on (ordinary) representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a finite group G, we denote by [G, •] the 1-groupoid with object-set G and no non- trivial morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us consider the category Fun([G, •], Vec) of functors from [G, •] to the category Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By definition, an object V in Fun([G, •], Vec) assigns to every g ∈ G a vector space Vg in Vec, and thus amounts to a G-graded vector space of the form V = À g∈G Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Natural transformations in Fun([G, •], Vec) then correspond to grading preserving linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The convolution product of functors [G, •] → Vec, which descends from the multiplication rule of G, further endows Fun([G, •], Vec) with the structure of a fusion (1-)category according to (V d W)g := à x∈G Vx b Wx−1g , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='48) with unit 1 satisfying 1g = δg,1G C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Henceforth, we denote this fusion category by VecG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' There are |G|-many simple objects in VecG provided by the one-dimensional vector spaces Cg, for every g ∈ G, such that Cg dCh ≃ Cgh and HomVecG(Cg, Ch) ≃ δg,h C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Notice that VecG can be equivalently defined as the fusion category Mod(CG) of modules over the algebra CG of functions on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Fusion category VecG is merely the lower categorical analogue of the fusion 2-category 2VecG we have been considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More specifically, we shall think of 2VecG as a categorification of VecG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Another way to treat a finite group G as a 1-category is to consider the delooping of G defined as the 1-groupoid [•, G] with a single object • and Hom[•,G](•, •) = G such that the composition of morphisms is given by the multiplication rule of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As before, we can consider the category Fun([•, G], Vec) of functors [•, G] → Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By definition, an object ρ in Fun([•, G], Vec) assigns to the unique object • a vector space V := ρ(•), and to every morphism g : • → • a linear map ρ(g) : V → V fulfilling ρ(g) ◦ ρ(h) = ρ(gh) for every g, h ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In other words, ρ is a representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows that natural transformations in Fun([•, G], Vec) correspond to intertwiners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The symmetric monoidal structure of Vec endows Fun([•, G], Vec) with the structure of a fusion 1-category according to (ρ d ϱ)(•) := ρ(•) b ϱ(•) ≡ V b W , (ρ d ϱ)(g) := ρ(g) b ϱ(g) ∈ End(V b W) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='49) where the tensor products on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' are that in Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Henceforth, we denote by Rep(G) this fusion category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note that the simple objects are provided by the irreducible representations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given the equivalence between representations of G and modules over the group algebra C[G], we also have Rep(G) ∼= Mod(C[G]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the following, we shall find that Morita duals of 2VecG are often related to ‘higher’ notions of group representation obtained by following the ethos categorification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We remarked above that a group representation is equivalent to a module over the group algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us adopt this viewpoint and 10Already in (1+1)d systems, it is easy to construct models with Rep(G)-symmetries for instance, which are very tedious to confirm without a systematic approach analogous to the one employed in this manuscript [LDV22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 31 ∼ categorify it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall that C[G] is the associative algebra whose elements are given by formal linear combinations of group elements over C and multiplication rule descends from that of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Loosely speaking, categorifying the notion of group algebra requires in particular to loosen the associativity condition so that it only holds up to isomorphisms [BD98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' But this would be inconsistent with having coefficients valued in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' One solution is to consider instead coefficients valued in Vec, where we should think of Vec as being a categorification of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The result is the fusion category VecG, whose definition was reviewed above, thought as a group ‘2-algebra’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This incites us to consider a notion of ‘2-representation’ of a group G as a module category over VecG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We shall refine this notion of 2-representation later, but let us accept it for the moment and proceed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The notion of VecG-module category is defined in close analogy with that of 2VecG-module 2- category reviewed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, a (left) VecG-module category can be defined as a triple (N, ▷, α▷) consisting of a (C-linear finite semisimple) category N, a binary action functor ▷ : VecG × N → N and a natural isomorphism α▷ : (− d −) ▷ − ∼ −→ − ▷ (− ▷ −) referred to as the left module associator, which is required to satisfy a ‘pentagon axiom’ akin to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='11 Indecomposable module categories over VecG are obtained following the same recipe as in the 2VecG case: Given a subgroup B ⊆ G and a normalised 2-cocycle ψ in H2(B, U(1)), let N(B, ψ) be a category with object-set the set G/B of left cosets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A left VecG-module structure can be defined on N(B, ψ) via Cg▷N := (gr(N))B for any g ∈ G and N ∈ G/B, where r : G/B → G assigns to every left coset its representative element in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As before, we notate via bg,N the group element in B satisfying gr(N) = r(Cg ▷ N)bg,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Associativity of the multiplication in G imposes bg1g2,N = bg1,Cg2▷N bg2,N, for every g1, g2 ∈ G and N ∈ G/B, in exact analogy with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Thinking of the abelian group Hom(G/B, U(1)) as a left G-module, let us consider the 2-cochain α▷ ∈ C2(G, Hom(G/A, U(1))) defined as α▷(g1, g2)(N) := ψ(bg1,Cg2▷N, bg2,N) for any g1, g2 ∈ G and N ∈ G/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In virtue of the cocycle condition dψ = 1 and the equation above, we have α▷(g2, g3)(N) α▷(g1, g2g3)(N) = α▷(g1g2, g3)(N) α▷(g1, g2)(Cg3 ▷ N) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='50) for every g1, g2, g3 ∈ G and N ∈ G/B, so that α▷ is a Hom(G/B, U(1))-valued 2-cocycle of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Defining the natural isomorphism α▷ with components α▷ Cg1,Cg2,N := α▷(g1, g2)(N) · 1(g1g2r(N))B : (Cg1 d Cg2) ▷ N ∼ −→ Cg1 ▷ (Cg2 ▷ N) for every g1, g2 ∈ G and N ∈ G/B, we find that the triple (N(B, ψ), ▷, α▷) does define a left VecG-module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It is a result of Ostrik that all indecomposable module categories over VecG are of this form [Ost01, Ost02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' If VecG-module categories admit an interpretation as 2-representations of the group G, then VecG- module functors should be understood as 1-intertwiners between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As previously, the notion of VecG-module functor is defined in immediate analogy with that of 2VecG-module 2-functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, given a pair of (left) VecG-module categories (N, ▷, α▷) and (N ′, ·▷, α·▷), we define a module functor between them as a pair (F, ω) consisting of a functor F : N → N ′ and a natural isomorphism ω : F(− ▷ −) ∼ −→ − ·▷ F(−), which is required to satisfy a pentagon axiom involving both α▷ and α·▷ akin to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' There is also a notion of map between module functors, which within our context shall be interpreted as ‘2-intertwiners’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More precisely, given a pair of left VecG-module functors (F, ω) and ( ˜F, ˜ω), we define a module natural transformation between them as a natural transformation θ : F ⇒ ˜F satisfying (1Cg ·▷ θM) ◦ ωCg,M = ˜ωCg,M ◦ θCg▷M for all g ∈ G and M ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows from the definitions that, given a pair (N, N ′) of VecG-module categories, 1- and 2- intertwiners form a category that we denote by FunVecG(N, N ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Special attention is paid to Morita dual fusion categories (VecG)⋆ N (B,ψ) := FunVecG(N(B, ψ), N(B, ψ)) of VecG with respect to N(B, ψ) 11Since we shall encounter both VecG-module categories and 2VecG-module 2-categories at the same time in the following, we notate them via N and M, respectively, to facilitate the distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 32 ∼ [M¨u03, EGNO16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, these categories inherit a fusion structure from the composition of module functors [ENO08], so that considering module endofunctors of indecomposable module categories is a way to construct new fusion categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Treating VecG as a module category over itself, we have for instance (VecG)⋆ VecG ∼= VecG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Since the composition of module functors is a well- defined operation, we can further consider the 2-category Mod(VecG) consisting of (left) VecG-module categories and hom-categories of VecG-module functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This is another example of a fusion 2-category, still in the sense of Douglas and Reutter [DR18], where the fusion structure is obtained by defining a VecG-module structure on N b N ′ via Cg ▷ (N b N ′) := (Cg ▷ N) b (Cg ▷ N ′) for every g ∈ G, N ∈ N and N ′ ∈ N ′ [Gre10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Explicit formulae for the fusion of indecomposable VecG-module categories N(B, ψ) can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [ENO10] for the abelian case and in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Gre10] for the non-abelian one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We are now ready to refine the notion of 2-representation alluded to above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Firstly, we require a categorification of the notion of vector space, a natural candidate being the notion of 2-vector space introduced in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Secondly, let [•, G, •] be the 2-groupoid with unique simple object •, 1- morphisms labelled by group elements in G, and no non-trivial 2-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Mimicking the definition of Rep(G), we would like to consider the 2-category 2Rep(G) := 2Fun([•, G, •], 2Vec) of pseudofunctors, pseudonatural transformations and modifications between [•, G, •] and 2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Unpacking the definition, one finds that an object ρ in 2Rep(G) is a map ρ : [•, G, •] → 2Vec : �→ ρ(•) =: V : • g−→ • �→ V ρ(g) −−→ V ∈ End(V) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='51) assigning to the unique object • a 2-vector space V := ρ(•) and to every 1-morphism g : • → • a linear functor ρ(g) : V → V, in sush a way that composition of 1-morphisms is only preserved up to natural 2-isomorphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ρ further assigns to every pair of 1-morphisms labelled by g1, g2 ∈ G a natural 2-isomorphism ρg1,g2 : ρ(g1) ◦ ρ(g2) ∼ =⇒ ρ(g1g2) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='52) which is required to fulfill12 ρg1,g2g3 · [1ρ(g1) ◦ ρg2,g3] = ρg1g2,g3 · [ρg2,g3 ◦ 1ρ(g3)] , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='53) for any g1, g2, g3 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Introducing the notations ▷ : VecG × V → V, whereby Cg ▷ M := ρ(g)(M) for every M ∈ V, and α▷ Cg1,Cg2,M := (ρg1,g2)M, it follows from the 2-cocycle condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='53) that [1Cg1 ▷ α▷ Cg2,Cg3,M] ◦ α▷ Cg1,Cg2dCg3,M ◦ 1Cg1g2g3▷M = α▷ Cg1,Cg2,Cg3▷M ◦ α▷ Cg1dCg2,Cg3,M (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='54) holds for any g1, g2, g3 ∈ G and M ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Consequently, the triple (V, ▷, α▷) thus constructed defines a left VecG-module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, we can readily check that pseudonatural transformations and modifications in 2Rep(G) corresponds to VecG-module functors and VecG-module natural transfor- mations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Putting everything together, we have an equivalence 2Rep(G) ∼= Mod(VecG), thereby justifying referring to VecG-module categories as 2-representations of the group [GK06, Bar09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The symmetric monoidal structure of 2Vec endows 2Rep(G) with the expected fusion structure ac- cording to (ρ d ϱ)(•) := ρ(•) b ρ(•) ≡ V b ˜V , (ρ d ϱ)(g) := ρ(g) b ϱ(g) ∈ End(V b ˜V) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='55) 12As customary, we notate via ◦ and · the horizontal and vertical compositions of 2-morphisms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 33 ∼ The main reason to define 2-representations of G as pseudofunctors [•, G, •] → 2Vec is that it is readily generalisable to other scenarios relevant to our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We present two such scenarios below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let G be a 2-group with homotopy groups Q and L in degree one and two, respectively [BL03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Succinctly, a 2-group is a monoidal groupoid such that every object has a weak inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, the 2-group G has object-set Q, hom-sets HomG(q, q) = L with composition rule13 (q l1 −→ q) ◦ (q l2 −→ q) = (q l1+l2 −−−→ q) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='56) and monoidal structure (q1 l1 −→ q1) d (q2 l2 −→ q2) = (q1q2 l1+φq1(l2) −−−−−−−→ q1q2) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='57) for any q1, q1 ∈ Q and l1, l2 ∈ L, where φ− : Q → Aut(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As for a group, we distinguish two ways to treat a 2-group as a 2-groupoid, namely [Q, L, •] and [•, Q, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us focus for now on the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We are interested in pseudofunctors between [Q, L, •] and 2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Unpacking the definition, one finds that such a pseudofunctor is a map ρ : [Q, L, •] → 2Vec : q �→ ρ(q) =: Vq : q l−→ q �→ Vq ρ(l) −−→ Vq ∈ End(Vq) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='58) assigning to every group element q ∈ Q a 2-vector space Vq := ρ(q) and to every 1-morphism l : q → q a linear functor ρ(l) : Vq → Vq in such a way that composition of the 1-morphisms is only preserved up to natural 2-isomorphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ρ further assigns to every pair of 1-morphisms labelled by l1, l2 ∈ L a natural 2-isomorphism ρl1,l2 : ρ(l1) ◦ ρ(l2) ∼ =⇒ ρ(l1 + l2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='59) which is required to fulfil eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In close analogy with the constructions presented so far, we deduce that ρ amounts to a Q-graded 2-vector space V := Ð q∈Q Vq such that every homogeneous component Vq has the structure of a (left) VecL-module category, or alternatively of a 2-representation of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Pseudonatural transformations between pseudofunctors [Q, L, •] → 2Vec provide the corresponding 1-morphisms, which amount to Q-grading preserving VecL-module functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More concretely, every group element q ∈ Q together with an indecomposable VecL-module category V furnishes a simple object Vq in 2Fun([Q, L, •], 2Vec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The hom-category between two such simple objects Vq1 and ˜Vq2 is then provided by δq1,q2 FunVecQ(Vq1, ˜Vq2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', it is terminal unless q1 = q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, the convolu- tion product of pseudofunctors [Q, L, •] → 2Vec endows 2Fun([Q, L, •], 2Vec) with a fusion structure, whereby Vq1 d ˜Vq2 is the L-graded 2-vector space with homogeneous components (Vq1 d ˜Vq2)q = ð x∈Q (Vq1)x b (˜Vq2)x−1q = δq,q1q2 Vq1 b ˜Vq2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='60) equipped with a VecL-module structure defined by Cl ▷ (N b N ′) := (Cl ▷ N) b (Cφq−1 1 (l) ▷ N ′) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='61) for any N ∈ Vq1, N ′ ∈ V′ q2 and l ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Henceforth, we denote by 2VecG this fusion 2-category and refer to it as the 2-category of G-graded 2-vector spaces [DR18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For our applications, the monoidal product 13Notice that we write the product rule in L as an addition to emphasise the fact that it is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 34 ∼ of simple objects will not play a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We shall rather be interested in the monoidal product of simple 1-morphisms, which is of the same form as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Consider for instance the monoidal product of the identity 1-endomorphism 1Vq of a 1-simple object Vq with any simple 1-endomorphism of Vec1Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By virtue of (VecL)⋆ Vec ∼= Rep(L), which establishes the Morita equivalence between VecL and Rep(L) [EGNO16], a simple 1-endomorphism of the simple object Vec1Q in 2VecG is labelled by a character ρ(−) of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows in particular from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='61) that 1Vq d ρ(−) = ρ(φq−1(−)) d 1Vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the notation of the above paragraph, let us finally consider the 2-category 2Fun([•, Q, L], 2Vec), where we recall that [•, Q, L] is the 2-groupoid with single object • and hom-category Hom[•,Q,L](•, •) = G such that horizontal and vertical compositions are provided by the monoidal product and the com- position in G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This 2-category was investigated in detail in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Elg04, BM04, BBFW08], or more recently in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [BBFP22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As such, we shall keep our exposition brief and merely re- view the salient features of this 2-category following the description in terms of module categories and module functors proposed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Del22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Unpacking the definition we find that an object in 2Fun([•, Q, L], 2Vec) is a map ρ : [•, Q, L] → 2Vec : �→ ρ(•) =: V : q−→ • �→ V ρ(q) −−→ V ∈ End(V) : • q q l �→ V V ρ(q) ρ(q) ρ(l) ∈ EndEnd(V)(ρ(q)) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='62) assigning to the unique simple object • a 2-vector space V, to every morphism q : • → • a linear functor ρ(q) : V → V, and to every 2-morphism l : q ⇒ q a natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Moreover, vertical and horizontal compositions of 2-morphisms are strictly preserved, whereas the composition of 1-morphisms is only preserved up to natural 2-isomorphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ρ further assigns to every pair of 1-morphisms labelled by q1, q2 ∈ Q a natural 2-isomorphism ρq1,q2 : ρ(q1) ◦ ρ(q2) ∼ =⇒ ρ(q1q2) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='63) which is required to fulfil eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given two objects ρ and ϱ, a 1-morphism θ : ρ → ϱ between them is a pseudonatural transformation that assigns to • an object θ• in Fun(ρ(•), ϱ(•)) and to every morphism q : • → • a natural 2-isomorphism defined by θq : θ• ◦ ρ(q) ∼ =⇒ ϱ(q) ◦ θ• (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='64) such that θ1Q = 1θ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Compatibility with the composition of 1-morphisms in G requires the following coherence relation to be satisfied: [ϱq1,q2 ◦ 1θ•] · [1ϱ(q1) ◦ θq2] · [θq1 ◦ 1ϱ(q2)] = θq1q2 · [1θ• ◦ ρq1,q2] , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='65) for every q1, q2 ∈ Q, whereas naturality stipulates that θq · [1θ• ◦ ρ(l)] = [ϱ(l) ◦ 1θ•] · θq , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='66) for every 2-morphism l : q ⇒ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the same vein as the previous derivations, we find that a 2- vector space V together with endofunctors ρ(q) : V → V and natural 2-isomorphisms ρq1,q2 satisfying ∼ 35 ∼ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='53) amounts to a (left) VecQ-module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As a natural transformation, ρ(l) : ρ(q) ⇒ ρ(q) assigns to every simple object N ∈ V an endomorphism ρ(q)(N) → ρ(q)(N), which, together with ρ(l1) · ρ(l2) = ρ(l1 · l2), implies that ρ further assigns to every simple object N ∈ V a representation ρ(−)N : L → EndV(ρ(q)(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Crucially, ρ(l1) ◦ ρ(l2) = ρ(l1 ◦ l2) requires the following condition: ρ(φq(−))N = ρ(−)ρ(q)(N) , ∀ q ∈ Q and N ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='67) Given the above, a 1-morphism ρ → ϱ in 2Fun([•, Q, L], 2Vec) amounts to a VecQ-module functor (θ•, (θ−)−) between the corresponding VecQ-module categories, which, in virtue of the naturality condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='66), must satisfy the additional requirement (θq)N ◦ θ•(ρ(l)N) = ϱ(l)θ•(N) ◦ (θq)N , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='68) for every 2-morphism l : q ⇒ q and N ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, the symmetric monoidal structure of 2Vec endows 2Fun([•, Q, L], 2Vec) with a fusion structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Henceforth, we denote by 2Rep(G) this fusion 2-category and refer to it as the 2-category of 2-representations of the 2-group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7 Morita duals Guided by the derivations above, we shall now compute Morita duals of 2VecG with respect to various choices of 2VecG-module 2-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the context of this manuscript, this will establish that given a generic two-dimensional G-symmetric Hamiltonian, the models obtained by gauging the G symmetry, or sub-symmetries thereof, are left invariant by symmetry operators organised into fusion 2-categories of some higher representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Combined with the results of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5, this provides an answer to the question, what does it mean for a lattice model to commute with symmetry operators labelled by higher representations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Choosing G as a subgroup of itself and the trivial cocycle in H3(G, U(1)) yields the module 2- category 2Vec via the forgetful functor 2VecG → 2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The Morita dual (2VecG)⋆ Vec was found in in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Del21, D´ec22] to be equivalent as a monoidal 2-category to 2Rep(G), whose definition was given in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us briefly review this derivation for completeness, we encourage the reader to consult ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Del21] for detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By definition, an object in 2Fun2VecG(2Vec, 2Vec) consists of a 2-functor F : 2Vec → 2Vec, which is fully determined by a 2-vector space V := F(2Vec), an adjoint natural 2-equivalence ω prescribed by ωg : Vecg ▷ F(Vec) ∼ −→ F(Vecg ▷ Vec) ∈ Fun(V, V) , ∀ g ∈ G , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='69) as well as an invertible modification Ω defined as per eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='19) with components Ωg1,g2 ∈ HomFun(V,V)(ωg1 ◦ ωg2, ωg1g2) ∀ g1, g2 ∈ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='70) Isomorphisms ωg provide an action 2-functor ▷ : VecG × V → V via Cg ▷ M := ωg(M) for any M ∈ V, whereas maps Ωg1,g2 yields natural isomorphisms α▷ Cg1,Cg2,M := (Ωg1,g2)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows from the asso- ciahedron axiom (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20) that the triple (V, ▷, α▷) defines a left VecG-module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a pair (F, ω, Ω) and ( ˜F, ˜ω, ˜Ω) of 2VecG-module 2-endofunctors of 2Vec, a 2VecG-module natural transforma- tion between them is given by a choice of natural transformation θ : F ⇒ ˜F specified by a choice of functor ˆF ∈ Fun(V, ˜V) between the corresponding VecG-module categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The invertible modification Θ defined as per eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='36) is prescribed by a collection of natural transformations Θg ∈ HomFun(V,˜V)( ˆF ◦ ωg, ˜ωg ◦ ˆF) , ∀ g ∈ G , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='71) ∼ 36 ∼ endowing ˆF with a VecG-module structure ˆωCg,M := (Θg)M, so that 1-morphisms are given by VecG- module functors ( ˆF, ˆω) between the corresponding VecG-module categories, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, we can show that 2VecG-module natural 2-transformations are identified with VecG-module natural trans- formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, it follows from the composition of 2VecG-module functors (F, ω, Ω) and ( ˜F, ˜ω, ˜Ω) that the monoidal structure is obtained by defining a VecG-module structure on (F ◦ ˜F)(Vec) ≡ V b ˜V via ωg b ˜ωg : V b ˜V → V b ˜V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Putting everything together, this shows the monoidal equivalence (2VecG)⋆ 2Vec ∼= Mod(VecG) ∼= 2Rep(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the context of our work, this computation together with the results of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 shows that gauging the G symmetry of (2+1)d quantum theory results in a theory with a 2Rep(G) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This result appeared in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Del21] and was recovered in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [BBFP22a, BSNW22] in terms of separable algebras in fusion 2-categories [D´ec22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, this means that a theory with a gauged G symmetry host non-trivial topological surface operators labelled by indecomposable VecG-module categories as well as topological line operators labelled by VecG-module functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Our construction further teaches us how these operators explicitly act on a lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We have already seen an example in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2, and we shall see further examples below, but in general the surface operator associated with the indecomposable left VecG-module category (N ≡ N(B, ψ), ▷, α▷) is proportional to14 ÿ g∈Z1(Σ△,G) n∈C0 g(Σ△,N ) � ź (v1v2v3)⊂Σ△ α▷(g[v1v2], g[v2v3])(n[v1])ϵ(v1v2v3) � |g⟩⟨g| , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='72) such that α▷ Cg1,Cg2,N ≡ α▷(g1, g2)(N)·1(g1g2r(N))B and C0 g(Σ△, N) refers to the collection of assignments n of simple objects in N at every vertex of Σ△ such that n[v1] = Cg[v1v2] ▷ n[v2] for every (v1v2) ⊂ Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us emphasise that the conditions n[v1] = Cg[v1v2] ▷ n[v2] are with respect to the module structure of the VecG-module 1-category N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The same operators appeared in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Del21] in the context of the (3+1)d gauge models of topological phases of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the case where B = {1G}, it is convenient to think of assignments n ∈ C0 g(Σ△, N) as (virtual) matter fields n ∈ C0(Σ△, G) fulfilling dn = g, as we did in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note finally that we recover through Hom2Rep(G)(Vec, Vec) = FunVecG(Vec, Vec) ∼= Rep(G) that Wilson lines generate a 1-form symmetry for the G-gauged theory (see sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More generally, given a surface operator labelled by a VecG-module category N(B, ψ), we can insert topological lines on it that organise into the Morita dual fusion 1-category (VecG)⋆ N (B,ψ) = FunVecG(N(B, ψ), N(B, ψ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Combining the construction of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 and the computations above, these line operators can be implemented on the lattice in terms of matrix product operators whose building blocks evaluate to the module structure of the functors in (VecG)⋆ N (B,ψ) [LFH+20, LDOV21, LDV22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We commented above that (VecG)⋆ Vec ∼= Rep(G) signifies that the fusion 1-categories VecG and Rep(G) are Morita equivalent, which implies in particular the equivalence Mod(VecG) ∼= Mod(Rep(G)) [EGNO16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Interestingly, the fusion 2-category Mod(Rep(G)) can be thought of as the idempotent completion of the delooping of the braided fusion 1-category Rep(G), which encodes the line operators of the trivial surface operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Physically, this idempotent completion amounts to including surface operators obtained by condensing suitable algebras of line operators in Rep(G), and as such these surface operators are often referred to as condensation defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In this context, the surface operators in 2Rep(G) are labelled by algebra objects in Rep(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By definition, such an algebra object in Rep(G) is a G-algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' an associative unital algebra equipped with a G-action by algebra automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We know from the Morita equivalence between Rep(G) and VecG that Morita classes of indecomposable 14In the notations of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5, we are using the fact that simple morphisms f[v′ 1v1] are given by simple objects in hom-categories that are equivalent N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 37 ∼ algebra objects in Rep(G) are labelled by pairs (B, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Davydov then provided in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Dav09] a recipe to explicitly construct the corresponding G-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We already showed in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='26) for the case of G = Z2 how to construct the non-trivial surface operator from this condensation perspective, and we shall provide additional comments along these lines in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note finally that, more generally, picking a representative of a non-trivial cohomology class in H3(G, U(1)) yields a 2VecG-module 2-category M(G, λ) ≡ 2Vecλ that only differ from 2Vec in the choice of module pentagonator π▷, which is such that π▷ Vecg1,Vecg2,Vecg3,C = λ(g1, g2, g3) for any g1, g2, g3 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Importantly, it follows immediately from the definitions that we still have (2VecG)⋆ 2Vecλ ∼= 2Rep(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The subsequent examples require the group G to be isomorphic to a semi-direct product Q ⋉φ L with L abelian, in which the multiplication is given by (q1, l1)(q1, l2) = (q1q2, l1 + φq1(l2)) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='73) for any q1, q2 ∈ Q and l1, l2 ∈ L, where we are using the notation of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Introducing projection maps ϖQ : G → Q and ϖL : G → L, every group element in G admits a decomposition of the form g ≡ (ϖQ(g), ϖL(g)) ∈ Q ⋉φ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Consider the 2VecG-module 2-category M(L, 1) ∼= 2VecQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15 We find that the Morita dual (2VecG)⋆ 2VecQ is equivalent as a monoidal 2-category to the fusion 2-category 2VecG := 2Fun([Q, L, •], 2Vec) of G-graded 2-vector spaces defined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We sketch below the main steps of this derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Any 2VecG-module 2-endofunctor of 2VecQ is in particular an object in (2VecQ)⋆ 2VecQ ∼= 2VecQ so that an object Vq1 ∈ 2VecQ determines a 2VecG-module endofunctor of 2VecQ of the form F(−) = − d Vq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The 2VecG-module structure on F is then prescribed by a collection of functors ωg,q ∈ Hom2VecQ � Vecg▷F(Vecq), F(Vecg▷Vecq) � = Hom2VecQ(VecϖQ(g)qdVq1, VecϖQ(g)qdVq1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='74) satisfying a coherence relation up to an invertible modification Ω defined as per eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='19) with com- ponents Ωg1,g2,q ∈ Hom2VecQ � ωg1,ϖQ(g2)q ◦ ωg2,q , ωg1g2,q � ∀ g1, g2 ∈ G and q ∈ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='75) As before, 1-morphisms between such objects correspond to 2VecG-module natural transformations between the corresponding 2-endofunctors of 2VecQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Before analysing more carefully this 2-category (2VecG)⋆ 2VecQ, let us immediately consider its fusion structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall that the monoidal product of objects in (2VecG)⋆ 2VecQ is provided by the composition of the corresponding module 2-endofunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let (F, ω, Ω) and ( ˜F, ˜ω, ˜Ω) be two 2VecG-module 2-endofunctors of 2VecQ such that F(−) = − d Vq1 and ˜F(−) = − d ˜Vq2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Composition yields a new module 2-endofunctor of 2VecQ given by ( ˜F ◦ F)(−) = − d (Vq1 d ˜Vq2), whose 2VecG-module structure is provided by (˜ω ◦ ω)g,q : Vecg ▷ ( ˜F ◦ F)(Vecq) ˜ωg,qq1 −−−−→ ˜F(Vecg ▷ F(Vecq)) ˜ F (ωg,q) −−−−−→ ( ˜F ◦ F)(Vecg ▷ Vecq) = ˜F(ωg,q) ◦ ˜ωg,qq1 ∈ Hom2VecQ(VecϖQ(g)q d (Vq1 d ˜Vq2), VecϖQ(g)q d (Vq1 d ˜Vq2)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='76) 15By definition, L ⊆ G is a normal subgroup so that G/L isomorphic to Q with the isomorphism being provided by the composition of the natural embedding L → G and the natural projection G → G/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows that we can identify M(L, 1) with 2VecQ with the 2VecG-module structure being provided by Vecg ▷ Vecq := VecϖQ(g)q, for all g ∈ G and q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 38 ∼ Let us now consider the monoidal equivalence given by16 (Vq1, ωg,q, Ωg1,g2,q) �→ (Vq1, ωl := ω(1Q,l),1Q, Ωl1,l2 := Ω(1Q,l1),(1Q,l2),1Q) (Vq1, ωl, Ωl1,l2) �→ (Vq1, ωg,q := ωag,q, Ωg1,g2,q := Ωag1,ϖQ(g2)q,ag2,q) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='77) where ag,q ∈ L was defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Invoking this equivalence, we find that the Q-graded 2- vector space Vq1 has the structure of a VecL-module category with the module action and the module associator being provided by the collection of maps ωl and Ωl1,l2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Furthermore, the fusion structure is now obtained by endowing Vq1 d ˜Vq2 with the VecL-module structure (˜ω ◦ ω)l = (˜ω ◦ ω)(1Q,l),1Q = ω(1Q,l),1Q d ˜ω(1Q,l),q1 = ωl d ˜ωφq−1 1 (l) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='78) where we used the fact that a(1Q,l),q1 = φq−1 1 (l), which agrees with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We can now check that 1-morphisms in (2VecG)⋆ 2VecQ amounts to Q-grading preserving VecL-module functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Putting everything together, this motivates the equivalence (2VecG)⋆ 2VecQ ∼= 2VecG, where G is the 2-group defined in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Together with previous results, this computation shows that gauging the L sub-symmetry of a (Q ⋉φ L)-symmetric (2+1)d quantum theory results in a theory with a 2VecG symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Although it is a little bit tedious to explicitly write down the lattice realisations of the corresponding topological surfaces and lines in general—but these can be obtained from the construction in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5—we shall consider specific examples in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Still assuming G ≃ Q⋉φ L, let us now consider the 2VecG-module 2-category M(Q, 1) ∼= 2VecG/Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17 Even though G/Q is not isomorphic to L as a group, we have |G/Q| = |L| and thus we label simple objects in M(Q, 1) by group elements in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We thus write the 2VecG-structure of M(Q, 1) as Vecg ▷ Vecl := VecϖL(g)+φϖQ(g)(l) for every g ∈ G and l ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Analogously to the previous computation, objects in (2VecG)⋆ 2VecG/Q are functors F(−) = − d V with V = Ð l1∈L Vl1 ∈ 2VecG/Q equipped with a 2VecG-module structure provided by ωg,l ∈ Hom2VecG/Q(Vecg ▷ (Vecl d V), VecϖL(g)+φϖQ(l) d V) = Hom2VecG/Q(V, V) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='79) satisfying a coherence relation to up an invertible modification Ω defined as per eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='19) with com- ponents Ωg1,g2,l ∈ Hom2VecG/Q(ωg1,ϖL(g1)+φϖQ(g1)(l) ◦ ωg2,l, ωg!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='g2,l) , ∀ g1, g2 ∈ G and l ∈ L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='80) Still in the same vein as the previous computation, let us consider the equivalence provided by (V, ωg,l, Ωg1,g2,l) �→ (V, ωq := ω(q,0L),0L, Ωq1,q2 := Ω(q1,0L),(q2,0L),0L) (V, ωq, Ωq1,q2) �→ (V, ωg,l := ωag,l, Ωg1,g2,l := Ωag1,ϖL(g1)+φϖQ(g1)(l),ag2,l) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='81) where ag,l ∈ Q was defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, invoking this equivalence, we find that the L-graded 2-vector space V has the structure of a VecQ-module category with the module action ▷ and the module associator α▷ being provided by the collections of maps ωq and Ωq1,q2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 16This equivalence essentially follows the isomorphism Hn(G, Hom(G/L, U(1))) ≃ Hn(L, U(1)) provided by Shapiro’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 17Note the slight abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Since Q is not a normal subgroup, the quotient G/Q is not equipped with a group structure and thus the 2-category 2VecG/Q is not monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 39 ∼ Moreover, going back to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='79), we find that ωq = À l1∈L ωg|Vl1 where ωq|Vl1 : Vφq(l1) → Vl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Associating to every object N ∈ V a group element l1(N) in such a way that N ∈ Vl1(N), we must have Cq ▷ N := ωq|Vl1 (N) ∈ Vl1 for every N ∈ Vφq(l1) and thus l1(Cq ▷ N) = φq−1(l1(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, invoking the isomorphism L ≃ L∨ = Hom(L, U(1)) and defining a Q-action on L∨ via q ▷ ρ(−) = ρ(φq−1(−)), we can equivalently state that an object in (2VecG)⋆ 2VecQ/G corresponds to a VecQ-module category V such that for every object N ∈ V, we assign a character ρ(−)N such that ρ(φq(−))N = ρ(−)Cq▷N, for every q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This is precisely the defining condition given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Analysing 1- and 2-morphisms under the same scope, it is then fairly immediate to obtain (2VecG)⋆ 2VecG/Q ∼= 2Rep(G), where G is the 2-group defined in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We summarise in the diagram below the Morita equivalences evoked in this section for a finite group G ≃ Q ⋉φ L: 2VecG 2Rep(G) 2VecG 2Rep(G) 2Vec 2Vec 2VecQ 2Rep(Q) 2VecG/Q 2Rep(G/Q) with 2VecG := 2Fun([G, •, •], 2Vec) , 2Rep(G) := 2Fun([•, G, •], 2Vec) , 2VecG := 2Fun([Q, L, •], 2Vec) , 2Rep(G) := 2Fun([•, Q, L], 2Vec) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='82) where fusion 2-categories connected by a double arrow are Morita equivalent with respect to the module 2-category labelling the arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note that the equivalence (2VecG)⋆ 2Vec ∼= 2Rep(G) holds for arbitrary G as demonstrated at the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Although we have not explicitly constructed all the Morita equivalences displayed above, we included them in this diagram for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18 We leave to future work a more systematic and general treatment of such Morita equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8 Gauging the transverse-field G-Ising model Let us now illustrate some of the concepts presented in this section with a series of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Starting from a finite group generalisation of the transverse-field Ising model, we shall construct various dual models obtained by gauging sub-symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Within our approach, this amounts to writing the initial model in terms of local operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17), and then simply replacing the initial module 2-category by another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By virtue of our construction, we already know that the resulting Hamiltonians will commute with symmetry operators encoded into Morita duals with respect to the corresponding module 2-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For now, the focus will be on deriving the various dual models using effective local operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='30) obtained after resolving kinematical constraints of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the following sections, we shall choose specific groups and analyse in detail the dual symmetries by translating the symmetry operators defined in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 into explicit spin operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a finite group G, let M ≡ M(A, λ) be an indecomposable 2VecG-module 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We are interested in Hamiltonians of the form HM = ÿ v⊂Σ△ 4ÿ n=1 hM v,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='83) 18Equivalence (2VecG)⋆ 2Vec ∼ = 2Rep(G) was established in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [D´ec22] ∼ 40 ∼ For any vertex v ⊂ Σ△ and gauge field g ∈ Z1( I v, G), the defining U(1)-coefficients hv,n(g) are chosen to be hv,1(g) := −Jδg[v′v],1δg[v v+ˆu],1 , hv,2(g) := −Jδg[v′v],1δg[v v+ˆv],1 , hv,3(g) := −Jδg[v′v],1δg[v v+ ˆ w],1 , hv,4(g) := − Jκ |G| , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='84) where the branching structure of I v is that given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This is all the data required to define a Morita class of Hamiltonian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Specific matrix realisations of this Hamiltonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' representatives of the Morita class, are obtained by choosing specific 2VecG-module 2-categories M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We consider below four different choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us begin with the choice M(A = {1G}, 1) ∼= 2VecG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' the module 2-category 2VecG over itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall that local operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='30) act on the effective Hilbert space obtained after resolving the kinematical constraints (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Since the kinematical constraints are such that degrees of freedom assigned to edges are fully determined by those assigned to vertices, we are left with a tensor product Hilbert space of the form  v C[G] ∋ |m⟩, where m ∈ C0(Σ△, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the notation of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4, this is the statement that the assignment ag,m is such that ag,m[v1v2] = 1G for every edge (v1v2) ⊂ I v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It immediately follows from the definition of the effective local operators and the choice of coefficients (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='84) that h2VecG v,4 acts as h2VecG v,4 = −Jκ |G| ÿ x∈G Lx v , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='85) where we recall that Lx v : |m[v]⟩ �→ |xm[v]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, we find that local operators h2VecG v,n=1,2,3 act as −JΠ1G v,v+ˆu, −JΠ1G v,v+ˆv and −JΠ1G v,v+ ˆ w, respectively, where the vectors (ˆu, ˆv, ˆw) were introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2) and Π1G v1,v2 := ř m δm[v1]−1m[v2],1G|m⟩⟨m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Putting everything together, we obtain H2VecG = −J ÿ e Π1G s(e),t(e) − Jκ |G| ÿ v ÿ x∈G Lx v , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='86) which we recognise as the finite group generalisation of the transverse-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We can readily check that this Hamiltonian possesses a G symmetry, which is consistent with (2VecG)⋆ 2VecG ∼= 2VecG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now construct dual models resulting from gauging sub-symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Within our framework, gauging the whole G symmetry—or rather 2VecG symmetry—amounts to choosing the 2VecG-module 2-category M(G, 1) ∼= Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Clearly, with this choice, there are no degrees of freedom left at vertices so the effective microscopic Hilbert space is spanned by states |g⟩ ∈  e C[G], where g ∈ Z1(Σ△, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The condition dg = 1G imposes kinematical constraints g[v1v2]g[v2v3] = g[v1v3] for every triangle (v1v2v3) ⊂ Σ△ so that g defines a G-gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the notation of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4, we have ag,m = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Going back to the definition of the local operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='30), it readily follows from the flatness condition of g that h2Vec v,4 = − Jκ |G| ÿ x∈G � ź e→v Rx e �� ź e←v Lx e � ≡ − Jκ |G| ÿ x∈G Ax v ≡ −JκAv , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='87) where e → v and e ← v refer to edges e ⊂ Σ△ such that t(e) = v and s(e) = v, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, we find that the local operators h2VecG v,n=1,2,3 read −JΠ1G (v v+ˆu), −JΠ1G (v v+ˆv) and −JΠ1G (v v+ ˆ w), respectively, where Π1G e = ř g δg[e],1G|g⟩⟨g|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Putting everything together, we obtain H2Vec = −J ÿ e Π1G e − Jκ ÿ v Av , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='88) ∼ 41 ∼ which we recognise as the pure Ising G-gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We know from the general construction that this model has a (2VecG)⋆ 2Vec ∼= 2Rep(G) symmetry and we provided in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='72) a formula for constructing the corresponding surface and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the following section, we shall study these symmetry operators on the lattice in more detail for specific choices of input group G, but let us make a few general comments in the meantime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Straightforward examples of surface operators are those associated with simple objects of the form Vecψ ∈ 2Rep(G), where Vecψ is the VecG-module category that only differs from Vec in the choice of module associator α▷, which is such that α▷ Cg1,Cg2,C = ψ(g1, g2) for every g1, g2 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Going back to the general definition provided in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 we find these surface operators act diagonally by multiplication by the evaluation of the 3-cocycle characterising the corresponding module associator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In symbols, these are proportional to ÿ g∈Z1(Σ△,G) � ź (v1v2v3) ψ(g[v1v2], g[v2v3])ϵ(v1v2v3)� |g⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='89) In particular, the commutation relation with operators Av introduced in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='88) follows from the 2-cocycle condition dψ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Topological lines living on such a surface operator were shown in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 to be labelled by 1-endomorphisms of Vecψ in 2Rep(G), which correspond by definition to VecG- module endofunctors of Vecψ in FunVecG(Vecψ, Vecψ) ∼= Rep(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Therefore, these amount to ordinary Wilson lines labelled by representations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Explicitly, the closed Wilson line operator labelled by ρ ∈ Rep(G) with support the closed path ℓ reads ÿ g∈Z1(Σ△,G) tr � ź e⊂ℓ ρ(g[e]) � |g⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='90) The fact that these commute with the Hamiltonian eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='86), and in particular with operators Av, follows from the gauge invariance of Wilson loop operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These generate the 1-form Rep(G) sym- metry of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We could also consider an open version of the surface operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='89) labelled by Vecψ, together with topological lines living at the interface of this surface operator and the trivial one labelled by Vec ∈ 2Rep(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Mimicking the derivation of (VecG)⋆ Vec ∼= Rep(G) [EGNO16], we immedi- ately find that such topological lines are labelled by simple objects in FunVecG(Vec, Vecψ) ∼= Repψ(G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' projective representations of the group G with Schur’s multiplier ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Explicit examples of such symmetry operators are presented in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note finally that instead of considering the 2VecG-module 2-category 2Vec, we could have consid- ered instead M(G, λ) ∼= 2Vecλ where λ is a non-trivial 3-cocycle in H3(G, U(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall from sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7 that 2Vecλ only differs from 2Vec in the choice of module pentagonator π▷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Physically, this amounts to the λ-twisted gauging of the G symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the case of Z2, and given the only non-trivial 3-cocycle in H3(Z2, U(1)) ≃ Z2, the resulting model would precisely correspond to that considered in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We already commented on the fact that (2VecG)⋆ 2Vecλ ∼= 2Rep(G) so the resulting twisted G-gauge theory would have the symmetry operators as H2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In order to proceed with the next two cases, we further assume that the group G is a semi-direct product of the form G ≃ Q ⋉φ L with L abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall that we write group elements as g ≡ (ϖQ(g), ϖL(g)) ∈ Q ⋉φ L and the multiplication is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now consider the gauging of the L sub-symmetry, which amounts to choosing the 2VecG-module 2-category M(A = L, 1) ∼= 2VecQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We know from sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4 that the effective microscopic Hilbert space is spanned by states |l, m⟩ ∈  e C[L]  v C[Q], where l ∈ Z1(Σ△, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now work out how the local operators hM(L,1) v,n ∼ 42 ∼ act on this Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Going back to the definition, we have hM(L,1) v,4 = − Jκ |G| ÿ g∈Z1( I v,G) m∈C0 g( I v,M) ��(ag,m, m) � 1 v ��� (ag,m, m) � 0 v ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='91) We shall first consider the operator associated with a fixed pair (g, m) ∈ Z1( I v, G) × C0 g( I v, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Up to the scalar prefactor −Jκ |G| , this operator acts on the state |m[v]⟩ as |m[v]⟩ �→ |m[v′]⟩ = |Vecg[v′v] ▷ m[v]⟩ = |ϖQ(g[v′v]) m[v]⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='92) while it acts on a state |l[vv1]⟩ ≡ |ag,m[vv1]⟩ = |ag[vv1],m[v1]⟩ as |l[vv1]⟩ �→ |ag[v′v]g[vv1],m[v1]⟩ = |l[vv1] + ag,m[v′v]⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='93) where we made use of (the abelian version of) eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, it acts on a state |l[v1v]⟩ ≡ |ag,m[v1v]⟩ = |ag[v1v],m[v]⟩ as |l[v1v]⟩ �→ |ag[v1v]g[v′v]−1,g[v′v]▷m[v]⟩ = |l[v1v] + ag,m[vv′]⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='94) Invoking eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='28), we have ag,m[v1v2] = φm[v1]−1� ϖL(g[v1v2]) � so that ag,m[v′v] = φm[v′]−1� ϖL(g[v′v]) � , ag,m[vv′] = −φm[v′]−1� ϖL(g[v′v]) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='95) where we used the fact that g[vv′] = g[v′v]−1 ≡ � ϖQ(g[v′v])−1, −φϖQ(g[v′v])−1� ϖL(g[v′v]) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These expressions in turn allow us to rewrite the actions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='93) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='94) more explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Keeping in mind that m[v′] = ϖQ(g[v′v]) m[v], we obtain the following expression for the local operator hM(L,1) v,4 : hM(L,1) v,4 = − Jκ |G| ÿ x∈G � ź e→v φRx e,v �� ź e←v φLx e,v � LϖQ(x) v ≡ − Jκ |G| ÿ x∈G φAx v ≡ −JκφAv , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='96) where φLx e,v : |l[e]⟩ �→ |l[e] + φm[v]−1(ϖL(x))⟩ , φRx e,v : |l[e]⟩ �→ |l[e] − φm[v]−1(ϖL(x))⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='97) Moreover, it immediately follows from the definitions that that local operators hM(L,1) v,n=1,2,3 act as −JΠ1Q,0L (v v+ˆu), −JΠ1Q,0L (v v+ˆv) and −JΠ1Q,0L (v v+ ˆ w), respectively, where Π1Q,0L (v1v2) := ÿ m,l δm[v1]−1m[v2],1Q δl[v1v2],0L |l, m⟩⟨l, m| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='98) Putting everything together, we obtain19 HM(L,1) = −J ÿ e Π1Q,0L e − Jκ ÿ v φAv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='99) 19An alternative Hamiltonian resulting from the gauging of a normal subgroup sub-symmetry of H2VecG is often found in the literature, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [WBV17, TJV21, TVV22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The Hamiltonian found in these references is related to ours via unitary transformation |φm(l), m⟩⟨l, m|, where φm(l)[e] = φm[s(e)](l[e]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The point of this additional unitary is for the remaining 0-form 2VecQ to be on-site so it can be subsequently gauged following the canonical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' However, the resulting model then possesses a twisted 1-form Rep(L) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 43 ∼ We know from the general construction of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 that this Hamiltonian must commute with symmetry operators encoded into the Morita dual (2VecG)⋆ M(L,1), which was shown in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7 to be equivalent to the fusion 2-category 2VecG of G-graded 2-vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, the model possesses a 1-form Rep(L) symmetry, which is acted upon by a 0-form 2VecQ symmetry that is not on-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Instead of explaining the lattice implementations of these symmetry operators in the general case, we shall provide explicit parametrisations in terms of spin operators in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 5 for the case of the symmetric group S3 of degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, we consider gauging the Q sub-symmetry, which amounts to choosing the 2VecG-module 2-category M(A = Q, 1) ∼= 2VecG/Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The effective microscopic Hilbert space is spanned by states |q, m⟩ ∈  e C[Q]  v C[L] where q ∈ Z1(Σ△, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20 Mimicking the previous derivation, given a fixed pair (g, m) ∈ Z1( I v, G) × C0 g( I v, M) and up to the scalar prefactor −Jκ |G| , we have an operator that acts on the state |m[v]⟩ as |m[v]⟩ �→ |m[v′]⟩ = |Vecg[v′v] ▷ m[v]⟩ = ��ϖL(g[v′v]) + φϖQ(g[v′v])(m[v]) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='100) while it acts on states |q[vv1]⟩ ≡ |ag,m[vv1]⟩ and |q[v1v] ≡ ||ag,m[vv1]⟩ as |q[vv1]⟩ �→ ��ϖQ(g[v′v])q[vv1] � and |q[v1v]⟩ �→ ��q[v1v]ϖQ(g[v′v])−1� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='101) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the latter equations, we used the fact ag,m[v1v2] = ag[v1v2],m[v2] = ϖQ(g[v1v2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We thus obtain the following expression for the local operators hM(Q,1) v,4 : hM(Q,1) v,4 = − Jκ |G| ÿ x∈G � ź e→v RϖQ(x) e � φLx v � ź e←v LϖQ(x) e � ≡ − Jκ |G| ÿ x∈G φ�Ax v ≡ −Jκφ�Av (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='102) where φLx v : |m[v]⟩ �→ ��ϖL(x) + φϖQ(x)(m[v]) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='103) Similarly, we find that local operators hM(Q,1) v,n=1,2,3 acts as −JΠ0L,1Q (v v+ˆu), −JΠ0L,1Q (v v+ˆv) and −JΠ0L,1Q (v v+ ˆ w), re- spectively, where Π0L,1Q (v1v2) := ÿ m,q δm[v1],m[v2] δq[v1v2],1Q |q, m⟩⟨q, m| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='104) Putting everything together, we obtain HM(Q,1) = −J ÿ e Π0L,1Q e − Jκ ÿ v φ�Av .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='105) We know from the general construction of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 that this Hamiltonian must commute with symmetry operators encoded into the Morita dual (2VecG)⋆ M(Q,1), which was shown in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7 to be equivalent to the fusion 2-category 2Rep(G) of 2-representations of the 2-group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As for the previous example, we shall refrain from describing the lattice implementations of the corresponding topological surfaces and topological lines in the general case, and shall rather focus in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 5 on the specific case of the symmetric group S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 20Recall that even though G/Q is not isomorphic to L as a group, we can still identify objects M in M(Q, 1) with group elements in l ∈ L such that M = lQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 44 ∼ Back to the transverse-field Ising model We conclude this section by specialising once more to the case of the transverse-field (Z2-)Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us focus on Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10) obtained by choosing the 2VecZ2-module category 2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We es- tablished that by construction this model has a 2Rep(Z2) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' There are two simple objects in 2Rep(Z2) provided by the two indecomposable VecZ2-module categories, namely Vec and VecZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The corresponding surface operators were notated via Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' and UZ2 in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows from the alternative definition provided in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='22) and the preceding paragraph that UZ2 indeed corresponds to surface operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='72) for N = VecZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Line operators living on the surface operator Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' are now identified with simple 1-morphisms in Hom2Rep(Z2)(Vec, Vec) ∼= Rep(Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Line oper- ators labelled by the non-trivial representation of Z2 as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='90) readily correspond to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, we recover line operators on the surface operator UZ2 as simple 1-morphisms in Hom2Rep(Z2)(VecZ2, VecZ2) ∼= VecZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' What about line operators at the junctions of surface operators UZ2 and Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We established in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2 that such lines are unique up to isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Within the framework of this section, these correspond to the unique simple objects in the hom-categories Hom2Rep(Z2)(VecZ2, Vec) = FunVecZ2 (VecZ2, Vec) ∼= Vec and FunVecZ2 (Vec, VecZ2) ∼= Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Moreover, composition of VecZ2-module functors FunVecZ2 (VecZ2, Vec) × FunVecZ2 (Vec, VecZ2) → Rep(Z2) informs us that composing the corresponding line operators yields a line operator living on Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' labelled by the regular representation in Rep(Z2), which is compatible with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, composition of VecZ2-module functors FunVecZ2 (Vec, VecZ2) × FunVecZ2 (VecZ2, Vec) → VecZ2 informs us that compos- ing the corresponding line operators yields a line operator living on UZ2 labelled by the object C0 ‘C1 in VecZ2, which is compatible with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, the monoidal structure of 2Rep(Z2) is such that VecZ2 d VecZ2 ∼= VecZ2 ‘ VecZ2, which amounts to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20) when Σ is the two-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' SECTION 4 Example: doubled transverse-field Ising model In this section, we study in detail the symmetry structure of the model obtained by gauging the Z2 2 symmetry of the doubled transverse-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1 Symmetric Hamiltonian and gauging The starting point is the doubled transverse-field Ising model on a triangulation Σ△ of a closed oriented surface Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Pairs of qubit degrees of freedom are assigned to vertices v ⊂ Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We identify such an assignment with a choice of 0-cochain m ∈ C0(Σ△, Z2 2) so the microscopic Hilbert space is provided by the tensor product  v C[Z2 2] ≃ C4, on which two sets of Pauli operators denoted as σµ,I v with I = 1, 2 act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The doubled transverse-field Ising model is then defined via the Hamiltonian H 2VecZ2 2 = − 2ÿ I=1 � JI,1 ÿ e σz,I s(e)σz,I t(e) + JI,2 ÿ v σx,I v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1) The model has a (0-form) global Z2 2 symmetry implemented by surface operators Og = ź v (σx,1 v )g1(σx,2 v )g2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2) for every g ≡ (g1, g2) ∈ Z2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Fusion rules of these surface operators are dictated by the multiplication rule in Z2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 45 ∼ In the language of the previous section, the symmetry structure of this model is encapsulated in the fusion 2-category 2VecZ2 2, whose four simple objects correspond to the surface operators O(0,0), O(0,1), O(1,0) and O(1,1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Moreover, recall that for any simple object Vecg in 2VecZ2 2, its endo-category is equivalent to Vec, whose unique simple object corresponds to the identity line operator living on Og.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, since there are no 1-morphisms in 2VecZ2 2 between distinct simple objects, there are no topological lines between distinct surface operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Note that for conciseness we only consider a minimal Z2 2-symmetric transverse-field Ising model, which realises the symmetric paramagnetic and symmetry-broken gapped phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, this Hamiltonian does not realise any SPT phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='21 However, as was explained in the previous section, details of the Hamiltonian are irrelevant to the ensuing analysis of the gauging procedure and the symmetry structure of the gauged model, so that the following derivations hold for any model with the same Z2 2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given the input fusion 2-category 2VecZ2 2, Hamiltonian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1) is implicitly defined with respect to the module 2-category 2VecZ2 2 over itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In this case, gauging the Z2 2 symmetry simply amounts to choosing instead the 2VecZ2 2-module 2-category 2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' That being said, since this Hamiltonian is merely a doubled version of that considered in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2, we can immediately infer from the procedure outlined there the resulting dual Hamiltonian: H2Vec = − 2ÿ I=1 � JI,1 ÿ e σz,I e + JI,2 ÿ v ź e⊃v σx,I e � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) which acts on the physical Hilbert space spanned by states |g⟩, where g ≡ (g1, g2) ∈ Z1(Σ△, Z2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall that we chose the basis such that σz,I e |g⟩ = (−1)gI[e]|g⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5) In this basis, the first term in the Hamiltonian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) acts diagonally, while an arbitrary combination of operators ś e⊃v σx,I e indexed by a 0-cochain x ≡ (x1, x2) ∈ C0(Σ△, Z2 2) acts as Ax := ź v⊂Σ△ ź e⊃v 2 â I=1 (σx,I e )xI[v] = ÿ g |g + dx⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6) We know from the results of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 that Hamiltonian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) must commute with various surface and line operators that are organised into the Morita dual fusion 2-category (2VecZ2 2)⋆ Vec ∼= 2Rep(Z2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall from sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6 that simple objects in 2Rep(Z2 2) are provided by indecomposable VecZ2 2-module categories N(B, ψ), which are conveniently labelled by tuples (B, ψ) consisting of a subgroup B ⊆ Z2 2 and a 2-cocycle ψ in H2(B, U(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Therefore, we count six simple objects in 2Rep(Z2 2) labelled by the tuples (Z2 2, 1), (Z2 2, ψ), (Z(1) 2 , 1), (Z(2) 2 , 1), (Z(diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=') 2 , 1) and (Z1, 1), respectively, where ψ refers here to a normalised representative of the non-trivial cohomology class in H2(Z2 2, U(1)) ≃ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Each such 21The classification of SPT phases with 0-form global Z2 2 symmetry is given by the cohomology group H3(Z2 2, U(1)) ≃ Z3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Therefore, there are eight distinct topological phases which can be labelled by p ≡ (p1, p2, p3) ∈ Z3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The corre- sponding fixed-point Hamiltonians, which we denote as Hp, have the form H(1,0,0) = − ÿ v σx,1 v ź (v v1v2) exp � iπ 4 (1 − σz,1 v1 σz,1 v2 ) � , H(0,1,0) = − ÿ v σx,2 v ź (v v1v2) exp � iπ 4 (1 − σz,2 v1 σz,2 v2 ) � , H(0,0,1) = − ÿ v σx,1 v ź (v v1v2) exp � iπ 4 (1 − σz,1 v1 σz,1 v2 ) � − ÿ v σx,2 v ź (v v1v2) exp � iπ 4 (1 − σz,2 v1 σz,2 v2 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3) ∼ 46 ∼ simple object provides a surface operator commuting with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Furthermore, there are various line operators within each surface operator as well as at interfaces between surface operators associated with distinct simple objects in 2Rep(Z2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='22 The remainder of this section is dedicated to explicitly constructing these various operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 2Rep(Z2 2) symmetry: invertible surface operators We begin our detailed analysis of the symmetry structure of Hamiltonian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) by enumerating the invertible surface operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Firstly, there is of course the identity operator23 Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' = ź e ide, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7) which corresponds to the identity object N(Z2 2, 1) ∼= Vec in 2Rep(Z2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As explained in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8, line operators living on this trivial operator form the hom-category Hom2Rep(Z2 2)(Vec, Vec) ∼= FunVecZ2 2 (Vec, Vec) ∼= Rep(Z2 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8) We provided in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='90) a general formula for such line operators but we can make it more explicit by specialising to G = Z2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a 1-cycle ℓ on Σ△ and an irreducible representation (ρ1, ρ2) ∈ Rep(Z2 2), we may define such a line operator as ÿ g � ź e⊂ℓ ź I ρI(gI[e]) � |g⟩⟨g| , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='9) which readily commutes with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For instance, choosing both ρ1 and ρ2 to be the non-trivial irreducible representation of Z2, the operator above can be equivalently defined as ś e⊂ℓ σz,1 e σz,2 e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More generally, an operator corresponding to a network of lines in Rep(Z2 2) can be defined as Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f) = ÿ g (−1) � Σ△(f1⌣g1+f2⌣g2)|g⟩⟨g| , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10) where f = (f1, f2) ∈ Z1(Σ△, Z2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows directly from the definition that the composition such lines satisfies Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f1 ◦ f2) = Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f1 + f2) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='11) which does amount to the monoidal structure in Rep(Z2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, the fusion of lines read Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f1) d Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f2) = Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f1 + f2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12) as predicted by the monoidal structure in 2Rep(Z2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The second and final invertible surface corresponds to the simple object N(Z2 2, ψ) ∼= Vecψ in 2Rep(Z2 2), which as a VecZ2 2-module category only differs from Vec in the choice of module associator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We provided in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='89) a general expression for the corresponding type of surface operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Writing ψ(g, g′) := (−1)g1g′ 2, we find it is a non-trivial operator that acts diagonally on basis states as Uψ[Σ△] = ÿ g (−1) � Σ△g1⌣g2|g⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='13) 22Note that two objects that have a non-trivial 1-morphism between them are said to belong to the same Schur component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Physically, this means that there is a condensation process relating both objects [GJF19, DR18, Reu22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 23Notice that, in a way that is reminiscent of the construction of the corresponding module categories, we notate the surface operator associated with the tuple (B, ψ) via UG/B,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 47 ∼ This operator can be shown to commute with the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' On the one hand, it is diagonal in the chosen computational basis, thus it clearly commutes with the first term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' On the other hand, the SPT being non-anomalous is gauge invariant, and thus commutes with the second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The operator can also be defined with non-trivial line insertions which are also labelled by f ∈ Z1(Σ△, Z2 2) as Uψ(f)[Σ△] = ÿ g (−1) � Σ△g1⌣g2+f1⌣g1+f2⌣g2|g⟩⟨g| , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='14) which satisfy composition rules analogous to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows that such lines also encoded into Rep(Z2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As evoked in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8, this is explained by the fact that FunVecZ2 2 (Vecψ, Vecψ) ∼= Rep(Z2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The operators Uψ[Σ△] and Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' satisfy Z2 fusion rules of the form � Uψ d Uψ� [Σ△] = Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' , � Uψ d Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='� [Σ△] = Uψ[Σ△] = � Uψ d Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='� [Σ△] , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15) which readily follows from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, composition rules for surface operators with line operators inserted take the form � Uψ(f1) d Uψ(f2) � [Σ△] = Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f1 + f2)[Σ△] , � Uψ(f1) d Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f2) � [Σ△] = Uψ(f1 + f2)[Σ△] , � Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (f1) d Uψ(f2) � [Σ△] = Uψ(f1 + f2)[Σ△] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='16) Interestingly, one may also define the operator Uψ on an open sub-complex Ξ△ ⊆ Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Equivalently, this is the statement that there is a topological line operator between the operators Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' and Uψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Naively, the operator Uψ[Ξ△] simply defined by restricting definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='13) to the open sub-complex Ξ△ does not commute with the Hamiltonian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) since AxUψ[Ξ△] = ÿ g exp � iπ � Ξ△ g1 ⌣g2 � |g + dx⟩⟨g| , Uψ[Ξ△]Ax = ÿ g exp � iπ � Ξ△ g1 ⌣g2 + iπ � ∂Ξ△ ζ(g, x) � |g + dx⟩⟨g| , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17) where dζ(g, x) = (g1+dx)⌣(g2+dx)−g1 ⌣ g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Such a lack of commutation is remedied by appending a line operator on the boundary ∂Ξ△, which has the form Uψ|triv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [∂Ξ△] = ÿ g Zanom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (g)[∂Ξ△]|g⟩⟨g| , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18) where the amplitude Zanom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (g)[∂Ξ△] can be understood as the partition function of a quantum me- chanical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', (0 + 1)-dimensional) system with an anomalous Z2 2 symmetry encoded into the (unique) irreducible projective representation with Schur multiplier ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, this operator can be con- structed by considering auxiliary Z2 2-valued vertex degrees of freedom p, q on ∂Ξ△ such that Zanom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (g)[∂Ξ△] = ÿ b,n exp � iπ � ∂Ξ△ � δI,JbI ⌣(dnJ + gJ) + dn1 ⌣dn2 �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='19) which has the required property Zanom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (g + dx)[∂Ξ△] = exp � iπ � ∂Ξ△ ζ(g, x) � Zanom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (g)[∂Ξ△] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20) ∼ 48 ∼ It follows that the combined operator Uψ→triv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Ξ△] := Uψ[Ξ△] · Uψ|triv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [∂Ξ△] , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='21) commutes with the Hamiltonian and is therefore a symmetry operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Graphically, we depict such a configuration as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='22) where the gray area depicts Ξ△ and the blue coloured edges the support of the line operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' From a category theoretic standpoint, recall that line operators at the interface of Uψ and Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' are organised into the fusion category FunVecZ2 2 (Vec, Vecψ) ∼= Repψ(Z2 2) of ψ-projective representations of Z2 2, which is compatible with the above construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, we define an operator Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='→ψ[Ξ△].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, the composition of the topological line Uψ|triv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [∂Ξ△] follows from the tensor product of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Since the line Uψ|triv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [∂Ξ△] carries a ψ-projective representation of Z2 2, composing it with itself must yield an object labelled by the trivial representation in Rep(Z2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We can compose this line between the trivial surface operators Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' and the non-trivial operator Uψ in two ways so as to recover the trivial representation line either within the trivial surface or within the non-trivial surface operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Uψ→triv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Ξ△] ◦ Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='→ψ[Σ△/Ξ△] = Uψ[Σ△] , Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='→ψ[Ξ△] ◦ Uψ→triv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Σ△/Ξ△] = Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='23) which is mathematically encoded into the composition of the corresponding VecZ2 2-module functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This concludes our analysis of the invertible surface operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3 2Rep(Z2 2) symmetry: non-invertible surface operators We continue our analysis of the symmetry structure of Hamiltonian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) with the study of surface operators that have non-invertible fusion rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In terms of simple objects in 2Rep(Z2 2), these are the ones labelled by the tuples (Z(1) 2 , 1), (Z(2) 2 , 1), (Zdiag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2 , 1) and (Z1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us focus for now on the first three surface operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Mimicking the definition of the non-invertible surface operator described in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2, one finds: UZ(1) 2 [Σ△] = 1 2#(Σ△) ÿ g,n,b (−1) � Σ△b⌣(dn+g1)|g⟩⟨g| = 1 2χ(Σ△) ÿ g,n δdn,g1|g⟩⟨g| , UZ(2) 2 [Σ△] = 1 2#(Σ△) ÿ g,n,b (−1) � Σ△b⌣(dn+g2)|g⟩⟨g| = 1 2χ(Σ△) ÿ g,n δdn,g2|g⟩⟨g| , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='24) UZ(diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=') 2 [Σ△] = 1 2#(Σ△) ÿ g,n,b (−1) � Σ△b⌣(dn+g1+g2)|g⟩⟨g| = 1 2χ(Σ△) ÿ g,n δdn,g1+g2|g⟩⟨g| , where the summation variables are n ∈ C0(Σ△, Z2) and b ∈ C1(Σ△, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2, we have defined #(Σ△) = 2|Σ0 △|+|Σ2 △| and χ(Σ△) is the Euler characteristic of Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Summing over b, imposes ∼ 49 ∼ a constraint that pins dn on each edge of the triangulation to be g1, g2 and g1 + g2 for the three operators UZ(1) 2 , UZ(2) 2 and UZ(diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=') 2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These operators should be thought of as an explicit version of the general operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' There, n refers to an assignment of simple objects in the VecZ2 2-module categories N(Z(1) 2 , 1), N(Z(2) 2 , 1) and N(Z(diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=') 2 ), which are all equivalent to VecZ2 as categories, satisfying n[v1] = Cg[v1v2] ▷ n[v2] for every edge (v1v2) ⊂ Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, the VecZ2 2- module structure on N(Z(diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=') 2 , 1) is given by Cg ▷N := (g1 +g2)+N mod 2, for any g ≡ (g1, g2) ∈ Z2 2 and N ∈ VecZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This amounts to the condition dn = g1 + g2 in the equation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Summing over n instead of b in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='24) imposes db = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Summing over equivalence classes of 1-cocycles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', f ∈ H1(Σ△, Z2), one finds UZ(1) 2 [Σ△] = 1 |H0(Σ△, Z2)| ÿ g,f (−1) � Σ△f⌣g1|g⟩⟨g| , UZ(2) 2 [Σ△] = 1 |H0(Σ△, Z2)| ÿ g,f (−1) � Σ△f⌣g2|g⟩⟨g| , UZ(diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=') 2 [Σ△] = 1 |H0(Σ△, Z2)| ÿ g,f (−1) � Σ△f⌣(g1+g2)|g⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='25) From these expressions, it is clear that these operators are condensation defects of the Rep(Z2 2) lines described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It turns out that this alternative form of the operators is particularly convenient to demonstrate the commutativity of these operators with the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows from the operators being diagonal in the chosen basis and invariance under the action of Ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Still in analogy with the non-invertible surface operator considered in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2, the surfaces (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='24) can be defined with topological lines inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall that these must be encoded into Morita duals of VecZ2 2 with respect to the corresponding module categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For instance, lines living on UZ(1) 2 form the fusion 1-category FunVecZ2 2 (VecZ(1) 2 , VecZ(1) 2 ) ∼= VecZ(1) 2 b Rep(Z(2) 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='26) The corresponding surface operator with a network of lines inserted can be constructed by choosing 1-cocycles ˜f1, f2 ∈ Z1(Σ△, Z2) as UZ(1) 2 (˜f1, f2)[Σ△] = 1 2#(Σ△) ÿ g,n,b (−1) � Σ△b⌣(dn+g1+˜f1)+f2⌣g2|g⟩⟨g| = 1 2χ(Σ△) ÿ g,n δdn,g1+˜f1(−1) � Σ△ f2∪g2|g⟩⟨g| , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='27) where ˜f1, which twists the cocycle condition on n, corresponds to the VecZ(1) 2 lines, whereas f2 cor- responds to the usual Wilson lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By analogy, the topological lines living on the surface operators UZ(2) 2 [Σ△] and UZ(diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=') 2 [Σ△] form the fusion 1-categories VecZ(2) 2 bRep(Z(1) 2 ) and VecZ(1) 2 bRep(Z(diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=') 2 ), respectively, where a networks of lines within these two operators take the form UZ(2) 2 (f1,˜f2)[Σ△] = 1 2#(Σ△) ÿ g,n,b (−1) � Σ△f1⌣g1+b⌣(dn+g2+˜f2)|g⟩⟨g| , UZ(diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=') 2 (f,˜f)[Σ△] = 1 2#(Σ△) ÿ g,n,b (−1) � Σ△b⌣(dn+(g1+g2)+˜f)+f⌣(g1+g2)|g⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='28) ∼ 50 ∼ Let us now describe the final symmetry surface operator in the fusion 2-category 2Rep(Z2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It is that labelled by the VecZ2 2-module category N(Z1, 1) ∼= VecZ2 2: UZ2 2[Σ△] = 1 22#(Σ△) 2 ź I=1 ÿ g ÿ nI,bI (−1) � Σ△δI,JbI⌣(dnJ+gJ)|g⟩⟨g| = 1 2χ(Σ△) 2 ź I=1 ÿ nI δdnI,gI|g⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='29) Similarly, UZ2 2[Σ△] can be defined with line operator insertions: UZ2 2(˜f1,˜f2)[Σ△] = 1 22#(Σ△) ź I ÿ g,nI,bI (−1) � Σ△δI,JbI⌣(dnJ+gJ+˜fJ)|g⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='30) The commutation of UZ2 2(˜f1,˜f2)[Σ△] with the Hamiltonian can be demonstrated as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Having described all the surface operators associated with simple objects in 2Rep(Z2 2), let now compute their fusion rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For instance, we have (UZ(1) 2 d UZ(1) 2 )[Σ△] = 1 22#(Σ△) ÿ g,b,n g′,b′,n′ (−1) � Σ△b⌣(dn+g1)+b′⌣(dn′+g′ 1)|g⟩⟨g|g′⟩⟨g′| = � 1 2#(Σ△) ÿ b′,n+ (−1) � Σ△b′⌣dn+� 1 2#(Σ△) ÿ g,b+,n (−1) � Σ△b+⌣(dn+g1)|g⟩⟨g| = Z2d[Σ△] · UZ(1) 2 [Σ△] , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='31) where in the second line, we have defined b+ = b + b′ and n+ = n + n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The pre-factor Z2d[Σ△] in the fusion rule outcome is the partition function of the pure two-dimensional Z2 gauge theory on Σ△ as in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2 and app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the case where Σ is a two-torus, we recover exactly the fusion structure of 2Rep(Z2 2) according to which VecZ(1) 2 d VecZ(1) 2 ∼= VecZ(1) 2 ‘ VecZ(1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='32) The remaining fusion rules can be computed analogously: � UZ(I) 2 d UZ(J) 2 � [Σ△] = � Z2d[Σ△] · UZ(I) 2 [Σ△] if I = J , UZ2 2[Σ△] otherwise , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='33) where I, J ∈ {1, 2, diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, the fusion rules between the symmetry operator UZ2 2[Σ△] and the three other non-invertible surfaces are given by � UZ2 2 d UZI 2� [Σ△] = � UZJ 2 d UZ2 2� [Σ△] = Z2d[Σ] × UZ2 2[Σ△] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='34) Finally, fusing UZ2 2 with itself yields � UZ2 2 d UZ2 2� [Σ△] = (Z2d[Σ△])2 · UZ2 2[Σ△] , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='35) where the coefficient (Z2d[Σ△])2 amounts to the partition function of the pure two-dimensional Z2 2 topological gauge theory on Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In order to conclude our analysis of the symmetry structure of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) as encoded into 2Rep(Z2 2), we are left to consider topological lines between distinct surfaces as well as the corresponding composition ∼ 51 ∼ rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It largely mimics the case presented in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A topological surface operator can be defined on a triangulation of the form Σ△ = (Σ△\\Ξ△) ⊔∂Ξ△ Ξ△, which locally looks like UZ(I) 2 and UZ(J) 2 in the regions Σ△\\Ξ△ and Ξ△, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Such an operator has the form UZ(I) 2 ,Z(J) 2 [Σ△\\Ξ△, Ξ△] = 1 2#(Σ△) ÿ g,n,b ˜n,˜b (−1) � Σ△\\Ξ△b⌣(dn+gI)+ � Ξ△ ˜b⌣(d˜n+gJ)|g⟩⟨g| , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='36) where I, J ∈ {1, 2, diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='} and gdiag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' := g1 + g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Moreover, we imposed in the previous equation Dirichlet boundary conditions b[∂Ξ△] = ˜b[∂Ξ△] = 0 along the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, one may define a topological surface operator that interpolates between UZ2 2 and UZ(I) 2 by defining UZ2 2 on Σ△\\Ξ△, UZ(I) 2 on Ξ△, and imposing suitable Dirichlet boundary conditions along the interface ∂Ξ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now compute the composition rules between topological interfaces separating regions with locally distinct symmetry operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' To do so, we consider a setup closely resembling that of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let Ξ△ be a thin annular strip of single lattice spacing width supporting a surface operator UZ(J) 2 , while the rest of the lattice Σ△\\Ξ△ supports UZ(I) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We denote the left and right boundaries of Ξ△ by ∂LΞ△ and ∂RΞ△, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The corresponding composition of lines is then given by the operator à f UZ(I) 2 (f)[Σ△] , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='37) where the sum is over the four simple topological lines of UZ(I) 2 [Σ△] traversing the (relative) homology cycle of Ξ△ with Dirichlet conditions b[∂LΞ△] = b[∂RΞ△] = 0 imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These fusion rules are reminiscent of the Z2 2 Tambara-Yamagami fusion category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' SECTION 5 Example: transverse-field S3-Ising model In this section, we consider the higher-categorical symmetry structures of the models obtained by gaug- ing various sub-symmetries of the transverse-field S3-Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We shall focus on features specific to dealing with a non-abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1 Symmetric Hamiltonian For our final series of examples, we consider a transverse-field Ising model with a non-abelian symmetry group, namely the symmetric group S3 of degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8, we explained how to perform the gauging of various sub-symmetries of this model for an arbitrary group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Moreover, we elucidated there the symmetry structures of the resulting models in terms of fusion 2-categories of higher representations of groups, and categorifications thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Although the construction of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 provides a general recipe to realise on the lattice operators commuting with dual Hamiltonians, it remains somewhat formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The goal of this section is to describe these symmetry operators more explicitly in the spirit of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us begin by reviewing the group structure of S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The permutation group S3 is the group with presentation S3 = ⟨r, s | r2 = s3 = (rs)2 = 1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1) Both the cyclic groups Z2 = ⟨r | r2 = 1⟩ and Z3 = ⟨s | s3 = 1⟩ are subgroups of S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Due to the action of Z2 on Z3 given by φ− : Z2 → Aut(Z3) such that φr(s) = s2, we have an isomorphism S3 ≃ Z2⋉φZ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Therefore, it is a group of the form considered in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 52 ∼ The microscopic Hilbert space of the transverse-field S3-model is given by the tensor product  v C[S3] ∋ |m⟩, where m is an assignment of group elements in S3 to every vertex of Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Due to the semi-direct product structure of S3, the local Hilbert space can be rather spanned by states |m[v]⟩ ≡ |ϖZ2(m[v]), ϖZ3(m[v])⟩, that is a pair of qubit and qutrit degrees of freedom at every vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given the above, it is useful to define the following operators σx = �0 1 1 0 � , σz = �1 0 0 −1 � , Σx = � � 0 0 1 1 0 0 0 1 0 � � , Σz = � � 1 0 0 0 ω 0 0 0 ω2 � � , Γ = � � 1 0 0 0 0 1 0 1 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2) The σ matrices satisfy the Pauli algebra σxσz = −σzσx as before, while the Σ matrices represent the Z3 clock and shift operators, which satisfy ΣzΣx = ωΣxΣz and (Σx)3 = (Σz)3 = id with ω = exp( 2πi 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Additionally, the operator Γ implements the action of Z2 on Z3 by automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This operator satisfies the relations ΓΣxΓ = Σx† and ΓΣzΓ = Σz†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We work in the basis such that (id b Σz v)|m[v]⟩ = ωϖZ3(m[v])|m[v]⟩ , (σz v b id)|m[v]⟩ = (−1)ϖZ2(m[v])|m[v]⟩ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3) effectively identifying group elements in Z2 with {0, 1} and those in Z3 with {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The Lg v operators which act by left multiplication (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' below eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='85)) have the following explicit form in this basis: Lr v : |m[v]⟩ �→ (σx b Γ)v|m[v]⟩ = |r · m[v]⟩ , Ls v : |m[v]⟩ �→ (id b Σx)v|m[v]⟩ = |s · m[v]⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) The qubit and qutrit degrees of freedom are subject to the 2VecS3-symmetric Hamiltonian whose expression we reproduce below: H2VecG = −J ÿ e Π 1S3 s(e),t(e) − Jκ 6 ÿ v ÿ g∈G Lg v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5) Both terms appearing in this Hamiltonian can be rewritten more explicitly in terms of the matrices introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' On the one hand, we have Π 1S3 s(e),t(e) = 1 6 � id + σz s(e)σz t(e) � b � id + Σz s(e)(Σz t(e))† + (Σz s(e))†Σz t(e) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6) which implicitly makes use of the fact that 1 + ω + ¯ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Such a term is analogous to the ferro- magnetic term in the Z2-symmetric transverse field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It energetically favors homogenous configurations in the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the limit κ → 0, the ground state spontaneously breaks the S3 global symmetry with m[v] = m0, for all v, and there are |S3| ground states associated with different choices of m0 ∈ S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' On the other hand, the term proportional to κ is a combination of operators Lg v which act on the basis by left multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The linear combination of Lg v operators has the following explicit form: 1 6 ÿ g∈S3 Lg v = 1 6 ÿ g∈S3 (σx b Γ)ϖZ2(g)(id b Σx)ϖZ3(g) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7) which acts as a projector onto the one-dimensional subspace of C[S3] (at the vertex v) transforming in the trivial representation of S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This term is analogous to the paramagnetic term in the Z2-symmetric transverse field Ising model and favours a unique ground state that preserves the full S3 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 53 ∼ One can readily check that this model has a global 0-form S3 symmetry implemented by surface operators Og = ź v Rg v , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8) where Rg v acts on the basis state at vertex v by right multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These operators satisfy the fusion rules provided by the multiplication rules in S3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', Og1 d Og2 = Og1g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In order to rewrite operators Rg v more explicitly, it is convenient to introduce the following controlled gate: cΣx : C2 b C3 → C2 b C3 : |q, l⟩ �→ � id b (Σx)1+q� |q, l⟩ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='9) where the qubit plays the role of the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These controlled gates satisfy in particular the following commutation relation cΣx(σx b Γ) = (σx b Γ)cΣx , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10) and adaptations thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The explicit action of right multiplication operators Rg v now reads: Rr v : |m[v]⟩ �→ (σx b id)v|m[v]⟩ = |m[v] · r⟩ , Rs v : |m[v]⟩ �→ (cΣx)† v|m[v]⟩ = |m[v] · s2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='11) Given this action, commutation of Og with the Hamiltonian is ensured by the various commutation relations listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 Gauging sub-symmetries Given a finite group isomorphic to a semi-direct product, we explained in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8 how to obtain three dual Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the present context, these are obtained by gauging the S3, Z3 and Z2 sub- symmetries of the S3-symmetric Hamiltonian, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the language of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8, these amount to choosing the 2VecS3-module 2-categories M(S3, 1) ∼= 2Vec, M(Z3, 1) ∼= 2VecZ2 and M(Z2, 1) ∼= 2VecS3/Z2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In preparation for the analysis of the symmetry structures, we provide below more explicit expressions for the terms appearing in the definitions of these Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us first consider gauging the full S3 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The resulting Hamiltonian was found in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='88) to be of the form H2Vec = −J ÿ e Π 1S3 e − Jκ ÿ v Av .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12) The edge term explicitly reads Π 1S3 e = 1 6 � id + σz e � b � id + Σz e + (Σz e)†� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='13) whereas the contribution of the generators of S3 to the vertex term Av = 1 6 ř g∈S3 Ag v can be depicted as Ar v ≡ σx σx σxΓ σxΓ σxΓ σx , As v ≡ cΣx†cΣx† Σx Σx Σx cΣx† , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='14) where we omitted b symbols for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The operators Ag v for the remaining g ∈ S3 can be obtained by suitably composing Ar v and As v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The model (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12) describes an S3 lattice gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 54 ∼ The first term suppresses the S3 fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the limit κ → 0, the model is in the confined phase, which is the dual analogue of the symmetry-broken phase in the pre-gauged model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Instead, the second term is responsible for gauge fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the κ → ∞ limit, the model is in the deconfined phase whose renormalisation group fixed point is provided by the Hamiltonian realisation of S3 Dijkgraaf- Witten theory with trivial cohomological twist [DW90, dWP95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the same vein, let us now consider gauging the Z3 sub-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The resulting Hamiltonian was found in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='99) to be of the form H2VecZ2 = −J ÿ e Π 0Z2,0Z3 e − Jκ ÿ v φAv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15) The edge term explicitly reads Π 0Z2,0Z3 e = 1 6 � id + σz s(e)σz t(e) � b � id + Σz e + (Σz e)†� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='16) In order to rewrite the vertex term more explicitly, we require the controlled gates introduced pre- viously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We shall apply here these gates between a qutrit assigned to an edge and a qubit assigned to a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It is convenient to graphically depict such controlled gates by means of a dotted line connecting a control qubit identified by ‘c’ and a target qutrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The contribution of the generators of S3 to the vertex term φAv = 1 6 ř g∈S3 φAg v can now be depicted as φAr v = σx v ≡ σx , φAs v = ź e←v cΣx e ź e→v (cΣx e )† ≡ Σx† Σx† Σx Σx Σx Σx† c , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17) where all the gates on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' are controlled by the qubit living at the vertex v the operator acts on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The model (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15) describes a Z3 lattice gauge theory coupled to a Z2-Ising matter model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the limit κ → 0, the Z3 gauge sector is in the confined phase while the Z2 matter sector is in the ferromagnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Conversely, in the κ → ∞ limit, the gauge sector is in the deconfined phase while the matter sector is in the paramagnetic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In this limit the model describes up to a unitary (see footnote 19) a Z3 topological gauge theory enriched by a global Z2 symmetry [WBV17, TJV21, TVV22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, let us consider gauging the Z2 sub-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The resulting Hamiltonian was found in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='105) to be of the form H2VecS3/Z2 = −J ÿ e Π 0Z3,0Z2 e − Jκ ÿ v φ�Av .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18) The edge term explicitly reads Π 0Z3,0Z2 e = 1 6 � id + σz e � b � id + Σz s(e)(Σz t(e))† + (Σz s(e))†Σz t(e) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='19) whereas the contribution of the generators of S3 to the vertex term φ�Av = 1 6 ř g∈S3 φ�Ag v can be depicted as φ�As v = Σx , φ�Ar v = σx σx σx σx σx σx Γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20) ∼ 55 ∼ The two phases of this Hamiltonian can be interpreted in the same vein as for the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Having described the models (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18), which are obtained by gauging the different subgroups of S3, we detail below the symmetry structures corresponding to each of these (partially) gauged models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3 2Rep(S3) symmetry Let us study the symmetry structure of Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12) acting on a Hilbert space spanned by states |g⟩ ∈  e C[G], where g ∈ Z1(Σ△, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Following the general discussions in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8, we know that Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12) must have a symmetry structure embodying the fusion 2-category 2Rep(S3) of 2-representations of S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, this means that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12) hosts topological surfaces associated with simple objects in 2Rep(S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall from sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6 that simple objects in 2Rep(S3) are provided by indecomposable VecS3-module categories N(B, ψ), which are conveniently labelled by tuples (B, ψ) consisting of a subgroup B ⊆ S3 and a 2-cocycle ψ in H2(B, U(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Since H2(B, U(1)) is trivial for any B ⊆ S3, we count four simple objects in 2Rep(S3) associated with each subgroup of S3, namely N(S3, 1) ∼= Vec, N(Z3, 1) ∼= VecZ2, N(Z2, 1) ∼= VecS3/Z2 and N(Z1, 1) ∼= VecS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We provided in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='72) a general formula to construct the corresponding surface operators, but let us unpack it further here in the spirit of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Generally speaking, defining topological surfaces associated with simple objects in 2Rep(S3) re- quires introducing virtual degrees of freedom n[v] at every vertex v ⊂ Σ△, whose configuration space is given by the set of isomorphism classes of simple objects in the corresponding VecS3-module cat- egory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, given the topological surface associated with N(B, 1), virtual degrees of freedom are valued in the collection G/B of left cosets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These virtual degrees of freedom are then coupled to the physical degrees of freedom via the conditions spelt out below (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='72) involving the VecS3-module structure of N(B, 1), which we recall is given by the natural action of S3 on the left cosets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Suppose for instance that the subgroup B is the whole group S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' There is a single left coset in S3/S3 ≃ Z1, namely S3 itself, on which S3 acts invariantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows that the resulting operator acts as the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We denote it by Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' in accordance with sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2 and 4: Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' = ź e ide .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='21) This topological surface, which is associated with the VecS3-module category Vec, is the only invertible one for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now consider non-invertible topological surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We first focus on that associated with the VecS3-module category N(Z3, 1) ∼= VecZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By definition, simple objects in N(Z3, 1) are left cosets in S3/Z3 ≃ � {1, s, s2} , {r, sr, s2r} � ≃ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The corresponding surface operator amounts to summing over configurations n ∈ C0(Σ△, Z2) of virtual degrees of freedom, which are coupled to the physical degrees of freedom by imposing conditions n[v1] = Cg[v1v2] ▷ n[v2] = ϖZ2(g[v1v2]) + n[v2] at every edge (v1v2) ⊂ Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These conditions can be enforced explicitly by introducing Lagrange multiplier b ∈ C1(Σ△, Z2) as follows: UZ2[Σ△] = 1 2#(Σ△) ÿ g,n,b exp � πi � Σ△ b⌣ � dn − ϖZ2(g) �� |g⟩⟨g| , = 1 2χ(Σ△) ÿ g,n ź (v1v2)⊂Σ△ δCg[v1v2]▷n[v2],n[v1] |g⟩⟨g| , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='22) ∼ 56 ∼ where (dn)[v1v2] = n[v1] − n[v2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The topological surface associated with the VecS3-module category N(Z2, 1) ∼= VecS3/Z2 is con- structed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' By definition, simple objects in N(Z2, 1) are provided by left cosets in S3/Z2 ≃ � {1, r}, {r, sr}, {s2, s2r} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The non-normal subgroup Z2 ⊂ S3 acts trivially on S3/Z3, while the re- maining elements permute the elements in S3/Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The corresponding surface operator amounts to sum- ming over configurations n ∈ C0(Σ△, Z3) of virtual degrees of freedom, which are coupled to the physi- cal degrees of freedom by imposing conditions n[v1] = Cg[v1v2]▷n[v2] = ϖZ3(g[v1v2])+φϖZ2(g[v1v2])(n[v2]) at every edge (v1v2) ⊂ Σ△, where we are using that S3/Z2 ≃ Z3 as a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These conditions can be enforced explicitly by introducing Lagrange multiplier b ∈ C1(Σ△, Z3) as follows: US3/Z2[Σ△] = 1 3#(Σ△) ÿ g,n,b exp �2πi 3 � Σ△ b⌣ � dgn − ϖZ3(g) �� |g⟩⟨g| , = 1 3χ(Σ△) ÿ g,n ź (v1v2)⊂Σ△ δCg[v1v2]▷n[v2],n[v1] |g⟩⟨g| , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='23) where we have used the twisted differential (dgn)[v1v2] := n[v1] − φϖZ2(g[v1v2])(n[v2]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='24) Finally, the remaining topological surface associated with the VecS3-module category N(Z1, 1) ∼= VecS3 can be simply expressed as US3[Σ△] = 1 |S3|χ(Σ△) ÿ g,n ź (v1v2)⊂Σ△ δg[v1v2]n[v2],n[v1] |g⟩⟨g| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='25) Let us now briefly confirm that all these surface operators do commute with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12), and are thus part of the symmetry structure of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Firstly, edge terms Π 1S3 e straightforwardly commute with all the topological surfaces since all the operators are diagonal in the chosen computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Vertex operators Av perform averagings over the group of gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' An arbitrary combination of gauge transformations indexed by an assignment x ∈ C0(Σ△, S3) acts as |g⟩ �→ |xg⟩, where xg[e] = x[s(e)] · g[e] · x[t(e)]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Commutation with the surface operators is simply obtained by absorbing the gauge transformation of g into a redefinition of n, which is summed over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' To summarise, we have thus far obtained symmetry surface operators associated with each simple of object of 2Rep(S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now compute the corresponding fusion rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We now by construction that these must correspond to the monoidal structure of 2Rep(S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Briefly, fusion rules follow from the fact that acting consecutively with two surface operators amounts to taking a Cartesian product of the local configuration space assigned to each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This Cartesian product can be subsequently decomposed into disjoint unions of isomorphism classes of S3-sets according to the Burnside ring multiplication rule [Gre10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, let N(B, 1) and N(B′, 1) be two indecomposable VecS3-module categories, the fusion of the corresponding topological surfaces read � UG/B d UG/B′� [Σ△] = ���� B × B′ G × G ���� χ(Σ△) ÿ g,g′,n,n′ ź (v1v2)⊂Σ△ δCg[v1v2]▷n[v2],n[v1] δCg′[v1v2]▷n′[v2],n′[v1] |g⟩⟨g|g′⟩⟨g′| = ���� B × B′ G × G ���� χ(Σ△) ÿ g,n,n′ ź (v1v2)⊂Σ△ δCg[v1v2]▷(n,n′)[v2],(n,n′)[v1] |g⟩⟨g| , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='26) ∼ 57 ∼ which is a topological surface with virtual degrees of freedom valued in the Cartesian product S3/B × S3/B′ that is acted upon diagonally by S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Decomposing this Cartesian product into a disjoint union of S3-sets then yields the decomposition of the topological surface into simple ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For instance, when taking the fusion of topological surfaces US3/Z2 d US3/Z2, one has S3/Z2 × S3/Z2 ≃ � {1, r}, {s, sr}, {s2, s2r} � × � {1, r}, {s, sr}, {s2, s2r} � ≃ �� {1, r}, {1, r} � , � {s, sr}, {s, sr} � , � {s2, s2r}, {s2, s2r} �� ⊔ �� {1, r}, {s, sr} � , � {1, r}, {s2, s2r} � , � {s, sr}, {1, r} � , � {s, sr}, {s2, s2r} � , � {s2, s2r}, {1, r} � , � {s2, s2r}, {s, sr} �� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='27) which decomposes into a three dimensional (diagonal) orbit and a six-dimensional off-diagonal orbit under S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Then it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='26) that � US3/Z2 d US3/Z2� [Σ△] = 1 3χ(Σ△) · US3/Z2[Σ△] ‘ �2 3 �χ(Σ△) US3[Σ△] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='28) Similarly, the fusion rules for all the remaining operators on a general (path-connected) Σ read � US3 d US3� [Σ△] = 61−χ(Σ△) · US3[Σ△] , � US3/Z2 d US3� [Σ△] = 31−χ(Σ△) · US3[Σ△] , � UZ2 d UZ2� [Σ△] = 21−χ(Σ△) · UZ2[Σ△] , � US3/Z2 d UZ2� [Σ△] = US3[Σ△] , � UZ2 d US3� [Σ△] = 21−χ(Σ△) · US3[Σ△] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='29) Note that p1−χ(Σ△) is the partition function for the Zp gauge theory on a general path connected manifold Σ with Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Choosing Σ to be a two-torus, we recover the monoidal structure of 2Rep(S3) [Gre10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us finally describe the topological lines living on the various surface operators constructed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We established in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7 that give a surface operator associated with a VecS3-module category N(B, 1), we can insert topological lines that form the Morita dual fusion 1-category (VecS3)⋆ N (B,1) = FunVecS3 (N(B, 1), N(B, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, we have (VecS3)⋆ Vec ∼= Rep(S3) , (VecS3)⋆ VecZ2 ∼= VecS3 , (VecS3)⋆ VecS3/Z2 ∼= Rep(S3) , (VecS3)⋆ VecS3 ∼= VecS3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='30) Generally speaking, the topological line associated with a simple object of such a Morita dual fusion 1-category can be realised on the lattice in terms of matrix product operators whose building blocks evaluate to the matrix elements of the module structure of the corresponding VecS3-module functors [LDOV21, Del21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, we have FunVecS3 (Vec, Vec) ∼= Rep(S3), which indicates topological lines of the identity surface operator are labelled by irreducible representations of S3 and amount to ordinary Wilson lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recall that Rep(S3) has three simple objects, namely the trivial 0, the sign 1 and the standard representation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We provided in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='90) the corresponding lattice operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These implement a non-invertible 1-form symmetry of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Indeed, commutation with edge terms Π 1S3 e follows from the fact that both operators act diagonally on basis states |g⟩, while commutation with vertex terms Av amounts to the gauge invariance of Wilson lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally, it follows straightforwardly from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='90) that composition of lines amounts to the monoidal structure of Rep(S3) with 0 the unit and 1 b 1 ≃ 0, 1 b 2 ≃ 2 and 2 b 2 ≃ 0 ‘ 1 ‘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 58 ∼ Finally, we mentioned in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 that we could recover the topological surfaces considered above as condensation defects obtained by condensing suitable algebras of topological lines in Rep(S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Specif- ically, topological surfaces in 2Rep(S3) can be identified with (separable) algebra objects in Rep(S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We further commented in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 that these algebra objects are given by S3-algebras, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=', associative unital algebras equipped with an S3-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Since none of the subgroups of S3 has a non-trivial second cohomology group, these admit a simple definition: Given a subgroup B ⊆ S3, the corresponding S3- algebra is provided by the permutation representation C[G/B] with pointwise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Besides, we have C[S3/S3] ≃ 0, C[S3/Z3] ≃ 0 ‘ 1, C[S3/Z2] ≃ 0 ‘ 2 and C[S3] ≃ 0 ‘ 1 ‘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, this means for instance that we can reconstruct the topological surface US3/Z2 by inserting a network of topological lines labelled by 0 ‘ 2 ∈ Rep(S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4 2VecG symmetry We now turn to the symmetry structure of Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15) acting on a Hilbert space spanned by states |l, m⟩ ∈  e C[Z3]  v C[Z2], where l ∈ Z1(Σ△, Z3) and m ∈ C0(Σ△, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Following the general discussions in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8, we know that Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12) must have a symmetry structure embodying the fusion 2-category 2VecG of 2-vector spaces graded by the 2-group G with homotopy groups Z2 and Z3 in degree one and two, respectively, as defined in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15) must commute with topological surfaces labelled by Z2-graded vector spaces of the form Vq, where q ∈ Z2 and Vq has the structure of a VecZ3-module category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We thus count four topological surfaces identified with Vec0, Vec1, (VecZ3)0 and (VecZ3)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Firstly, as always, there is the identity operator Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' = ź e ide , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='31) which is here identified with Vec0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Next, there is a non-invertible surface operator defined as UZ3[Σ△] = 1 3#(Σ△) ÿ l,m,n,b exp �2πi 3 � Σ△ b⌣(dn − l) � |l, m⟩⟨l, m| , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='32) where b ∈ Z1(Σ△, Z3) and n ∈ C0(Σ△, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This is the surface operator identified with (VecZ2)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Summing over n, one obtains the presentation of this operator as a condensation defect of Rep(Z3) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The third operator, which is identified with Vec1, implements the 0-form Z2 symmetry that is left over from the initial S3 0-form symmetry after gauging of the Z3 sub-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This Z2 operator is somewhat unconventional owing to the fact that Z2 acted non-trivially on Z3 via the outer automorphism φ in S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, the 0-form Z2 symmetry operator is O1 = ź v σx v ź e Γe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='33) Commutation with Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15), and more specifically with the vertex terms, then follows from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This operator was obtained by applying the general recipe of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5: Recall that 2VecG arises as the Morita dual (2VecS3)⋆ 2VecZ2 of 2VecS3 with respect to 2VecZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In this context, the operator O1 is associated with the module 2-endofunctor − d Vec1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As such, it acts on degrees of freedom at vertices valued in the set of isomorphism classes of simple objects of 2VecZ2 by mulitplication by the non-trivial element in Z2 (hence σx v ), whereas the action on edge degrees of freedom simply follows from the identification of the effective degrees of freedom according to the formula ag,m[v1v2] = φm[v1]−1� ϖL(g[v1v2]) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The fourth and final operator denoted by UZ3,1[Σ△], which is identified with the simple object (VecZ2)1, can be simply defined as the fusion (O1 d UZ2)[Σ△].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The fusion rules of ∼ 59 ∼ these various topological surfaces can be computed using the explicit formulas above following methods employed for the other examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We write below the non-trivial fusion rules: � UZ3 d UZ3� [Σ△] = 31−χ(Σ△) · UZ3[Σ△] , � UZ3 d O1� [Σ△] = UZ3,r[Σ△] , � UZ3 d UZ3,r� [Σ△] = 31−χ(Σ△) · UZ3[Σ△] , � O1 d UZ3,r� [Σ△] = UZ3[Σ△] , � UZ3,r d UZ3,r� [Σ△] = 31−χ(Σ△) · UZ3[Σ△] , Or d O1 = Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='34) which matches the monoidal structure of 2VecG as defined in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now analyse the topological lines living on the four topological surfaces described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We know that these are labelled by simple objects in the endo-categories of 2VecG, and we explained in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6 that these amount to Z2-grading preserving VecZ3-module functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, this means that the operator O1 must act on the whole space and cannot have support on an (open) sub-region of Σ△, while the topological lines on Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' are labelled by simple objects (VecZ3)⋆ Vec ∼= Rep(Z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The characterisation in terms of module functors also indicates that topological lines living on UZ3 and UZ3,1 are labelled by simple objects in (VecZ3)⋆ VecZ3 ∼= VecZ3 and can be constructed mimicking previous constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us focus on the topological (Wilson) lines of Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='. Given a 1-cycle ℓ on Σ△ and an irreducible representation ρ ∈ Rep(Z3), we may define such a line operator as ÿ l,m � ź e⊂ℓ ρ(l[e]) � |l, m⟩⟨l, m| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='35) For instance, given the non-trivial irreducible representation such that ρ(s) = ω, this operator can be equivalently defined as ś e⊂ℓ Σz e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It is then straightforward to confirm that these commute with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Furthermore, composition of these lines is provided by the monoidal structure of Rep(Z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Interestingly, the 0-form operator O1 acts non-trivially on these topological lines via the action of Z2 on Z∨ 3 , mapping a topological line labelled by ρ to one labelled by the dual representation ρ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Indeed, due to the presence of the Γ matrices in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='33), we have for instance O1� ź e⊂ℓ Σz e � = � ź e⊂ℓ Σz e †� O1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='36) and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We explained this feature below eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='61) in terms of the monoidal structure of 2VecG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, the operator O1 acts non-trivially on the topological lines living on the topological surfaces UZ3 and UZ3,1, which can for instance be traced back to the fact that these surfaces results from condensing topological lines in Rep(Z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This concludes our analysis of the symmetry structure of Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='15) as encoded into 2VecG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5 2Rep(G)-symmetry We finally describe the symmetry structure of Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18), obtained by gauging the non- normal Z2 subgroup of S3 in the transverse field S3-Ising model (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This model acts on a Hilbert space spanned by states |q, m⟩ ∈  e C[Z2]  v C[Z3], where q ∈ Z1(Σ△, Z2) and m ∈ C0(Σ△, Z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Following the general discussions in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='8, we know that Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='12) must have a symmetry structure embodying the fusion 2-category 2Rep(G) of 2-representations of the same 2-group G considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Invoking the results of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='7, Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18) must commute in particular with topological surfaces labelled by tuples (V, {l(N)}N∈V) consisting of an indecomposable VecZ2- module category V and a collection {l(N)}N∈V of group elements in Z3 for each simple object in V such ∼ 60 ∼ that l(Cq ▷ N) = φq−1(l(N)) for every q ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, we count three simple topological surfaces identified with tuples (Vec, {0}), (VecZ2, (0, 0)) and (VecZ2, (1, −1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='24 Note that by exchanging the roles of 1 and 2 in Z3, we find a simple object (VecZ2, (−1, 1)) that is equivalent to (VecZ2, (1, −1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As usual, we begin with the identity operator Utriv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' = ź e ide , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='37) which is now identified with (Vec, {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Next, there is a non-invertible surface operator defined as UZ2,0[Σ△] = 1 2#(Σ△) ÿ q,m,b,n � iπ � Σ△ b⌣(dn + q) � |q, m⟩⟨q, m| , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='38) where n ∈ C0(Σ△, Z2) and b ∈ C1(Σ△, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This is the surface operator identified with (VecZ2, (0, 0)), which is an ordinary condensation defect as we described for the transverse field Z2-Ising model in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The third and fourth operators, which are identified with (VecZ2, (1, −1)) and (VecZ2, (−1, 1)), respectively, implement the 0-form Z3 symmetry that is left over from the initial S3 0-form symmetry after gauging the Z2 sub-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Conventionally, a Z3 0-form symmetry generator would take the form ś v Σx v , but this operator clearly does not commute vertex terms depicted eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20) due to the presence of the Γ matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Instead, one may define the following topological surface operators UZ2,±1[Σ△] = 1 2#(Σ△) ÿ q,m,b,n exp � iπ � Σ△ b⌣(dn + q) � |q, m ± φn(1)⟩⟨q, m| , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='39) where φn(1)[v] := φn[v](1) for all v ⊂ Σ△, which is identified with (VecZ2, (±1, ∓1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In contrast, the conventional 0-form Z3 operators would be given by O±1 = ř q,m |q, m ± 1⟩⟨q, m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows that UZ2,±1 locally acts like O±1 or O∓1 depending on the configuration n of virtual degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More precisely, UZ2,1 is a Z2 condensation defect that additionally acts as Σx v at a vertex v ⊂ Σ△ if n[v] = 0 and Σx v † if n[v] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Commutation of this surface operator with Hamiltonian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18) can be demonstrated as follows: The only non-trivial commutation to check is that with vertex terms φ�Ar v as depicted in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This operator acts on the computational basis as φ�Ar v : |q, m⟩ �→ |q + dxv, φxv(m)⟩ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='40) where xv ∈ C0(Σ△, Z2) is trivial everywhere except at the vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us now separately evaluate φ�Ar v d UZ2,±1[Σ△] and UZ2,±1[Σ△] d φ�Ar v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' On the one hand, we have UZ2,±1[Σ△] d φ�Ar v = 1 2#(Σ△) ÿ q,m,b,n q′,m′ (−1) � Σ△b⌣(dn+q)|q, m ± φn(1)⟩⟨q, m|q′ + dxv, φxv(m′)⟩⟨q′, m′| = 1 2#(Σ△) ÿ q,m,b,n (−1) � Σ△b⌣(dn+q+dxv)|q + dxv, φxv(m) ± φn(1)⟩⟨q, m| = 1 2#(Σ△) ÿ q,m,b,n (−1) � Σ△b⌣(dn+q)|q + dxv, φxv(m) ± φn−xv(1)⟩⟨q, m| , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='41) 24Recall that we write group elements in Z3 as {0, 1, 2} so that the mutliplication is given by the addition modulo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 61 ∼ where in the last line we performed the change of variable n �→ n − xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' On the other hand, we have φ�Ar v d UZ2,±1[Σ△] = 1 2#(Σ△) ÿ q,m,b,n q′,m′ (−1) � Σ△b⌣(dn+q)|q′ + dxv, φxv(m′)⟩⟨q′, m′|q, m ± φn(1)⟩⟨q, m| = 1 2#(Σ△) ÿ q,m,b,n (−1) � Σ△b⌣(dn+q)|q + dxv, φxv(m) ± φxvφn(1)⟩⟨q, m| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='42) Since φn−xv(1) = φxvφn(1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='41) equals (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='42), establishing that UZ2,±1 is indeed a symmetry operator of the model (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Guided by the presentation of 2Rep(G) provided in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6, topological lines living on the surfaces described above can be constructed mimicking previous examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us rather focus on the fusion rules of these surface operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The surface operator UZ2,0 being identical to that encountered in the study of the transverse-field Z2-Ising model, we already know that it satisfies fusion rules (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' However, here we present an alternative derivation, which is more suitable to compute fusion rules of the remaining operators: � UZ2,0 d UZ2,0� [Σ△] = 1 22#(Σ△) ÿ q1,2,m1,2 n1,2,b1,2 (−1) � Σ△δI,JbI⌣(dnJ+qJ)|q1, m1⟩⟨q1, m1|q2, m2⟩⟨q2, m2| = 1 22#(Σ△) ÿ q,m b1,2,n1,2 (−1) � Σ△b1⌣(dn1+q)+b2⌣(dn2+q)|q, m⟩⟨q, m| = 1 22χ(Σ△) ÿ q,m,n1,2 δdn1,qδdn2,q|q, m⟩⟨q, m| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='43) At this point, notice that the configuration space of virtual degrees of freedom at each vertex is given by the Cartesian product Z2 × Z2, which can be decomposed into the disjoint union of two Z2-sets, namely Z(1) 2 ≡ {(0, 0), (1, 1)} and Z(2) 2 ≡ {(0, 1), (1, 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Therefore, we can decompose the summation over n1,2 into ÿ n1,2 = ÿ n∈C0(Σ△,Z(1) 2 ) ‘ ÿ n′∈C0(Σ△,Z(2) 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='44) Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='44) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='43), we immediately obtain � UZ2,0 d UZ2,0� [Σ△] = 1 2χ(Σ△) · UZ2 0 [Σ△] ‘ 1 2χ(Σ△) · UZ2 0 [Σ△] = 21−χ(Σ△) · UZ2 0 [Σ△] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='45) Note that the prefactor 21−χ(Σ△) is precisely the partition function Z2d[Σ△] of the pure Z2 gauge theory for a path-connected manifold Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Whenever Σ is a torus so that χ(Σ△) = 0, the fusion rules reproduce the monoidal product of VecZ2-module categories, namely VecZ2 d VecZ2 ∼= VecZ2 ‘ VecZ2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='46) where the first copy of VecZ2 on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' has simple objects C0 bC0 and C1 bC1, whereas the second copy has simple objects C0 b C1 and C1 b C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Guided by the derivation above, we can compute the fusion of two surface operators UZ2,l1 and UZ2,l2 with l1, l2 ∈ {0, ±1}: ∼ 62 ∼ � UZ2,l1 d UZ2,l2� [Σ△] = 1 22χ(Σ△) ÿ q,m,n1,2 δdn1,qδdn2,q|q, m + φn1(l1) + φn2(l2)⟩⟨q, m| = 1 22χ(Σ△) ÿ q,m n∈C0(Σ△,Z(1) 2 ) δdn,q|q, m + φn1(l1) + φn2(l2)⟩⟨q, m| ‘ 1 22χ(Σ△) ÿ q,m n′∈C0(Σ△,Z(2) 2 ) δdn′,q1|q, m + φn′ 1(l1) + φn′ 2(l2)⟩⟨q, m| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='47) Let us describe various choices of l1, l2 in some detail: First of all, choosing l1 = l2 = 0, we recover the fusion rules (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Similarly, choosing l2 = 0, we find � UZ2,l d UZ2,0� [Σ△] = 1 2χ(Σ△) · � UZ2,l[Σ△] ‘ UZ2,l[Σ△] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='48) Let us now suppose that l1 = l2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given n ∈ C0(Σ△, Z(1) 2 ) so that n1 = n2, we find that the first operator appearing in the decomposition of the fusion product is a Z2 condensation defect that additionally acts as Σx v † = (Σx v )2 at a vertex v ⊂ Σ△ if n[v] = (0, 0) and Σx v = (Σx v †)2 if n[v] = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Conversely, given n′ ∈ C0(Σ△, Z(2) 2 ) so that n′ 1 ̸= n′ 2, we find that the second operator appearing in the decomposition is a plain Z2 condensation defect since it acts as Σx v Σx v † = id at a vertex v ⊂ Σ△ if n′[v] = (0, 1) and Σx v †Σx v = id if n′[v] = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Putting everything together, we find � UZ2,1 d UZ2,1 1 � [Σ△] = 1 2χ(Σ△) · � UZ2,−1[Σ△] ‘ UZ2,0[Σ△] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='49) More generally, fusion rules of arbitrary topological surfaces read � UZ2,l1 d UZ2,l2 1 � [Σ△] = 1 2χ(Σ△) · � UZ2,l1+l2[Σ△] ‘ UZ2,l1−l2[Σ△] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='50) Choosing Σ to be the two-torus, we recover the fusion rules provided by the monoidal structure of 2Rep(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' These can be immediately inferred from the treatment of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='46) [Del22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' SECTION 6 Discussion We conclude with a discussion of extensions and generalisations of the results presented in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1 Further examples The focus of this manuscript was on developing a general framework for gauging invertible symmetries of two-dimensional quantum models and studying the resulting higher categorical symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In order to illustrate our constructions, we considered finite group generalisations of the transverse-field Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It will be very interesting to employ the present formalism in order to tackle more challenging as well as physically more relevant models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Indeed, using the framework presented in this manuscript, the entire parameter space of symmetric Hamiltonians generated by the local operators described in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1 can be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 63 ∼ Symmetries strongly constrain various aspects of the low-energy or infra-red phase diagrams of symmetric quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For instance, the kinds of phases realised, the spectrum of excitations within each phase, universality classes of phase transitions and dualities acting on the parameter space of symmetric models, can all be studied from the lens of the symmetry structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Therefore it is natural to study the phase diagrams of the quantum spin models introduced in this work from the perspective of their higher categorical symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It is also possible to extend the current framework so as to consider more general models as well as more general higher categorical symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For instance, given a group G ≃ Q ⋉φ L, our framework readily accommodates models with 2Vecπ G-symmetry where π is a (non-trivial) 4-cocycle in H4(G, U(1)) characterising the monoidal pentagonator of the fusion 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='25 Such a monoidal pentagonator encapsulates an anomaly revealing an obstruction to gauging the whole symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For certain choices of 4-cocycle π, gauging the L-sub-symmetry would result in a model with a 2VecG- symmetry, where G is a 2-group with a non-trivial Postnikov class [Tho20], thereby going beyond the examples considered in the current manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More generally, the framework employed in this manuscript can be extended so as to accommo- date arbitrary fusion 2-categories generalising further the one-dimensional framework presented in [LDOV21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For instance, given an input fusion 2-category and a choice of module 2-category over it, local operators can be defined of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17) evaluating to matrix entries of the corresponding module pentagonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, it would be interesting to consider models built from the data of fusion 2-categories obtained by idempotent completions of deloopings of braided fusion 1-categories [DR18, GJF19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The corresponding module 2-categories were discussed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [D´e21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2 Self-dual models A celebrated result in low-dimensional condensed matter physics is the exact localisation of the critical point of the (1+1)d transverse-field Ising model by invoking its self-duality [KW41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The relevant duality in this case is the Kramers-Wannier duality, which, up to a local unitary, is obtained by gauging its Z2-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As we reviewed in this manuscript, this phenomenon is specific to (1+1)d since the (2+1)d transverse-field Ising model that possesses an ordinary Z2-symmetry is dual to a model that possesses in particular a 1-form Z∨ 2 -symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This begs the question, how to construct (non-trivially) self-dual symmetric spin systems in two spatial dimensions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It turns out that the mathematical framework developed in this manuscript allows us to rule out many possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Indeed, a necessary condition for self-duality is that the fusion 2-categories of symmetry operators associated with a model and its dual are monoidally equivalent, in addition to being Morita equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This is the case of the (1+1)d Ising model, where the initial VecZ2- symmetry is monoidally equivalent to the dual Rep(Z2)-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In contrast, 2VecZ2 and 2Rep(Z2) are clearly not monoidally equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As a matter of fact, starting from a G-symmetric theory, any duality involving the gauging of a non-trivial subgroup of G would result in a model whose symmetry is not monoidally equivalent to 2VecG, making it impossible for the model to be self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Recent computations of the Brauer-Picard group of 2Rep(G), which informs us about auto-equivalences of the algebraic structure encoding the super-selection sectors of a G-symmetric model, suggests that the only candidate dualities may be of the form 2Vec → 2Vecλ, which only involve a change of module pentagonator amounting to the pasting of a (2+1)d symmetry-protected topological phase [KLW+20b, D´e22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This special type of duality, and the possibility of defining self-dual models with respect to it, will be investigated elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 25Within our framework, this is simply accomplished by choosing 2Vecπ G as a module 2-category over itself when defining the local operators (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='17), so that the module pentagonator coincides with the monoidal pentagonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 64 ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3 Symmetry-twisted boundary conditions Throughout this manuscript, we have purposefully been somewhat vague regarding the role of bound- ary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' For instance, given a two-dimensional surface with the topology of a torus, our results would implicitly assume periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' But our framework can be extended so as to accommodate symmetry-twisted boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the case of a G-symmetric model, we expect symmetry-twisted boundary conditions along the pair of non-contractible cycles of the torus to be labelled by commuting group elements in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Commutativity in G should be required so as to preserve translation invariance of the model up to local unitary transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Importantly, the original G-symmetry interacts with these boundary conditions in such a way that one is typically left with a smaller symmetry in the presence of non-trivial symmetry twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, given symmetry twists (g1, g2) ∈ G2 such that g1g2 = g2g1, the leftover symmetry group is given by the stabiliser subgroup of group elements x ∈ G satisfying (xg1x−1, xg2x−1) = (g1, g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It follows in particular that every pair of commuting twists (g1, g2) and (g′ 1, g′ 2) for which there exists x ∈ G such that (g′ 1, g′ 2) = (xg1x−1, xg2x−1) possess the same stabiliser subgroup, thereby defining equivalence classes of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given such an equivalence class and one of its representatives, the resulting Hamiltonian would decompose into symmetry charge sectors labelled by irreducible representations of the stabiliser subgroup of the representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Interestingly, the same data labelling the super-selection sectors described above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' symmetry- twisted boundary conditions together with twisted symmetry charge sectors, appeared before in a different context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Indeed, these correspond to the simple modules of tube algebras, which were first considered in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [Del17] and generalised in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [BD19a, BD19b], that classify and characterise loop-like excitations in (3+1)d Hamiltonian realisations of Dijkgraaf-Witten theory [DW90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' More specifically, such a simple module labels a loop-like flux, to which a point-like charge may be attached, while being threaded by an auxiliary string-like flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A complimentary field-theoretic approach was developed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [CTR15, TCR16, CTNR17, TCSR17] to extract topological line and surface operators and the braiding phases of (3+1)d topological finite group gauge theories from their gapless surface theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' A similar interplay between super-selection sectors of symmetric models and topological excita- tions of a higher-dimensional topological model exist in (1+1)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Indeed, super-selection sectors of a symmetric model are known to be labelled by simple objects in the monoidal centre of the symmetry fusion 1-category [AFM20, KZ22, MMT22, LOST22, LDV22], and the interplay between dualities and super-selection sectors was recently studied in detail for arbitrary one-dimensional quantum lat- tice models in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [MMT22, LDV22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' But the monoidal center of a fusion 1-category encodes the anyonic excitations of the corresponding Hamiltonian realisation of the Turaev-Viro-Barrett-Westbury state-sum invariant [TV92, BW93, LW05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Interestingly, the notion of monoidal centre of a fusion 2-category also exists [BN96, Cra98] and was computed explicitly in a few cases [KTZ19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Perhaps surprisingly, given a topological model with input datum a certain (spherical) fusion 2-category, the excitation content encoded into its monoidal centre differs from that described by the tube algebras mentioned above [BD20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' The precise relation between both algebraic structures together with the corresponding physical implications were clarified in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [BD21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Concretely, this means that in con- trast to the one-dimensional scenario, super-selection sectors of a G-symmetric model on a torus are not labelled by simple objects in the monoidal centre of 2VecG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In light of the results obtained in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' [BD20, BD21], we rather conjecture that the monoidal centre in the higher dimensional case would encode super-selection sectors of a model defined on a cylinder hosting a combination of open and closed boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 65 ∼ Acknowledgements CD would like to thank Thibault D´ecoppet for numerous discussions on Morita equivalence of fusion 2-categories, as well as Laurens Lootens, Frank Verstraete and Gerardo Ortiz for collaborations on the lower-dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' AT would like to thank Lakshya Bhardwaj, Lea Bottini, Heidar Moradi, Faroogh Moosavian and Sakura Sch¨afer Nameki for numerous discussions on related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' This work has received funding from the Research Foundation Flanders (FWO) through postdoctoral fellowship No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' 1228522N awarded to CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' AT is supported by the Swedish Research Council (VR) through grants number 2019-04736 and 2020-00214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' APPENDIX A Two-dimensional Zp gauge theory In this appendix, we collect some basic facts about different presentations of the two-dimensional Zp topological gauge theory, which appear in the definition of condensation defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In particular, the case p = 2 appears throughout the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Given a closed oriented surface Σ endowed with a triangulation Σ△, the partition function of the theory reads26 Z(p) 2d [Σ△] = 1 p#(Σ△) ÿ b,n exp �2πi p � Σ△ b⌣dn � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1) where #(Σ△) := |Σ0 △| + |Σ2 △| and |Σj △| is the number of j-simplices in the triangulation Σ△.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' In the above partition sum, b ∈ C1(Σ△, Zp) and n ∈ C0(Σ△, Zp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As in the main text, the symbols d and ⌣ denote the simplicial codifferential and the cup product, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Let us begin by listing a couple of important identities that are used in several occurences in the main text ÿ n exp �2πi p � Σ△ n⌣(db − q2) � = p|Σ2 △|δdb,q2 , ÿ b exp �2πi p � Σ△ b⌣(dn − q1) � = p|Σ1 △|δdn,q1 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2) where qj ∈ Cj(Σ△, Zp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' We can now evaluate the partition function by summing over n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' First we perform an integration by parts such that the codifferential acts on b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Then summing over n imposes a Zp delta function on each 2-simplex, giving an overall factor of p|Σ2 △|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Putting everything together, one obtains Z(p) 2d [Σ△] = p|Σ2 △| p#(Σ△) ÿ b δdb,0 = |Z1(Σ△, Zp)| p|Σ0 △| = |H1(Σ△, Zp)| × |B1(Σ△, Zp)| p|Σ0 △| = |H1(Σ△, Zp)| |H0(Σ△, Zp)| = pb1(Σ)−b0(Σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3) Notice that the sum over b gives a factor of |Z1(Σ△, Zp)| due to the Zp cocycle constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Moreover, we used H1(Σ△, Zp) = Z1(Σ△, Zp)/B1(Σ△, Zp), where Z1(Σ△, Zp) and B1(Σ△, Zp) are the set of 1-cocycles and 1-coboundaries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Finally we employed the expression B1(Σ△, Zp) ≃ C0(Σ△, Zp)/Z0(Σ△, Zp) ≃ C0(Σ△, Zp)/H0(Σ△, Zp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='4) 26In the main text, we denote the partition function of the two dimensional Z2 gauge theory Z2d[Σ△] := Z(2) 2d [Σ△].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 66 ∼ Notice that the final expression in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='3), which is a topological invariant, can equivalently be expressed in terms of the 1st and 2nd Betti numbers b1,2(Σ) of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' It is also instructive to compute (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='1) by summing over b instead of summing over n: Z(p) 2d = p|Σ1 △| p#(Σ△) ÿ n δdn,0 = 1 pχ(Σ) ÿ n δdn,0 = |H0(Σ△, Zp)| pχ(Σ) = pb1(Σ)−b2(Σ) = pb1(Σ)−b0(Σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='5) We first used eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='2), as well as the expression χ(Σ) := |Σ0 △| − |Σ1 △| + |Σ2 △| = b0(Σ) − b1(Σ) + b2(Σ) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content='6) defining the Euler characteristic χ(Σ) of the surface Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Then, we evaluated the sum over n, producing a factor |Z0(Σ△, Zp)| = |H0(Σ△, Zp)| = pb0(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Lastly, in going to the final expression, we used that for any 2-manifold b2(Σ) = b0(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' As expected, the two ways of computing the partition function give the same topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' ∼ 67 ∼ References [AFM20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfT_x1/content/2301.01259v1.pdf'} +page_content=' Aasen, P.' 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Marius Z¨ollner†‡ +†FZI Research Center for Information Technology, Germany +bogdoll@fzi.de +‡Karlsruhe Institute of Technology, Germany +Abstract—Autonomous Driving (AD), the area of robotics with +the greatest potential impact on society, has gained a lot of +momentum in the last decade. As a result of this, the number +of datasets in AD has increased rapidly. Creators and users of +datasets can benefit from a better understanding of developments +in the field. While scientometric analysis has been conducted in +other fields, it rarely revolves around datasets. Thus, the impact, +attention, and influence of datasets on autonomous driving +remains a rarely investigated field. In this work, we provide a +scientometric analysis for over 200 datasets in AD. We perform +a rigorous evaluation of relations between available metadata +and citation counts based on linear regression. Subsequently, we +propose an Influence Score to assess a dataset already early on +without the need for a track-record of citations, which is only +available with a certain delay. +Index Terms—Robotics, Autonomous Driving, Datasets, Influ- +ence, Impact, Attention, Scientometrics, Bibliometrics +I. INTRODUCTION +Autonomous driving technology does not only affect ur- +ban transportation [1] and delivery of goods [2], but also +farming [3] or warehouse logistics [4]. With the progress +of deep learning and this growing interest in AD in many +fields of robotic, the number of related datasets is consistently +increasing. The datasets have also increased in size and many +have become increasingly specialized [5]. The most extensive +collection of datasets known to us, ad-datasets, currently +lists 231 datasets in the domain [6]. However, not all of +them are being equally used in the robotics community, the +distribution of their citations is heavily skewed. As part of +the more impactful works, well known datasets for the core +tasks perception and prediction dominate [7]–[9]. As part of +the long tail, many datasets for niche research areas exist [10]– +[12]. Well known datasets tend to bring many advantages with +them: They enable comparison between works, have higher +quality, advanced tooling, and often community knowledge +and support is available. The increasing number of datasets, +which are potentially interesting but lack reputation, leads to a +lot of untapped potential: Many researchers are hesitant to use +such datasets and stick to old, but established ones instead [6]. +This is why we asked ourselves the question: Given a novel +dataset without a multi-year track record of citations, is there +a way to estimate its future development? Datasets with a high +potential might be more appealing already in their early days. +* These authors contributed equally +2008 +2010 +2012 +2014 +2016 +2018 +2020 +2022 +year +5 +10 +15 +20 +25 +30 +35 +40 +publications +0 +2000 +4000 +6000 +8000 +10000 +citations +Fig. 1. +Course of published datasets and citations of the accompanying +publications in the domain of AD. This growing number of datasets, initially +without reputation, holds a great deal of untapped potential as researchers +struggle to use new datasets for their research. Datasets as listed on ad- +datasets [6], citation counts from Semantic Scholar [13]. +Research Gap. To date, citations are mostly used to as- +sess datasets, which are not available early on. Thus, new +datasets can have a hard time gaining traction, which results +in untapped potential. It is yet not well understood if and +how metadata of datasets relate to future impact or how they +can be utilized to assess datasets early on. To the best of our +knowledge, such an analysis has not yet been performed. +Contribution. In order to analyze the field of dataset, we +first assembled the largest collection of datasets with enriched +metadata available, including over 200 datasets with metadata +from three different sources. We then applied linear regression +to evaluate factors which relate to the future impact of datasets, +measured in citations. Finally, we propose the Influence Score +(IS), which is a mean to assess datasets early on without the +need of a multi-year track record of citations. The IS can be +used to assess any datasets at any given year, which also allows +for later analysis. Our work aims to help researchers from +the robotics community to better understand and assess the +performance of datasets. This can lead to the design of better +and thus more influential datasets as well as an actionable +analysis of new datasets to assess their potential. All data used +in this work is as of January 04, 2023. All code is available +on GitHub. +arXiv:2301.02200v1 [cs.DL] 5 Jan 2023 + +II. RELATED WORK +Here, we give an introduction to the scientometrics, biblio- +metrics, and altmetrics, followed by dataset analysis. +A. Scientometrics, Bibliometrics, and Altmetrics +Scientometrics, Bibliometrics, and Altmetrics are highly +intertwined fields that focus on the analysis of science and +its processes as a whole, written works of science, and online +communication of science, respectively [14]. +Scientometrics. Ravenscroft et al. [15] examined the impact +of research by comparing citation-based metrics, such as +citation count or h-index [16], with altmetrics and impact other +than citations, e.g., societal and economic impact. However, +they found no strong relationship between the fields. Hicks +et al. [17] suggest using multiple factors to portray multiple +aspects. +Leydesdorff et al. [18] claim that citations are equated to im- +pact and evaluate the relationship between impact and research +quality. They found that short-term citations signify the invest- +ment in a current discourse, while long-term citations signify +acceptance as reliable scientific knowledge. However, some +researchers question if or to what extent citations measure +scientific impact and point to issues, e.g., inconsistent reasons +for citations [19]. Problems include the cumulative advantages +already successful papers experience [20], self-citations, which +men do more often [21], negative citations, and citing out +of reasons that do not reflect actual use or relevance [22]. +Valenzuela et al. [23] presented a method to identify four types +of citations: ”Related work, Comparison, Using the work, +Extending the work” [23], which is used by Semantic Scholar +to determine “Highly Influential Citations” [24]. However, it +shows a high correlation with citations. +The field of trend detection analyzes large corpora of works +to detect upcoming patterns [25]–[28]. Lopez Belmonte et +al. [29] analyzed publications in Machine Learning and Big +Data and found exponential growth of publications. They +compared the popularity of keywords and the h-index. +Bibliometrics. Citations can be aggregated on different +levels, e.g., for the papers of one author as the h-index does, +or on the journal level, like the journal impact factor (JIF), +which is the two-year average ratio of citations to articles +published. The JIF is ill-suited for evaluating individual papers +by means of the journal it was published in [30]. This is due +to the heavy skewness of the distribution of citation counts +within journals [31]. The Hirsch-index, usually referred to as +the h-index, combines the productivity of an author with the +impact of their individual papers. Using the h-index increases +robustness compared to simply counting the total number of +citations, as few highly cited papers have little effect on the +h-index. In addition, there have been efforts to recommend +papers and citations [32], [33], predict future citation counts +of papers [34]–[37] and the impact of scientists [38]. Such +approaches remain challenging and are often domain-specific. +Bornmann and Marx [39] have proposed to expand the +bibliometric analysis by not only considering citations but also +references. Following this idea, reference analysis has been +used to identify influential references [40]. +Altmetrics. Online interactions with papers are referred to +as altmetrics and are usually available earlier than citations, +which gives altmetrics an advantage over bibliometrics [41]. +Bornmann and Marx [42] examined if Altmetrics can be used +to predict paper quality which was measured through peer as- +sessments and found that both tweets and readers do, with the +latter having a stronger relationship. Lamb et al. [43] showed +that the Altmetric Attention Score is a predictor of the citations +of a paper in ecology and conservation. Zavrel et al. [44] +clustered papers released at the International Conference on +Machine Learning (ICML) in 2022 and calculated a score +for their impact. They used Twitter mentions, citations, the +authors’ average h-index, and an award for outstanding papers +rewarded by the conference itself. They claim to do a “sim- +ple combination of these four scores to calculate an impact +score” [44] but do not reveal the formula. F¨arber analyzed +GitHub repositories of papers, mostly from the field of AI, and +found a power-law distribution of stars and forks [45]. While +Haustein et al. claim that ”Altmetrics measures scientific +impact based on online references and activity” [46], many +disagree with equating altmetrics with impact. For example, +Sugimoto states that ”attention is not impact” and calls online +interaction with scientific works ”attention” [47]. Altmetrics +might reflect broader or societal impact [41]. +B. Dataset Analysis +Bogdoll et al. [5] gathered metadata of over 200 datasets +in the field of autonomous driving. Similarly, F¨arber and +Lamprecht released the data set knowledge graph, which is +a collection of over 2,000 datasets with added metadata [48]. +D’Ulizia et al. [49] analyzed the metadata of datasets for fake +news detection. Utamachant and Anutariya [50] analyzed the +datasets of Thailand’s national open data portal, but relied +on domain experts to assess impact. Nguyen and Weller +proposed FAIRnets, a service to search for neural networks +and their related datasets [51] published on GitHub. They build +upon the Findable, Accessible, Interoperable, Reusable (FAIR) +principles [52], which ”put specific emphasis on enhancing +the ability of machines to automatically find and use the +data” [52]. Khan et al. [53] analyzed datasets from the Global +Biodiversity Information Facility (GBIF) which publishes +datasets with a DOI and indexes datasets in biodiversity. +They promote data standards and the reuse of datasets [54] +as well as accompanying publications, which they call ”data +papers”, that describe a dataset thoroughly [55]. Khan et +al. [53] report a strong correlation between dataset download +numbers and citation counts, and suggest that downloads and +citations signify a similar kind of impact. They also find +correlations between altmetrics and citations. Moreover, they +question whether every citation means the usage of a dataset +and point to differences in citation behavior. F¨arber et al. +proposed an approach to find methods and datasets which +authors actually used when citing the related paper [56]. +However, unrealistically few dataset usages were identified. + +AD-Datasets +Altmetric +Semantic Scholar +List of Datasets +with Enriched +Metadata +Regression Analysis +of Citations +and Metadata +Influence +Score +Fig. 2. Overview: First, we collect data from various sources and combine them to a single list of datasets. Based on this, we perform a regression analysis +to determine which metadata correlate with future prediction counts. Based on the metadata, we compute our Influence Score (IS). +III. REGRESSION ANALYSIS +Here, we first introduce our taxonomy of terms related to the +assessment of datasets. Subsequently, we introduce our data +sources. Based on these, we describe the regression analysis of +citations and metadata. In Section IV, we present the resulting +Influence Score. Figure 2 gives an overview over this process. +A. Taxonomy +As became clear in Section II, no common language for +specific aspects in the domain has evolved yet. Thus, we +introduce a taxonomy to clearly describe different aspects with +respect to the development of a dataset or paper. As general +terms for this, we utilize success, progress, performance, or +potential. For concrete aspects, we establish the following +terms, where each one can be applied to any single paper: +Impact: We use the number of citations to measure the sci- +entific impact of a paper, which is common in Scientometrics, +but not without criticism [19]. +Attention: The online reception, such as tweets and +Wikipedia articles mentioning a paper, represents the attention +by researchers and the public. +Influence: We refer to the resulting score of our proposed +method, which combines a multitude of aspects, as the influ- +ence, or IS, of a dataset. We deem this term appropriate for +any method that goes beyond purely impact-based assessment. +B. Data Sources and Selection +We used three sources for our data: ad-datasets.com [6], +the +Semantic +Scholar +Academic +Graph +API +[13], +and +altmetric.com [57]. Based on the DOI and arXiv-Id from +ad-datasets, we automatically extracted the metadata of papers +from Semantic Scholar and altmetric.com. Based on these +papers, all of which describe datasets, we performed data +exploration, regression, and the computation of the IS. +AD-Datasets: This web tool offers an overview of over +200 data sets in AD +[5]. It includes a detailed breakdown +of most dataset entries by 20 different meta categories, +provided by the authors and the research community. This +way, relations between datasets, accompanying papers and +further metadata are available. The underlying data is stored +in the JSON format and can be accessed accordingly. In this +work, we utilize the nframes and nsensors metadata, which +indicate the size of datasets in different dimensions, which is +a potential aspect of the relevance of a dataset. +Altmetric: We used the API by altmetric.com [57], which +provides insight into online attention and readership. These +properties are provided by the following categories: +Attention Score: The aascurr aggregates different sources +into a single score [58]. It is a weighted count of different +online sources. For example, the weight for a reference on +Wikipedia is 3, while Twitter and Reddit mentions are both +weighted with 0.25. Unfortunately, the history of this score is +only provided for the most recent year. +Attention Score after three months: The aas3m is the +percentile of the papers’ Attention Score three months after +publication. The percentile is calculated in comparison to +papers that have been released at a similar time. +Readers: The number of people nreaders that have saved +a paper in their reference management software. Reading +a paper is less significant than citing it, but the number of +readers might imply interest in a paper early on. The number +of readers is provided individually for multiple reference +management services, which we sum into a single count for +online readers. Altmetric.com cannot verify the number of +readers, thus, it is not included in the attention score. This +is a relevant attribute, as it decouples the metrics. However, +there are no historic data available. +Semantic Scholar: For every accompanying paper of a +dataset, we pulled data from Semantic Scholar. Sometimes, +multiple datasets are described in the same paper, which will +lead to the same information for those datasets. We extracted +the following nested data: +• List of referenced papers, including for each a list of all +citing papers and the year of citation. +• List of authors and their respective publications, including +for each publication a list of citing papers and the year +of citation. +• List of citing papers, including for each a list of citing +papers and the year of citation. +Wherever possible, we collected associated timestamps, +including the publication year apub of each paper. The first + +two categories, while dynamic, are directly available. We use +the citations of references as a measure of the impact of +references. Having impactful references might indicate that a +paper is covering popular topics within AD or that the authors +are knowledgeable in the field. +The performance of authors can be estimated by evaluating +their paper count and how many citations they have received, +which becomes only meaningful over time. As discussed +earlier, not every citation means usage of a dataset. While it +would have been interesting to take into account, in which +section a paper has been cited, this data was not available for +most papers. Based on the ncit3 citations from the previous +three years, citations of works that cited a dataset signify the +value created by working with the dataset, which is why we +included those. +A critical aspect of the collected data is that oftentimes, no +historic information was available. Also, oftentimes, data was +not available due to limitations, e.g., Altmetric is incompatible +with DOIs from IEEE publications, which are common in the +fields of robotics, autonomous driving, and machine learning. +Similarly, Ravenscroft et al. [15] expressed concerns about +Altmetric, as they were unable to find 40 % of the papers +they analyzed, all from the field of computer science. +C. Data Aggregation +To further utilize the raw data we collected, we aggregated +some of it with the aim to assemble a finite list of features +that describe a dataset. We aggregated some of our data +sources using the concept of the h-index formula, as it is +widely known, transparent, and easy to reproduce. In order +to analyze smaller timespans, we deviated from the typical 5- +year duration and calculated multiple 3-year indexes ourselves. +For authors, we applied the h3-index for each individual. +We then aggregated the h-indices of all authors of a paper via +the arithmetic mean in autµh3. Respectively, we applied the +h-index formula to references and citations. For the references +of a paper, the refh3 is calculated identically to the way it +is utilized for authors. Just like an author has papers with +citations, a paper has references with citations. A high h-index +for references would signify that several of the referenced +papers gained lots of attraction. We also applied the h3-index +formula to the citations and their citations to get the h3-index +of citations cith3, following Schubert et al. [59]. The final list +of all extracted and calculated features can be found in Table I. +D. Cluster Analysis and Regression Setup +We evaluated our computed features with respect to their +ability to predict future citations. Therefor, we performed +linear regression. For this, we first computed clusters of +the datasets to determine a meaningful time horizon. Subse- +quently, we defined our regression setup. +Cluster Analysis: To show that there are meaningful vari- +ations between clusters, we looked at the impact of papers +for up to 2 years after publication in a journal or conference +1 +0 +1 +2 +years after publication +0 +50 +100 +150 +200 +250 +300 +350 +400 +ncits +Fig. 3. Development of the number of citations for publication-clusters over +a dynamic 3-year window. Papers are clustered based on similar performance. +Semantic Scholar also tracks citations of pre-prints, which leads to citations +prior to the publication date of the final work. +proceedings. As visualized in Fig. 3, clear clusters are visi- +ble, where line-thickness indicates cluster size. For k-means +clustering, we used six clusters based on the elbow plot, +which shows which additional cluster provides a non-marginal +reduction of the total variation within clusters. The growth of +citation counts behaves exponentially for the top performing +works. A clear differentiation between all clusters becomes +apparent already after one year, which we chose as the time +horizon for the regression. This allowed us to include more +recent papers, which would have been excluded otherwise due +to their missing track record of citations. +Regression Setup: As independent variables, we included +the features nsensors, apub, refh3, autµh3, ncit3, and aas3m, +as shown in Table I, in order to estimate the citation count after +one year. Preliminary data exploration suggested a curvilinear +relationship between aas3m and the number of citations. +Therefore, a quadratic term was added. All predictors were +standardized by subtracting the mean and dividing by the +standard deviation prior to the analysis. The feature ncit3 was +log(x+1)-transformed to ensure a normal distribution of the +residuals, which are the error terms of the regression. +For the regression, we were able to utilize 111 datasets, as +values for all included features were available, and they had +been released at least one year prior. Residuals and collinearity, +the ability to linearly predict one independent variable with +other independent variables, were checked. The collinearity +was quantified through the variance inflation factor of each +regressor which all were lower than three. We performed the +Breusch-Pagan and White test for heteroskedasticity, which is +the inconsistency of the variance of residuals at different levels +of the dependent variable. Both tests indicated that we do not +have sufficient evidence for the presence of heteroskedasticity. +Still, robust standard errors were used to ensure the standard +errors are calculated correctly in the presence of heteroskedas- +ticity which at worst leads to standard errors being estimated +larger. + +Feature +Description +Availability +Standardized +Log(x+1) +Influence Score +Source +nframes +Number of frames in the dataset +At publication +– +– +✓ +AD-Datasets [6] +nsensors +Number of sensor types +At publication +✓ +✗ +✓ +AD-Datasets [6] +apub +Year of publication +At publication +✓ +✗ +✗ +Semantic Scholar [13] +refh3 +3 year h-index of references +At publication +✓ +✗ +✓ +Semantic Scholar [13] +autµh3 +Mean 3 year h-index of authors papers +At publication +✓ +✗ +✓ +Semantic Scholar [13] +ncit3 +Number of citations within past 3 years +Anytime after publication +✓ +✓ +✓ +Semantic Scholar [13] +cith3 +3 year h-index of citations +>3 years after publication +– +– +✓ +Semantic Scholar [13] +aascurr +Altmetric Attention Score +Anytime after publication +– +– +✓ +Altmetric [57] +aas3m +Altmetric Attention Score at 3 mos +After 3 months +✓ +✗ +✗ +Altmetric [57] +nreaders +Number of readers +Anytime after publication +– +– +✓ +Altmetric [57] +TABLE I +OVERVIEW OF METADATA USED FOR THE REGRESSION ANALYSIS AND THE INFLUENCE SCORE. +We chose not to include nframes for the regression because +numerous of the datasets did not contain this meta-information. +However, we examined a model in which the feature was +included, which did not lead to new findings. +E. Regression Analysis +With the explained regression setup, we were now interested +in finding statistically significant predictor variables for the +citation count at the end of the year after publication. +The aas3m and aas2 +3m were positively related to the number +of citations and both relationships were significant at <0.0001. +Both coefficients were positive. All other features were not +significantly related to the number of citations. The results are +reported in Table II. +coef +std err +z +P>|z| +[0.025 +0.975] +refh3 +0.0987 +0.103 +0.96 +0.337 +-0.103 +0.3 +autµh3 +0.071 +0.086 +0.824 +0.41 +-0.098 +0.24 +apub +0.0281 +0.084 +0.337 +0.736 +-0.136 +0.192 +nsensors +0.1383 +0.108 +1.287 +0.198 +-0.072 +0.349 +aas3m +0.803 +0.116 +6.895 +0 +0.575 +1.031 +aas2 +3m +0.3375 +0.072 +4.72 +0 +0.197 +0.478 +intercept +2.6402 +0.117 +22.48 +0 +2.41 +2.87 +TABLE II +REGRESSION FOR CITATIONS AFTER ONE YEAR. REGRESSION +COEFFICIENTS AND 95% CONFIDENCE INTERVAL ARE REPRESENTED ON +THE LOG SCALE. +Since only one feature showed a relationship with the +number of citations, we do not consider a stable prediction +of citations possible with the available data. In order to still +perform an early evaluation of datasets, in the following we +present our Influence Score (IS). +IV. INFLUENCE SCORE +We propose the Influence Score (IS), which includes a +variety of features that are available early on. These are +weighted dynamically in order to receive an indication of the +relative performance of any given dataset at any given time. +The calculation compares each data set with all existing ones +from the domain, so that relative differences and trends are +immediately recognizable. +Percentiles are used to allow relative scoring within the +surrounding group of datasets. The data sets roughly follow a +normal distribution in their IS scores. As shown in Table I, we +utilize eight different features for the IS: nframes, nsensors, +refh3, autµh3, ncit3, cith3, aascurr and nreaders. This way, +we consider more than just the citations, but do not exclude +them: If early citations are already available, they become a +meaningful part of the score, as the relation to other datasets +of the peer group is relevant. This way, citation velocity is +included. The IS is defined as follows: +IS(paper) = 1/n ∗ +n +� +i=0 +percentile(featurei) +(1) +where: +i = Feature Index +n = Number of available features +Only features, which are available, are dynamically included +in the IS. As we used percentiles of each feature to facil- +itate the understanding of the features, common issues are +mitigated. E.g., typical feature values change over time: For +example, with the growth of AD, the ncit value of a paper +today is likely higher than a decade ago, which becomes +clearly visible in Figure 1. Furthermore, commonly observed +values for features might differ between different fields. This +helps people who are not familiar with AD or the features to +easily assess if the score a dataset achieved is high or low. +A. Qualitative Demonstration +To showcase the IS, we compare exemplary the development +of the five most and least cited papers with a latest publication +in 2019, by their IS and visualize the results in Fig. 4. +It becomes clearly visible, that the two groups are easily +distinguishable by their IS, but also that differences within +the groups are visible. +The individual features show different pictures: For refh3, +also papers with only a few citations can have meaningful +references in their works. ncitt3 and cith3 only confirm what +was known by our data selection, as we selected the datasets +by citation count. autµh3 shows, how successful datasets can +also boost personal careers, as some authors became professors +and remained active in their field. nsensors and nframes show + +2014 +2016 +2018 +2020 +2022 +years +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +IS +2014 +2016 +2018 +2020 +2022 +years +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +percentile refh3 +2014 +2016 +2018 +2020 +2022 +years +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +percentile ncit3 +2014 +2016 +2018 +2020 +2022 +years +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +percentile cith3 +2014 +2016 +2018 +2020 +2022 +years +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +percentile aut h3 +2014 +2016 +2018 +2020 +2022 +years +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +percentile nsensors +2014 +2016 +2018 +2020 +2022 +years +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +percentile nframes +KITTI +nuImages +Cars +Synthia +Waymo Open Perception +Daimler Stereo Pedestrian Detection Benchmark +TRoM +DriveSeg (Semi-auto) +DriveSeg (MANUAL) +WZ-traffic +Fig. 4. Influence Score and individual features for different datasets. We show exemplary results for the five best and worst performing datasets of all time, +measured by citations, with a latest release in 2019 for historical data. We also show six individual features of the IS, where historical data was available. +rather static results, with a trend towards larger datasets being +more successful. +B. Quantitative Demonstration +In order to show the quantitative performance of the IS, we +showcase all datasets released in 2022 in a detailed overview +in Table III. Such a pre-filtering process is useful in order +to explore novel datasets. Here, it becomes clear that the +IS captures a wide variety of different aspects of a dataset. +Of particular interest is the fact that even low-performing +data sets can lead in certain features. Thus, if a researcher +is interested in certain aspects of a dataset, they can simply +focus on the features they are interested in and omit the others, +which enables less-known datasets to be discovered and used. +Figure 5 shows an overview of the IS distributions. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +IS +0 +2 +4 +6 +8 +10 +number of datasets from 2022 +Fig. 5. Distribution of the Influence Score (IS) of all datasets from 2022. +V. CONCLUSION +In this paper, we addressed the lack of knowledge with +respect to the scientific impact, attention, and influence of +datasets in robotics. Our focus was on an early assessment +of datasets, given a flood of new datasets published every +year. We analyzed impact measured by citations and evalu- +ated relations of metadata and features which we extracted +from multiple online sources. Our regression analysis showed +no strong relation between future citations and our selected +features. Subsequently, we presented our developed Influence +Score (IS). This score utilizes a set of eight features to assess +any dataset also early on. This is based on an analysis within +the peer group of all datasets, which allows for the early +detection of relative trends. +Our work contributes to a better understanding of datasets, +which enables researchers to find and assess published +datasets in the domain of autonomous driving without the +need of waiting for a track record of citations. +Limitations and Outlook: For our work, we evaluated the +paper accompanying the dataset assuming that the paper is a +good representation of the dataset. When measuring scientific +impact through citations, we think this holds because the +paper is actually the cited scientific work. However, not every +citation might be meaningful, positive, or indicate the usage +of a dataset. Ideally, large language models could evaluate +if a dataset is actually used, if cited. Khan et al. [53], who +analyzed datasets in biodiversity, suggested that the correlation +between the number of downloads and citations signifies +that these two measures are comparable representations of +impact. However, in the domain of AD, download numbers +are typically not available, but this might change. As some +datasets are presented in the same paper, a further decoupling +of accompanying papers and the respective datasets would be +helpful. We found, that the quality and availability of metadata +in AD provided by the creators of datasets varies strongly. +Thus, standards should be established [90]. While we focussed +on dataset and paper specific features for this work, we are +also interested in the venue or journal of publication, which +can be considered as an additional feature in future work. + +IS +ncit3 +cith3 +refh3 +autµh3 +nframes +nsensors +aascurr +nreaders +Waymo Block-NeRF [60] +0.82 +0.7 +0.53 +0.95 +0.87 +– +– +1.0 +0.89 +SHIFT [61] +0.62 +0.25 +0.2 +0.99 +0.88 +0.94 +0.77 +0.65 +0.42 +Street Hazards [62] +0.62 +0.67 +0.58 +0.83 +0.92 +0.16 +0.25 +0.73 +0.45 +KITTI-360-APS [63] +0.48 +0.23 +0.06 +0.68 +0.67 +0.56 +0.25 +0.97 +0.21 +ScribbleKITTI [64] +0.45 +0.2 +0.15 +0.83 +0.97 +0.32 +0.25 +0.63 +0.02 +BDD100K-APS [63] +0.42 +0.23 +0.06 +0.68 +0.67 +0.1 +0.25 +0.97 +0.21 +Ithaca365 [65] +0.4 +0.15 +0.15 +0.68 +0.22 +0.82 +0.58 +0.53 +0.26 +CODA [66] +0.39 +0.2 +0.15 +0.75 +0.36 +– +0.25 +0.53 +0.33 +Rope3D [67] +0.37 +0.15 +0.15 +0.83 +0.42 +0.53 +0.58 +0.25 +0.25 +Comma2k19 LD [68] +0.37 +0.12 +0.15 +0.71 +0.56 +– +– +0.64 +0.02 +RoadSaW [69] +0.29 +0.09 +0.06 +0.27 +0.19 +0.83 +0.25 +– +– +K-Radar [70] +0.26 +0.09 +0.06 +0.47 +0.11 +0.44 +0.93 +0.4 +0.26 +CARLA-WildLife [71] +0.26 +0.03 +0.06 +0.92 +0.12 +– +0.25 +0.34 +0.08 +AugKITTI [72] +0.25 +0.03 +0.06 +0.88 +0.16 +– +– +0.25 +0.1 +WildDash 2 [73] +0.24 +0.17 +0.2 +0.52 +0.08 +– +0.25 +– +– +MONA [74] +0.23 +0.03 +0.06 +0.36 +0.48 +– +0.25 +– +– +Street Obstacle Sequences [71] +0.23 +0.03 +0.06 +0.92 +0.12 +0.07 +0.25 +0.34 +0.08 +HDBD [75] +0.22 +0.03 +0.06 +0.16 +0.64 +– +0.58 +– +– +GLARE [76] +0.22 +0.03 +0.06 +0.47 +0.29 +– +– +0.42 +0.06 +Boreas [77] +0.22 +0.29 +0.25 +0.14 +0.2 +– +– +– +– +Autonomous Platform Inertial [78] +0.21 +0.2 +0.25 +0.36 +0.03 +– +– +– +– +aiMotive [79] +0.2 +0.03 +0.06 +0.33 +0.04 +0.41 +0.93 +0.44 +0.13 +CarlaScenes [80] +0.2 +0.03 +0.06 +0.52 +0.2 +– +0.77 +– +– +LUMPI [81] +0.19 +0.09 +0.06 +0.02 +0.07 +0.74 +0.58 +– +– +A9 [82] +0.19 +0.25 +0.25 +0.14 +0.1 +– +0.58 +0.29 +0.12 +Amodal Cityscapes [83] +0.19 +0.09 +0.06 +0.27 +0.43 +0.11 +0.25 +0.29 +0.08 +R-U-MAAD [84] +0.16 +0.03 +0.06 +0.33 +0.35 +– +0.25 +0.15 +0.06 +TJ4DRadSet [85] +0.15 +0.12 +0.15 +0.14 +0.05 +– +– +0.42 +0.02 +OpenMPD [86] +0.14 +0.18 +0.15 +0.02 +0.07 +0.27 +0.58 +– +– +I see you [87] +0.12 +0.03 +0.06 +0.09 +0.01 +– +– +0.44 +0.08 +SceNDD [88] +0.11 +0.09 +0.06 +0.16 +0.19 +– +– +0.15 +0.02 +exiD [89] +0.11 +0.12 +0.06 +0.02 +0.24 +– +0.25 +– +– +TABLE III +INFLUENCE SCORE AND FEATURES FOR DATASETS RELEASED IN 2022. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Marius Z¨ollner†‡ †FZI Research Center for Information Technology, Germany bogdoll@fzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='de ‡Karlsruhe Institute of Technology, Germany Abstract—Autonomous Driving (AD), the area of robotics with the greatest potential impact on society, has gained a lot of momentum in the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' As a result of this, the number of datasets in AD has increased rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Creators and users of datasets can benefit from a better understanding of developments in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' While scientometric analysis has been conducted in other fields, it rarely revolves around datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Thus, the impact, attention, and influence of datasets on autonomous driving remains a rarely investigated field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' In this work, we provide a scientometric analysis for over 200 datasets in AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We perform a rigorous evaluation of relations between available metadata and citation counts based on linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Subsequently, we propose an Influence Score to assess a dataset already early on without the need for a track-record of citations, which is only available with a certain delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Index Terms—Robotics, Autonomous Driving, Datasets, Influ- ence, Impact, Attention, Scientometrics, Bibliometrics I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' INTRODUCTION Autonomous driving technology does not only affect ur- ban transportation [1] and delivery of goods [2], but also farming [3] or warehouse logistics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' With the progress of deep learning and this growing interest in AD in many fields of robotic, the number of related datasets is consistently increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The datasets have also increased in size and many have become increasingly specialized [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The most extensive collection of datasets known to us, ad-datasets, currently lists 231 datasets in the domain [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' However, not all of them are being equally used in the robotics community, the distribution of their citations is heavily skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' As part of the more impactful works, well known datasets for the core tasks perception and prediction dominate [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' As part of the long tail, many datasets for niche research areas exist [10]– [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Well known datasets tend to bring many advantages with them: They enable comparison between works, have higher quality, advanced tooling, and often community knowledge and support is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The increasing number of datasets, which are potentially interesting but lack reputation, leads to a lot of untapped potential: Many researchers are hesitant to use such datasets and stick to old, but established ones instead [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This is why we asked ourselves the question: Given a novel dataset without a multi-year track record of citations, is there a way to estimate its future development?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Datasets with a high potential might be more appealing already in their early days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' These authors contributed equally 2008 2010 2012 2014 2016 2018 2020 2022 year 5 10 15 20 25 30 35 40 publications 0 2000 4000 6000 8000 10000 citations Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Course of published datasets and citations of the accompanying publications in the domain of AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This growing number of datasets, initially without reputation, holds a great deal of untapped potential as researchers struggle to use new datasets for their research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Datasets as listed on ad- datasets [6], citation counts from Semantic Scholar [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Research Gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' To date, citations are mostly used to as- sess datasets, which are not available early on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Thus, new datasets can have a hard time gaining traction, which results in untapped potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' It is yet not well understood if and how metadata of datasets relate to future impact or how they can be utilized to assess datasets early on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' To the best of our knowledge, such an analysis has not yet been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' In order to analyze the field of dataset, we first assembled the largest collection of datasets with enriched metadata available, including over 200 datasets with metadata from three different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We then applied linear regression to evaluate factors which relate to the future impact of datasets, measured in citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Finally, we propose the Influence Score (IS), which is a mean to assess datasets early on without the need of a multi-year track record of citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The IS can be used to assess any datasets at any given year, which also allows for later analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Our work aims to help researchers from the robotics community to better understand and assess the performance of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This can lead to the design of better and thus more influential datasets as well as an actionable analysis of new datasets to assess their potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' All data used in this work is as of January 04, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' All code is available on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='02200v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='DL] 5 Jan 2023 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' RELATED WORK Here, we give an introduction to the scientometrics, biblio- metrics, and altmetrics, followed by dataset analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Scientometrics, Bibliometrics, and Altmetrics Scientometrics, Bibliometrics, and Altmetrics are highly intertwined fields that focus on the analysis of science and its processes as a whole, written works of science, and online communication of science, respectively [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Scientometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Ravenscroft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [15] examined the impact of research by comparing citation-based metrics, such as citation count or h-index [16], with altmetrics and impact other than citations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=', societal and economic impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' However, they found no strong relationship between the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Hicks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [17] suggest using multiple factors to portray multiple aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Leydesdorff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [18] claim that citations are equated to im- pact and evaluate the relationship between impact and research quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' They found that short-term citations signify the invest- ment in a current discourse, while long-term citations signify acceptance as reliable scientific knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' However, some researchers question if or to what extent citations measure scientific impact and point to issues, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=', inconsistent reasons for citations [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Problems include the cumulative advantages already successful papers experience [20], self-citations, which men do more often [21], negative citations, and citing out of reasons that do not reflect actual use or relevance [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Valenzuela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [23] presented a method to identify four types of citations: ”Related work, Comparison, Using the work, Extending the work” [23], which is used by Semantic Scholar to determine “Highly Influential Citations” [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' However, it shows a high correlation with citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The field of trend detection analyzes large corpora of works to detect upcoming patterns [25]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Lopez Belmonte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [29] analyzed publications in Machine Learning and Big Data and found exponential growth of publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' They compared the popularity of keywords and the h-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Bibliometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Citations can be aggregated on different levels, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=', for the papers of one author as the h-index does, or on the journal level, like the journal impact factor (JIF), which is the two-year average ratio of citations to articles published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The JIF is ill-suited for evaluating individual papers by means of the journal it was published in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This is due to the heavy skewness of the distribution of citation counts within journals [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The Hirsch-index, usually referred to as the h-index, combines the productivity of an author with the impact of their individual papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Using the h-index increases robustness compared to simply counting the total number of citations, as few highly cited papers have little effect on the h-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' In addition, there have been efforts to recommend papers and citations [32], [33], predict future citation counts of papers [34]–[37] and the impact of scientists [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Such approaches remain challenging and are often domain-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Bornmann and Marx [39] have proposed to expand the bibliometric analysis by not only considering citations but also references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Following this idea, reference analysis has been used to identify influential references [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Altmetrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Online interactions with papers are referred to as altmetrics and are usually available earlier than citations, which gives altmetrics an advantage over bibliometrics [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Bornmann and Marx [42] examined if Altmetrics can be used to predict paper quality which was measured through peer as- sessments and found that both tweets and readers do, with the latter having a stronger relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Lamb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [43] showed that the Altmetric Attention Score is a predictor of the citations of a paper in ecology and conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Zavrel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [44] clustered papers released at the International Conference on Machine Learning (ICML) in 2022 and calculated a score for their impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' They used Twitter mentions, citations, the authors’ average h-index, and an award for outstanding papers rewarded by the conference itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' They claim to do a “sim- ple combination of these four scores to calculate an impact score” [44] but do not reveal the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' F¨arber analyzed GitHub repositories of papers, mostly from the field of AI, and found a power-law distribution of stars and forks [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' While Haustein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' claim that ”Altmetrics measures scientific impact based on online references and activity” [46], many disagree with equating altmetrics with impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' For example, Sugimoto states that ”attention is not impact” and calls online interaction with scientific works ”attention” [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Altmetrics might reflect broader or societal impact [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Dataset Analysis Bogdoll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [5] gathered metadata of over 200 datasets in the field of autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Similarly, F¨arber and Lamprecht released the data set knowledge graph, which is a collection of over 2,000 datasets with added metadata [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' D’Ulizia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [49] analyzed the metadata of datasets for fake news detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Utamachant and Anutariya [50] analyzed the datasets of Thailand’s national open data portal, but relied on domain experts to assess impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Nguyen and Weller proposed FAIRnets, a service to search for neural networks and their related datasets [51] published on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' They build upon the Findable, Accessible, Interoperable, Reusable (FAIR) principles [52], which ”put specific emphasis on enhancing the ability of machines to automatically find and use the data” [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [53] analyzed datasets from the Global Biodiversity Information Facility (GBIF) which publishes datasets with a DOI and indexes datasets in biodiversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' They promote data standards and the reuse of datasets [54] as well as accompanying publications, which they call ”data papers”, that describe a dataset thoroughly [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [53] report a strong correlation between dataset download numbers and citation counts, and suggest that downloads and citations signify a similar kind of impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' They also find correlations between altmetrics and citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Moreover, they question whether every citation means the usage of a dataset and point to differences in citation behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' F¨arber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' proposed an approach to find methods and datasets which authors actually used when citing the related paper [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' However, unrealistically few dataset usages were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' AD-Datasets Altmetric Semantic Scholar List of Datasets with Enriched Metadata Regression Analysis of Citations and Metadata Influence Score Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Overview: First, we collect data from various sources and combine them to a single list of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Based on this, we perform a regression analysis to determine which metadata correlate with future prediction counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Based on the metadata, we compute our Influence Score (IS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' REGRESSION ANALYSIS Here, we first introduce our taxonomy of terms related to the assessment of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Subsequently, we introduce our data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Based on these, we describe the regression analysis of citations and metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' In Section IV, we present the resulting Influence Score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Figure 2 gives an overview over this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Taxonomy As became clear in Section II, no common language for specific aspects in the domain has evolved yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Thus, we introduce a taxonomy to clearly describe different aspects with respect to the development of a dataset or paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' As general terms for this, we utilize success, progress, performance, or potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' For concrete aspects, we establish the following terms, where each one can be applied to any single paper: Impact: We use the number of citations to measure the sci- entific impact of a paper, which is common in Scientometrics, but not without criticism [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Attention: The online reception, such as tweets and Wikipedia articles mentioning a paper, represents the attention by researchers and the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Influence: We refer to the resulting score of our proposed method, which combines a multitude of aspects, as the influ- ence, or IS, of a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We deem this term appropriate for any method that goes beyond purely impact-based assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Data Sources and Selection We used three sources for our data: ad-datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='com [6], the Semantic Scholar Academic Graph API [13], and altmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='com [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Based on the DOI and arXiv-Id from ad-datasets, we automatically extracted the metadata of papers from Semantic Scholar and altmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Based on these papers, all of which describe datasets, we performed data exploration, regression, and the computation of the IS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' AD-Datasets: This web tool offers an overview of over 200 data sets in AD [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' It includes a detailed breakdown of most dataset entries by 20 different meta categories, provided by the authors and the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This way, relations between datasets, accompanying papers and further metadata are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The underlying data is stored in the JSON format and can be accessed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' In this work, we utilize the nframes and nsensors metadata, which indicate the size of datasets in different dimensions, which is a potential aspect of the relevance of a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Altmetric: We used the API by altmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='com [57], which provides insight into online attention and readership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' These properties are provided by the following categories: Attention Score: The aascurr aggregates different sources into a single score [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' It is a weighted count of different online sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' For example, the weight for a reference on Wikipedia is 3, while Twitter and Reddit mentions are both weighted with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Unfortunately, the history of this score is only provided for the most recent year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Attention Score after three months: The aas3m is the percentile of the papers’ Attention Score three months after publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The percentile is calculated in comparison to papers that have been released at a similar time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Readers: The number of people nreaders that have saved a paper in their reference management software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Reading a paper is less significant than citing it, but the number of readers might imply interest in a paper early on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The number of readers is provided individually for multiple reference management services, which we sum into a single count for online readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Altmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='com cannot verify the number of readers, thus, it is not included in the attention score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This is a relevant attribute, as it decouples the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' However, there are no historic data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Semantic Scholar: For every accompanying paper of a dataset, we pulled data from Semantic Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Sometimes, multiple datasets are described in the same paper, which will lead to the same information for those datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We extracted the following nested data: List of referenced papers, including for each a list of all citing papers and the year of citation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' List of authors and their respective publications, including for each publication a list of citing papers and the year of citation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' List of citing papers, including for each a list of citing papers and the year of citation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Wherever possible, we collected associated timestamps, including the publication year apub of each paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The first two categories, while dynamic, are directly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We use the citations of references as a measure of the impact of references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Having impactful references might indicate that a paper is covering popular topics within AD or that the authors are knowledgeable in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The performance of authors can be estimated by evaluating their paper count and how many citations they have received, which becomes only meaningful over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' As discussed earlier, not every citation means usage of a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' While it would have been interesting to take into account, in which section a paper has been cited, this data was not available for most papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Based on the ncit3 citations from the previous three years, citations of works that cited a dataset signify the value created by working with the dataset, which is why we included those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' A critical aspect of the collected data is that oftentimes, no historic information was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Also, oftentimes, data was not available due to limitations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=', Altmetric is incompatible with DOIs from IEEE publications, which are common in the fields of robotics, autonomous driving, and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Similarly, Ravenscroft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [15] expressed concerns about Altmetric, as they were unable to find 40 % of the papers they analyzed, all from the field of computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Data Aggregation To further utilize the raw data we collected, we aggregated some of it with the aim to assemble a finite list of features that describe a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We aggregated some of our data sources using the concept of the h-index formula, as it is widely known, transparent, and easy to reproduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' In order to analyze smaller timespans, we deviated from the typical 5- year duration and calculated multiple 3-year indexes ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' For authors, we applied the h3-index for each individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We then aggregated the h-indices of all authors of a paper via the arithmetic mean in autµh3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Respectively, we applied the h-index formula to references and citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' For the references of a paper, the refh3 is calculated identically to the way it is utilized for authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Just like an author has papers with citations, a paper has references with citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' A high h-index for references would signify that several of the referenced papers gained lots of attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We also applied the h3-index formula to the citations and their citations to get the h3-index of citations cith3, following Schubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The final list of all extracted and calculated features can be found in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Cluster Analysis and Regression Setup We evaluated our computed features with respect to their ability to predict future citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Therefor, we performed linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' For this, we first computed clusters of the datasets to determine a meaningful time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Subse- quently, we defined our regression setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Cluster Analysis: To show that there are meaningful vari- ations between clusters, we looked at the impact of papers for up to 2 years after publication in a journal or conference 1 0 1 2 years after publication 0 50 100 150 200 250 300 350 400 ncits Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Development of the number of citations for publication-clusters over a dynamic 3-year window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Papers are clustered based on similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Semantic Scholar also tracks citations of pre-prints, which leads to citations prior to the publication date of the final work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' As visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' 3, clear clusters are visi- ble, where line-thickness indicates cluster size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' For k-means clustering, we used six clusters based on the elbow plot, which shows which additional cluster provides a non-marginal reduction of the total variation within clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The growth of citation counts behaves exponentially for the top performing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' A clear differentiation between all clusters becomes apparent already after one year, which we chose as the time horizon for the regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This allowed us to include more recent papers, which would have been excluded otherwise due to their missing track record of citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Regression Setup: As independent variables, we included the features nsensors, apub, refh3, autµh3, ncit3, and aas3m, as shown in Table I, in order to estimate the citation count after one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Preliminary data exploration suggested a curvilinear relationship between aas3m and the number of citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Therefore, a quadratic term was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' All predictors were standardized by subtracting the mean and dividing by the standard deviation prior to the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The feature ncit3 was log(x+1)-transformed to ensure a normal distribution of the residuals, which are the error terms of the regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' For the regression, we were able to utilize 111 datasets, as values for all included features were available, and they had been released at least one year prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Residuals and collinearity, the ability to linearly predict one independent variable with other independent variables, were checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The collinearity was quantified through the variance inflation factor of each regressor which all were lower than three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We performed the Breusch-Pagan and White test for heteroskedasticity, which is the inconsistency of the variance of residuals at different levels of the dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Both tests indicated that we do not have sufficient evidence for the presence of heteroskedasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Still, robust standard errors were used to ensure the standard errors are calculated correctly in the presence of heteroskedas- ticity which at worst leads to standard errors being estimated larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Availability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Standardized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Log(x+1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Influence Score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Source ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='nframes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Number of frames in the dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='At publication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='AD-Datasets [6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='nsensors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Number of sensor types ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='At publication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='AD-Datasets [6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='apub ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Year of publication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='At publication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Semantic Scholar [13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='refh3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='3 year h-index of references ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='At publication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Semantic Scholar [13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='autµh3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Mean 3 year h-index of authors papers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='At publication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Semantic Scholar [13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='ncit3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Number of citations within past 3 years ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Anytime after publication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Semantic Scholar [13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='cith3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='3 year h-index of citations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='>3 years after publication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Semantic Scholar [13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='aascurr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Altmetric Attention Score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Anytime after publication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Altmetric [57] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='aas3m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Altmetric Attention Score at 3 mos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='After 3 months ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Altmetric [57] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='nreaders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Number of readers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Anytime after publication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='Altmetric [57] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='TABLE I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='OVERVIEW OF METADATA USED FOR THE REGRESSION ANALYSIS AND THE INFLUENCE SCORE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We chose not to include nframes for the regression because numerous of the datasets did not contain this meta-information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' However, we examined a model in which the feature was included, which did not lead to new findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Regression Analysis With the explained regression setup, we were now interested in finding statistically significant predictor variables for the citation count at the end of the year after publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The aas3m and aas2 3m were positively related to the number of citations and both relationships were significant at <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Both coefficients were positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' All other features were not significantly related to the number of citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The results are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' coef std err z P>|z| [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='025 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='337 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='736 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='192 nsensors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='1383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='108 1.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='575 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='031 aas2 3m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='3375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='072 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='72 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='478 intercept 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='6402 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='117 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='48 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='87 TABLE II REGRESSION FOR CITATIONS AFTER ONE YEAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' REGRESSION COEFFICIENTS AND 95% CONFIDENCE INTERVAL ARE REPRESENTED ON THE LOG SCALE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Since only one feature showed a relationship with the number of citations, we do not consider a stable prediction of citations possible with the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' In order to still perform an early evaluation of datasets, in the following we present our Influence Score (IS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' INFLUENCE SCORE We propose the Influence Score (IS), which includes a variety of features that are available early on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' These are weighted dynamically in order to receive an indication of the relative performance of any given dataset at any given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The calculation compares each data set with all existing ones from the domain, so that relative differences and trends are immediately recognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Percentiles are used to allow relative scoring within the surrounding group of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The data sets roughly follow a normal distribution in their IS scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' As shown in Table I, we utilize eight different features for the IS: nframes, nsensors, refh3, autµh3, ncit3, cith3, aascurr and nreaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This way, we consider more than just the citations, but do not exclude them: If early citations are already available, they become a meaningful part of the score, as the relation to other datasets of the peer group is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This way, citation velocity is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The IS is defined as follows: IS(paper) = 1/n ∗ n � i=0 percentile(featurei) (1) where: i = Feature Index n = Number of available features Only features, which are available, are dynamically included in the IS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' As we used percentiles of each feature to facil- itate the understanding of the features, common issues are mitigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=', typical feature values change over time: For example, with the growth of AD, the ncit value of a paper today is likely higher than a decade ago, which becomes clearly visible in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Furthermore, commonly observed values for features might differ between different fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This helps people who are not familiar with AD or the features to easily assess if the score a dataset achieved is high or low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Qualitative Demonstration To showcase the IS, we compare exemplary the development of the five most and least cited papers with a latest publication in 2019, by their IS and visualize the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' It becomes clearly visible, that the two groups are easily distinguishable by their IS, but also that differences within the groups are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' The individual features show different pictures: For refh3, also papers with only a few citations can have meaningful references in their works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' ncitt3 and cith3 only confirm what was known by our data selection, as we selected the datasets by citation count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' autµh3 shows, how successful datasets can also boost personal careers, as some authors became professors and remained active in their field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' nsensors and nframes show 2014 2016 2018 2020 2022 years 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 IS 2014 2016 2018 2020 2022 years 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 percentile refh3 2014 2016 2018 2020 2022 years 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 percentile ncit3 2014 2016 2018 2020 2022 years 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 percentile cith3 2014 2016 2018 2020 2022 years 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 percentile aut h3 2014 2016 2018 2020 2022 years 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 percentile nsensors 2014 2016 2018 2020 2022 years 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 percentile nframes KITTI nuImages Cars Synthia Waymo Open Perception Daimler Stereo Pedestrian Detection Benchmark TRoM DriveSeg (Semi-auto) DriveSeg (MANUAL) WZ-traffic Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Influence Score and individual features for different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We show exemplary results for the five best and worst performing datasets of all time, measured by citations, with a latest release in 2019 for historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We also show six individual features of the IS, where historical data was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' rather static results, with a trend towards larger datasets being more successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Quantitative Demonstration In order to show the quantitative performance of the IS, we showcase all datasets released in 2022 in a detailed overview in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Such a pre-filtering process is useful in order to explore novel datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Here, it becomes clear that the IS captures a wide variety of different aspects of a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Of particular interest is the fact that even low-performing data sets can lead in certain features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Thus, if a researcher is interested in certain aspects of a dataset, they can simply focus on the features they are interested in and omit the others, which enables less-known datasets to be discovered and used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Figure 5 shows an overview of the IS distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='0 IS 0 2 4 6 8 10 number of datasets from 2022 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Distribution of the Influence Score (IS) of all datasets from 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' CONCLUSION In this paper, we addressed the lack of knowledge with respect to the scientific impact, attention, and influence of datasets in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Our focus was on an early assessment of datasets, given a flood of new datasets published every year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We analyzed impact measured by citations and evalu- ated relations of metadata and features which we extracted from multiple online sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Our regression analysis showed no strong relation between future citations and our selected features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Subsequently, we presented our developed Influence Score (IS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This score utilizes a set of eight features to assess any dataset also early on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' This is based on an analysis within the peer group of all datasets, which allows for the early detection of relative trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Our work contributes to a better understanding of datasets, which enables researchers to find and assess published datasets in the domain of autonomous driving without the need of waiting for a track record of citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Limitations and Outlook: For our work, we evaluated the paper accompanying the dataset assuming that the paper is a good representation of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' When measuring scientific impact through citations, we think this holds because the paper is actually the cited scientific work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' However, not every citation might be meaningful, positive, or indicate the usage of a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Ideally, large language models could evaluate if a dataset is actually used, if cited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [53], who analyzed datasets in biodiversity, suggested that the correlation between the number of downloads and citations signifies that these two measures are comparable representations of impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' However, in the domain of AD, download numbers are typically not available, but this might change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' As some datasets are presented in the same paper, a further decoupling of accompanying papers and the respective datasets would be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We found, that the quality and availability of metadata in AD provided by the creators of datasets varies strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' Thus, standards should be established [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' While we focussed on dataset and paper specific features for this work, we are also interested in the venue or journal of publication, which can be considered as an additional feature in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' IS ncit3 cith3 refh3 autµh3 nframes nsensors aascurr nreaders Waymo Block-NeRF [60] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='82 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='05 – – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='02 OpenMPD [86] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='58 – – I see you [87] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='01 – – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='08 SceNDD [88] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='19 – – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='02 exiD [89] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='24 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='25 – – TABLE III INFLUENCE SCORE AND FEATURES FOR DATASETS RELEASED IN 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' SORTED BY IS, TOP 3 FEATURES BOLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' ACKNOWLEDGMENT This work results partly from the KIGLIS project supported by the German Federal Ministry of Education and Research (BMBF), grant number 16KIS1231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' We want to thank both Altmetric and Semantic Scholar, who have provided us with the necessary API accesses for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' REFERENCES [1] Waymo, “Waymo One,” https://waymo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='com/waymo-one/, 2022, ac- cessed: 2022-12-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content=' [2] Waabi, “Introducing the Waabi Driver,” https://waabi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE0T4oBgHgl3EQfRAAw/content/2301.02200v1.pdf'} +page_content='ai/introducing-the- waabi-driver/, 2022, accessed: 2022-12-14.' 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Brissenden, Konstantinos Dimopoulos and Samuel S´anchez +L´opez +Consortium for Fundamental Physics, Physics Department, +Lancaster University, Lancaster LA1 4YB, United Kingdom. +E-mail: l.brissenden@lancaster.ac.uk, k.dimopoulos1@lancaster.ac.uk, +s.sanchezlopez@lancaster.ac.uk +Abstract. +Early dark energy (EDE) is one of the most promising possibilities in order to +resolve the Hubble tension: the discrepancy between early and late-Universe measurements +of the Hubble constant. In this paper we propose a model of a scalar field which can explain +both EDE and late Dark Energy (DE) in a joined manner without additional fine-tuning. +The field features kinetic poles as with α-attractors. Our model provides an injection of EDE +near matter-radiation equality, and redshifts away shortly after via free-fall, later refreezing to +become late-time DE at the present day. Using reasonable estimates of the current constraints +on EDE from the literature, we find that the parameter space is narrow but viable. As such +our model is readily falsifiable. In contrast to other work in EDE, our model is non-oscillatory, +which causes its decay to be faster than that of the usual oscillatory EDE, thereby achieving +better agreement with observations. +arXiv:2301.03572v1 [astro-ph.CO] 9 Jan 2023 + +Contents +1 +Introduction +1 +1.1 +The Hubble tension +2 +1.2 +Early Dark Energy +2 +1.3 +α-attractors +3 +1.4 +Quintessence +4 +2 +The Model +5 +2.1 +Lagrangian and Field Equations +5 +2.2 +Shape of Potential and Expected Behaviour +5 +2.3 +Asymptotic forms of the scalar potential +5 +2.3.1 +Expected Field Behaviour +7 +2.4 +Tuning requirements +8 +3 +Numerical Simulation +9 +4 +Results and analysis +11 +4.1 +Parameter Space +11 +4.2 +Field Behaviour +13 +5 +Initial Conditions +14 +6 +Conclusions +18 +A Quintessential Inflation +19 +1 +Introduction +In the last few decades cosmological observations of the early and late Universe have con- +verged into a broad understanding of the history of our Universe from the very first seconds +of its existence until today. Thus, cosmology has developed a standard model called the +concordance model, or in short ΛCDM. +However, the latest data might imply that the celebrated ΛCDM model is not that +robust after all. In particular, there is a 5-σ discrepancy between the measurements of the +current expansion rate, the Hubble constant H0, as inferred by early Universe observations +compared with late Universe observations. This Hubble tension has undermined our confi- +dence in ΛCDM and as such it is investigated intensely at present. +In this work we study a toy model that can simultaneously solve the Hubble tension +and explain the current accelerated expansion with no more tuning that in ΛCDM. Our +model introduces a scalar field which plays both the role of early dark energy (EDE) and +quintessence. In contrast to most other works in the literature which consider scalar fields +as EDE, ours is not an oscillating scalar field. +We use natural units with c = ¯h = 1, the reduced Planck mass mP = 1/ +√ +8πG = +2.43 × 1018GeV and consider a positive signature metric (−1, +1, +1, +1) throughout the +present work. +– 1 – + +1.1 +The Hubble tension +Measurements in observational cosmology can broadly be classified into two groups. These +are measurements of quantities which depend only on the early-time history of our Universe +(such as the cosmic microwave background (CMB) radiation at redshift z ≃ 1100, or Baryon +Acoustic Oscillations (BAO)) and measurements of quantities which depend on present-day +observations (the primary example of this is the cosmic distance ladder, which measures the +redshift of observable astrophysical objects such as Cepheid stars and type-1a supernovae, +at redshift z = O(1)). +The value of the Hubble constant H0 can in principle be inferred from both early and +late-time measurements. However, it has been found that while early-time measurements are +in good agreement with each other, they disagree with current late-time data. Latest analysis +of the CMB temperature anisotropies’ data gives the value inferred from Planck satellite [1], +H0 = 67.44 ± 0.58 km s−1Mpc−1, +(1.1) +and a distance scale measurement using Cepheid-SN 1a data from the SH0ES collaboration +[2] as +H0 = 73.04 ± 1.04 km s−1Mpc−1. +(1.2) +This is a 5σ tension which includes estimates of all systematic errors and which the SH0ES +team conclude has “no indication of arising from measurement uncertainties or analysis varia- +tions considered to date”. It is becoming increasingly apparent with successive measurements +that this tension is likely to have a theoretical resolution [3, 4], which can have many possible +sources [5, 6]. +1.2 +Early Dark Energy +One proposed class of solutions to the Hubble tension is models of Early Dark Energy (EDE), +whose early works include references [7–10], followed by many others, e.g. see Refs. [5, 11–32]. +These involve an injection of energy in the dark energy sector at around the time of matter- +radiation equality, which then dilutes or otherwise decays away faster than the background +energy density, such that it becomes negligible before it can be detected in the CMB. As +briefly reviewed below, such models result in a slight change in the expansion history of the +Universe, bumping up the value of the Hubble parameter at the present day. +It has previously been concluded [3, 5, 6] that EDE models are most likely to source +a theoretical resolution to the Hubble tension. One reason for this is that EDE can effect +substantial modifications to H0 without significant effect on other cosmological parameters +which are tightly constrained by observations.1 In particular, EDE models can be incorpo- +rated into existing scalar-field models of inflation and late-time dark energy; one example of +the latter is the model detailed in this work. +However, precisely because EDE models exist so close in time to existing observational +data, they have significant constraints; the primary consideration being that EDE must be +subdominant at all times and must decay away fast enough to be essentially negligible at +the time of last scattering translating to a redshift rate that is faster than radiation [8]. So +far, in previous works in EDE, this has been achieved by considering first or second-order +phase transitions (e.g. [23], [29]). These abrupt events might have undesirable side-effects +1Models which modify other cosmological parameters are often unable to reconcile their changes with +current observational constraints on said parameters (see Ref. [5] for a comprehensive review). +– 2 – + +such as inhomogeneities from bubble collisions or topological defects. Other proposed models +[5, 7, 8, 23–30] typically feature oscillatory behaviour to achieve the rapid decay rate necessary +for EDE to be negligible at last scattering. As with the original proposal in Ref. [7], the +EDE field is taken to oscillate around its Vacuum Expectation Value (VEV) in a potential +minimum which is tuned to be of order higher than quartic. As a result, its energy density +decays on average as ∝ a−n, with 4 < n < 6. In contrast, in our model, the EDE scalar field +experiences a period of kinetic domination, where the field is in non-oscillatory free-fall and +its density decreases as ∝ a−6, exactly rather than approximately. +Before continuing, we briefly explain how EDE manages to increase the value of H0 +as from CMB observations. Measurements of the CMB temperature anisotropies provide +very tight constraints on the cosmological parameters. One would therefore think that this +severely limits models which alter the Universe content and dynamics at this time. However, +there are certain classes of models for which this is not the case. These are models that affect +both the Hubble parameter and rs, the comoving sound horizon2 (in this case during the +drag epoch, shortly after recombination), given by +rs = +� ∞ +zd +cs(z) +H(z)dz, +(1.3) +where cs(z) is the sound speed and H(z) is the Hubble parameter, both as a function of +redshift. +An additional amount of dark energy in the Universe increases the total density, which in +turn increases the Hubble parameter because of the Friedmann equation ρ ∝ H2. Therefore, +EDE considers such a brief increase at or before decoupling, which lowers the value of the +sound horizon because it increases H(z) in Eq. (1.3). However, there is a way to avoid this +being evident in and therefore disproved by current CMB measurements. This is because +BAO and CMB measurements do not constrain the value of the sound horizon directly. +For example, BAO measurements do not constrain the sound horizon alone, but the com- +bination H(z)rs [33]. The observations of the Planck satellite measure the quantity θ∗ ≡ r∗ +D∗ +[34], the angular scale of the sound horizon; given by ratio of the comoving sound horizon to +the angular diameter distance at which we observe fluctuations. Both of these measurements +entail an assumption of ΛCDM cosmology and can be shown to be equally constrained by +other models, provided that they make only small modifications which simultaneously lower +the value of rs and increase H0. +EDE may have a significant drawback, however, in that it does not alleviate the σ8 +tension (associated with matter clustering) and may in fact exacerbate it [3, 35]. As with +many others, our model does not attempt to solve this problem. +1.3 +α-attractors +Our model unifies EDE with late DE in the context of α-attractors. An earlier attempt +for such unification in the same theoretical context can be seen in Ref. [30]. However, this +proposal is also of oscillatory EDE. +α-attractors [36–44], which appear naturally in conformal field theory or supergravity +theories, are a class of models whose inflationary predictions continuously interpolate between +those of chaotic inflation [45] and those of Starobinsky [46] and Higgs inflation [47]. +In +2This is the characteristic scale of BAO, typically approximately proportional to the value of the cosmo- +logical horizon at that point by rs = +1 +√ +3rH assuming spatial flatness. +– 3 – + +supergravity, introducing curvature to the internal field-space manifold can give rise to a +non-trivial K¨ahler metric, which results in kinetic poles for some of the scalar fields of the +theory. The free parameter α is inversely proportional to said curvature. It is also worth +clarifying what is meant by the word “attractor”. It is not only used in the usual sense (i.e., +field trajectories during inflation flowing to a unique one, regardless of the initial conditions), +but also to refer to the fact that the inflationary predictions are largely insensitive of the +specific characteristics of the model under consideration. Such an attractor behaviour is seen +for sufficiently large curvature (small α) in the internal field-space manifold. +In practical terms, the scalar field has a non-canonical kinetic term, featuring two poles, +which the field cannot transverse. To aid our intuition, the field can be canonically normalised +via a field redefinition, such that the finite poles for the non-canonical field are transposed +to infinity for the canonical one. As a result, the scalar potential is “stretched” near the +poles, resulting in two plateau regions, which are useful for modelling inflation, see e.g. Refs. +[48–53] or quintessence [54], or both, in the context of quintessential inflation [54–56]. +Following the standard recipe, we introduce two poles at ϕ = ± +√ +6α mP by considering +the Lagrangian +L = +− 1 +2(∂ϕ)2 +(1 − +ϕ2 +6α m2 +P )2 − V (ϕ) , +(1.4) +where ϕ is the non-canonical scalar field and we use the short-hand notation (∂ϕ)2 ≡ +gµν∂µϕ ∂νϕ. We then redefine the non-canonical field in terms of the canonical scalar field φ +as +dφ = +dϕ +1 − +ϕ2 +6αm2 +P +⇒ +ϕ = mP +√ +6α tanh +� +φ +√ +6α mP +� +. +(1.5) +It is obvious that the poles ϕ = ± +√ +6α are transposed to infinity. +In terms of the canonical field, the Lagrangian now reads +L = −1 +2(∂φ)2 − V (φ). +(1.6) +1.4 +Quintessence +“Early” Dark Energy is so named in order to make it distinct from “late” Dark Dnergy, +which is the original source of the name (and often just called Dark Energy (DE)). In cos- +mological terms the latter is just beginning to dominate the Universe at present, making up +approximately 70% of the Universe’s energy density [57]. This is the mysterious unknown +substance that is responsible for the current accelerating expansion of the Universe and has +equation-of-state (barotropic) parameter of w = −1.03 ± 0.03 [1]. +Late DE that is due to an (as-yet-undiscovered) scalar field is called quintessence [58], +so-named because it is the “fifth element” making up the content of the Universe 3. In this +case, the Planck-satellite bound on the barotropic parameter of DE is −1 ≤ w < −0.95 [1]. +Quintessence is distinct from other explanations for DE because a scalar field has a variable +barotropic parameter and can therefore exhibit completely different behaviour in different +periods of the Universe’s history. In order to get it to look like late-time DE, a scalar field +should be dominated by its potential density, making its barotropic parameter sufficiently +3After baryonic matter, dark matter, photons and neutrinos. +– 4 – + +close to −1. It is useful to consider the CPL parametrization, which is obtained by Taylor +expanding w(z) near the present as [59, 60] +w(z) = w0 + wa +z +z + 1 , +(1.7) +where wa ≡ −(dw/da)0. The Planck satellite observations impose the bounds [1] +−1 ≤ w < −0.95 +wa = −0.29+0.32 +−0.26 . +(1.8) +2 +The Model +2.1 +Lagrangian and Field Equations +Consider a potential of the form +V (ϕ) = VX exp +� +−λeκϕ/mP +� +, +with VΛ ≡ exp +� +−λeκ +√ +6α� +VX , +(2.1) +where α, κ, λ are dimensionless model parameters, VX is a constant energy density scale and +ϕ is the non-canonical scalar field with kinetic poles given by the typical alpha attractors form +(see [40]) with Lagrangian density given by Eq. (1.4).4 In the above, VΛ is the vacuum density +at present. To assist our intuition, we switch to the canonically normalised (canonical) scalar +field φ, using the transformation in Eq. (1.5). In terms of the canonical scalar field, the +Lagrangian density is then given by Eq. (1.6), where the scalar potential is +V (φ) = exp +� +λeκ +√ +6α� +VΛ exp +� +−λeκ +√ +6α tanh(φ/ +√ +6α mP)� +. +(2.2) +As usual, the Klein-Gordon equation of motion for the homogeneous canonical field is +¨φ + 3H ˙φ + V ′(φ) = 0 , +(2.3) +where the dot and prime denote derivatives with respect to the cosmic time and the scalar +field respectively, and we assumed that the field was homogenised by inflation, when the +latter overcame the horizon problem. +2.2 +Shape of Potential and Expected Behaviour +Henceforth we will discuss the behaviour of the field in terms of the variation, i.e. movement +in field space, of the canonical field. +2.3 +Asymptotic forms of the scalar potential +We are interested in two limits for the potential above: φ → 0 (ϕ → 0) and φ → +∞ (ϕ → +√ +6α mP ). The first limit would correspond to matter-radiation equality. In this limit, the +potential is +4The model parameter is VX and not VΛ, the latter being generated by VX and the remaining model +parameters as shown in Eq. (2.1). +– 5 – + +Veq ≃ exp +� +λ(eκ +√ +6α − 1) +� +VΛ exp(−κλ φeq/mP) , +(2.4) +where the subscript ‘eq’ denotes the time of matter-radiation equality when the field un- +freezes. It is assumed that the field was originally frozen there. We discuss and justify this +assumption in Sec. 5. After unfreezing, it is considered that the field has not varied much, +for the above approximation to hold, i.e. +0 ≲ φeq ≪ +√ +6αmP . +(2.5) +This is a reasonable assumption given that the field begins shortly before matter-radiation +equality frozen at the origin, unfreezing at some point during this time 5. +At large φ (φ → ∞), the non-canonical field is near the kinetic pole (ϕ → +√ +6α mP). +Then the potential in this limit is +V0 ≃ VΛ +� +1 + 2κλeκ +√ +6α√ +6α exp +� +− +2φ0 +√ +6α mP +�� +, +(2.6) +which, even for sub-Planckian total field excursion in φ, should be a good approximation for +sufficiently small α. The subscript ‘0’ denotes the present time. +The above approximations describe well the scalar potential near equality and the +present time, as shown in Fig. 1. As we exlain below, in between these regions, the scalar +field free-falls and becomes oblivious of the scalar potential as the term V ′(φ) in its equation +of motion (2.3) becomes negligible. +Canonical Potential +Approximation at Low Field Values +Approximation at High Field Values +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +-120 +-119 +-118 +-117 +-116 +-115 +ϕ +mP √(6 α) +log V(ϕ) +mP +4 +VΛ +mP +4 = 10-120.068 +α =0.0002 +κ=200 +λ=0.01 +Figure 1: Graph of the canonical potential and its two approximations for small and large +field values, given in Eqs. (2.4) and (2.6) respectively. +These approximations are useful +because they are simple exponential potentials with known attractors, so we know the type +of behaviour the field should exhibit when each approximation is valid. It can be readily seen +that, after leaving the origin the field jumps off a potential plateau and is free-falling as a +result. +5There is no suggestion in the EDE literature [5, 7, 8, 23–30] that the field has to unfreeze at any particular +time, as long as it does not grow to larger than the allowed fraction and its energy density is essentially +negligible by the time of decoupling. +– 6 – + +2.3.1 +Expected Field Behaviour +Here we explain the rationale behind the mechanism envisaged. We make a number of crude +approximations, which enable us to follow the evolution of the scalar field, but which need +to be carefully examined numerically. We do so in the next section. +First, we consider that originally the field is frozen at zero (for reasons explained in +Sec. 5). Its energy density is such that it remains frozen there until equality, when it thaws +following the appropriate exponential attractor, since Veq in Eq. (2.4) is approximately ex- +ponential [61]. Assuming that this is the subdominant attractor requires that the strength +of the exponential is [62, 63] +Z ≡ κλ > +√ +3 . +(2.7) +The subdominant exponential attractor dictates that the energy density of the rolling scalar +field mimics the dominant background energy density. Thus, the density parameter of the +field is constant, given by the value [61–63] +Ωeq +φ ≃ 3 +Z2 = +3 +(κλ)2 < 1 +(2.8) +This provides an estimate of the moment when the originally frozen scalar field, unfreezes and +begins rolling down its potential. Unfreezing happens when Ωφ (which is growing while the +field is frozen, because the background density decreases with the expansion of the Universe) +obtains the above value. +However, after unfreezing, the field soon experiences the full exp(exp) steeper than +exponential potential so, it does not follow the subdominant attractor any more but it free- +falls,6 such that its density scales as ρφ ≃ 1 +2 ˙φ2 ∝ a−6, until it refreezes at a larger value φF . +This value is estimated as follows. +In free-fall, the slope term in the equation of motion (2.3) of the field is negligible, so +that the equation is reduced to ¨φ + 3H ˙φ ≃ 0, where H = 2/3t after equality. The solution is +φ(t) = φeq + C +teq +� +1 − teq +t +� +, +(2.9) +where C is an integration constant. +From the above, it is straightforward to find that +˙φ = Ct−2. Thus, the density parameter at equality is +Ωeq +φ = ρφ +ρ +���� +eq += +1 +2C2t−4 +eq +4 +3( mP teq)2 = 3 +8 +C2 +(mP teq)2 +⇒ C = +� +8 +3Ωeq +φ mP teq = +√ +8 +κλ mP teq , +(2.10) +where we used Eq. (2.8), ρφ ≃ 1 +2 ˙φ2 and that ρ = 1/6πGt2 = 4 +3(mP /t)2. Thus, the field freezes +at the value +φ0 = φeq + C/teq = φeq + +√ +8 +κλ mP , +(2.11) +where we considered that teq ≪ tfreeze < t0 . +Using that teq ∼ 104 y and t0 ∼ 1010 y, we can estimate +Veq +V0 +≃ +Ωeq +φ ρeq +0.7 ρ0 +≃ +30 +7(κλ)2 +� t0 +teq +�2 +≃ +3 +7(κλ)2 × 1013 . +(2.12) +6i.e. its energy density is dominated by its kinetic energy density only. +– 7 – + +Now, from Eqs. (2.4) and (2.6) we find +Veq +V0 +≃ +eλ(eκ +√ +6α−1) exp(−κλ φeq/mP ) +1 + 2κλ eκ +√ +6α√ +6α exp +� +−2φ0/ +√ +6α mP +� . +(2.13) +In view of Eqs. (2.5) and (2.11), the above can be written as +Veq +V0 +≃ +eλ(eκ +√ +6α−1) +1 + 2κλ eκ +√ +6α√ +6α e−2 +√ +8/κλ +√ +6α . +(2.14) +Taking Ωeq +φ ≃ 0.1 as required by EDE, Eq. (2.8) suggests +κλ ≃ +√ +30 . +(2.15) +Combining this with Eq. (2.12) we obtain +e +√ +30 +κ (eκ +√ +6α−1) ∼ 1012/7 , +(2.16) +where we have ignored the 2nd term in the denominator of the right-hand-side of Eq. (2.14). +From the above we see that, κ is large when α is small. Taking, as an example, α = 0.01 +we obtain κ ≃ 18 and λ ≃ 0.30 (from Eq. (2.15)). With these values, the second term in the +denominator of the right-hand-side of Eq. (2.14), which was ignored above, amounts to the +value 3.2. This forces a correction to the ratio Veq/V0 of order unity, which means that the +order-of-magnitude estimate in Eq. (2.16) is not affected. +Using the selected values, Eq. (2.11) suggests that the total excursion of the field is +∆φ = φ0 − φeq = +√ +8 +κλ mP ≃ 0.5 mP , +(2.17) +i.e. it is sub-Planckian. In the approximation of Eq. (2.4), we see that the argument of the +exponential becomes κλ∆φ/mP ≃ 2.7 > 1, where we used Eq. (2.15). This means that the +approximation breaks down and the exp(exp) potential is felt as considered, as depicted also +in Fig. 1. +For small α the eventual exponential potential in Eq. (2.6) is steep, which suggests that +field rushes towards the minimum at infinity and the barotropic parameter is w ≈ −1 because +the potential is dominated by the constant VΛ. +2.4 +Tuning requirements +Our model addresses in a single shot two cosmological problems: firstly, the Hubble tension +between inferences of H0 using early and late-time data; and secondly, the reason for the +late-time accelerated expansion of the Universe; late DE. However, it is subject to some +tuning. Namely, the two free parameters κ and λ, the intrinsic field-space curvature dictated +by α, and the scale of the potential introduced by VΛ. +As we have seen κ and λ seem to take natural values, not too far from order unity. +Regarding α we only need that it is small enough to lead to rapid decrease of the exponential +contribution in the scalar potential in Eq. (2.6), leaving the constant VΛ to dominate at +present. We show in the next section that α ∼ 10−4 is sufficient for this task. This leaves +VΛ itself. The required tuning of this parameter is given by VΛ = +� HPlanck +0 +HSH0ES +0 +�2 +V Planck +Λ +, where +– 8 – + +V Planck +Λ +is given by the Planck 2018 [1] estimate of ρ0, the density today, multiplied by ΩΛ, +the estimate of the density parameter of dark energy today, i.e. +V Planck +Λ += ΩΛρ0. +Since +� HPlanck +0 +HSH0ES +0 +�2 +≃ ( 67.44 +73.04)2 = 0.8525 we see that the required fine-tuning of our VΛ is not different +from the fine-tuning introduced in ΛCDM, but, in contrast to ΛCDM, our proposal addresses +two cosmological problems; not only late DE but also the Hubble tension.7 +3 +Numerical Simulation +In order to numerically solve the dynamics of the system, it is enough to solve for the scale +factor a(t), the field φ(t) and the background fluid densities ρm(t) and ρr(t), as every other +quantity depends on these. +They are governed by the Friedmann equations, the Klein- +Gordon equation and the continuity equations respectively. Of course, the Klein-Gordon +equation is a second order ODE, while the continuity equations are first order so that we +need the initial value and velocity of φ and just the initial value of ρm and ρr as initial +conditions. As described above, the field starts frozen and unfreezes around matter-radiation +equality. Effectively, this means using φini = 0 and ˙φini = 0 as initial conditions, a few e- +folds before matter-radiation equality, while the initial radiation and matter energy densities +are chosen to satisfy the bounds obtained by Planck [1] at matter-radiation equality, i.e., +ρm(teq) = ρr(teq) = 1.27 × 10−110m4 +P. +For convenience, we rewrite the equations in terms of the logarithmic energy densities +˜ρm(t) = ln (ρm(t)/m4 +P) and ˜ρr(t) = ln (ρr(t)/m4 +P). Plugging the first Friedmann equation in +the Klein-Gordon equation, gives +¨φ(t) + +� +3ρ(t) +mP +˙φ(t) + dV +dφ = 0, +(3.1) +˙˜ρm(t) + +� +3ρ(t) +mP += 0, +(3.2) +˙˜ρr(t) + 4 +3 +� +3ρ(t) +mP += 0, +(3.3) +where 3m2 +PH2(t) = ρ(t) = [ exp(˜ρm(t))+exp(˜ρr(t))]m4 +P+ρφ(t) and ρφ(t) = K(φ(t))+V (φ(t)) +where K(φ(t)) = 1 +2( ˙φ(t))2 and V (φ(t)) is given by Eq. (2.2). +As mentioned above, we assume the field to be frozen at an ESP, such that it could have +been the inflaton or a spectator field at earlier times. The time of unfreezing is then controlled +only by the parameters of the model’s potential.8 The densities of matter and radiation are +scaled back to find some initial conditions at some arbitrary redshift, zini = 104, before +equality. +The differential solver records three “events” during solving: matter-radiation equality, +triggered by the obvious condition; decoupling, triggered by the total energy density taking +the correct value; and the present day, triggered by the field making up the correct fraction of +the total energy density (as estimated by the Planck satellite [1]). These values are saved to +an association so that they can later be searched to identify points which fulfill the necessary +7In our simulations we use VΛ = 10−120.068 m4 +P as assumed also in Fig. 1. +8 Although we could use an estimate for the initial time, it turns out that it makes no difference to the +numerical results or the behaviour of the field and simply offsets the differential equations. +– 9 – + +Initial Densities +Calculation +Value +Matter +ρm = 3ΩPlanck +m,0 +m2 +P (HSH0ES +0 +)2 +3.84 × 10−121m4 +P +Radiation +π2 +30g∗(T Planck +CMB, 0)4 +9.56 × 10−125m4 +P +Table 1: Table of present-day densities, where the present matter density parameter is +ΩPlanck +m,0 += 0.3111, T Planck +CMB, 0 = 2.7255 K and the effective relativistic degrees of freedom of +radiation are g∗ = 3.36, calculated by taking the photon and neutrino contribution into +account (see section 5 of [64]). +Variable +Initial Value +Source +Redshift +zinitial = 104 +chosen to be shortly before +matter-radiation equality +Time +tini = 0.1m−1 +P +chosen to be close to zero +(see footnote 8) +Field Value +φ(tini) = 0 +simplified initial conditions +Rate of change of Field +Value +˙φ(tini) = 0 +simplified initial conditions +Density of Matter +ρm(tini) = 3.84 × 10−109m4 +P +ρm(t0)Planck(zini + 1)3 +Density of Radiation +ρr(tini) = 1.24 × 10−108m4 +P +ρr(t0)Planck(zini + 1)4 +E-folds elapsed +Nini = 0 +chosen for convenience +Table 2: Table detailing the initial conditions for the differential equations. +constraints, in order to find a viable parameter space. Once the final event is recorded, the +solver is terminated. +Event +Criteria +Justification +Matter-Radiation +Equality +ρm(teq) = ρr(teq) +Theoretical Definition +Last Scattering +ρm(tls) = 4.98 × 10−112 m4 +P +Extrapolation +from +ΛCDM +initial conditions (see Table 2) +using Planck results ρm(zeq) +with zeq = 1089.80 [1] +Present Day +Ωφ = 0.6889 +Planck data [1] +Table 3: Table of events recorded during the numerical solving of equations and how. +If a field point does not meet the conditions for the final event (i.e. the present day), +this indicates that the field began the simulation as the dominant component and will never +reach the correct energy density. The point is thrown away. Finally, reasonable observational +and theoretical constraints to the parameter space are applied to the data collected, which +are outlined in Table 4. +– 10 – + +Parameter +to +be +constrained +Source +Description +Constraint +Density +parameter +of +the field at equality +EDE +literature +[25] +Upper limit governed by the +maximum value that does +not impede structure forma- +tion; lower limit is so that +EDE actually has an effect +0.015 ≤ Ωeq +φ < 0.107 +Density parameter +of the field at +Last Scattering +EDE +literature +[8] +This is the upper limit that +ensures EDE cannot cur- +rently be detected in the +CMB +Ωls +φ < 0.015 +Density parameters of +the field at Last Scatter- +ing and Equality +Theoretical +Achieves desired behaviour +of the field +Ωeq +φ > Ωls +φ +Density +parameter +of +the field today +Planck 2018 +[1] +Observational constraint +0.6833 ≤ Ω0 +φ ≤ 0.6945 +Barotropic parameter of +the field today +Planck 2018 +Observational constraint +−1 ≤ w0 +φ ≤ −0.95 +Running of the +barotropic parameter +today +Planck 2018 +[1] +Observational constraint +−0.55 ≤ wa +φ ≤ 0.03 +Hubble constant +SH0ES +2021 [2] +Observational constraint +72.00≤ +H0 +km s−1 Mpc−1 ≤74.08 +Total Field Excursion +Theoretical +From analytical estimates, +the total excursion of the +field should ideally be sub- +Planckian +φ0 − φeq < mP +Table 4: Table describing and justifying constraints used to identify the viable parameter +space. In the above, wa +φ = − dwφ +da +��� +0, c.f. Eq. (1.8). +4 +Results and analysis +4.1 +Parameter Space +As evident from Figs. 2, 3 and 4, we find that κ ∼ 102 and λ ∼ 10−3, which are rather +reasonable values. In particular, the value of κ suggests that the mass-scale which suppresses +the non-canonical field ϕ in the original potential in Eq. (2.1) is near the scale of grand +unification ∼ 10−2 mP. Regarding the curvature of field space we find α ∼ 10−4, which again +is not unreasonable. +The viable parameter space suggests that κλ > +√ +3, which contradicts our assumption +in Eq. (2.7). This implies that, unlike the analytics in Sec. 2.3.1, the field does not adopt +the subdominant exponential scaling attractor but the slow-roll exponential attractor, which +leads to domination [61, 63]. +As the field thaws and starts following this attractor, the +approximation in Eq. (2.4) breaks down as the field experiences the full exp(exp) potential, +which is steeper that exponential (see Fig. 1). Consequently, instead of becoming dominant +the field free-falls. This contradiction with our discussion in Sec. 2.3.1 is not very important. +– 11 – + +0.0000 +0.0001 +0.0002 +0.0003 +0.0004 +0.0005 +0.0006 +0.0007 +0 +100 +200 +300 +400 +500 +600 +700 +α +κ +VΛ +mP +4 += 10-120.068, +0 < λ < 0.027 +Figure 2: +Parameter space slice in the κ − α plane with 0 < λ < 0.027 and VΛ = +10−120.068m4 +P. +The blue dotted line is the boundary of the region that produces non- +inflationary results (see below), while the orange region is constituted by the success- +ful points, i.e., those for which the constraints detailed in Table 4 are satisfied. +Note +that the region bounded in blue is not equal to the range of the scan, which goes from +0 ≤ κ ≤ 700, 0 ≤ α ≤ 0.00071. This is because points with potential larger than a certain +starting value result in the field beginning the simulation dominant, which means that the +Universe goes into inflation which cannot terminate and will never meet the numerical end +condition for the present day. These points are very close to the viable parameter space for +these two parameters and therefore must be thrown away. +The existence of the scaling attractor provided an easy analytic estimate for the moment +when the field unfreezes. It turns out that, because the scaling attractor has been substituted +by the slow-roll attractor, the field unfreezes because its potential energy density becomes +comparable to the total energy density, going straight into free-fall. In is much harder to +analytically estimate when exactly this takes place, but the eventual result (free-fall) is the +same. +The redshift of matter-radiation equality occurs earlier than usual at zeq ≃ 4000. How- +ever, equality occurs well before last scattering, zeq > zls and its redshift is only indirectly +inferred by observations. In contrast, the redshift of last scattering is where we would expect +it at zls ≃ 1087. Theoretical constraints suggest zls ≃ 1090 [65], and the observations of the +Planck satellite suggest zls = 1089.80 ± 0.21 [1]. +– 12 – + +0.0000 +0.0001 +0.0002 +0.0003 +0.0004 +0.0005 +0.0006 +0.0007 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +α +λ +VΛ +mP +4 += 10-120.068, +0 < κ < 700 +Figure 3: Parameter space slice in the λ−α plane with 0 < κ < 700 and VΛ = 10−120.068m4 +P. +The orange region is constituted by the successful points, i.e., those for which the constraints +detailed in Table 4 are satisfied. +4.2 +Field Behaviour +The field behaves as expected, with the mild modification of the attractor solution at un- +freezing (slow-roll instead of scaling), which leads to free-fall. The evolution is depicted in +Figs. 5, 6, 7 and 8 for the example point at α = 0.0005, κ = 145, λ = 0.008125, and Vλ +tuned to the SH0ES cosmological constant [2]. The observables obtained in this case (i.e. the +values of H0, w0 and wa) are shown in Table 5. The behaviour of the Hubble parameter is a +function of redshift as can be seen in Fig. 7. +As mentioned in Table 4, the maximum allowed value of the EDE density parameter +at equality is just over 0.1. +However, it is possible that this is too lenient a constraint +because unlike the models for which this constraint was developed, our model has a true +free-fall period, which means it redshifts away exactly as a−6 rather than below this rate +as in oscillatory behaviour (see Figs. 5 and 8). A full MCMC analysis may provide a more +accurate constraint for non-oscillatory models. +At present, the exponential contribution to the potential density in Eq. (2.6) is largely +subdominant to VΛ, so the contribution of the scalar field to the total density budget is +almost constant, as in ΛCDM. Its barotropic parameter is, therefore, wφ ≈ −1 (see Fig. 5). +Technically, it is not exactly -1 but its running is negligible, with the viable parameter space +for wa fitting easily within the constraint in Eq. (1.8) by some ten orders of magnitude (see +Table 5). +– 13 – + +0 +100 +200 +300 +400 +500 +600 +700 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +κ +λ +VΛ +mP +4 += 10-120.068, +0 < α < 0.00071 +Figure 4: Parameter space slice in the λ − κ plane with 0 < α < 0.00071 and VΛ = +10−120.068m4 +P. The orange region is constituted by the successful points, i.e., those for which +the constraints detailed in Table 4 are satisfied. +5 +Initial Conditions +Our model accounts for both EDE and late-time dark energy in a non-oscillatory manner +(in contrast to Ref. [30]). The field is frozen at early times, thawing just before matter- +radiation equality when its density grows to nearly 0.1 of the total value (see Fig. 6), as set +by constraints in Ref. [25]. A steep exp(− exp) potential then forces the field into free-fall, +causing its energy density to dilute away as ρφ ∝ a−6. After this, the field hits the asymptote +of the exponential decay and refreezes, becoming dominant at present (see Fig. 8). +Thus, we achieve DE-like behaviour at the present day by ensuring that the field re- +freezes after its period of free-fall, therefore remaining at a constant energy density equal to +the value of the potential density at that point. Although this constant potential density is +initially negligible, the expansion of the Universe causes the density of matter to decrease. +Because the field refreezes at a potential density that is comparable to the density of matter +at present, the field starts to become dominant at the present day. Once it begins to domi- +nate the Universe, the field thaws again, but the density of the Universe is dominated by a +constant contribution VΛ, as with ΛCDM. +The obvious question is why our scalar field finds itself frozen at the origin in the first +place. One compelling explanation is the following. We assume that the origin is an enhanced +symmetry point (ESP) such that, at very early times, an interaction of ϕ with some other +scalar field χ traps the rolling of ϕ at zero. The idea follows the scenario explored in Ref. [66]. +– 14 – + +wϕ +wm+r +wUniverse +0 +2 +4 +6 +8 +-1.0 +-0.5 +0.0 +0.5 +1.0 +3671 +1350 +496 +182 +66 +24 +8 +2 +N +z +VΛ +mP +4 = 10-120.068 +α =0.0005 +κ=145 +λ=0.008125 +Figure 5: Barotropic parameter of the scalar field (dotted green), of the background perfect +fluid (full blue) and of the sum of both components (full black), for α = 0.0005, κ = 145, λ = +0.008125, and VΛ = 10−120.068m4 +P. +Density parameter of field Ωϕ +0 +2 +4 +6 +8 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +3671 +1350 +496 +182 +66 +24 +8 +2 +N +z +VΛ +mP +4 = 10-120.068 +α =0.0005 +κ=145 +λ=0.008125 +Figure 6: The density parameter of the scalar field, for α = 0.0005, κ = 145, λ = 0.008125, +and VΛ = 10−120.068m4 +P, as a function of the redshift (top) and e-folds (bottom) elapsed since +the beginning of the simulation. +In this scenario, the scalar potential includes the interaction +∆V = 1 +2g2ϕ2χ2 , +(5.1) +– 15 – + +HϕCDM +HCDM only +HΛCDM +8.0 +8.2 +8.4 +8.6 +8.8 +9.0 +9.2 +50 +100 +150 +200 +250 +2.35 2.03 1.74 1.48 1.24 1.03 0.84 0.66 0.50 0.36 0.23 0.11 0.01 +N +z +VΛ +mP +4 = 10-120.068 +α =0.0005 +κ=145 +λ=0.008125 +Figure 7: The Hubble parameter (in units of km s−1Mpc−1) of a Universe with the modelled +scalar field (green), a classical ΛCDM simulation (black), and one with only matter and +radiation (blue), as a function of the redshift (top) and the e-folds (bottom) elapsed since +the beginning of the simulation. +log[ρm/mP +4] +log[ρr/mP +4] +log[ρϕ/mP +4] +log[(ρm+ρr)/mP +4] +0 +2 +4 +6 +8 +-125 +-120 +-115 +-110 +-105 +3671 +1350 +496 +182 +66 +24 +8 +2 +N +z +VΛ +mP +4 = 10-120.068 +α =0.0005 +κ=145 +λ=0.008125 +Figure 8: The logarithmic densities of matter (dot-dashed red), radiation (dotted orange), +the sum of both (solid blue) and the scalar field (dashed green), as a function of the redshift +(top) and the e-folds (bottom) elapsed since the beginning of the simulation. The horizontal +full line represents the (SH0ES) energy density of the Universe at present. +where the coupling g < 1 parametrises the strength of the interaction. +– 16 – + +Constraint +Field Value +0.015 ≤ Ωeq +φ < 0.107 +0.05178 +Ωls +φ < 0.015 +0.001722 +Ωeq +φ > Ωls +φ +YES +0.6833 ≤ Ω0 +φ ≤ 0.6945 +0.6889 +−1 ≤ w0 +φ ≤ −0.95 +-1.000 +−0.55 ≤ wa +φ ≡ − dwφ +da +��� +0 ≤ 0.03 +−4.850 × 10−11 +72.00 ≤ +H0 +km s−1 Mpc−1 ≤ 74.08 +73.27 +κλ +1.178 +(φ0 − φeq)/mP < 1 +0.4274 +Table 5: Table giving the constraints and their corresponding values for an example point, +α = 0.0005, κ = 145, λ = 0.008125, and VΛ tuned to the SH0ES cosmological con- +stant, in the viable parameter space. +The Hubble constant obtained in this example is +H0 = 73.27 km/s Mpc. +We assume that initially ϕ is rolling down its steep potential.9 Then, the interaction +in Eq. (5.1) provides a modulated effective mass-squared m2 +eff = g2ϕ2 to the scalar field χ. +When ϕ crosses the origin, this effective mass becomes momentarily zero. If the variation of +the ϕ field (i.e. the speed | ˙ϕ| in field space) is large enough, then there is a window around +the origin when | ˙meff| ≫ m2 +eff (because, | ˙ϕ| ≫ ϕ2 ≃ 0). This violates adiabaticity and leads +to copious production of χ-particles [66].10 +As the field moves past the ESP, the produced χ particles become heavy, which takes +more energy from the ϕ field, producing an effective potential incline in the direction the +ϕ field is moving. Indeed, the particle production generates an additional linear potential +∼ g|ϕ|nχ [66], where nχ is the number density of the produced χ-particles. This number +density is constant because the duration of the effect is much smaller than a Hubble time, +so that we can ignore dilution from the Universe expansion. The rolling ϕ field climbs up +the linear potential until its kinetic energy density is depleted. Then the field momentarily +stops and afterwards reverses its motion (variation) back to the origin. When crossing the +origin again, there is another bout of χ-particle production, which increases nχ and makes the +linear potential steeper to climb. This time, ϕ variation halts at a value closer to the origin. +Then, the field reverses its motion and rushes through the origin again. Another outburst of +χ-particle production steepens the linear potential further. The process continues until the +9For away from the origin, the scalar potential V (ϕ) does not have to be of the form in Eq. (2.1). In fact, +it is conceivable that ϕ might play the role of the inflaton field too (see Appendix). +10Near the origin, when ϕ ≃ 0, the ϕ-field is approximately canonically normalised, as suggested by Eq. (1.5), +so the considerations of Ref. [66] are readily applicable. +– 17 – + +ϕ-field is trapped at the origin [63, 66]. +The trapping of a rolling scalar field at an ESP can take place only if the χ-particles do +not decay before trapping occurs. If they did, the nχ would decrease and the potential g|ϕ|nχ +would not be able to halt the motion (variation) of the ϕ-field. The end result of this process is +that all the kinetic energy density of the rolling ϕ has been given to the χ-particles. Now, since +ϕ is trapped at the origin, the effective mass of the χ-particles is zero, which means that they +are relativistic matter, with density scaling as ρχ ∝ a−4. As far as ϕ is concerned, it is trapped +at the origin and its density is only ρϕ = V (ϕ = 0) = e−λVX = constant (cf. Eq. (2.1)). +After some time, it may be assumed that the χ-particles do eventually decay into the +standard model particles, which comprise the thermal bath of the hot Big Bang. The con- +fining potential, which is proportional to nχ, disappears but, we expect the ϕ-field to remain +frozen at the origin because the scalar potential V (ϕ) in Eq. (2.1) is flat enough there. As we +have discussed, the ϕ-field unfreezes again in matter-radiation equality. The above scenario +is depicted in Fig. 9 +For simplicity, we have considered that, apart from the obvious violation of adiabacity at +the ESP, the χ direction is otherwise approximately flat and the χ-field has a negligible bare +mass compared to the ϕ field. It would be more realistic to consider a non-zero bare mass for +the χ-particles, which when they become non-relativistic (much later than the trapping of +ϕ) can safely decay to the thermal bath of the hot Big Bang, reheating thereby the Universe, +e.g. in a manner not dissimilar to Ref. [67]. +The above scenario is one possible explanation of the initial condition considered and +not directly relevant to the scope of this work - numerical simulations simply assume that +the field begins frozen at the origin. Other possibilities to explain our initial condition exist, +for example considering a thermal correction of the form δV ∝ T 2ϕ2, which would make the +origin an effective minimum of the potential at high temperatures and drive the ϕ-field there. +6 +Conclusions +In conclusion, we have studied in detail a non-oscillatory model of unified early and late dark +energy, which resolves the Hubble tension and simultaneously explains the observed current +accelerated expansion with no more fine tuning than ΛCDM. Our model considers a single +scalar field in the context of α-attractors, as in Ref. [30], but in our case the field is not +oscillating; instead after equality, it free-falls with energy density decreasing as a−6, faster +than most early dark energy (EDE) proposals and the fastest possible. +In our proposed scenario, the scalar field lies originally frozen at the origin, until it +thaws near the time of equal matter-radiation densities, when it becomes EDE. Afterwards +it free-falls until it refreezes at a lower potential energy density value, which provides the +vacuum density of ΛCDM. We showed that the total excursion of the field in configuration +space is sub-Planckian, which implies that our potential is stable under radiative corrections. +One explanation of our initial conditions is that the origin is an enhanced symmetry +point (ESP). Our scalar field is originally kinetically dominated until it is trapped at the ESP +when crossing it.11 As we discuss in Appendix A, the scalar field could even be the inflaton, +which after inflation rolls down its runaway potential until it becomes trapped at the ESP. +Our potential in Eq. (2.1) really serves to demonstrate that a model unifying EDE with +ΛCDM can be achieved with a suitably steep runaway potential. With the parameters of +our model assuming rather natural values, thereby not introducing fine-tuning additional to +11A thermal correction to the scalar potential can have a similar effect. +– 18 – + +ln ρ +ln a +ρφ +ρr + ρm +today +equality +ESP +e−λVX +VΛ +Figure 9: Schematic log-log plot depicting the evolution of the density of the scalar field +ρφ (solid blue line) and the density of radiation and matter ρr + ρm (dashed red line) in +the case when the decay of the kinetic energy density of the trapped scalar field generates +the thermal bath of the hot Big Bang (as in Ref. [REF]). Originally the φ-field is rushing +towards the minimum of the potential, dominated by its kinetic density, so that ρφ ∝ a−6 +(free-fall). When it crosses the enhanced symmetry point (ESP) its interaction to the χ- +field (cf. Eq. (5.1)) traps the rolling φ-field at the ESP while all its kinetic energy is given +to χ-particles, which soon decay into the radiation and matter of the hot Big Bang (the +decay is assumed to be quick, just after trapping). Afterwards, the φ-field stays frozen, with +energy density V (φ = 0) = e−λVX (cf. Eq. (2.1)) until much later, when its potential density +is comparable to the background. Then it unfreezes before dominating, acting as early dark +energy at the time near matter-radiation equality, and subsequently free-falls to its value φ0, +with potential density approximately VΛ = constant. The field stays there until the present +when it dominates the Universe and becomes late dark energy. +that of ΛCDM, we show that this is indeed possible with a simple design. The challenge lies +in constructing a concrete theoretical framework for such a potential. +Acknowledgements: LB is supported by STFC. KD is supported (in part) by the Lancaster- +Manchester-Sheffield Consortium for Fundamental Physics under STFC grant: ST/T001038/1. +SSL is supported by the FST of Lancaster University. +A +Quintessential Inflation +Is it possible that our scalar field can not only be early and late dark energy, but also be the +inflaton field, responsible for accelerated expansion in the early Universe? +– 19 – + +The α-attractors construction leads to two flat regions in the scalar potential of the +canonical field, as the kinetic poles of the non-caninical field are displaced to infinity. This +idea has been employed in the construction of quintessential inflation models in Refs. [54–56], +where the low-energy plateau was the quintessential tail, responsible for quintessence and the +high-energy plateau was responsible for inflation. +However, if we inspect the potential in Eq. (2.1) at the poles ϕ = ± +√ +6α mP, we find that +the potential for the positive pole is V (ϕ+) = VΛ as expected, while for the negative pole we +have V (ϕ−) = VΛ exp +� +2λ sinh +� +κ +√ +6α +�� +. For the values of the parameters obtained (κ ∼ 102, +λ ∼ 10−3 and α ∼ 10−4) it is easy to check that V (ϕ−) is unsuitable for the inflationary +plateau. Thus, our model needs to be modified to lead to quintessential inflation. +The first modification is a shift in field space such that our new field is +˜ϕ = ϕ + Φ , +(A.1) +where Φ is a constant. The α-attractors construction applies now on the new field ˜ϕ for +which the Lagrangian density is given by the expression in Eq. (1.4) with the substitution +ϕ → ˜ϕ. The poles of our new field lie at ˜ϕ± = ± +√ +6˜α mP, where ˜α is the new α-attractors +parameter. +We want all our results to remain unaffected, which means that, for the positive pole, +Eq. (A.1) suggests +ϕ+ = +√ +6α mP = ˜ϕ+ − Φ = +√ +6˜α mP − Φ ⇒ ˜α = 1 +6 +� Φ +mP ++ +√ +6α +�2 +. +(A.2) +The above, however, is not enough. It turns out we need to modify the scalar potential +as well. +This modification must be such that near the positive pole the scalar potential +reduces to the one in Eq. (2.1). A simple proposal is +V ( ˜ϕ) = VX exp{−2λ sinh[κ( ˜ϕ − Φ)/mP]} , +(A.3) +which indeed reduces to Eq. (2.1) when κ( ˜ϕ − Φ) = κϕ > mP Note that κ +√ +6α > 1 is implied +from the requirement that near the positive pole we have κ +√ +6α mP = κϕ+ > mP. The ESP +discussed in Sec. 5 is now located at ˜ϕ = Φ, such that Eq. (5.1) is now ∆V = 1 +2g2( ˜ϕ − Φ)2χ2.12 +We are interested in investigating the inflationary plateau. This is generated for the +canonical field near the negative pole ˜ϕ− = − +√ +6˜α mP, where the scalar potential of the +canonical field “flattens out” [40]. +Assuming that Φ > +√ +6α mP, we have that ˜ϕ− − Φ = −2Φ − +√ +6α mP ≃ −2Φ, where we +used Eq. (A.2). Hence, for the potential energy density of the inflationary plateau we obtain +Vinf = V ( ˜ϕ−) ≃ VX exp[−2λ sinh(−2κΦ/mP)] +≃ exp +� +λ eκ +√ +6α� +VΛ exp[λ exp(2κΦ/mP)] += exp +� +λ(eκ +√ +6α + e2κΦ/mP) +� +VΛ ≃ VΛ exp +� +λ e2κΦ/mP +� +, +(A.4) +where we used Eq. (2.1) and that in −2 sinh(−x) ≃ ex, when x ≫ 1. +12Near the ESP the potential does not approximate Eq. (2.1). However, we assume that, after unfreezing, +the field rolls away fast from the ESP, such that soon the exp(exp) form of the potential becomes valid and +the evolution is the one discussed in the main text of our paper. +– 20 – + +With α-attractors, the inflationary predictions are ns = 1 − 2/N and r = 12˜α/N2 [40], +where ns is the spectral index of the scalar curvature perturbation and r is the ratio of +the spectrum of the tensor curvature perturbation to the spectrum of the scalar curvature +perturbation, with N being the number of inflationary efolds remaining after the cosmo- +logical scales exit the horizon. +Typically, N = 60 − 65 for quintessential inflation, which +means that ns = 0.967 − 0.969, in excellent agreement with the observations [68]. For the +tensor-to-scalar ratio the observations provide the bound r < 0.036 [69], which suggests +˜α < 0.003 N2 = 10.8 − 12.7. +The COBE constraint requires Vinf ∼ 10−10 m4 +P. Using that VΛ ∼ 10−120 m4 +P, Eq. (A.4), +suggests that κΦ/mP = 1 +2 ln(110 ln 10/λ). Hence. the conditions Φ > +√ +6α mP and κ +√ +6α > 1 +suggest +1 < κ +√ +6α < κΦ/mP = 1 +2 ln(110 ln 10/λ) . +(A.5) +Our findings in Sec. 4 are marginally in agreement with the above requirements. +For +example, taking α = 0.0006 and κ = 100 we find κ +√ +6α = 6 and then Eq. (A.5) suggests +λ < 1.556 × 10−3. +We also find Φ/mP > +√ +6α = 0.06, which is rather reasonable. +Then, +Eq. (A.2) implies ˜α > 12α = 7.2 × 10−3, which comfortably satisfies the observational con- +straint on r. In fact, taking N ≃ 60, we find r = 12˜α/N2 > α/25 = 2.4 × 10−5. +The above should be taken with a pinch of salt because the approximations employed +are rather crude. However, they seem to suggest that our augmented model in Eq. (A.3) +may lead to successful quintessential inflation while also resolving the Hubble tension, with no +more fine-tuning than that of ΛCDM. 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Lett. 127 (2021), no. 15 151301, [arXiv:2110.00483]. +– 24 – + diff --git a/FdE1T4oBgHgl3EQf-wYy/content/tmp_files/load_file.txt b/FdE1T4oBgHgl3EQf-wYy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..762bae9f6ebef264406f9e7b7f0445b2f4c5588c --- /dev/null +++ b/FdE1T4oBgHgl3EQf-wYy/content/tmp_files/load_file.txt @@ -0,0 +1,1283 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf,len=1282 +page_content='Non-oscillating Early Dark Energy and Quintessence from α-Attractors Lucy Brissenden, Konstantinos Dimopoulos and Samuel S´anchez L´opez Consortium for Fundamental Physics, Physics Department, Lancaster University, Lancaster LA1 4YB, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' E-mail: l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='brissenden@lancaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='uk, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='dimopoulos1@lancaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='uk, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='sanchezlopez@lancaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='uk Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Early dark energy (EDE) is one of the most promising possibilities in order to resolve the Hubble tension: the discrepancy between early and late-Universe measurements of the Hubble constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In this paper we propose a model of a scalar field which can explain both EDE and late Dark Energy (DE) in a joined manner without additional fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The field features kinetic poles as with α-attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Our model provides an injection of EDE near matter-radiation equality, and redshifts away shortly after via free-fall, later refreezing to become late-time DE at the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Using reasonable estimates of the current constraints on EDE from the literature, we find that the parameter space is narrow but viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As such our model is readily falsifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In contrast to other work in EDE, our model is non-oscillatory, which causes its decay to be faster than that of the usual oscillatory EDE, thereby achieving better agreement with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='03572v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='CO] 9 Jan 2023 Contents 1 Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 The Hubble tension 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2 Early Dark Energy 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3 α-attractors 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4 Quintessence 4 2 The Model 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 Lagrangian and Field Equations 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2 Shape of Potential and Expected Behaviour 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3 Asymptotic forms of the scalar potential 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 Expected Field Behaviour 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4 Tuning requirements 8 3 Numerical Simulation 9 4 Results and analysis 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 Parameter Space 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2 Field Behaviour 13 5 Initial Conditions 14 6 Conclusions 18 A Quintessential Inflation 19 1 Introduction In the last few decades cosmological observations of the early and late Universe have con- verged into a broad understanding of the history of our Universe from the very first seconds of its existence until today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Thus, cosmology has developed a standard model called the concordance model, or in short ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, the latest data might imply that the celebrated ΛCDM model is not that robust after all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In particular, there is a 5-σ discrepancy between the measurements of the current expansion rate, the Hubble constant H0, as inferred by early Universe observations compared with late Universe observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This Hubble tension has undermined our confi- dence in ΛCDM and as such it is investigated intensely at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In this work we study a toy model that can simultaneously solve the Hubble tension and explain the current accelerated expansion with no more tuning that in ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Our model introduces a scalar field which plays both the role of early dark energy (EDE) and quintessence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In contrast to most other works in the literature which consider scalar fields as EDE, ours is not an oscillating scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' We use natural units with c = ¯h = 1, the reduced Planck mass mP = 1/ √ 8πG = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='43 × 1018GeV and consider a positive signature metric (−1, +1, +1, +1) throughout the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 1 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 The Hubble tension Measurements in observational cosmology can broadly be classified into two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' These are measurements of quantities which depend only on the early-time history of our Universe (such as the cosmic microwave background (CMB) radiation at redshift z ≃ 1100, or Baryon Acoustic Oscillations (BAO)) and measurements of quantities which depend on present-day observations (the primary example of this is the cosmic distance ladder, which measures the redshift of observable astrophysical objects such as Cepheid stars and type-1a supernovae, at redshift z = O(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The value of the Hubble constant H0 can in principle be inferred from both early and late-time measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, it has been found that while early-time measurements are in good agreement with each other, they disagree with current late-time data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Latest analysis of the CMB temperature anisotropies’ data gives the value inferred from Planck satellite [1], H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='58 km s−1Mpc−1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) and a distance scale measurement using Cepheid-SN 1a data from the SH0ES collaboration [2] as H0 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='04 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='04 km s−1Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2) This is a 5σ tension which includes estimates of all systematic errors and which the SH0ES team conclude has “no indication of arising from measurement uncertainties or analysis varia- tions considered to date”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' It is becoming increasingly apparent with successive measurements that this tension is likely to have a theoretical resolution [3, 4], which can have many possible sources [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2 Early Dark Energy One proposed class of solutions to the Hubble tension is models of Early Dark Energy (EDE), whose early works include references [7–10], followed by many others, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [5, 11–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' These involve an injection of energy in the dark energy sector at around the time of matter- radiation equality, which then dilutes or otherwise decays away faster than the background energy density, such that it becomes negligible before it can be detected in the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As briefly reviewed below, such models result in a slight change in the expansion history of the Universe, bumping up the value of the Hubble parameter at the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' It has previously been concluded [3, 5, 6] that EDE models are most likely to source a theoretical resolution to the Hubble tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' One reason for this is that EDE can effect substantial modifications to H0 without significant effect on other cosmological parameters which are tightly constrained by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 In particular, EDE models can be incorpo- rated into existing scalar-field models of inflation and late-time dark energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' one example of the latter is the model detailed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, precisely because EDE models exist so close in time to existing observational data, they have significant constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' the primary consideration being that EDE must be subdominant at all times and must decay away fast enough to be essentially negligible at the time of last scattering translating to a redshift rate that is faster than radiation [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' So far, in previous works in EDE, this has been achieved by considering first or second-order phase transitions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [23], [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' These abrupt events might have undesirable side-effects 1Models which modify other cosmological parameters are often unable to reconcile their changes with current observational constraints on said parameters (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [5] for a comprehensive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 2 – such as inhomogeneities from bubble collisions or topological defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Other proposed models [5, 7, 8, 23–30] typically feature oscillatory behaviour to achieve the rapid decay rate necessary for EDE to be negligible at last scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As with the original proposal in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [7], the EDE field is taken to oscillate around its Vacuum Expectation Value (VEV) in a potential minimum which is tuned to be of order higher than quartic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As a result, its energy density decays on average as ∝ a−n, with 4 < n < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In contrast, in our model, the EDE scalar field experiences a period of kinetic domination, where the field is in non-oscillatory free-fall and its density decreases as ∝ a−6, exactly rather than approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Before continuing, we briefly explain how EDE manages to increase the value of H0 as from CMB observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Measurements of the CMB temperature anisotropies provide very tight constraints on the cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' One would therefore think that this severely limits models which alter the Universe content and dynamics at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, there are certain classes of models for which this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' These are models that affect both the Hubble parameter and rs, the comoving sound horizon2 (in this case during the drag epoch, shortly after recombination), given by rs = � ∞ zd cs(z) H(z)dz, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3) where cs(z) is the sound speed and H(z) is the Hubble parameter, both as a function of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' An additional amount of dark energy in the Universe increases the total density, which in turn increases the Hubble parameter because of the Friedmann equation ρ ∝ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Therefore, EDE considers such a brief increase at or before decoupling, which lowers the value of the sound horizon because it increases H(z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, there is a way to avoid this being evident in and therefore disproved by current CMB measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This is because BAO and CMB measurements do not constrain the value of the sound horizon directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' For example, BAO measurements do not constrain the sound horizon alone, but the com- bination H(z)rs [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The observations of the Planck satellite measure the quantity θ∗ ≡ r∗ D∗ [34], the angular scale of the sound horizon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' given by ratio of the comoving sound horizon to the angular diameter distance at which we observe fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Both of these measurements entail an assumption of ΛCDM cosmology and can be shown to be equally constrained by other models, provided that they make only small modifications which simultaneously lower the value of rs and increase H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' EDE may have a significant drawback, however, in that it does not alleviate the σ8 tension (associated with matter clustering) and may in fact exacerbate it [3, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As with many others, our model does not attempt to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3 α-attractors Our model unifies EDE with late DE in the context of α-attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' An earlier attempt for such unification in the same theoretical context can be seen in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, this proposal is also of oscillatory EDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' α-attractors [36–44], which appear naturally in conformal field theory or supergravity theories, are a class of models whose inflationary predictions continuously interpolate between those of chaotic inflation [45] and those of Starobinsky [46] and Higgs inflation [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In 2This is the characteristic scale of BAO, typically approximately proportional to the value of the cosmo- logical horizon at that point by rs = 1 √ 3rH assuming spatial flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 3 – supergravity, introducing curvature to the internal field-space manifold can give rise to a non-trivial K¨ahler metric, which results in kinetic poles for some of the scalar fields of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The free parameter α is inversely proportional to said curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' It is also worth clarifying what is meant by the word “attractor”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' It is not only used in the usual sense (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=', field trajectories during inflation flowing to a unique one, regardless of the initial conditions), but also to refer to the fact that the inflationary predictions are largely insensitive of the specific characteristics of the model under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Such an attractor behaviour is seen for sufficiently large curvature (small α) in the internal field-space manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In practical terms, the scalar field has a non-canonical kinetic term, featuring two poles, which the field cannot transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' To aid our intuition, the field can be canonically normalised via a field redefinition, such that the finite poles for the non-canonical field are transposed to infinity for the canonical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As a result, the scalar potential is “stretched” near the poles, resulting in two plateau regions, which are useful for modelling inflation, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [48–53] or quintessence [54], or both, in the context of quintessential inflation [54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Following the standard recipe, we introduce two poles at ϕ = ± √ 6α mP by considering the Lagrangian L = − 1 2(∂ϕ)2 (1 − ϕ2 6α m2 P )2 − V (ϕ) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4) where ϕ is the non-canonical scalar field and we use the short-hand notation (∂ϕ)2 ≡ gµν∂µϕ ∂νϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' We then redefine the non-canonical field in terms of the canonical scalar field φ as dφ = dϕ 1 − ϕ2 6αm2 P ⇒ ϕ = mP √ 6α tanh � φ √ 6α mP � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5) It is obvious that the poles ϕ = ± √ 6α are transposed to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In terms of the canonical field, the Lagrangian now reads L = −1 2(∂φ)2 − V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4 Quintessence “Early” Dark Energy is so named in order to make it distinct from “late” Dark Dnergy, which is the original source of the name (and often just called Dark Energy (DE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In cos- mological terms the latter is just beginning to dominate the Universe at present, making up approximately 70% of the Universe’s energy density [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This is the mysterious unknown substance that is responsible for the current accelerating expansion of the Universe and has equation-of-state (barotropic) parameter of w = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='03 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Late DE that is due to an (as-yet-undiscovered) scalar field is called quintessence [58], so-named because it is the “fifth element” making up the content of the Universe 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In this case, the Planck-satellite bound on the barotropic parameter of DE is −1 ≤ w < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='95 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Quintessence is distinct from other explanations for DE because a scalar field has a variable barotropic parameter and can therefore exhibit completely different behaviour in different periods of the Universe’s history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In order to get it to look like late-time DE, a scalar field should be dominated by its potential density, making its barotropic parameter sufficiently 3After baryonic matter, dark matter, photons and neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 4 – close to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' It is useful to consider the CPL parametrization, which is obtained by Taylor expanding w(z) near the present as [59, 60] w(z) = w0 + wa z z + 1 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='7) where wa ≡ −(dw/da)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The Planck satellite observations impose the bounds [1] −1 ≤ w < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='95 wa = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='29+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='26 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='8) 2 The Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 Lagrangian and Field Equations Consider a potential of the form V (ϕ) = VX exp � −λeκϕ/mP � , with VΛ ≡ exp � −λeκ √ 6α� VX , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) where α, κ, λ are dimensionless model parameters, VX is a constant energy density scale and ϕ is the non-canonical scalar field with kinetic poles given by the typical alpha attractors form (see [40]) with Lagrangian density given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4 In the above, VΛ is the vacuum density at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' To assist our intuition, we switch to the canonically normalised (canonical) scalar field φ, using the transformation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In terms of the canonical scalar field, the Lagrangian density is then given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6), where the scalar potential is V (φ) = exp � λeκ √ 6α� VΛ exp � −λeκ √ 6α tanh(φ/ √ 6α mP)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2) As usual, the Klein-Gordon equation of motion for the homogeneous canonical field is ¨φ + 3H ˙φ + V ′(φ) = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3) where the dot and prime denote derivatives with respect to the cosmic time and the scalar field respectively, and we assumed that the field was homogenised by inflation, when the latter overcame the horizon problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2 Shape of Potential and Expected Behaviour Henceforth we will discuss the behaviour of the field in terms of the variation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' movement in field space, of the canonical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3 Asymptotic forms of the scalar potential We are interested in two limits for the potential above: φ → 0 (ϕ → 0) and φ → +∞ (ϕ → √ 6α mP ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The first limit would correspond to matter-radiation equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In this limit, the potential is 4The model parameter is VX and not VΛ, the latter being generated by VX and the remaining model parameters as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 5 – Veq ≃ exp � λ(eκ √ 6α − 1) � VΛ exp(−κλ φeq/mP) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4) where the subscript ‘eq’ denotes the time of matter-radiation equality when the field un- freezes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' It is assumed that the field was originally frozen there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' We discuss and justify this assumption in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' After unfreezing, it is considered that the field has not varied much, for the above approximation to hold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 0 ≲ φeq ≪ √ 6αmP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5) This is a reasonable assumption given that the field begins shortly before matter-radiation equality frozen at the origin, unfreezing at some point during this time 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' At large φ (φ → ∞), the non-canonical field is near the kinetic pole (ϕ → √ 6α mP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Then the potential in this limit is V0 ≃ VΛ � 1 + 2κλeκ √ 6α√ 6α exp � − 2φ0 √ 6α mP �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6) which, even for sub-Planckian total field excursion in φ, should be a good approximation for sufficiently small α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The subscript ‘0’ denotes the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The above approximations describe well the scalar potential near equality and the present time, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As we exlain below, in between these regions, the scalar field free-falls and becomes oblivious of the scalar potential as the term V ′(φ) in its equation of motion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3) becomes negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Canonical Potential Approximation at Low Field Values Approximation at High Field Values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0 120 119 118 117 116 115 ϕ mP √(6 α) log V(ϕ) mP 4 VΛ mP 4 = 10-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068 α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0002 κ=200 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='01 Figure 1: Graph of the canonical potential and its two approximations for small and large field values, given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' These approximations are useful because they are simple exponential potentials with known attractors, so we know the type of behaviour the field should exhibit when each approximation is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' It can be readily seen that, after leaving the origin the field jumps off a potential plateau and is free-falling as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 5There is no suggestion in the EDE literature [5, 7, 8, 23–30] that the field has to unfreeze at any particular time, as long as it does not grow to larger than the allowed fraction and its energy density is essentially negligible by the time of decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 6 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 Expected Field Behaviour Here we explain the rationale behind the mechanism envisaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' We make a number of crude approximations, which enable us to follow the evolution of the scalar field, but which need to be carefully examined numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' We do so in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' First, we consider that originally the field is frozen at zero (for reasons explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Its energy density is such that it remains frozen there until equality, when it thaws following the appropriate exponential attractor, since Veq in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4) is approximately ex- ponential [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Assuming that this is the subdominant attractor requires that the strength of the exponential is [62, 63] Z ≡ κλ > √ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='7) The subdominant exponential attractor dictates that the energy density of the rolling scalar field mimics the dominant background energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Thus, the density parameter of the field is constant, given by the value [61–63] Ωeq φ ≃ 3 Z2 = 3 (κλ)2 < 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='8) This provides an estimate of the moment when the originally frozen scalar field, unfreezes and begins rolling down its potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Unfreezing happens when Ωφ (which is growing while the field is frozen, because the background density decreases with the expansion of the Universe) obtains the above value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, after unfreezing, the field soon experiences the full exp(exp) steeper than exponential potential so, it does not follow the subdominant attractor any more but it free- falls,6 such that its density scales as ρφ ≃ 1 2 ˙φ2 ∝ a−6, until it refreezes at a larger value φF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This value is estimated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In free-fall, the slope term in the equation of motion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3) of the field is negligible, so that the equation is reduced to ¨φ + 3H ˙φ ≃ 0, where H = 2/3t after equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The solution is φ(t) = φeq + C teq � 1 − teq t � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='9) where C is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' From the above, it is straightforward to find that ˙φ = Ct−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Thus, the density parameter at equality is Ωeq φ = ρφ ρ ���� eq = 1 2C2t−4 eq 4 3( mP teq)2 = 3 8 C2 (mP teq)2 ⇒ C = � 8 3Ωeq φ mP teq = √ 8 κλ mP teq , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='10) where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='8), ρφ ≃ 1 2 ˙φ2 and that ρ = 1/6πGt2 = 4 3(mP /t)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Thus, the field freezes at the value φ0 = φeq + C/teq = φeq + √ 8 κλ mP , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='11) where we considered that teq ≪ tfreeze < t0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Using that teq ∼ 104 y and t0 ∼ 1010 y, we can estimate Veq V0 ≃ Ωeq φ ρeq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='7 ρ0 ≃ 30 7(κλ)2 � t0 teq �2 ≃ 3 7(κλ)2 × 1013 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='12) 6i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' its energy density is dominated by its kinetic energy density only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 7 – Now, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6) we find Veq V0 ≃ eλ(eκ √ 6α−1) exp(−κλ φeq/mP ) 1 + 2κλ eκ √ 6α√ 6α exp � −2φ0/ √ 6α mP � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='13) In view of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='11), the above can be written as Veq V0 ≃ eλ(eκ √ 6α−1) 1 + 2κλ eκ √ 6α√ 6α e−2 √ 8/κλ √ 6α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='14) Taking Ωeq φ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 as required by EDE, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='8) suggests κλ ≃ √ 30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='15) Combining this with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='12) we obtain e √ 30 κ (eκ √ 6α−1) ∼ 1012/7 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='16) where we have ignored the 2nd term in the denominator of the right-hand-side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' From the above we see that, κ is large when α is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Taking, as an example, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='01 we obtain κ ≃ 18 and λ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='30 (from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' With these values, the second term in the denominator of the right-hand-side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='14), which was ignored above, amounts to the value 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This forces a correction to the ratio Veq/V0 of order unity, which means that the order-of-magnitude estimate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='16) is not affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Using the selected values, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='11) suggests that the total excursion of the field is ∆φ = φ0 − φeq = √ 8 κλ mP ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5 mP , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='17) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' it is sub-Planckian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In the approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4), we see that the argument of the exponential becomes κλ∆φ/mP ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='7 > 1, where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This means that the approximation breaks down and the exp(exp) potential is felt as considered, as depicted also in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' For small α the eventual exponential potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6) is steep, which suggests that field rushes towards the minimum at infinity and the barotropic parameter is w ≈ −1 because the potential is dominated by the constant VΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4 Tuning requirements Our model addresses in a single shot two cosmological problems: firstly, the Hubble tension between inferences of H0 using early and late-time data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' and secondly, the reason for the late-time accelerated expansion of the Universe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' late DE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, it is subject to some tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Namely, the two free parameters κ and λ, the intrinsic field-space curvature dictated by α, and the scale of the potential introduced by VΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As we have seen κ and λ seem to take natural values, not too far from order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Regarding α we only need that it is small enough to lead to rapid decrease of the exponential contribution in the scalar potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6), leaving the constant VΛ to dominate at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' We show in the next section that α ∼ 10−4 is sufficient for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This leaves VΛ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The required tuning of this parameter is given by VΛ = � HPlanck 0 HSH0ES 0 �2 V Planck Λ , where – 8 – V Planck Λ is given by the Planck 2018 [1] estimate of ρ0, the density today, multiplied by ΩΛ, the estimate of the density parameter of dark energy today, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' V Planck Λ = ΩΛρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Since � HPlanck 0 HSH0ES 0 �2 ≃ ( 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='44 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='04)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='8525 we see that the required fine-tuning of our VΛ is not different from the fine-tuning introduced in ΛCDM, but, in contrast to ΛCDM, our proposal addresses two cosmological problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' not only late DE but also the Hubble tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='7 3 Numerical Simulation In order to numerically solve the dynamics of the system, it is enough to solve for the scale factor a(t), the field φ(t) and the background fluid densities ρm(t) and ρr(t), as every other quantity depends on these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' They are governed by the Friedmann equations, the Klein- Gordon equation and the continuity equations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Of course, the Klein-Gordon equation is a second order ODE, while the continuity equations are first order so that we need the initial value and velocity of φ and just the initial value of ρm and ρr as initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As described above, the field starts frozen and unfreezes around matter-radiation equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Effectively, this means using φini = 0 and ˙φini = 0 as initial conditions, a few e- folds before matter-radiation equality, while the initial radiation and matter energy densities are chosen to satisfy the bounds obtained by Planck [1] at matter-radiation equality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=', ρm(teq) = ρr(teq) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='27 × 10−110m4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' For convenience, we rewrite the equations in terms of the logarithmic energy densities ˜ρm(t) = ln (ρm(t)/m4 P) and ˜ρr(t) = ln (ρr(t)/m4 P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Plugging the first Friedmann equation in the Klein-Gordon equation, gives ¨φ(t) + � 3ρ(t) mP ˙φ(t) + dV dφ = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) ˙˜ρm(t) + � 3ρ(t) mP = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2) ˙˜ρr(t) + 4 3 � 3ρ(t) mP = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3) where 3m2 PH2(t) = ρ(t) = [ exp(˜ρm(t))+exp(˜ρr(t))]m4 P+ρφ(t) and ρφ(t) = K(φ(t))+V (φ(t)) where K(φ(t)) = 1 2( ˙φ(t))2 and V (φ(t)) is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As mentioned above, we assume the field to be frozen at an ESP, such that it could have been the inflaton or a spectator field at earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The time of unfreezing is then controlled only by the parameters of the model’s potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='8 The densities of matter and radiation are scaled back to find some initial conditions at some arbitrary redshift, zini = 104, before equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The differential solver records three “events” during solving: matter-radiation equality, triggered by the obvious condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' decoupling, triggered by the total energy density taking the correct value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' and the present day, triggered by the field making up the correct fraction of the total energy density (as estimated by the Planck satellite [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' These values are saved to an association so that they can later be searched to identify points which fulfill the necessary 7In our simulations we use VΛ = 10−120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068 m4 P as assumed also in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 8 Although we could use an estimate for the initial time, it turns out that it makes no difference to the numerical results or the behaviour of the field and simply offsets the differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 9 – Initial Densities Calculation Value Matter ρm = 3ΩPlanck m,0 m2 P (HSH0ES 0 )2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='84 × 10−121m4 P Radiation π2 30g∗(T Planck CMB, 0)4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='56 × 10−125m4 P Table 1: Table of present-day densities, where the present matter density parameter is ΩPlanck m,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3111, T Planck CMB, 0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='7255 K and the effective relativistic degrees of freedom of radiation are g∗ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='36, calculated by taking the photon and neutrino contribution into account (see section 5 of [64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Variable Initial Value Source Redshift zinitial = 104 chosen to be shortly before matter-radiation equality Time tini = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1m−1 P chosen to be close to zero (see footnote 8) Field Value φ(tini) = 0 simplified initial conditions Rate of change of Field Value ˙φ(tini) = 0 simplified initial conditions Density of Matter ρm(tini) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='84 × 10−109m4 P ρm(t0)Planck(zini + 1)3 Density of Radiation ρr(tini) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='24 × 10−108m4 P ρr(t0)Planck(zini + 1)4 E-folds elapsed Nini = 0 chosen for convenience Table 2: Table detailing the initial conditions for the differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' constraints, in order to find a viable parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Once the final event is recorded, the solver is terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Event Criteria Justification Matter-Radiation Equality ρm(teq) = ρr(teq) Theoretical Definition Last Scattering ρm(tls) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='98 × 10−112 m4 P Extrapolation from ΛCDM initial conditions (see Table 2) using Planck results ρm(zeq) with zeq = 1089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='80 [1] Present Day Ωφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6889 Planck data [1] Table 3: Table of events recorded during the numerical solving of equations and how.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' If a field point does not meet the conditions for the final event (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' the present day), this indicates that the field began the simulation as the dominant component and will never reach the correct energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The point is thrown away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Finally, reasonable observational and theoretical constraints to the parameter space are applied to the data collected, which are outlined in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 10 – Parameter to be constrained Source Description Constraint Density parameter of the field at equality EDE literature [25] Upper limit governed by the maximum value that does not impede structure forma- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' lower limit is so that EDE actually has an effect 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='015 ≤ Ωeq φ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='107 Density parameter of the field at Last Scattering EDE literature [8] This is the upper limit that ensures EDE cannot cur- rently be detected in the CMB Ωls φ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='015 Density parameters of the field at Last Scatter- ing and Equality Theoretical Achieves desired behaviour of the field Ωeq φ > Ωls φ Density parameter of the field today Planck 2018 [1] Observational constraint 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6833 ≤ Ω0 φ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6945 Barotropic parameter of the field today Planck 2018 Observational constraint −1 ≤ w0 φ ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='95 Running of the barotropic parameter today Planck 2018 [1] Observational constraint −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='55 ≤ wa φ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='03 Hubble constant SH0ES 2021 [2] Observational constraint 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='00≤ H0 km s−1 Mpc−1 ≤74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='08 Total Field Excursion Theoretical From analytical estimates, the total excursion of the field should ideally be sub- Planckian φ0 − φeq < mP Table 4: Table describing and justifying constraints used to identify the viable parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In the above, wa φ = − dwφ da ��� 0, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 4 Results and analysis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 Parameter Space As evident from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 2, 3 and 4, we find that κ ∼ 102 and λ ∼ 10−3, which are rather reasonable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In particular, the value of κ suggests that the mass-scale which suppresses the non-canonical field ϕ in the original potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) is near the scale of grand unification ∼ 10−2 mP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Regarding the curvature of field space we find α ∼ 10−4, which again is not unreasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The viable parameter space suggests that κλ > √ 3, which contradicts our assumption in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This implies that, unlike the analytics in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1, the field does not adopt the subdominant exponential scaling attractor but the slow-roll exponential attractor, which leads to domination [61, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As the field thaws and starts following this attractor, the approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4) breaks down as the field experiences the full exp(exp) potential, which is steeper that exponential (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Consequently, instead of becoming dominant the field free-falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This contradiction with our discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 is not very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 11 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0007 0 100 200 300 400 500 600 700 α κ VΛ mP 4 = 10-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068, 0 < λ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='027 Figure 2: Parameter space slice in the κ − α plane with 0 < λ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='027 and VΛ = 10−120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068m4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The blue dotted line is the boundary of the region that produces non- inflationary results (see below), while the orange region is constituted by the success- ful points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=', those for which the constraints detailed in Table 4 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Note that the region bounded in blue is not equal to the range of the scan, which goes from 0 ≤ κ ≤ 700, 0 ≤ α ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='00071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This is because points with potential larger than a certain starting value result in the field beginning the simulation dominant, which means that the Universe goes into inflation which cannot terminate and will never meet the numerical end condition for the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' These points are very close to the viable parameter space for these two parameters and therefore must be thrown away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The existence of the scaling attractor provided an easy analytic estimate for the moment when the field unfreezes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' It turns out that, because the scaling attractor has been substituted by the slow-roll attractor, the field unfreezes because its potential energy density becomes comparable to the total energy density, going straight into free-fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In is much harder to analytically estimate when exactly this takes place, but the eventual result (free-fall) is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The redshift of matter-radiation equality occurs earlier than usual at zeq ≃ 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' How- ever, equality occurs well before last scattering, zeq > zls and its redshift is only indirectly inferred by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In contrast, the redshift of last scattering is where we would expect it at zls ≃ 1087.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Theoretical constraints suggest zls ≃ 1090 [65], and the observations of the Planck satellite suggest zls = 1089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='21 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 12 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='025 α λ VΛ mP 4 = 10-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068, 0 < κ < 700 Figure 3: Parameter space slice in the λ−α plane with 0 < κ < 700 and VΛ = 10−120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068m4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The orange region is constituted by the successful points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=', those for which the constraints detailed in Table 4 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2 Field Behaviour The field behaves as expected, with the mild modification of the attractor solution at un- freezing (slow-roll instead of scaling), which leads to free-fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The evolution is depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 5, 6, 7 and 8 for the example point at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0005, κ = 145, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='008125, and Vλ tuned to the SH0ES cosmological constant [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The observables obtained in this case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' the values of H0, w0 and wa) are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The behaviour of the Hubble parameter is a function of redshift as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As mentioned in Table 4, the maximum allowed value of the EDE density parameter at equality is just over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, it is possible that this is too lenient a constraint because unlike the models for which this constraint was developed, our model has a true free-fall period, which means it redshifts away exactly as a−6 rather than below this rate as in oscillatory behaviour (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 5 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' A full MCMC analysis may provide a more accurate constraint for non-oscillatory models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' At present, the exponential contribution to the potential density in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6) is largely subdominant to VΛ, so the contribution of the scalar field to the total density budget is almost constant, as in ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Its barotropic parameter is, therefore, wφ ≈ −1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Technically, it is not exactly -1 but its running is negligible, with the viable parameter space for wa fitting easily within the constraint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='8) by some ten orders of magnitude (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 13 – 0 100 200 300 400 500 600 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='025 κ λ VΛ mP 4 = 10-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068, 0 < α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='00071 Figure 4: Parameter space slice in the λ − κ plane with 0 < α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='00071 and VΛ = 10−120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068m4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The orange region is constituted by the successful points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=', those for which the constraints detailed in Table 4 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 5 Initial Conditions Our model accounts for both EDE and late-time dark energy in a non-oscillatory manner (in contrast to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The field is frozen at early times, thawing just before matter- radiation equality when its density grows to nearly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 of the total value (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 6), as set by constraints in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' A steep exp(− exp) potential then forces the field into free-fall, causing its energy density to dilute away as ρφ ∝ a−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' After this, the field hits the asymptote of the exponential decay and refreezes, becoming dominant at present (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Thus, we achieve DE-like behaviour at the present day by ensuring that the field re- freezes after its period of free-fall, therefore remaining at a constant energy density equal to the value of the potential density at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Although this constant potential density is initially negligible, the expansion of the Universe causes the density of matter to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Because the field refreezes at a potential density that is comparable to the density of matter at present, the field starts to become dominant at the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Once it begins to domi- nate the Universe, the field thaws again, but the density of the Universe is dominated by a constant contribution VΛ, as with ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The obvious question is why our scalar field finds itself frozen at the origin in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' One compelling explanation is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' We assume that the origin is an enhanced symmetry point (ESP) such that, at very early times, an interaction of ϕ with some other scalar field χ traps the rolling of ϕ at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The idea follows the scenario explored in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 14 – wϕ wm+r wUniverse 0 2 4 6 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0 3671 1350 496 182 66 24 8 2 N z VΛ mP 4 = 10-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068 α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0005 κ=145 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='008125 Figure 5: Barotropic parameter of the scalar field (dotted green), of the background perfect fluid (full blue) and of the sum of both components (full black), for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0005, κ = 145, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='008125, and VΛ = 10−120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068m4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Density parameter of field Ωϕ 0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='7 3671 1350 496 182 66 24 8 2 N z VΛ mP 4 = 10-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068 α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0005 κ=145 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='008125 Figure 6: The density parameter of the scalar field, for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0005, κ = 145, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='008125, and VΛ = 10−120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068m4 P, as a function of the redshift (top) and e-folds (bottom) elapsed since the beginning of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In this scenario, the scalar potential includes the interaction ∆V = 1 2g2ϕ2χ2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) – 15 – HϕCDM HCDM only HΛCDM 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2 50 100 150 200 250 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='01 N z VΛ mP 4 = 10-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068 α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0005 κ=145 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='008125 Figure 7: The Hubble parameter (in units of km s−1Mpc−1) of a Universe with the modelled scalar field (green), a classical ΛCDM simulation (black), and one with only matter and radiation (blue), as a function of the redshift (top) and the e-folds (bottom) elapsed since the beginning of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' log[ρm/mP 4] log[ρr/mP 4] log[ρϕ/mP 4] log[(ρm+ρr)/mP 4] 0 2 4 6 8 125 120 115 110 105 3671 1350 496 182 66 24 8 2 N z VΛ mP 4 = 10-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='068 α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0005 κ=145 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='008125 Figure 8: The logarithmic densities of matter (dot-dashed red), radiation (dotted orange), the sum of both (solid blue) and the scalar field (dashed green), as a function of the redshift (top) and the e-folds (bottom) elapsed since the beginning of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The horizontal full line represents the (SH0ES) energy density of the Universe at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' where the coupling g < 1 parametrises the strength of the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 16 – Constraint Field Value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='015 ≤ Ωeq φ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='05178 Ωls φ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='001722 Ωeq φ > Ωls φ YES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6833 ≤ Ω0 φ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='6889 −1 ≤ w0 φ ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='55 ≤ wa φ ≡ − dwφ da ��� 0 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='03 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='850 × 10−11 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='00 ≤ H0 km s−1 Mpc−1 ≤ 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='08 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='27 κλ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='178 (φ0 − φeq)/mP < 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4274 Table 5: Table giving the constraints and their corresponding values for an example point, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0005, κ = 145, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='008125, and VΛ tuned to the SH0ES cosmological con- stant, in the viable parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The Hubble constant obtained in this example is H0 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='27 km/s Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' We assume that initially ϕ is rolling down its steep potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='9 Then, the interaction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) provides a modulated effective mass-squared m2 eff = g2ϕ2 to the scalar field χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' When ϕ crosses the origin, this effective mass becomes momentarily zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' If the variation of the ϕ field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' the speed | ˙ϕ| in field space) is large enough, then there is a window around the origin when | ˙meff| ≫ m2 eff (because, | ˙ϕ| ≫ ϕ2 ≃ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This violates adiabaticity and leads to copious production of χ-particles [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='10 As the field moves past the ESP, the produced χ particles become heavy, which takes more energy from the ϕ field, producing an effective potential incline in the direction the ϕ field is moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Indeed, the particle production generates an additional linear potential ∼ g|ϕ|nχ [66], where nχ is the number density of the produced χ-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This number density is constant because the duration of the effect is much smaller than a Hubble time, so that we can ignore dilution from the Universe expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The rolling ϕ field climbs up the linear potential until its kinetic energy density is depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Then the field momentarily stops and afterwards reverses its motion (variation) back to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' When crossing the origin again, there is another bout of χ-particle production, which increases nχ and makes the linear potential steeper to climb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This time, ϕ variation halts at a value closer to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Then, the field reverses its motion and rushes through the origin again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Another outburst of χ-particle production steepens the linear potential further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The process continues until the 9For away from the origin, the scalar potential V (ϕ) does not have to be of the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In fact, it is conceivable that ϕ might play the role of the inflaton field too (see Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 10Near the origin, when ϕ ≃ 0, the ϕ-field is approximately canonically normalised, as suggested by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5), so the considerations of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [66] are readily applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 17 – ϕ-field is trapped at the origin [63, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The trapping of a rolling scalar field at an ESP can take place only if the χ-particles do not decay before trapping occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' If they did, the nχ would decrease and the potential g|ϕ|nχ would not be able to halt the motion (variation) of the ϕ-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The end result of this process is that all the kinetic energy density of the rolling ϕ has been given to the χ-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Now, since ϕ is trapped at the origin, the effective mass of the χ-particles is zero, which means that they are relativistic matter, with density scaling as ρχ ∝ a−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As far as ϕ is concerned, it is trapped at the origin and its density is only ρϕ = V (ϕ = 0) = e−λVX = constant (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' After some time, it may be assumed that the χ-particles do eventually decay into the standard model particles, which comprise the thermal bath of the hot Big Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The con- fining potential, which is proportional to nχ, disappears but, we expect the ϕ-field to remain frozen at the origin because the scalar potential V (ϕ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) is flat enough there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' As we have discussed, the ϕ-field unfreezes again in matter-radiation equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The above scenario is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 9 For simplicity, we have considered that, apart from the obvious violation of adiabacity at the ESP, the χ direction is otherwise approximately flat and the χ-field has a negligible bare mass compared to the ϕ field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' It would be more realistic to consider a non-zero bare mass for the χ-particles, which when they become non-relativistic (much later than the trapping of ϕ) can safely decay to the thermal bath of the hot Big Bang, reheating thereby the Universe, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' in a manner not dissimilar to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The above scenario is one possible explanation of the initial condition considered and not directly relevant to the scope of this work - numerical simulations simply assume that the field begins frozen at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Other possibilities to explain our initial condition exist, for example considering a thermal correction of the form δV ∝ T 2ϕ2, which would make the origin an effective minimum of the potential at high temperatures and drive the ϕ-field there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 6 Conclusions In conclusion, we have studied in detail a non-oscillatory model of unified early and late dark energy, which resolves the Hubble tension and simultaneously explains the observed current accelerated expansion with no more fine tuning than ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Our model considers a single scalar field in the context of α-attractors, as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [30], but in our case the field is not oscillating;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' instead after equality, it free-falls with energy density decreasing as a−6, faster than most early dark energy (EDE) proposals and the fastest possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In our proposed scenario, the scalar field lies originally frozen at the origin, until it thaws near the time of equal matter-radiation densities, when it becomes EDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Afterwards it free-falls until it refreezes at a lower potential energy density value, which provides the vacuum density of ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' We showed that the total excursion of the field in configuration space is sub-Planckian, which implies that our potential is stable under radiative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' One explanation of our initial conditions is that the origin is an enhanced symmetry point (ESP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Our scalar field is originally kinetically dominated until it is trapped at the ESP when crossing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='11 As we discuss in Appendix A, the scalar field could even be the inflaton, which after inflation rolls down its runaway potential until it becomes trapped at the ESP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Our potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) really serves to demonstrate that a model unifying EDE with ΛCDM can be achieved with a suitably steep runaway potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' With the parameters of our model assuming rather natural values, thereby not introducing fine-tuning additional to 11A thermal correction to the scalar potential can have a similar effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 18 – ln ρ ln a ρφ ρr + ρm today equality ESP e−λVX VΛ Figure 9: Schematic log-log plot depicting the evolution of the density of the scalar field ρφ (solid blue line) and the density of radiation and matter ρr + ρm (dashed red line) in the case when the decay of the kinetic energy density of the trapped scalar field generates the thermal bath of the hot Big Bang (as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [REF]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Originally the φ-field is rushing towards the minimum of the potential, dominated by its kinetic density, so that ρφ ∝ a−6 (free-fall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' When it crosses the enhanced symmetry point (ESP) its interaction to the χ- field (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1)) traps the rolling φ-field at the ESP while all its kinetic energy is given to χ-particles, which soon decay into the radiation and matter of the hot Big Bang (the decay is assumed to be quick, just after trapping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Afterwards, the φ-field stays frozen, with energy density V (φ = 0) = e−λVX (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1)) until much later, when its potential density is comparable to the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Then it unfreezes before dominating, acting as early dark energy at the time near matter-radiation equality, and subsequently free-falls to its value φ0, with potential density approximately VΛ = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The field stays there until the present when it dominates the Universe and becomes late dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' that of ΛCDM, we show that this is indeed possible with a simple design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The challenge lies in constructing a concrete theoretical framework for such a potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Acknowledgements: LB is supported by STFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' KD is supported (in part) by the Lancaster- Manchester-Sheffield Consortium for Fundamental Physics under STFC grant: ST/T001038/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' SSL is supported by the FST of Lancaster University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' A Quintessential Inflation Is it possible that our scalar field can not only be early and late dark energy, but also be the inflaton field, responsible for accelerated expansion in the early Universe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 19 – The α-attractors construction leads to two flat regions in the scalar potential of the canonical field, as the kinetic poles of the non-caninical field are displaced to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This idea has been employed in the construction of quintessential inflation models in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' [54–56], where the low-energy plateau was the quintessential tail, responsible for quintessence and the high-energy plateau was responsible for inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, if we inspect the potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) at the poles ϕ = ± √ 6α mP, we find that the potential for the positive pole is V (ϕ+) = VΛ as expected, while for the negative pole we have V (ϕ−) = VΛ exp � 2λ sinh � κ √ 6α �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' For the values of the parameters obtained (κ ∼ 102, λ ∼ 10−3 and α ∼ 10−4) it is easy to check that V (ϕ−) is unsuitable for the inflationary plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Thus, our model needs to be modified to lead to quintessential inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The first modification is a shift in field space such that our new field is ˜ϕ = ϕ + Φ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) where Φ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The α-attractors construction applies now on the new field ˜ϕ for which the Lagrangian density is given by the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4) with the substitution ϕ → ˜ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The poles of our new field lie at ˜ϕ± = ± √ 6˜α mP, where ˜α is the new α-attractors parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' We want all our results to remain unaffected, which means that, for the positive pole, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) suggests ϕ+ = √ 6α mP = ˜ϕ+ − Φ = √ 6˜α mP − Φ ⇒ ˜α = 1 6 � Φ mP + √ 6α �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2) The above, however, is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' It turns out we need to modify the scalar potential as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This modification must be such that near the positive pole the scalar potential reduces to the one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' A simple proposal is V ( ˜ϕ) = VX exp{−2λ sinh[κ( ˜ϕ − Φ)/mP]} , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3) which indeed reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) when κ( ˜ϕ − Φ) = κϕ > mP Note that κ √ 6α > 1 is implied from the requirement that near the positive pole we have κ √ 6α mP = κϕ+ > mP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The ESP discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 5 is now located at ˜ϕ = Φ, such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) is now ∆V = 1 2g2( ˜ϕ − Φ)2χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='12 We are interested in investigating the inflationary plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' This is generated for the canonical field near the negative pole ˜ϕ− = − √ 6˜α mP, where the scalar potential of the canonical field “flattens out” [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Assuming that Φ > √ 6α mP, we have that ˜ϕ− − Φ = −2Φ − √ 6α mP ≃ −2Φ, where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Hence, for the potential energy density of the inflationary plateau we obtain Vinf = V ( ˜ϕ−) ≃ VX exp[−2λ sinh(−2κΦ/mP)] ≃ exp � λ eκ √ 6α� VΛ exp[λ exp(2κΦ/mP)] = exp � λ(eκ √ 6α + e2κΦ/mP) � VΛ ≃ VΛ exp � λ e2κΦ/mP � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4) where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1) and that in −2 sinh(−x) ≃ ex, when x ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 12Near the ESP the potential does not approximate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, we assume that, after unfreezing, the field rolls away fast from the ESP, such that soon the exp(exp) form of the potential becomes valid and the evolution is the one discussed in the main text of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' – 20 – With α-attractors, the inflationary predictions are ns = 1 − 2/N and r = 12˜α/N2 [40], where ns is the spectral index of the scalar curvature perturbation and r is the ratio of the spectrum of the tensor curvature perturbation to the spectrum of the scalar curvature perturbation, with N being the number of inflationary efolds remaining after the cosmo- logical scales exit the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Typically, N = 60 − 65 for quintessential inflation, which means that ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='967 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='969, in excellent agreement with the observations [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' For the tensor-to-scalar ratio the observations provide the bound r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='036 [69], which suggests ˜α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='003 N2 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='8 − 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The COBE constraint requires Vinf ∼ 10−10 m4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Using that VΛ ∼ 10−120 m4 P, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4), suggests that κΦ/mP = 1 2 ln(110 ln 10/λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Hence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' the conditions Φ > √ 6α mP and κ √ 6α > 1 suggest 1 < κ √ 6α < κΦ/mP = 1 2 ln(110 ln 10/λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5) Our findings in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 4 are marginally in agreement with the above requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' For example, taking α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='0006 and κ = 100 we find κ √ 6α = 6 and then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='5) suggests λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='556 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' We also find Φ/mP > √ 6α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='06, which is rather reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2) implies ˜α > 12α = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='2 × 10−3, which comfortably satisfies the observational con- straint on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' In fact, taking N ≃ 60, we find r = 12˜α/N2 > α/25 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='4 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' The above should be taken with a pinch of salt because the approximations employed are rather crude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' However, they seem to suggest that our augmented model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content='3) may lead to successful quintessential inflation while also resolving the Hubble tension, with no more fine-tuning than that of ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' A full numerical investigation is needed to confirm this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' References [1] Planck Collaboration, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=', Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Cosmological parameters, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQf-wYy/content/2301.03572v1.pdf'} +page_content=' 641 (2020) A6, [arXiv:1807.' metadata={'source': 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radar. The radar sys- +tem is based on a transceiver module with about one milli- +Watt output power and more than 30-GHz bandwidth. The +front-end optics consists of an off-axis parabola fed by a horn +antenna from the transceiver unit, resulting in a collimated radar +beam. The digital radar waveform generation allows for coherent +and arbitrary FMCW pulse waveforms. The performance in +terms of sensitivity and resolution (range/cross-range/velocity) is +demonstrated, and the system’s ability to detect and map single +particles (0.1–10 mm diameter), as well as clouds of particles, at +a 5-m distance, is presented. A range resolution of ∼1 cm and +a cross-range resolution of a few centimeters (3-dB beam-width) +allow for the characterization of the dynamics of particle clouds +with a measurement voxel size of a few cubic centimeters. The +monitoring of particle dynamics is of interest in several industrial +applications, such as in the manufacturing of pharmaceuticals +and the control/analysis of fluidized bed combustion reactors. +Index Terms—FMCW, pulse-Doppler, radar, remote sensing, +sensors, submillimeter waves, terahertz systems, transceivers +I. INTRODUCTION +F +OR many industrial applications, such as in the manu- +facturing of pharmaceuticals [1], or energy conversion +using fluidized bed reactors [2], the industrial process involves +particles or powders dispersed in a process reactor. It is neces- +sary to monitor the particle dynamics to maintain the process +quality and to gain insights regarding the process. Therefore, +measuring the particle concentration and the local particle +velocities at a high update rate and high spatial resolution +is desirable. Ideally, these quantities should be measured ex +vivo without inserting any physical probes into the reactors +so that the introduction of measurement sensors does not +alter the processes. In particular, this is required in harsh +process environments [3]. Frequency-modulated continuous- +wave (FMCW) range-Doppler radar operating at center fre- +quencies (fc) within the submillimeter wave range [4] of +the electromagnetic spectrum offers a realistic opportunity to +provide the desired information. +Compared to other contactless measurement methods using +visible or infrared light [5], [6], the submillimeter wavelength +range allows more penetration depth into dense particle clouds +[7] and is less sensitive to contaminations on the reactor +access windows. The radar technique also allows for Doppler +Manuscript received January 1st, 2023. This work was supported in part +by the Swedish Foundation for Strategic Research (SSF) under the contract +ITM17-0265. +Tomas Bryllert, Marlene Bonmann, and Jan Stake are with the Tera- +hertz and Millimetre Wave Laboratory, Chalmers University of Technol- +ogy, SE-412 96 Gothenburg, Sweden. (e-mail: tomas.bryllert@chalmers.se; +marbonm@chalmers.se; jan.stake@chalmers.se) +processing, which reveals information about the velocities +of the particles [8]. Compared with more traditional radar +techniques in the microwave and millimeter wave region [9], +there are a few properties that favor submillimeter waves [10]: +• Short wavelengths (λ) result in higher sensitivity for +detecting smaller particles since the radar cross-section +of particles in the Rayleigh regime scales as λ−4; +• Wide bandwidth and, thereby, a higher range resolution. +For example, a 30-GHz bandwidth results in a theoretical +range resolution of 5 mm; +• The cross-range resolution for a fixed antenna size, +typically limited by the access window size in an ac- +tual application, improves with high frequency since the +diffraction-limited resolution scales with λ. +Several FMCW radars for high-resolution, 3D imaging have +been presented with center frequencies above 300 GHz [11]– +[14]. These systems use ranging to produce 3D static images +and are not using pulse-Doppler processing [15]. FMCW +radars using MMIC-transceivers based on SiGe technology +have been demonstrated in the millimeter wave region [16], +including promising performance up to 480 GHz [17]. Still, +submillimeter-wave transceivers, with a high dynamic range at +room temperature, require diode technology [18], [19]. Cooper +et al. [10] reported a FMCW range-Doppler radar system at +660 GHz, demonstrating the range-Doppler concept’s feasibil- +ity at submillimeter wave frequencies, but with few details. +This work presents the implementation of a FMCW pulse- +Doppler radar based on a 340-GHz transceiver module with +30-GHz bandwidth [20]. A digital waveform generator con- +trols the system. The transceiver module provides an accept- +able trade-off between performance and hardware complexity, +resulting in a relatively compact tripod-mounted radar design, +as shown in Fig. 1. The form factor allows easy implementa- +tion in industrial scenarios. The performance of the transceiver +modules and their application in a 3D imaging radar was pre- +sented in [13]. Here the implementation of the coherent pulse +generation and signal processing to realize range-Doppler +radar operation are explained, together with the resulting radar +system’s noise- and resolution performance. Furthermore, the +ability of the radar to detect single particles with diameters +ranging from 100 µm – 500 µm is demonstrated. The accuracy +of the velocity measurements is validated by comparing the +measured range-Doppler profile of a falling metal sphere with +known weight and diameter to the standard free-fall model. +The results demonstrate that the performance of the radar +system is highly suitable for the suggested industrial scenarios. +arXiv:2301.00558v1 [physics.ins-det] 2 Jan 2023 + +2 +Fig. 1. Photograph of the radar system. The front-end optics and electronics +are mounted on a base plate together with analog and digital baseband +circuitry. +II. METHOD +A. Radar electronics and optics +Fig. 2 shows a schematic block diagram of the 340-GHz +FMCW range-Doppler radar. The system architecture is a +frequency up-converted, frequency multiplied FMCW radar. +A few hardware details deserve to be highlighted: The digital +waveform generator is an FPGA-controlled arbitrary waveform +card with 4 Gb of useful memory and a maximum sampling +rate of >6 Gs/s of which 4 Gs/s is used in the current work. +The card can write >100 ms of 1-GHz bandwidth waveform +data directly from memory. This means that, in a coher- +ent pulse-Doppler processing interval (CPI), typically much +shorter than 100 ms, an arbitrary pulse train of FMCW pulses +can be transmitted – and then repeated. Multiple FMCW +waveforms can therefore be interleaved, addressing different +parts of the system bandwidth (323 – 357 GHz) within a +coherent processing interval. This capability can be used to +extract frequency-resolved (spectroscopic) information from +the scene in an efficient way. This feature is not used in the +presented performance demonstrations. The baseband chirp, +typically 1-GHz bandwidth, generated by the digital hardware, +is centered at 1 GHz. This signal is up-converted to X-band +using frequency mixing and a 9.6-GHz local oscillator (LO) +and is then passed on to the transceiver unit. The transceiver +unit multiplies the X-band chirp by a factor of 32 for a +total final bandwidth of 32 GHz and transmits the signal, now +centered at ∼340 GHz. The radar echoes are received back in +the transceiver and are mixed on the outgoing signal straight +down to the baseband using a balanced configuration [21]. +The front-end 340-GHz Schottky diode circuit is designed +to operate as a frequency multiplier (x2) and sub-harmonic +mixer - thereby simultaneously operating as a transmitter and +receiver. The transceiver’s LO chain consists of an InGaAs +pHEMT active frequency multiplier MMIC (x8) developed by +Gotmic AB and a 170-GHz Schottky diode frequency doubler. +The GaAs Schottky barrier diode circuits were fabricated +in the Nanofabrication Laboratory at Chalmers university of +technology. Originally, the complete transceiver module was +developed for a 16-channel, high frame-rate, imaging radar +[13] by Wasa Millimeter Wave AB, and is described in detail +in [20]. +At the output of the transceiver unit, a circular horn from +Custom Microwave Inc is used as a feed antenna for the optical +system. This feedhorn illuminates a 4” off-axis parabolic +mirror from Edmund Optics with an effective focal length of +6”. The optical system results in a collimated radar beam. +The digital hardware on the receiver side consists of an +eight-channel, 250-Ms/s digitizer from National Instruments +(1 channel is used). The digitizer is controlled by an FPGA +which gives deterministic timing control. The digitizer card +(PXIe format) integrates with a PC controller via a PXIe +bus allowing for real-time signal processing and display. The +waveform card, the analog-to-digital converter (ADC), and +the local oscillator run from a common 10-MHz reference +resulting in a fully coherent system. +B. Radar signal processing +Typical radar parameters used in the experiments presented +in this work are: +• Pulse bandwidth = 32 GHz; +• Pulse time = 41 µs; +• Pulse repetition interval (PRI) = 102.4 µs or 51.2 µs; +• Number of pulses coherently processed (nP RI) = 128; +• Target distance 4 – 6 m. +Fig. 3 shows a block diagram of the signal processing. +The data matrix format that is coherently processed is of +the form: (nr of samples per pulse, ns) × (nP RI). After +down-conversion in the transceiver, the received baseband (IF) +signal is in the frequency range of 21 – 31 MHz, which is +digitized. The data is digitally filtered with a finite impulse +response bandpass filter (FIR BPF), converted to IQ format +with the help of the Hilbert transform, down-converted to +complex baseband, and decimated by a factor of 16 to 1.5 × +Nyquist limited sampling (15.625 Ms/s IQ), with: n′ +s = 640. +In reality, several samples at the beginning and the end of +each waveform are discarded (due to low-frequency ringing), +leaving 590 samples instead of 640. This also reduces the +used bandwidth from 32 GHz to 29.5 GHz. Both the pulse +compression in range and the Doppler processing can be +done using Fourier transforms in FMCW pulse-Doppler radar, +which means that the signal processing can be done with a 2D +fast Fourier transform (FFT) over the coherent data matrix – +with appropriate windowing functions and digital filters. The +output displayed for the radar user is the logarithm of the +squared amplitude of the radar signal in a range-Doppler map. + +45 cm +30 cm3 +Fig. 2. Schematic block diagram of the 340-GHz FMCW pulse-Doppler radar. +Fig. 3. Schematic block diagram of the digital signal processing steps. +C. Radar characterization and evaluation +To demonstrate the performance of the radar system in +terms of the noise floor, range and velocity resolution, and +small particle detection, the following measurements were +conducted: noise floor measurements, range and Doppler reso- +lution, detection of small particles, and velocity measurements +of a free-falling metal sphere. To study the origin of the +noise floor in zero-Doppler and at finite Doppler frequency, +the noise floor was measured without a target under four +different conditions: First, with ADC only; second, with ADC +together with IF amplifiers; third, ADC with IF amplifiers +and a 10.1 GHz continuous wave (CW) signal driving the +transceiver, fourth, ADC with IF amplifier and a chirp signal +driving the transceivers. Additionally, the noise floor as a +function of target strength was measured. Different radar cross +sections (RCSs) were achieved by placing a corner reflection +at different positions in the radar beam. +Increasing the number of pulses per CPI, with other radar +parameters fixed, the S/N for a target should increase linearly +with the number of pulses (integration time) if the target +and the radar system remain coherent and if the noise is +uncorrelated with the radar signal. To verify this, a radar +measurement on a static, corner reflector target was performed +with nP RI= 16, 32, 64, 128, 256, and 512 per CPI. +Three metal beads with a 2-mm diameter were glued onto +a string and positioned at a 5 m distance to demonstrate the +radar system’s range resolution. The target with three beads +on a string was positioned so that all beads were illuminated +by the radar beam and angled so that the beads were separated +in range by approximately 3 cm. Another radar measurement +was performed to display the velocity resolution while gently +tapping the string to make it vibrate. +To investigate the radar system’s ability to detect small +particles, the radar beam is folded with a flat metallic mirror +to be directed vertically upwards. A transparent plastic box +was placed directly above the folding mirror to collect the +particles. This way different test materials could be dropped +straight into the radar beam. This experiment used 2-mm and +10-mm diameter metal beads, 500-µm diameter quartz sand, +and 100-µm spherical glass beads. +The velocity measurement of the radar system was validated +by comparing the measured velocity of a free-falling metal +sphere of known diameter (1.27 cm) and weight (m = 8.44 g) +with an analytical free-fall model. Letting the metal bead +drop towards the radar it moves vertically under gravity and +quadratic air resistance. Solving Newton’s second law of +motion, the velocity (v) and position (x) with time (t) are +then described by +v = vt tanh (t/τ) +(1a) +x = x0 − vtτ ln (cosh (t/τ)) +(1b) +with the terminal velocity vt = +� +(2mg/(Aρaircd)) and the +characteristic time τ = vt/g, where g is the gravity of Earth, m +is the mass of the metal bead, ρair is the air density at normal +temperature pressure, A is the metal beads cross-section, cd +is the drag coefficient (here 0.47 for a sphere [22]), and x0 is +the initial position. + +Ref.10MHz +LO +Mixer +BPF +RFAmplifier +TxRx +9.6 GHz +10.2-11 GHz +RF in +x32 +IF out +326.4-352GHz +Software for frequency +DAC +0.6-1.4 GHz +>6Gs/swaveform +LPF +control +generator +Software for data +IF Amplifier +BPF +processinganddata +ADC +display +250 Ms/s digitizerns +Hilbert Down conversion +Range +Data decimation Windowing +FIR BPF +LPF +n. +processing +transform +n +_npRI +FFT +Coherent +data matrix +Doppler +Windowing +processing +Display +10log1o(I Ampl. 12) +npRI +FFT4 +Fig. 4. Noise floor in the range-Doppler map. (a) General view of the noise +floor with the cuts that are presented in (b-d) indicated. (b) Constant range +cut. (c) Constant velocity cut at zero-Doppler. (d) Constant velocity cut at +finite Doppler. +III. RESULTS +A. Noise performance +Fig. 4 shows the noise floor at different hardware settings +and at different cuts through the range-Doppler map as indi- +cated in Fig 4(a). No target is used in these measurements +which have the purpose of demonstrating the origins of the +noise floor for the radar. +Ideally, the noise floor in the whole range-Doppler map +should be set by thermal noise, deteriorated by the loss and +noise figure of the front-end electronics, and scaled by the IF +amplification. The transceiver unit trades noise performance +for simplicity though. Using the same balanced pair of Schot- +tky diode circuits for the final stage frequency multiplication +and subharmonic homodyne down-conversion to baseband +[20], the transceiver unit can be made quite compact – at +the cost of excess noise. The excess noise comes in two +shapes – through a conversion loss in the subharmonic mixing +that is worse than would be the case in a dedicated mixer +and through excess amplitude modulated noise from the LO +(FMCW chirp) that mix into the IF side of the transceiver +(despite the balanced configuration). In addition, Fig. 4(c) +shows that excess noise is generated in zero-Doppler from +driving the RF hardware with short (40 µs) chirps with high +bandwidth. The cost of the excess noise is acceptable, though, +since S/N is generally sufficient in the application scenarios +that are evaluated. +Fig. 5 shows how the noise floor for zero-Doppler and for +finite Doppler is affected by the strength (RCS) of a static +target. The noise floor is calculated as the mean when aver- +aging over relevant range bins within the IF filter bandwidth +(excluding the range-bin with the target response). The noise +floor in zero-Doppler is not random noise but the result of +sidelobes and amplitude/phase modulation of the waveform, +as well as multiple reflections in the RF hardware. These +effects are not seen at a finite Doppler frequency since the +sidelobe/modulation/reflection pattern is identical from pulse +Fig. 5. The noise floor in zero-Doppler and at a finite Doppler frequency as +a function of target strength (RCS). +Fig. 6. +S/N as a function of the number of pulses used in the coherent +processing. The target was a static corner cube. +to pulse and therefore, only appear in the zero-Doppler bin. At +strong target returns, the noise floor increases at finite Doppler +frequencies, but then as a general increase of the noise floor +in the whole range-Doppler plane - indicating that this noise +increase originates in the actual noise of the RF carrier. +Fig. 6 shows the radar signal of a static target and the +noise floor versus the number of pulses per CPI. The S/N, +when comparing the target signal with the Doppler noise +floor increased linearly as expected. Thus verifying that the +target and the radar system remain coherent and the noise is +uncorrelated with the radar signal. As discussed above, the +noise in zero-Doppler (the static noise floor) originates from +the radar signal, meaning no improvement in S/N in zero- +Doppler is seen at longer integration times. +B. Range resolution, small particle detection, and velocity +measurement +Fig. 7 shows that the three metal beads with 2-mm diameter +are clearly separated in the radar measurement with the signal + +-30 +-40 +(a) +(b) +6 +60 +-40 +Radar signal (dB) +5.5 +(p) +-80 +(w) +-50 +Radar signal ( +5 +-100 +ADCsonly +4.5 +(b) +-60 +-120 +ADCs +IFamps. +4 +ADCs + IF amps. + chirp +-70 +-140 +ADCs + IF amps. + 1GHz cw +3.5 +-80 +-160 +-4 +-2 +0 +2 +4 +-4 +-2 +0 +2 +4 +Velocity (m/s) +Velocity (m/s) +40 +-40 +(d) +60 +-60 +(dB) +(dB) +Radar signal ( +-80 +-80 + signal +100 +100 +ADCs only +ADCs only +Radar +120 +ADCs+IFampS. +120 +-ADCs+IFamps +ADCs + IF amps. + chirp +ADCs + IF amps.+ chirp +-140 +-140 +ADCs +IF amps.+ 1GHz cw +ADCs +IFamps.+1GHz cw +-160 +-160 +3.5 +4 +4.5 +5 +5.5 +6 +3.5 +4 +4.5 +5 +5.5 +6 +Range (m) +Range (m)-30 +Noise level (dB) +40 +Static.noise.floor. +50 +-60 +Doppler...noise...floor +-70 +-40 +-20 +0 +20 +40 +Target signal (dB)0 +-10 +Target +Radar signal (dB) +20 +30 +Static noise floor +40 +50 +Doppler noise floor +-60 +-70 +16 +32 +64 +128 +256 +512 +Number of PRl5 +Fig. 7. Range and velocity resolution. (a-c) Shows a radar measurement of +three beads with 2-mm diameter, demonstrating that the beads are resolved +in range when positioned 3 cm apart in the range direction. The S/N is +approximately 20 dB. (d) The string is vibrating, moving the beads in different +directions and resulting in small Doppler shifts. +peaks measured to be 3 cm and 3.1 cm apart and are visible +with a S/N of approximately 20 dB. When lightly tapping the +string, the beads are also separated in Doppler due to the fine +Doppler resolution of 0.04 m/s per Doppler bin. +Figures 8(a-b) show photographs of the materials used for +testing the radar system’s ability to detect small particles. For +each material, Figures 8(c-f) show the corresponding range- +Doppler maps integrated over several CPI. This way, one +can clearly see the acceleration of the 2-mm diameter metal +bead and the 10-mm diameter metal sphere toward the radar. +Each detection corresponds to a separate CPI, or “frame”, +of the radar with a frame rate of 6.2 frames/s. For 500-µm +diameter sand grains and 100-µm diameter glass spheres, the +integrated particle stream over several pinches of particles is +clearly visible. Fig. 8(e) shows clear detection of single sand +grains. At a 4.3 m distance, the sand grains hit the plastic +box and bounce to a stop. The deflection from the plastic +box appears as positive Doppler velocity. In conclusion, all +tested materials could be detected with significant S/N at a 5-m +distance, proving the radar instrument’s suitability to monitor +particle clouds’ dynamics. +Fig. 8(c) includes the predicted trajectory for the 10-mm +diameter metal sphere from the free-fall model (1), which +indicates that the measurement agrees very well with the +theory, thus supporting the velocity measurement of the radar. +IV. CONCLUSIONS +We +have +presented +a +340-GHz +frequency-modulated +continuous-wave pulse-Doppler radar. The performance of the +radar is described and shown to follow what is expected +from theoretical predictions. The instrument’s sensitivity and +Fig. 8. +Measurement of falling objects at a 5-m distance. (a-b) Show the +photographs of the tested materials. The time-integrated range-Doppler image +of (c) a 10-mm diameter falling metal bead, (d) a 2-mm diameter metal bead, +(e) a few pinches of 500 µm sand grains, and (f) a few pinches of 100 µm +glass spheres.(c) Shows the predicted trajectory from the free-fall model (1). +resolution, both in the spatial domain and in Doppler velocity, +are adequate to map the dynamics of particle clouds. This +is demonstrated by performing radar measurements on free- +falling particles with grain sizes down to 100-µm diameter. +The mapping of particle clouds is relevant in many industrial +applications, such as in the manufacturing of pharmaceuticals +or energy conversion using fluidized bed reactors. Future work +will demonstrate the radar technique in these applications. +ACKNOWLEDGMENT +The authors would like to thank Mats Myremark for ma- +chining mechanical parts for the measurement setup; Vladimir +Drakinskiy for his help with the fabrication of the front- +end terahertz circuits; Divya Jayasankar for valuable feedback +on the manuscript and help with LATEX. The devices were +fabricated and measured in the Nanofabrication Laboratory +and Kollberg Laboratory, respectively, at Chalmers University +of Technology, Gothenburg, Sweden. +REFERENCES +[1] P. Bawuah and J. A. 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Granstr¨om, M. Ferndahl, V. Drakinskiy, and +J. Stake, “Compact 340 ghz homodyne transceiver modules for fmwc +imaging radar arrays,” in 2016 IEEE MTT-S International Microwave +Symposium (IMS), 2016, pp. 1–4, doi: 10.1109/MWSYM.2016.7540113. +[21] T. Bryllert, V. Drakinskiy, K. B. Cooper, and J. Stake, “Integrated +200–240-GHz FMCW radar transceiver module,” IEEE Transactions on +Microwave Theory and Techniques, vol. 61, no. 10, pp. 3808–3815, Oct. +2013, doi: 10.1109/tmtt.2013.2279359. +[22] D. G. Miller and A. B. Bailey, “Sphere drag at mach numbers from 0·3 to +2·0 at reynolds numbers approaching 107,” Journal of Fluid Mechanics, +vol. 93, no. 3, p. 449–464, 1979, doi: 10.1017/S0022112079002597. +[23] T. Reck, C. Jung-Kubiak, J. V. Siles, C. Lee, R. Lin, G. Chattopad- +hyay, I. Mehdi, and K. Cooper, “A silicon micromachined eight-pixel +transceiver array for submillimeter-wave radar,” IEEE Transactions on +Terahertz Science and Technology, vol. 5, no. 2, pp. 197–206, 2015, +doi: 10.1109/TTHZ.2015.2397274. +[24] B. Baumann, B. Gashi, D. Meier, and C. Zech, “High-resolution 400 +ghz submillimeter-wave quasi-optical radar imaging system,” IEEE +Microwave and Wireless Components Letters, vol. 32, no. 3, pp. 226– +229, 2022, doi: 10.1109/LMWC.2022.3142354. +Tomas Bryllert was born in V¨axj¨o, Sweden, in +1974. He received an M.Sc. degree in physics and a +Ph.D. in semiconductor physics from Lund Univer- +sity, Lund, Sweden, in 2000 and 2005, respectively. +In 2006, he joined the Microwave Electronics +Laboratory, Chalmers University of Technology, +G¨oteborg, Sweden. From 2007 to 2009, he was +with the Submillimeter Wave Advanced Technology +(SWAT) group, Jet Propulsion Laboratory, California +Institute of Technology, Pasadena, CA, USA. He is +currently with the Terahertz and Millimetre Wave +Laboratory at Chalmers University of Technology, G¨oteborg, Sweden. He is +also the co-founder and Chief Executive Officer of Wasa Millimeter Wave AB, +a company that develops and fabricates millimeter wave products. Dr. Bryllert +also works part-time in the new concepts team at Saab AB. His research +interests include submillimeter wave electronic circuits and their applications +in imaging and radar systems. + +7 +Marlene Bonmann was born in Karlsruhe, Ger- +many, in 1988. She received an M.Sc. degree in +physics and astronomy and a Ph.D. in Microtechnol- +ogy and Nanoscience from the Chalmers University +of Technology, Gothenburg, Sweden, in 2014 and +2020, respectively. +She is currently with the Terahertz and Millimetre +Wave Laboratory at the Chalmers University of +Technology. +Jan Stake (S’95–M’00–SM’06) was born in Ud- +devalla, Sweden, in 1971. He received an M.Sc. +degree in electrical engineering and a Ph.D. in +microwave electronics from the Chalmers University +of Technology, Gothenburg, Sweden, in 1994 and +1999, respectively. +In 1997, he was a Research Assistant at the +University of Virginia, Charlottesville, VA, USA. +From 1999 to 2001, he was a Research Fellow +with the Millimetre Wave Group at the Rutherford +Appleton Laboratory, Didcot, UK. He then joined +Saab Combitech Systems AB, Link¨oping, Sweden, as a Senior RF/microwave +Engineer, until 2003. From 2000 to 2006, he held different academic positions +with the Chalmers University of Technology and from 2003 to 2006, he was +also the Head of the Nanofabrication Laboratory, Department of Microtech- +nology and Nanoscience (MC2). In 2007, he was a Visiting Professor with the +Sub-millimetre Wave Advanced Technology (SWAT) Group at Caltech/JPL, +Pasadena, CA, USA. In 2020, he was a Visiting Professor at TU Delft. +He is currently a Professor and the Head of the Terahertz and Millimetre +Wave Laboratory at the Chalmers University of Technology. He is also +the co-founder of Wasa Millimeter Wave AB, Gothenburg, Sweden. His +research interests include graphene electronics, high-frequency semiconductor +devices, terahertz electronics, submillimeter wave measurement techniques, +and terahertz systems. +Prof. Stake served as the Editor-in-Chief for the IEEE Transactions on +Terahertz Science and Technology between 2016 and 2018 and as Topical +Editor between 2012 and 2015. + +HAGLOFS \ No newline at end of file diff --git a/FtAyT4oBgHgl3EQfrPm5/content/tmp_files/load_file.txt b/FtAyT4oBgHgl3EQfrPm5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0fb5b8512c8f6c87e624f53a522709a2b38f329 --- /dev/null +++ b/FtAyT4oBgHgl3EQfrPm5/content/tmp_files/load_file.txt @@ -0,0 +1,648 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf,len=647 +page_content='1 A Submillimeter-Wave FMCW Pulse-Doppler Radar to Characterize the Dynamics of Particle Clouds Tomas Bryllert, Member, IEEE, Marlene Bonmann, and Jan Stake, Senior Member, IEEE Abstract—This work presents a 340-GHz frequency-modulated continuous-wave (FMCW) pulse-Doppler radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The radar sys- tem is based on a transceiver module with about one milli- Watt output power and more than 30-GHz bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The front-end optics consists of an off-axis parabola fed by a horn antenna from the transceiver unit, resulting in a collimated radar beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The digital radar waveform generation allows for coherent and arbitrary FMCW pulse waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The performance in terms of sensitivity and resolution (range/cross-range/velocity) is demonstrated, and the system’s ability to detect and map single particles (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='1–10 mm diameter), as well as clouds of particles, at a 5-m distance, is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' A range resolution of ∼1 cm and a cross-range resolution of a few centimeters (3-dB beam-width) allow for the characterization of the dynamics of particle clouds with a measurement voxel size of a few cubic centimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The monitoring of particle dynamics is of interest in several industrial applications, such as in the manufacturing of pharmaceuticals and the control/analysis of fluidized bed combustion reactors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Index Terms—FMCW, pulse-Doppler, radar, remote sensing, sensors, submillimeter waves, terahertz systems, transceivers I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' INTRODUCTION F OR many industrial applications, such as in the manu- facturing of pharmaceuticals [1], or energy conversion using fluidized bed reactors [2], the industrial process involves particles or powders dispersed in a process reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' It is neces- sary to monitor the particle dynamics to maintain the process quality and to gain insights regarding the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Therefore, measuring the particle concentration and the local particle velocities at a high update rate and high spatial resolution is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Ideally, these quantities should be measured ex vivo without inserting any physical probes into the reactors so that the introduction of measurement sensors does not alter the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' In particular, this is required in harsh process environments [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Frequency-modulated continuous- wave (FMCW) range-Doppler radar operating at center fre- quencies (fc) within the submillimeter wave range [4] of the electromagnetic spectrum offers a realistic opportunity to provide the desired information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Compared to other contactless measurement methods using visible or infrared light [5], [6], the submillimeter wavelength range allows more penetration depth into dense particle clouds [7] and is less sensitive to contaminations on the reactor access windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The radar technique also allows for Doppler Manuscript received January 1st, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This work was supported in part by the Swedish Foundation for Strategic Research (SSF) under the contract ITM17-0265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Tomas Bryllert, Marlene Bonmann, and Jan Stake are with the Tera- hertz and Millimetre Wave Laboratory, Chalmers University of Technol- ogy, SE-412 96 Gothenburg, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' (e-mail: tomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='bryllert@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='se;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' marbonm@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='se;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='stake@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='se) processing, which reveals information about the velocities of the particles [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Compared with more traditional radar techniques in the microwave and millimeter wave region [9], there are a few properties that favor submillimeter waves [10]: Short wavelengths (λ) result in higher sensitivity for detecting smaller particles since the radar cross-section of particles in the Rayleigh regime scales as λ−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Wide bandwidth and, thereby, a higher range resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' For example, a 30-GHz bandwidth results in a theoretical range resolution of 5 mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The cross-range resolution for a fixed antenna size, typically limited by the access window size in an ac- tual application, improves with high frequency since the diffraction-limited resolution scales with λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Several FMCW radars for high-resolution, 3D imaging have been presented with center frequencies above 300 GHz [11]– [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' These systems use ranging to produce 3D static images and are not using pulse-Doppler processing [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' FMCW radars using MMIC-transceivers based on SiGe technology have been demonstrated in the millimeter wave region [16], including promising performance up to 480 GHz [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Still, submillimeter-wave transceivers, with a high dynamic range at room temperature, require diode technology [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' [10] reported a FMCW range-Doppler radar system at 660 GHz, demonstrating the range-Doppler concept’s feasibil- ity at submillimeter wave frequencies, but with few details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This work presents the implementation of a FMCW pulse- Doppler radar based on a 340-GHz transceiver module with 30-GHz bandwidth [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' A digital waveform generator con- trols the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The transceiver module provides an accept- able trade-off between performance and hardware complexity, resulting in a relatively compact tripod-mounted radar design, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The form factor allows easy implementa- tion in industrial scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The performance of the transceiver modules and their application in a 3D imaging radar was pre- sented in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Here the implementation of the coherent pulse generation and signal processing to realize range-Doppler radar operation are explained, together with the resulting radar system’s noise- and resolution performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Furthermore, the ability of the radar to detect single particles with diameters ranging from 100 µm – 500 µm is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The accuracy of the velocity measurements is validated by comparing the measured range-Doppler profile of a falling metal sphere with known weight and diameter to the standard free-fall model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The results demonstrate that the performance of the radar system is highly suitable for the suggested industrial scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='00558v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='ins-det] 2 Jan 2023 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Photograph of the radar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The front-end optics and electronics are mounted on a base plate together with analog and digital baseband circuitry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Radar electronics and optics Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 2 shows a schematic block diagram of the 340-GHz FMCW range-Doppler radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The system architecture is a frequency up-converted, frequency multiplied FMCW radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' A few hardware details deserve to be highlighted: The digital waveform generator is an FPGA-controlled arbitrary waveform card with 4 Gb of useful memory and a maximum sampling rate of >6 Gs/s of which 4 Gs/s is used in the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The card can write >100 ms of 1-GHz bandwidth waveform data directly from memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This means that, in a coher- ent pulse-Doppler processing interval (CPI), typically much shorter than 100 ms, an arbitrary pulse train of FMCW pulses can be transmitted – and then repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Multiple FMCW waveforms can therefore be interleaved, addressing different parts of the system bandwidth (323 – 357 GHz) within a coherent processing interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This capability can be used to extract frequency-resolved (spectroscopic) information from the scene in an efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This feature is not used in the presented performance demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The baseband chirp, typically 1-GHz bandwidth, generated by the digital hardware, is centered at 1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This signal is up-converted to X-band using frequency mixing and a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='6-GHz local oscillator (LO) and is then passed on to the transceiver unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The transceiver unit multiplies the X-band chirp by a factor of 32 for a total final bandwidth of 32 GHz and transmits the signal, now centered at ∼340 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The radar echoes are received back in the transceiver and are mixed on the outgoing signal straight down to the baseband using a balanced configuration [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The front-end 340-GHz Schottky diode circuit is designed to operate as a frequency multiplier (x2) and sub-harmonic mixer - thereby simultaneously operating as a transmitter and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The transceiver’s LO chain consists of an InGaAs pHEMT active frequency multiplier MMIC (x8) developed by Gotmic AB and a 170-GHz Schottky diode frequency doubler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The GaAs Schottky barrier diode circuits were fabricated in the Nanofabrication Laboratory at Chalmers university of technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Originally, the complete transceiver module was developed for a 16-channel, high frame-rate, imaging radar [13] by Wasa Millimeter Wave AB, and is described in detail in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' At the output of the transceiver unit, a circular horn from Custom Microwave Inc is used as a feed antenna for the optical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This feedhorn illuminates a 4” off-axis parabolic mirror from Edmund Optics with an effective focal length of 6”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The optical system results in a collimated radar beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The digital hardware on the receiver side consists of an eight-channel, 250-Ms/s digitizer from National Instruments (1 channel is used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The digitizer is controlled by an FPGA which gives deterministic timing control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The digitizer card (PXIe format) integrates with a PC controller via a PXIe bus allowing for real-time signal processing and display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The waveform card, the analog-to-digital converter (ADC), and the local oscillator run from a common 10-MHz reference resulting in a fully coherent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Radar signal processing Typical radar parameters used in the experiments presented in this work are: Pulse bandwidth = 32 GHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Pulse time = 41 µs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Pulse repetition interval (PRI) = 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4 µs or 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 µs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Number of pulses coherently processed (nP RI) = 128;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Target distance 4 – 6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 3 shows a block diagram of the signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The data matrix format that is coherently processed is of the form: (nr of samples per pulse, ns) × (nP RI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' After down-conversion in the transceiver, the received baseband (IF) signal is in the frequency range of 21 – 31 MHz, which is digitized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The data is digitally filtered with a finite impulse response bandpass filter (FIR BPF), converted to IQ format with the help of the Hilbert transform, down-converted to complex baseband, and decimated by a factor of 16 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 × Nyquist limited sampling (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='625 Ms/s IQ), with: n′ s = 640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' In reality, several samples at the beginning and the end of each waveform are discarded (due to low-frequency ringing), leaving 590 samples instead of 640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This also reduces the used bandwidth from 32 GHz to 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Both the pulse compression in range and the Doppler processing can be done using Fourier transforms in FMCW pulse-Doppler radar, which means that the signal processing can be done with a 2D fast Fourier transform (FFT) over the coherent data matrix – with appropriate windowing functions and digital filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The output displayed for the radar user is the logarithm of the squared amplitude of the radar signal in a range-Doppler map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 45 cm 30 cm3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Schematic block diagram of the 340-GHz FMCW pulse-Doppler radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Schematic block diagram of the digital signal processing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Radar characterization and evaluation To demonstrate the performance of the radar system in terms of the noise floor, range and velocity resolution, and small particle detection, the following measurements were conducted: noise floor measurements, range and Doppler reso- lution, detection of small particles, and velocity measurements of a free-falling metal sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' To study the origin of the noise floor in zero-Doppler and at finite Doppler frequency, the noise floor was measured without a target under four different conditions: First, with ADC only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' second, with ADC together with IF amplifiers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' third, ADC with IF amplifiers and a 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='1 GHz continuous wave (CW) signal driving the transceiver, fourth, ADC with IF amplifier and a chirp signal driving the transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Additionally, the noise floor as a function of target strength was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Different radar cross sections (RCSs) were achieved by placing a corner reflection at different positions in the radar beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Increasing the number of pulses per CPI, with other radar parameters fixed, the S/N for a target should increase linearly with the number of pulses (integration time) if the target and the radar system remain coherent and if the noise is uncorrelated with the radar signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' To verify this, a radar measurement on a static, corner reflector target was performed with nP RI= 16, 32, 64, 128, 256, and 512 per CPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Three metal beads with a 2-mm diameter were glued onto a string and positioned at a 5 m distance to demonstrate the radar system’s range resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The target with three beads on a string was positioned so that all beads were illuminated by the radar beam and angled so that the beads were separated in range by approximately 3 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Another radar measurement was performed to display the velocity resolution while gently tapping the string to make it vibrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' To investigate the radar system’s ability to detect small particles, the radar beam is folded with a flat metallic mirror to be directed vertically upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' A transparent plastic box was placed directly above the folding mirror to collect the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This way different test materials could be dropped straight into the radar beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This experiment used 2-mm and 10-mm diameter metal beads, 500-µm diameter quartz sand, and 100-µm spherical glass beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The velocity measurement of the radar system was validated by comparing the measured velocity of a free-falling metal sphere of known diameter (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='27 cm) and weight (m = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='44 g) with an analytical free-fall model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Letting the metal bead drop towards the radar it moves vertically under gravity and quadratic air resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Solving Newton’s second law of motion, the velocity (v) and position (x) with time (t) are then described by v = vt tanh (t/τ) (1a) x = x0 − vtτ ln (cosh (t/τ)) (1b) with the terminal velocity vt = � (2mg/(Aρaircd)) and the characteristic time τ = vt/g, where g is the gravity of Earth, m is the mass of the metal bead, ρair is the air density at normal temperature pressure, A is the metal beads cross-section, cd is the drag coefficient (here 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='47 for a sphere [22]), and x0 is the initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='10MHz LO Mixer BPF RFAmplifier TxRx 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='6 GHz 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2-11 GHz RF in x32 IF out 326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4-352GHz Software for frequency DAC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='6-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4 GHz >6Gs/swaveform LPF control generator Software for data IF Amplifier BPF processinganddata ADC display 250 Ms/s digitizerns Hilbert Down conversion Range Data decimation Windowing FIR BPF LPF n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' processing transform n _npRI FFT Coherent data matrix Doppler Windowing processing Display 10log1o(I Ampl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 12) npRI FFT4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Noise floor in the range-Doppler map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' (a) General view of the noise floor with the cuts that are presented in (b-d) indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' (b) Constant range cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' (c) Constant velocity cut at zero-Doppler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' (d) Constant velocity cut at finite Doppler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Noise performance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 4 shows the noise floor at different hardware settings and at different cuts through the range-Doppler map as indi- cated in Fig 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' No target is used in these measurements which have the purpose of demonstrating the origins of the noise floor for the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Ideally, the noise floor in the whole range-Doppler map should be set by thermal noise, deteriorated by the loss and noise figure of the front-end electronics, and scaled by the IF amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The transceiver unit trades noise performance for simplicity though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Using the same balanced pair of Schot- tky diode circuits for the final stage frequency multiplication and subharmonic homodyne down-conversion to baseband [20], the transceiver unit can be made quite compact – at the cost of excess noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The excess noise comes in two shapes – through a conversion loss in the subharmonic mixing that is worse than would be the case in a dedicated mixer and through excess amplitude modulated noise from the LO (FMCW chirp) that mix into the IF side of the transceiver (despite the balanced configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' In addition, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 4(c) shows that excess noise is generated in zero-Doppler from driving the RF hardware with short (40 µs) chirps with high bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The cost of the excess noise is acceptable, though, since S/N is generally sufficient in the application scenarios that are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 5 shows how the noise floor for zero-Doppler and for finite Doppler is affected by the strength (RCS) of a static target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The noise floor is calculated as the mean when aver- aging over relevant range bins within the IF filter bandwidth (excluding the range-bin with the target response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The noise floor in zero-Doppler is not random noise but the result of sidelobes and amplitude/phase modulation of the waveform, as well as multiple reflections in the RF hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' These effects are not seen at a finite Doppler frequency since the sidelobe/modulation/reflection pattern is identical from pulse Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The noise floor in zero-Doppler and at a finite Doppler frequency as a function of target strength (RCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' S/N as a function of the number of pulses used in the coherent processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The target was a static corner cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' to pulse and therefore, only appear in the zero-Doppler bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' At strong target returns, the noise floor increases at finite Doppler frequencies, but then as a general increase of the noise floor in the whole range-Doppler plane - indicating that this noise increase originates in the actual noise of the RF carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 6 shows the radar signal of a static target and the noise floor versus the number of pulses per CPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The S/N, when comparing the target signal with the Doppler noise floor increased linearly as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Thus verifying that the target and the radar system remain coherent and the noise is uncorrelated with the radar signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' As discussed above, the noise in zero-Doppler (the static noise floor) originates from the radar signal, meaning no improvement in S/N in zero- Doppler is seen at longer integration times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Range resolution, small particle detection, and velocity measurement Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 7 shows that the three metal beads with 2-mm diameter are clearly separated in the radar measurement with the signal 30 40 (a) (b) 6 60 40 Radar signal (dB) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 (p) 80 (w) 50 Radar signal ( 5 100 ADCsonly 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 (b) 60 120 ADCs +IFamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 4 ADCs + IF amps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' + chirp 70 140 ADCs + IF amps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' + 1GHz cw 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 80 160 4 2 0 2 4 4 2 0 2 4 Velocity (m/s) Velocity (m/s) 40 40 (d) 60 60 (dB) (dB) Radar signal ( 80 80 signal 100 100 ADCs only ADCs only Radar 120 ADCs+IFampS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 120 ADCs+IFamps ADCs + IF amps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' + chirp ADCs + IF amps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='+ chirp 140 140 ADCs +IF amps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='+ 1GHz cw ADCs +IFamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='+1GHz cw 160 160 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 6 Range (m) Range (m)-30 Noise level (dB) 40 Static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 50 60 Doppler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='floor 70 40 20 0 20 40 Target signal (dB)0 10 Target Radar signal (dB) 20 30 Static noise floor 40 50 Doppler noise floor 60 70 16 32 64 128 256 512 Number of PRl5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Range and velocity resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' (a-c) Shows a radar measurement of three beads with 2-mm diameter, demonstrating that the beads are resolved in range when positioned 3 cm apart in the range direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The S/N is approximately 20 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' (d) The string is vibrating, moving the beads in different directions and resulting in small Doppler shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' peaks measured to be 3 cm and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='1 cm apart and are visible with a S/N of approximately 20 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' When lightly tapping the string, the beads are also separated in Doppler due to the fine Doppler resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='04 m/s per Doppler bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Figures 8(a-b) show photographs of the materials used for testing the radar system’s ability to detect small particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' For each material, Figures 8(c-f) show the corresponding range- Doppler maps integrated over several CPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This way, one can clearly see the acceleration of the 2-mm diameter metal bead and the 10-mm diameter metal sphere toward the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Each detection corresponds to a separate CPI, or “frame”, of the radar with a frame rate of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 frames/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' For 500-µm diameter sand grains and 100-µm diameter glass spheres, the integrated particle stream over several pinches of particles is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 8(e) shows clear detection of single sand grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' At a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='3 m distance, the sand grains hit the plastic box and bounce to a stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The deflection from the plastic box appears as positive Doppler velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' In conclusion, all tested materials could be detected with significant S/N at a 5-m distance, proving the radar instrument’s suitability to monitor particle clouds’ dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 8(c) includes the predicted trajectory for the 10-mm diameter metal sphere from the free-fall model (1), which indicates that the measurement agrees very well with the theory, thus supporting the velocity measurement of the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' CONCLUSIONS We have presented a 340-GHz frequency-modulated continuous-wave pulse-Doppler radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The performance of the radar is described and shown to follow what is expected from theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The instrument’s sensitivity and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Measurement of falling objects at a 5-m distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' (a-b) Show the photographs of the tested materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The time-integrated range-Doppler image of (c) a 10-mm diameter falling metal bead, (d) a 2-mm diameter metal bead, (e) a few pinches of 500 µm sand grains, and (f) a few pinches of 100 µm glass spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' (c) Shows the predicted trajectory from the free-fall model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' resolution, both in the spatial domain and in Doppler velocity, are adequate to map the dynamics of particle clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' This is demonstrated by performing radar measurements on free- falling particles with grain sizes down to 100-µm diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The mapping of particle clouds is relevant in many industrial applications, such as in the manufacturing of pharmaceuticals or energy conversion using fluidized bed reactors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Future work will demonstrate the radar technique in these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' ACKNOWLEDGMENT The authors would like to thank Mats Myremark for ma- chining mechanical parts for the measurement setup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Vladimir Drakinskiy for his help with the fabrication of the front- end terahertz circuits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Divya Jayasankar for valuable feedback on the manuscript and help with LATEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' The devices were fabricated and measured in the Nanofabrication Laboratory and Kollberg Laboratory, respectively, at Chalmers University of Technology, Gothenburg, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' REFERENCES [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Bawuah and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Zeitler, “Advances in terahertz time-domain spec- troscopy of pharmaceutical solids: A review,” TrAC Trends in Analytical Chemistry, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 139, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 116272, 2021, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='trac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='116272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4 30 (a) (b) 40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='35 40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 Range (m) beads (m) 50 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='3 60 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='25 70 70 80 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 0 80 2 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 Velocity (m/s) Velocity(m/s) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4 30 0 (c) d 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='35 40 40 (w) 50 Range 50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='3 60 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='25 70 80 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4 Range (m) Velocity (m/s)(a) (b) 10 mm ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5mm ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='1mm 2mm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='8 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='8 30 (c) (d) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4 40 40 Radar signal (dB) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4 Radar signal (dB) Range (m) Range (m) 50 50 5 5 free f ta 60 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='6 70 70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 80 4 2 0 2 4 2 0 2 Velocity(m/s) Velocity(m/s) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='8 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='8 30 (e) (f) 40 40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4 Radar signal (dB) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='4 Radar signal (dB) Range (m) ngle grains Range (m) 50 50 5 lastic box 5 60 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='6 70 70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 80 4 2 0 2 4 2 0 2 Velocity(m/s) Velocity (m/s)6 TABLE I COMPARISON OF SUBMILLIMETER-WAVE RADARS Center frequency Bandwidth Output power Comment Technology Reference (GHz) (GHz) (mW) 350 19 4 FMCW Schottky diode [11] 675 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='5 FMCW pulse-Doppler Schottky diode [10] 340 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='6 FMCW Schottky diode [23] 332 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2 FMCW, MIMO Schottky diode [12] 340 30 1 FMCW Schottky diode [13] 383 80 8 FMCW mHEMT [24] 480 55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='06 FMCW SiGe [17] 340 30 1 FMCW pulse-Doppler Schottky diode This work [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Koornneef, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Junginger, and A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Gashi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Meier, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Zech, “High-resolution 400 ghz submillimeter-wave quasi-optical radar imaging system,” IEEE Microwave and Wireless Components Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 226– 229, 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='1109/LMWC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='3142354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Tomas Bryllert was born in V¨axj¨o, Sweden, in 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' He received an M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' degree in physics and a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' in semiconductor physics from Lund Univer- sity, Lund, Sweden, in 2000 and 2005, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' In 2006, he joined the Microwave Electronics Laboratory, Chalmers University of Technology, G¨oteborg, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' From 2007 to 2009, he was with the Submillimeter Wave Advanced Technology (SWAT) group, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' He is currently with the Terahertz and Millimetre Wave Laboratory at Chalmers University of Technology, G¨oteborg, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' He is also the co-founder and Chief Executive Officer of Wasa Millimeter Wave AB, a company that develops and fabricates millimeter wave products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Bryllert also works part-time in the new concepts team at Saab AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' His research interests include submillimeter wave electronic circuits and their applications in imaging and radar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' 7 Marlene Bonmann was born in Karlsruhe, Ger- many, in 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' She received an M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' degree in physics and astronomy and a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' in Microtechnol- ogy and Nanoscience from the Chalmers University of Technology, Gothenburg, Sweden, in 2014 and 2020, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' She is currently with the Terahertz and Millimetre Wave Laboratory at the Chalmers University of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Jan Stake (S’95–M’00–SM’06) was born in Ud- devalla, Sweden, in 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' He received an M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' degree in electrical engineering and a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' in microwave electronics from the Chalmers University of Technology, Gothenburg, Sweden, in 1994 and 1999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' In 1997, he was a Research Assistant at the University of Virginia, Charlottesville, VA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' From 1999 to 2001, he was a Research Fellow with the Millimetre Wave Group at the Rutherford Appleton Laboratory, Didcot, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' He then joined Saab Combitech Systems AB, Link¨oping, Sweden, as a Senior RF/microwave Engineer, until 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' From 2000 to 2006, he held different academic positions with the Chalmers University of Technology and from 2003 to 2006, he was also the Head of the Nanofabrication Laboratory, Department of Microtech- nology and Nanoscience (MC2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' In 2007, he was a Visiting Professor with the Sub-millimetre Wave Advanced Technology (SWAT) Group at Caltech/JPL, Pasadena, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' In 2020, he was a Visiting Professor at TU Delft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' He is currently a Professor and the Head of the Terahertz and Millimetre Wave Laboratory at the Chalmers University of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' He is also the co-founder of Wasa Millimeter Wave AB, Gothenburg, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' His research interests include graphene electronics, high-frequency semiconductor devices, terahertz electronics, submillimeter wave measurement techniques, and terahertz systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' Stake served as the Editor-in-Chief for the IEEE Transactions on Terahertz Science and Technology between 2016 and 2018 and as Topical Editor between 2012 and 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} +page_content=' HAGLOFS' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAyT4oBgHgl3EQfrPm5/content/2301.00558v1.pdf'} diff --git a/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf b/FtE2T4oBgHgl3EQfTAfF/content/2301.03799v1.pdf new file mode 100644 index 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As long as +the multipartite system is concerned, the relation between the entanglement contained in different +partitions or different subsystems need to take into account. The complete multipartite entanglement +measure and the complete monogamy relation is a framework that just deals with such a issue. In this +paper, we put forward conditions to justify whether the multipartite entanglement monotone (MEM) +and genuine multipartite entanglement monotone (GMEM) are complete, completely monogamous, +and tightly complete monogamous according to the feature of the reduced function. Especially, +we proposed a class of complete MEMs and a class of complete GMEMs via the maximal reduced +function for the first time. +By comparison, it is shown that, for the tripartite case, this class +of GMEMs is better than the one defined from the minimal bipartite entanglement in literature +under the framework of complete MEM and complete monogamy relation. In addition, the relation +between monogamy, complete monogamy, and the tightly complete monogamy are revealed in light +of different kinds of MEMs and GMEMs. +PACS numbers: 03.67.Mn, 03.65.Db, 03.65.Ud. +I. +INTRODUCTION +Entanglement, as one of the most puzzling features in +quantum mechanics, has been widely used as an essen- +tial resource for quantum communication [1–3], quantum +cryptography [4, 5], and quantum computing [6, 7], etc. +The utility of an entangled state for these applications +is often directly related to the degree or type of entan- +glement contained in it. +Therefore, efficiently quanti- +fying and characterizing multipartite entanglement is of +paramount importance. +Especially, the genuine multi- +partite entanglement, as one of the important types of +entanglement, offers significant advantages in quantum +tasks compared with bipartite entanglement [8]. +The phenomenon becomes much more complex for +multipartite entanglement, particularly the genuinely +multipartite entanglement, entanglement shared between +all of the particles. +Over the years, many multipar- +tite entanglement measures have been proposed, such as +the “residual tangle” which reports the genuine three- +qubit entanglement [9], the genuinely multipartite con- +currence [10], the k-ME concurrence [11], the m con- +currence [12], the generalization of negativity [13], the +SL-invariant multipartite measure of entanglement [14– +19], and the α-entanglement entropy [20], concurrence +triangle [21], concentratable entanglement [22], geomet- +ric mean of bipartite concurrence [23], concurrence tri- +angle induced genuine multipartite entanglement mea- +sure [24], and a general way of constructing genuine mul- +tipartite entanglement monotone is proposed in Ref. [25]. +In Ref. [26], we proposed a framework of complete mul- +tipartite entanglement monotone from which the entan- +glement between any partitions or subsystems with the +coarsening relation could be compared with each other. +∗ guoyu3@aliyun.com +In the context of describing multipartite entanglement, +another fundamental task is to understand how entan- +glement is distributed over many parties since it reveals +fundamental insights into the nature of quantum correla- +tions [8] and has profound applications in both quantum +communication [27, 28] and other area of physics [29– +33]. +This characteristic trait of distribution is known +as the monogamy law of entanglement [27, 34], which +means that the more entangled two parties are, the +less correlated they can be with other parties. +Quan- +titatively, the monogamy of entanglement is described +by an inequality [9, 29, 34–36] or equality [26, 37, 38], +involving a bipartite entanglement monotone or multi- +partite entanglement monotone (MEM). Consequently, +considerable research has been undertaken in this direc- +tion [9, 26, 29, 34–39]. +Very recently, we discussed when the genuine multi- +partite entanglement measure is complete [40] with the +same spirit as in Ref. [26]. Under such a sense, the hierar- +chy structure of the entanglement in the system is clear. +Moreover, whether the multipartite entanglement mea- +sure is proper or not can be justified together with the +framework of complete monogamy relation for the mul- +tipartite system established in Ref. [26]. The framework +of complete monogamy relation is based on the complete +multipartite entanglement measure [26, 40, 41]. +With +this postulates, the distribution of entanglement appears +more explicitly. +Multipartite entanglement measure is always defined +via the bipartite entanglement measure. Let SX be the +set of all density matrices acting on the state space HX. +Recall that, a function E : SAB → R+ is called a mea- +sure of entanglement [42, 43] if (1) E(σAB) = 0 for any +separable density matrix σAB ∈ SAB, and (2) E be- +haves monotonically decreasing under local operations +and classical communication (LOCC). Moreover, convex +measures of entanglement that do not increase on average + +2 +under LOCC are called entanglement monotones [42, 44]. +By replacing SAB with SA1A2···An, it is just the mul- +tipartite entanglement measure/monotone, and denoted +by E(n). +Any bipartite entanglement monotone corre- +sponds to a concave function on the reduced state when +it is evaluated for the pure states [44]. For any entangle- +ment measure E, if +h +� +ρA� += E +� +|ψ⟩⟨ψ|AB� +(1) +is concave, i.e. +h[λρ1 + (1 − λ)ρ2] ≥ λh(ρ1) + (1 − +λ)h(ρ2) for any states ρ1, ρ2, and any 0 ≤ λ ≤ 1, +then the convex roof extension of E, i.e., EF +� +ρAB� +≡ +min �n +j=1 pjE +� +|ψj⟩⟨ψj|AB� +, is an entanglement mono- +tone, where the minimum is taken over all pure state +decompositions of ρAB = �n +j=1 pj|ψj⟩⟨ψj|AB. We call h +the reduced function of E and HA the reduced subsystem +throughout this paper. +An n-partite pure state |ψ⟩ ∈ HA1A2···An is called +biseparable if it can be written as |ψ⟩ += +|ψ⟩X ⊗ +|ψ⟩Y +for some bipartition of A1A2 · · · An (for exam- +ple, A1A3|A2A4 is a bipartition of A1A2A3A4). An n- +partite mixed state ρ is biseparable if it can be writ- +ten as a convex combination of biseparable pure states +ρ = � +i pi|ψi⟩⟨ψi|, wherein the contained {|ψi⟩} can be +biseparable with respect to different bipartitions (i.e., a +mixed biseparable state does not need to be separable +with respect to any particular bipartition). If ρ is not +biseparable, then it is called genuinely entangled. A mul- +tipartite entanglement measure E(n) is called a genuine +multipartite entanglement measure if (i) E(n)(σ) = 0 for +any biseparable state σ, (ii) E(n)(ρ) > 0 for any genuine +entangled state, and (iii) it is convex [10]. +A genuine +multipartite entanglement measure is called a genuine +multipartite entanglement monotone (GMEM) if it does +not increase on average under LOCC. +In Refs. [25, 26, 40], we present MEMs and GMEMs +that are defined by the sum of the reduced function on +pure states and then extended to mixed states via the +convex-roof structure. The aim of this paper is to give +a condition that can justify when the MEMs and the +GMEMs defined in this way is complete and completely +monogamous. Moreover, we give another way of defining +MEMs and the GMEMs from the maximal reduced func- +tion and then discuss when these quantities are complete +and completely monogamous. +The remainder of this paper is organized as follows. +In Sec. +II, we introduce some preliminaries. +Sec. +III +discusses the properties of the reduced functions of the +entanglement monotones so far in literature. +Sec. +IV +is divided into two subsections. +Subsec. +A discusses +the MEM defined by the sum of reduced functions, and +in Subsec. B, we give the MEMs defined by the max- +imal reduced function. +Both of theses two MEMs are +explored under the framework of the complete measure +and the complete monogamy relation. +In Sec. +V, we +consider three kinds of GMEMs which are defined by the +sum of reduced functions, the maximal reduced function, +and the minimal reduced function, respectively, under +the framework the complete measure and the complete +monogamy relation. We present a conclusion in Sec. VI. +II. +NOTATIONS AND PRELIMINARIES +The framework of the complete entanglement mea- +sure/monotone is closely related to the coarser relation +of multipartite partition. We first introduce three kinds +of coarser relation in Subsec. A, from which we then re- +view the complete MEM, complete GMEM, monogamy +relation and complete monogamy relation, respectively, +in the latter three subsections. +A. +Coarser relation of multipartite partition +Let X1|X2| · · · |Xk and Y1|Y2| · · · |Yl be two partitions +of A1A2 · · · An or subsystem of A1A2 · · · An (for instance, +partition AB|C|DE is a 3-partition of the 5-particle sys- +tem ABCDE with X1 = AB, X2 = C and X3 = DE). +We denote by [40] +X1|X2| · · · |Xk ≻a Y1|Y2| · · · |Yl, +(2) +X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl, +(3) +X1|X2| · · · |Xk ≻c Y1|Y2| · · · |Yl +(4) +if Y1|Y2| · · · |Yl can be obtained from X1|X2| · · · |Xk by +(a) discarding some subsystem(s) of X1|X2| · · · |Xk, +(b) combining some subsystems of X1|X2| · · · |Xk, +(c) discarding some subsystem(s) of some subsystem(s) +Xk provided that Xk = Ak(1)Ak(2) · · · Ak(f(k)) with +f(k) ⩾ 2, +respectively. For example, A|B|C|D ≻a A|B|D ≻a B|D, +A|B|C|D ≻b AC|B|D ≻b AC|BD, A|BC ≻c A|B. +Furthermore, if X1|X2| · · · |Xk ≻ Y1|Y2| · · · |Yl, we de- +note by Ξ(X1|X2| · · · |Xk −Y1|Y2| · · · |Yl) the set of all the +partitions that are coarser than X1|X2| · · · |Xk and either +exclude any subsystem of Y1|Y2| · · · |Yl or include some +but not all subsystems of Y1|Y2| · · · |Yl [40]. For exam- +ple, Ξ(A|B|CD|E − A|B) = {CD|E, A|CD|E, B|CD|E, +A|CD, B|CD, B|C|E, B|D|E, A|D|E, A|C|E, A|E, +B|E, A|C, A|D, B|C, B|D, C|E, D|E}. +B. +Complete MEM +A multipartite entanglement measure E(n) is called a +unified multipartite entanglement measure if it satisfies +the unification condition [26]: +(i) (additivity): +E(n)(A1A2 · · · Ak ⊗ Ak+1 · · · An) += E(k)(A1A2 · · · Ak) + E(n−k)(Ak+1 · · · An), +(5) + +3 +holds for all ρA1A2···An +∈ SA1A2···An, hereafter +E(n)(X) refers to E(n)(ρX); +(ii) (permutation invariance): +E(n)(A1A2 · · · An) += +E(n)(Aπ(1)Aπ(2) · · · Aπ(n)), +for all ρA1A2···An +∈ +SA1A2···An and any permutation π; +(iii) (coarsening monotone): +E(k)(X1|X2| · · · |Xk) ⩾ E(l)(Y1|Y2| · · · |Yl) +(6) +holds +for +all +ρA1A2···An +∈ +SA1A2···An +when- +ever +X1|X2| · · · |Xk +≻a +Y1|Y2| · · · |Yl, +where +X1|X2| · · · |Xk and Y1|Y2| · · · |Yl are two partitions +of A1A2 · · · An or subsystem of A1A2 · · · An, the +vertical bar indicates the split across which the en- +tanglement is measured.. +E(n) is called a complete multipartite entanglement mea- +sure if it satisfies both the conditions above and the hi- +erarchy condition [26]: +(iv) (tight coarsening monotone): +Eq. (6) holds for +all ρ ∈ SA1A2···An whenever X1|X2| · · · |Xk ≻b +Y1|Y2| · · · |Yl. +C. +Complete GMEM +Let E(n) +g +be a genuine multipartite entanglement mea- +sure. It is defined to be a unified genuine multipartite +entanglement measure if it satisfies the unification con- +dition [40], i.e., +(i) (permutation invariance): +E(n) +g +(A1A2 · · · An) += +E(n) +g +(Aπ(1)Aπ(2) · · · Aπ(n)), +for all ρA1A2···An +∈ +SA1A2···An +g +and any permutation π; +(ii) (coarsening monotone): +E(k) +g (X1|X2| · · · |Xk) > E(l) +g (Y1|Y2| · · · |Yl) +(7) +holds for all ρA1A2···An +∈ SA1A2···An +g +whenever +X1|X2| · · · |Xk ≻a Y1|Y2| · · · |Yl. +A unified GMEM E(n) +g +is call a complete genuine multi- +partite entanglement measure if E(n) +g +admits the hierar- +chy condition [40], i.e., +(iii) (tight coarsening monotone): +E(k) +g (X1|X2| · · · |Xk) ≥ E(l) +g (Y1|Y2| · · · |Yl) +(8) +holds +for +all +ρ +∈ +SA1A2···An +g +whenever +X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl. +D. +Monogamy Relation +For an bipartite entanglement measure E, E is said to +be monogamous if [9, 39] +E(A|BC) ⩾ E(AB) + E(AC). +(9) +However, Equation (9) is not valid for many entangle- +ment measures [9, 35, 37] but some power function of +Q admits the monogamy relation (i.e., Eα(A|BC) ⩾ +Eα(AB) + Eα(AC) for some α > 0). In Ref. [37], we +improved the definition of monogamy as: A bipartite +measure of entanglement E is monogamous if for any +ρ ∈ SABC that satisfies the disentangling condition, i.e., +E(ρA|BC) = E(ρAB), +(10) +we have that E(ρAC) = 0, where ρAB = TrCρABC. +With respect to this definition, a continuous measure E +is monogamous according to this definition if and only if +there exists 0 < α < ∞ such that +Eα(ρA|BC) ⩾ Eα(ρAB) + Eα(ρAC) +(11) +for all ρ acting on the state space HABC with fixed +dim HABC = d < ∞ (see Theorem 1 in Ref. [37]). +In Ref. [26], in order to characterize the distribution +of entanglement in a “complete” sense, the term “com- +plete monogamy” of the unified multipartite entangle- +ment measure is proposed. +For a unified multipartite +entanglement measure E(n), it is said to be completely +monogamous if for any ρ ∈ SA1A2···An that satisfies [26] +E(k)(X1|X2| · · · |Xk) = E(l)(Y1|Y2| · · · |Yl) +(12) +with X1|X2| · · · |Xk ≻a Y1|Y2| · · · |Yl we have that +E(∗) +g (Γ) = 0 +(13) +holds for all Γ ∈ Ξ(X1|X2| · · · |Xk − Y1|Y2| · · · |Yl), here- +after the superscript (∗) is associated with the partition +Γ, e.g., if Γ is a n-partite partition, then (∗) = (n). For +example, E(3) is completely monogamous if for any ρABC +that admits E(3)(ABC) = E(2)(AB) we get E(2)(AC) = +E(2)(BC) = 0. Let E(n) be a complete multipartite en- +tanglement measure. E(n) is defined to be tightly com- +plete monogamous if for any ρ ∈ SA1A2···An that satis- +fies [26] +E(k)(X1|X2| · · · |Xk) = E(l)(Y1|Y2| · · · |Yl) +(14) +with X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl we have that +E(∗) +g (Γ) = 0 +(15) +holds for all Γ ∈ Ξ(X1|X2| · · · |Xk − Y1|Y2| · · · |Yl). For +instance, E(3) is tightly complete monogamous if for any +ρABC that admits E(3)(ABC) = E(2)(A|BC) we have +E(2)(BC) = 0. +Let E(n) +g +be a genuine multipartite entanglement mea- +sure. We denote by SA1A2···Am +g +the set of all genuine en- +tangled states in SA1A2···Am. E(n) +g +is completely monog- +amous if it obeys Eq. (7) [40]. A complete genuine mul- +tipartite entanglement measure E(n) +g +is tightly complete + +4 +monogamous if it satisfies the genuine disentangling con- +dition, i.e., either for any ρ ∈ SA1A2···Am +g +that satis- +fies [40] +E(k) +g (X1|X2| · · · |Xk) = E(l) +g (Y1|Y2| · · · |Yl) +(16) +with X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl we have that +E(∗) +g (Γ) = 0 +(17) +holds for all Γ ∈ Ξ(X1|X2| · · · |Xk − Y1|Y2| · · · |Yl), or +E(k) +g (X1|X2| · · · |Xk) > E(l) +g (Y1|Y2| · · · |Yl) +(18) +holds for any ρ ∈ SA1A2···Am +g +. +In Ref. [26], we showed that the tightly complete +monogamy is stronger than the complete monogamy for +the complete MEMs that defined by the convex-roof ex- +tension. One can easily find that it is also true for any +complete GMEM defined by the convex-roof extension. +III. +STRICT CONCAVITY AND +SUBADDITIVITY OF THE REDUCED +FUNCTION +Any entanglement monotone, when evaluated on pure +states, is uniquely determined by its reduced function +and vice versa. Therefore, the feature of the entangle- +ment monotone defined via the convex-roof extension +rests with the quality of its reduced function. In Ref. [38], +we proved that the bipartite entanglement monotone is +monogamous whenever its reduced function is strictly +concave. In ths Section, we review all the reduced func- +tions of the entanglement monotones in literature so far +and then discuss the subadditivity of theses functions. As +what we will show in the next two Sections, the subaddi- +tivity is affinitive with the completeness of the measures +for some kind of MEM/GMEM. +A. +Strict concavity +The reduced functions of the entanglement of for- +mation Ef [45, 46], tangle τ [47], concurrence C [48– +50], negativity N [51], the Tsallis q-entropy of entangle- +ment Eq [52], and the R´enyi α-entropy of entanglement +Eα [44, 53] are +h(ρ) = S(ρ), +hτ(ρ) = h2 +C(ρ) = 2(1 − Trρ2), +hN(ρ) = 1 +2[(Tr√ρ)2 − 1], +hq(ρ) = 1 − Trρq +q − 1 +, +q > 0, +hα(ρ) = (1 − α)−1 ln(Trρα), +0 < α < 1, +respectively, where S is the von Neumann entropy. It has +been shown that h, hτ, hC, hN, hq, and hα are not only +concave but also strictly concave [38, 44, 54] (where the +strict concavity of hN is proved very recently in Ref. [55]). +The reduced functions of the entanglement monotones +induced by the fidelity-based distances EF, EF ′, and +EAF are [56] +hF(ρ) = 1 − Trρ3, +hF ′(ρ) = 1 − +� +Trρ2�2 , +hAF(ρ) = 1 − +� +Trρ3, +respectively. They are strictly concave [40]. +In Ref. [55], four kinds of partial norm of entangle- +ment are investigated: the partial-norm of entanglement +E2, the minimal partial-norm of entanglement Emin, the +reinforced minimal partial-norm of entanglement Emin′, +and the partial negativity ˆN. The reduced functions of +E2, Emin, E′ +min, and ˆN are +h2(ρ) = 1 − ∥ρ∥, +hmin(ρ) = ∥ρ∥min, +hmin′(ρ) = r(ρ)∥ρ∥min, +ˆh(ρ) = +� +δ1δ2, +where r(ρ) denotes the rank of ρ, ∥ · ∥ is the operator +norm, i.e., ∥X∥ = sup|ψ⟩ ∥A|ψ⟩∥, +∥ρ∥min = +� +λ2 +min, +λmin < 1, +0, +λmin = 1, +and δ1, δ2 are the two largest eigenvalues of ρ. All of them +are concave but not strictly concave (ˆh is only strictly +concave on qubit states), and these entanglement mono- +tones are not monogamous [55]. +B. +Subadditivity +We summarize the subadditivity of the reduced func- +tions in literature as following: +(i) S is additive and subadditive [54], i.e., +S(ρ ⊗ σ) = S(ρ) + S(σ) +(19) +and +S(ρAB) ≤ S(ρA) + S(ρB), +(20) +respectively. +(ii) Sq is subadditive iff q > 1, but not additive, and +for 0 < q < 1, Sq is neither subadditive nor super- +additive [57] (superadditivity refers to Sq(ρAB) ⩾ +Sq(ρA) + Sq(ρB)). In addition, +Sq(ρA ⊗ ρB) = Sq(ρA) + Sq(ρB) +(21) +iff ρA or ρB is pure [57]. + +5 +TABLE I. Comparing of the properties of the reduced func- +tions. +C, SC, SA, and A signify the function is concave, +strictly concave, subadditive, and additive, respectively. +E +h +C +SC +SA +A +Ef +S +✓ +✓ +✓ +✓ +C +� +2(1 − Trρ2) +✓ +✓ +✓ +× +τ +2(1 − Trρ2) +✓ +✓ +✓ +× +Eq +1−Trρq +q−1 +✓(q > 0) ✓(q > 1) ✓(q > 1) × +Eα +ln(Trρα) +1−α +, α ∈ (0, 1) +✓ +✓ +× +✓ +NF +(Tr√ρ)2−1 +2 +✓ +✓ +× +× +EF +1 − Trρ3 +✓ +✓ +✓ +× +EF′ +1 − (Trρ2)2 +✓ +✓ +✓a +× +EAF +1 − +� +Trρ3 +✓ +✓ +✓a +× +E2 +1 − ∥ρ∥ +✓ +× +✓ +× +Emin +∥ρ∥min +✓ +× +× +× +Emin′ +r(ρ)∥ρ∥min +✓ +× +× +× +ˆ +N +√ +δ1δ2 +✓b +× +✓a +× +a We conjecture that they are subadditive. +b We conjecture that it is concave. +(iii) hα is additive but not subadditive [58, 59]. +(iv) hτ is subadditive [60], i.e., +1 + Trρ2 +AB ≥ Trρ2 +A + Trρ2 +B. +(22) +In particular, the equality holds iff ρA or ρB is +pure [26]. +(v) hN is neither subadditive nor supperadditive [26]. +Item (iv) implies hC is subadditive and the equality holds +iff ρA or ρB is pure. hF is subadditive since it coincides +with Sq/2 (q = 3). We conjecture that hF ′ and hAF are +subadditive. +Proposition 1. h2 is subadditive, i.e., +1 + ∥ρAB∥ ⩾ ∥ρA∥ + ∥ρB∥ +(23) +holds for any ρAB ∈ SAB. +In particular, the equality +holds iff ρA or ρB is a pure state. +Proof. Note that partial trace is a quantum channel and +any quantum channel can be regarded as a operator on +the space of the trace-class operators. The norm of quan- +tum channel in such a sense is 1. Therefore ∥ρAB∥ ⩾ +∥ρA,B∥. Moreover, if 1 + ∥ρAB∥ = ∥ρA∥ + ∥ρB∥, then +∥ρA∥ = 1 or ∥ρB∥ = 1, which completes the proof. +Let +ρAB = 1 +2|ψ⟩⟨ψ| + 1 +2|φ⟩⟨φ| +with |ψ⟩ = +� +4 +5|00⟩+ +� +1 +5|11⟩ and |φ⟩ = +� +4 +5|22⟩+ +� +1 +5|33⟩. +It is clear that ∥ρAB∥min = 1 +2 > ∥ρA∥min + ∥ρB∥min = +1/10 + 1/10 = 1/5. +That is, ∥ · ∥min is not subaddi- +tive. Clearly, hmin′ is also not subadditive. According +to Proposition 1, hmin and hmin′ are subadditive on the +states that satisfies r(ρAB) = r(ρA) = r(ρB) = 2. One +can easily verifies that h2, hmin, hmin′, and ˆh are not +additive. +We conjecture that ˆh is subadditive, i.e., +ˆh(ρAB) ⩽ ˆh(ρA) + ˆh(ρB) +(24) +holds for any ρAB ∈ SAB. In what follows, we always +assume that hF ′, hAF, and ˆh are subadditive, and that +ˆh is concave. +The reduced functions of parametrized entanglement +monotones in Ref. [61] and Ref. [62] are +hq′(ρ) = 1 − Trρq, +q > 1, +and +hα′(ρ) = Trρα − 1, +0 < α < 1, +respectively. Obviously, the properties of these two func- +tions above are the same as that of hq, although they are +different from Eq [61, 62]. We summarize the properties +of theses reduced functions in Table I for more conve- +nience. +IV. +COMPLETE MEM +A. +Complete MEM from sum of the reduced +functions +In Ref. [26], we put forward several complete MEMs +defined by the sum of the reduced functions on all the +single subsystems. In fact, this scenario is valid for all en- +tanglement monotones. Let |ψ⟩A1A2···An be a pure state +in HA1A2···An and h be a non-negative concave function +on SX. We define +E(n)(|ψ⟩A1A2···An) = 1 +2 +� +i +h(ρAi) +(25) +and then extend it to mixed states by the convex-roof +structure. +We denote E(n) by E(n) +f +, C(n), τ (n), E(n) +q +, +E(n) +α , N (n) +F , E(n) +F , E(n) +F ′ , E(n) +AF, E(n) +2 +, E(n) +min, E(n) +min′, and ˆN (n) +whenever h = S, hC, hτ, hq, hα, hN, hF, hF ′, hAF, h2, +hmin, hmin′, and ˆh, respectively. Here, E(n) +f +, C(n), τ (n), +E(n) +q +, E(n) +α , and N (n) +F +have been discussed in Ref. [26] +for the first time. The coefficient “1/2” is fixed by the +unification condition when E(n) is regarded as a unified +MEM. One need note here that E(n) +F , E(n) +F ′ , and E(n) +AF are +different from E(n) +F,F , E(n) +F ′,F , and E(n) +AF,F respectively in +Ref. [56]. +Theorem 1. Let E(n) be a non-negative function defined +as in Eq. (25). Then the following statements hold true. + +6 +(i) E(n) is a unified MEM and is completely monoga- +mous; +(ii) E(n) is a complete MEM iff h is subadditive; +(iii) E(n) is tightly complete monogamous iff h is sub- +additive with +h(ρAB) = h(ρA) + h(ρB) ⇒ ρAB is separable. (26) +Proof. We only need to discuss the case of n = 3 with no +loss of generality. +(i) For any |ψ⟩ABC ∈ HABC, we let E(2)(ρAB) = +� +i piE(2)(|ψi⟩) = 1 +2 +� +i pi[h(ρA +i ) + h(ρB +i )]. Then +E(3)(|ψ⟩ABC) = 1 +2 +� +h(ρA) + h(ρB) + h(ρC) +� +≥ 1 +2 +� +h(ρA) + h(ρB) +� +≥ 1 +2 +� +i +pi[h(ρA +i ) + h(ρB +i )] += E(2)(ρAB). +That is, E(3) satisfies Eq. (6) for pure states and it is +completely monogamous on pure states. For any mixed +state ρABC, we let E(3)(ρABC) = � +j qjE(3)(|ψj⟩) and +E(2)(ρAB +j +) = � +i pi(j)E(2)(|ψi(j)⟩) = 1 +2 +� +i pi(j)[h(ρA +i(j)) + +h(ρB +i(j))]. Then +E(3)(ρABC) = 1 +2 +� +j +qj +� +h(ρA +j ) + h(ρB +j ) + h(ρC +j ) +� +≥ 1 +2 +� +j +qj +� +hj(ρA) + hj(ρB) +� +≥ 1 +2 +� +i,j +qjpi(j)[h(ρA +i(j)) + h(ρB +i(j))] ≥ E(2)(ρAB), +i.e., it is a unified MEM. If E(3)(ρABC) = E(2)(ρAB), +it yields h(ρC +j ) = 0 for any j, and thus |ψj⟩ABC = +|ψj⟩AB|ψj⟩C. Therefore it is completely monogamous. +(ii) If E(3) is a complete MEM, then E(3)(|ψ⟩ABC) ≥ +E(2)(|ψ⟩A|BC) for any |ψ⟩ABC, which implies h(ρBC) ≤ +h(ρB) + h(ρC). That is, h is subadditive since |ψ⟩ABC +is arbitrarily given. +Conversely, if h is subadditive, +then E(3)(|ψ⟩ABC) ≥ E(2)(|ψ⟩A|BC) for any pure state +|ψ⟩ABC. For any mixed state ρABC, we let E(3)(ρABC) = +� +j qjE(3)(|ψj⟩). Then +E(3)(ρABC) = 1 +2 +� +j +qj +� +h(ρA +j ) + h(ρB +j ) + h(ρC +j ) +� +≥ 1 +2 +� +j +qj +� +hj(ρA) + hj(ρBC) +� +≥ E(2)(ρA|BC), +i.e., it is a complete MEM. +(iii) It can be easily checked using the argument anal- +ogous to that of (ii) together with the fact that, if E(n) +is tightly complete monogamous, it is automatically a +complete MEM. +TABLE II. Comparing of E(n) with different different reduced +functions, and E (n). +CM and TCM signify the measure is +completely monogamous and tightly completel monogamous, +respectively. +MEM +Unified +Complete +CM +TCM +E(n) +f +✓ +✓ +✓ +✓ +C(n) +✓ +✓ +✓ +✓ +τ (n) +✓ +✓ +✓ +✓ +E(n) +q +✓ +✓ +✓ +✓a +E(n) +α +✓ +× +✓ +× +N (n) +F +✓ +× +✓ +× +E(n) +F +✓ +✓ +✓ +✓a +E(n) +F′ +✓ +✓b +✓ +✓a +E(n) +AF +✓ +✓b +✓ +✓a +E(n) +2 +✓ +✓ +✓ +✓ +E(n) +min +✓ +× +✓ +× +E(n) +min′ +✓ +× +✓ +× +ˆ +N (n) +✓ +✓b +✓ +✓a +E (n) (n ≥ 4) +✓ +✓ +✓ +✓ +a It is tightly complete monogamous under the assumption that +h is subadditive and Eq. (26) holds. +b It is complete under the assumption that h is subadditive. +By Theorem 1, we can conclude: (i) E(n) +f +, C(n), τ (n), +E(n) +q +, E(n) +α , N (n) +F , E(n) +F , E(n) +F ′ , E(n) +AF, E(n) +2 +, E(n) +min, E(n) +min′, +and ˆN (n) are unified MEMs and are completely monog- +amous; (ii) E(n) +f +, C(n), τ (n), E(n) +q +, E(n) +F , E(n) +F ′ , E(n) +AF, +E(n) +2 +, and ˆN (n) are complete MEMs; (iii) E(n) +α , N (n) +F , +E(n) +min, and E(n) +min′ are not complete MEMs since the asso- +ciated reduced functions are not subadditive which vio- +late the hierarchy condition for some states. (iv) E(n) +f +, +C(n), τ (n), and E(n) +2 +are tightly complete monogamous. +However E(n) +2 +, E(n) +min, E(n) +min′, and ˆN (n) are not monoga- +mous.Together with Theorem in Ref. [38], we obtain that, +for these MEMs, both monogamy and tightly complete +monogamy are stronger than the complete monogamy +under the frame work of the complete MEM, and that +monogamy is stronger than both complete monogamy +and tightly complete monogamy (e.g., E(n) +2 +). +In particular, if h is subadditive with h(ρAB) += +h(ρA) + h(ρB) implies ρAB = ρA ⊗ ρB, then E(n) is +tightly complete monogamous. S, hτ, hC, and h2 be- +long to such situations. We also conjecture that hq, hF, +hF ′, hAF, and ˆh belong to such situations as well. That +is, we conjecture that E(n) +q +, E(n) +F , E(n) +F ′ , E(n) +AF, and ˆN (n) +are tightly complete monogamous. +In Ref. [25], we put forward several multipartite en- +tanglement measures which are defined by the sum of all +bipartite entanglement. Let |ψ⟩A1A2···An be a pure state +in HA1A2···An and h be a non-negative concave function + +7 +on SX. We define [25] +E(n)(|ψ⟩A1A2···An) += + + + + + +1 +2 +� +i1≤···≤is,s 0 for any i, +0, +h(ρAi) = 0 for some i, +(36) +and then extend it to mixed states by the convex- +roof structure. By Proposition 1 and Proposition 4 in +Ref. [40], together with Theorem 1, we have the follow- +ing statement. +Proposition 3. Let E(n) +g +be a non-negative function de- +fined as in Eq. (36). Then the following statements hold +true. +(i) E(n) +g +is a unified GMEM and is completely monog- +amous; +(ii) E(n) +g +is a complete GMEM iff h is subadditive; +(iii) E(n) +g +is tightly complete monogamous iff h is sub- +additive with Eq. (26) holds. +We denote E(n) +g +in the previous Section by E(n) +g,f , C(n) +g +, +τ (n) +g +, E(n) +g,q , E(n) +g,α, N (n) +g,F , E(n) +g,F, E(n) +g,F ′, E(n) +g,AF, E(n) +g,2 , E(n) +g,min, +E(n) +g,min′, and ˆN (n) +g +, respectively. +By Proposition 3, we +can conclude: (i) All theses GMEMs are unified GMEMs +and are completely monogamous; (ii) E(n) +g,f , C(n) +g +, τ (n) +g +, +E(n) +g,q , E(n) +g,F, E(n) +g,F ′, E(n) +g,AF, E(n) +g,2 , and ˆN (n) +g +are complete +GMEMs; (iii) E(n) +g,α, N (n) +g,F , E(n) +g,min, and E(n) +g,min′ are not +complete GMEMs since the associated reduced functions +are not subadditive and thus they violate the hierarchy +condition for some states. (iv) E(n) +g,f , C(n) +g +, τ (n) +g +, and E(n) +g,2 +are tightly complete monogamous. Therefore, for these +GMEMs, tightly complete monogamy are stronger than +the complete monogamy under the frame work of the +complete GMEM. +By the assumption, we conjecture that E(n) +g,q , E(n) +g,F, +E(n) +g,F ′, E(n) +g,AF, and ˆN (n) +g +are tightly complete monoga- +mous. +That is, E(n) +g +is complete, completely monoga- +mous, tightly complete monogamous, if and only if E(n) +is complete, completely monogamous, tightly complete +monogamous, respectively. +A similar quantity, εg−12···n(2), is also put forward in +Ref. [25]. Let |ψ⟩A1A2···An be a pure state in HA1A2···An + +9 +TABLE IV. Comparing of E(n) +g +with different reduced func- +tions and E (n) +g +(n ≥ 4). +GMEM +Unified +Complete +CM +TCM +E(n) +g,f +✓ +✓ +✓ +✓ +C(n) +g +✓ +✓ +✓ +✓ +τ (n) +g +✓ +✓ +✓ +✓ +E(n) +g,q +✓ +✓ +✓ +✓a +E(n) +g,α +✓ +× +✓ +× +N (n) +g,F +✓ +× +✓ +× +E(n) +g,F +✓ +✓ +✓ +✓a +E(n) +g,F′ +✓ +✓b +✓ +✓a +E(n) +g,AF +✓ +✓b +✓ +✓a +E(n) +g,2 +✓ +✓ +✓ +✓ +E(n) +g,min +✓ +× +✓ +× +E(n) +g,min′ +✓ +× +✓ +× +ˆ +N (n) +g +✓ +✓b +✓ +✓a +E (n) +g +(n ≥ 4) +✓ +✓ +✓ +✓ +a Assume that h is subadditive and Eq. (26) holds. +b Assume that h is subadditive. +and h be a non-negative concave function on SX. We +define +E(n) +g +(|ψ⟩A1A2···An) += +� +E(n)(|ψ⟩A1A2···An), +h(ρAi) > 0 for any i, +0, +h(ρAi) = 0 for some i, (37) +and then extend it to mixed states by the convex-roof +structure. Notice here that E(n) +g +is slightly different than +εg−12···n(2) in which the factor “1/2” is ignored. +Clearly, +E(n) +g +≤ E(n) +g +, +(38) +and E(3) +g +coincides with E(3) +g +but E(n) +g +is different from +E(n) +g +whenever n ≥ 4. E(n) +g +is just Eg−12···n(2) in Ref. [25] +if the corresponding bipartite entanglement measure is +an entanglement monotone. The following Proposition +can be easily checked. +Proposition 4. Let E(n) be a non-negative function de- +fined as in Eq. (37), n ≥ 4. Then E(n) is a complete +MEM and it is completely monogamous and tightly com- +plete monogamous. +That is, for the case of n ≥ 4, all these MEMs E(n) +g +with +the reduced functions we discussed in Sec. III are com- +plete GMEMs, and are not only completely monogamous +but also tightly complete monogamous. For convenience, +we list all these MEMs in Table IV. In addition, it is ob- +vious that E(n) +g +< E(n) +g +whenever n ≥ 4 for any E(4) +g +and +E(4) +g +mentioned above. +B. +Complete GMEM from the maximal reduced +function +Let |ψ⟩A1A2···An be a pure state in HA1A2···An and h +be a non-negative concave function on the set of density +matrices. We define +E(n) +g′ (|ψ⟩A1A2···An) += +� +max +i +h(ρAi), +if h(ρAi) > 0 for any i, +0, +if h(ρAi) = 0 for some i, +(39) +and then extend it to mixed states by the convex-roof +structure. From Theorem 2, we have the following Propo- +sition. +Proposition 5. Let E(n) +g′ +be a GMEM defined as in +Eq. (39). Then (i) E(3) +g′ +is a complete GMEM but not +tightly complete monogamous, and if h is strictly con- +cave, E(3) +g′ +is completely monogamous, and (ii) E(n) +g′ +is +not complete whenever n ≥ 4. +We denote E(n) +g′ +the corresponding GMEMs men- +tioned in the previous Subsection by E(n) +g′,f, C(n) +g′ , τ (n) +g′ , +E(n) +g′,q, E(n) +g′,α, N (n) +g′,F , E(n) +g′,F, E(n) +g′,F ′, E(n) +g′,AF, E(n) +g′,2, E(n) +g′,min, +E(n) +g′,min′, and ˆN (n) +g′ , respectively. Then all these GMEMS +are complete GMEMs but not tightly complete monoga- +mous for the case of n = 3, E(3) +g′,f, C(3) +g′ , τ (3) +g′ , E(3) +g′,q, E(3) +g′,α, +N (3) +g′,F , E(3) +g′,F, E(3) +g′,F ′, and E(3) +g′,AF, are completely monog- +amous, all of these GMEMs are not complete GMEMs +whenever n ≥ 4. +One need note here that, when h is not strictly con- +cave, E(n) +g′ +is not a unified GMEM since it may hap- +pen that E(k) +g′ (X1|X2| · · · |Xk) = E(l) +g′ (Y1|Y2| · · · |Yl) for +some ρA1A2···An ∈ SA1A2···An +g +with X1|X2| · · · |Xk ≻a +Y1|Y2| · · · |Yl, namely, it violates Eq. (7). +In addition, +E(n) +g′ +also violates Eq. (17) or Eq. (18). For example, we +take the state in Eq. (31) with both |ψ⟩AB1 and |ψ⟩B2C +are entangled. We assume +E(3) +g′,min(|ψ⟩ABC) = E(2) +g′,min(|ψ⟩A|BC), +ˆN (3) +g′ (|ψ⟩ABC) = ˆN (2) +g′ (|ψ⟩A|BC), +then +E(3) +g′,min(|ψ⟩ABC) = E(2) +g′,min(|ψ⟩AB1) = E(2) +g′,min(ρAB), +ˆN (3) +g′ (|ψ⟩ABC) = ˆN (2) +g′ (|ψ⟩AB1) = ˆN (2) +g′ (ρAB), +and ρBC is entangled. +In addition, for the sate in +Eq. (32), we have +E(3) +g′,2(|φ⟩ABC) += E(2) +g′,2(|φ⟩A|BC) = E(2) +g′,2(|φ⟩AB|C) = E(2) +g′,2(|φ⟩B|AC) += E(2) +g′,2(ρAB) = E(2) +g′,2(ρAC) = E(2) +g′,2(ρBC) += 1 +3. + +10 +TABLE V. Comparing of E(3) +g′ +with different reduced func- +tions, E(4) +g′ , and E (n) +g′ . +GMEM +Unified +Complete +CM +TCM +E(3) +g′,f +✓ +✓ +✓ +× +C(3) +g′ +✓ +✓ +✓ +× +τ (3) +g′ +✓ +✓ +✓ +× +E(3) +g′,q +✓ +✓ +✓ +× +E(3) +g′,α +✓ +✓ +✓ +× +N (3) +g′,F +✓ +✓ +✓ +× +E(3) +g′,F +✓ +✓ +✓ +× +E(3) +g′,F′ +✓ +✓ +✓ +× +E(3) +g′,AF +✓ +✓ +✓ +× +E(3) +g′,2 +× +× +× +× +E(3) +g′,min +× +× +× +× +E(3) +g′,min′ +× +× +× +× +ˆ +N (3) +g′ +× +× +× +× +E(n) +g′ +(n ≥ 4) +? +× +? +× +E (n) +g′ +(n ≥ 4) +? +× +? +× +That is, whenever h is strictly concave, E(n) +g′ +is complete, +completely monogamous, tightly complete monogamous, +if and only if E′(n) is complete, completely monogamous, +tightly complete monogamous, respectively. +For the states that admit the form +|η⟩ABC = |η⟩AB1|η⟩B2C +(40) +where B1B2 refers to HB has a subspace isomorphic to +HB(x) +1 +⊗HB(x) +2 +such that up to local unitary on system B, +we have E(3) +g′ (|η⟩ABC) = E(2) +g′ (|η⟩B|AC) whenever h(ρ ⊗ +σ) ≥ h(ρ) and h(ρ ⊗ σ) ≥ h(σ) for any ρ and σ, and ρAC +is a product state. We therefore have the following fact. +Proposition 6. If h is strictly concave and h(ρ ⊗ σ) ≥ +h(ρ) and h(ρ⊗σ) ≥ h(σ) for any ρ and σ, E(3) +g′ defined as +in Eq. (39) is tightly complete monogamous on the states +that admit the form (40). +In fact, we always have h(ρ⊗σ) ≥ h(ρ) and h(ρ⊗σ) ≥ +h(ρ) if h ∈ {S, hC, hτ, hq, hα, hN, hF, hF ′, hAF}. So +E(n) +g′,f, C(n) +g′ , τ (n) +g′ , E(n) +g′,q, E(n) +g′,α, N (n) +g′,F , E(n) +g′,F, E(n) +g′,F ′, and +E(n) +g′,AF are tightly complete monogamous on the states +with the form as in Eq. (40). Proposition 6 is also valid +when we replacing E(3) +g′ +with E′(3). +Let |ψ⟩A1A2···An be a pure state in HA1A2···An and h +be a non-negative concave function on SX. We define +E(n) +g′ (|ψ⟩A1A2···An) += +� +E′(n)(|ψ⟩A1A2···An), +if h(ρAi) > 0 for any i, +0, +if h(ρAi) = 0 for some i,(41) +for pure states and for mixed states by the convex-roof +structure. By definition, +E(n) +g′ +≤ E(n) +g′ , +(42) +E(3) +g′ +coincides with E′(3) +g , and E(n) +g′ +satisfies the hierarchy +condition, but it violates the unification condition if n ≥ +4. It is easy to see that all these GMEMs E(n) +g′ +with the +reduced function we discussed are not complete GMEMs +whenever n ≥ 4. We give comparison for these GMEMs +in Table V. +For the case of n ≥ 4, it is possible that E(n) +g′ +< E(n) +g′ . +For example, for |W4⟩ and the state in Eq. (35) we have +E(4) +g′ < E(4) +g′ for any E(4) +g′ and E(4) +g′ mentioned above except +for h = hmin and h = ˆh. +C. +GMEM from the minimal reduced function +With h is a non-negative concave function on the set +of density matrices, when we define +E(n) +g′′ (|ψ⟩A1A2···An) = min +i +h(ρAi), +(43) +and then extend it to mixed states by the convex-roof +structure, it is a GMEM. Moreover, we can define +E(n) +g′′ (|ψ⟩A1A2···An) = +min +i1≤···≤is,s≤n/2 h(ρAi1 Ai2···Ais ), (44) +and then extend it to mixed states by the convex-roof +structure, it is also a GMEM. For example, GMC, de- +noted by Cgme [31], is defined as in Eq. (44). +Recall +that, +Cgme(|ψ⟩) := min +γi∈γ +� +2 +� +1 − Tr(ρAγi )2� +for pure state |ψ⟩ ∈ HA1A2···Am, where γ = {γi} repre- +sents the set of all possible bipartitions of A1A2 · · · Am, +and via the convex-roof extension for mixed states. +We denote E(n) +g′′ the corresponding GMEMs mentioned +in the previous Subsection by E(n) +g′′,f, C(n) +g′′ , τ (n) +g′′ , E(n) +g′′,q, +E(n) +g′′,α, N (n) +g′′,F , E(n) +g′′,F, E(n) +g′′,F ′, E(n) +g′′,AF, E(n) +g′′,2, E(n) +g′′,min, +E(n) +g′′,min′, and +ˆN (n) +g′′ , respectively, and denote E(n) +g′′ +by +E(n) +g′′,f, C(n) +g′′ +(or Cgme), ˆτ (n) +g′′ , E(n) +g′′,q, E(n) +g′′,α, N (n) +g′′,F , E(n) +g′′,F, +E(n) +g′′,F ′, E(n) +g′′,AF, E(n) +g′′,2, E(n) +g′′,min, E(n) +g′′,min′, and ˆ +N (n) +g′′ , respec- +tively. +By definition, +E(n) +g′′ ≤ E(n) +g′′ ≤ E(n) +g′ +≤ E(n) +g +(45) +for any h, and E(3) +g′′ = E(3) +g′′ . If n ≥ 4, there does exist +state such that E(n) +g′′ < E(n) +g′′ . For example, we take +|ψ⟩ABCD = |ψ⟩AB1|ψ⟩B2C1|ψ⟩C2D, + +11 +0 +0.2 +0.4 +0.6 +0.8 +1 +t +0 +0.5 +1 +1.5 +E +(a) Cg +0 +0.2 +0.4 +0.6 +0.8 +1 +t +0 +0.5 +1 +E +(b) Eg,F′ +0 +0.2 +0.4 +0.6 +0.8 +1 +t +0 +0.2 +0.4 +0.6 +0.8 +E +(c) Eg,2 +FIG. 1. (color online). Comparing (a) C(3) +g +and C(3) +g′ +for |Ψ⟩. +(b) E(3) +g,F′ and E(3) +g′,F′, and (c) Comparing E(3) +g,2 and E(3) +g′,2 for +|Ψ⟩. E(3) +g′ = E(3) +g′′ in such a case. +where X1X2 refers to HX has a subspace isomorphic +to HX(x) +1 +⊗ HX(x) +2 +such that up to local unitary on sys- +tem X. If h(ρB2) < h(ρA) and h(ρB2) < h(ρD), then +E(4) +g′′ (|ψ⟩ABCD) = h(ρB2) < E(4) +g′′ (|ψ⟩ABCD). In addition, +for the state in Eq. (35), +E(4) +g′′,min = 5 +16 < E(4) +g′′,min = 3 +8, +ˆ +N (4) +g′′ = +√ +15 +8 +√ +2 < ˆN (4) +g′′ = +√ +15 +8 . +Cgme is not a complete GMEM since it does not satisfy +the hierarchy condition (8) [40]: Let +|ξ⟩ = +√ +5 +4 |0000⟩ + 1 +4|1111⟩ + +√ +5 +4 |0100⟩ + +√ +5 +4 |1010⟩, (46) +then +Cgme(|ξ⟩) = C(|ξ⟩ABC|D) = +√ +15 +8 +< C(|ξ⟩AB|CD) = +√ +65 +8 . +In general, E(n) +g′′ and E(n) +g′′ do not obey the unification +condition (7) and the hierarchy condition (8). +For in- +stance, for the state as in Eq. (31), we have +E(3) +g′′,min(|ψ⟩ABC) = E(2) +g′′,min(|ψ⟩B|AC), +ˆN (3) +g′′ (|ψ⟩ABC) = ˆN (2) +g′′ (|ψ⟩B|AC), +and +E(3) +g′′,min(|ψ⟩ABC) < E(2) +g′′,min(|ψ⟩A|BC) += E(2) +g′′,min(|ψ⟩AB1) = E(2) +g′′,min(ρAB), +E(3) +g′′,min(|ψ⟩ABC) < E(2) +g′′,min(|ψ⟩C|AB) += E(2) +g′′,min(|ψ⟩B2C) = E(2) +g′′,min(ρBC), +ˆ +N (3) +g′′ (|ψ⟩ABC) < ˆN (2) +g′′ (|ψ⟩A|BC) += ˆ +N (2) +g′′ (|ψ⟩AB1) = ˆN (2) +g′′ (ρAB), +ˆ +N (3) +g′′ (|ψ⟩ABC) < ˆN (2) +g′′ (|ψ⟩AB|C) += ˆ +N (2) +g′′ (|ψ⟩B2C) = ˆN (2) +g′′ (ρBC). +In addition, +C(ρBD) ≈ 0.839 > Cgme(|ξ⟩) +for the pure state |ψ⟩ in Eq. (46). Let +|ζ⟩ABC = λ0|000⟩ + λ2|101⟩ + λ3|110⟩ +(47) +with λ0 ≥ λ2 ≥ λ3 > 0. If we take λ0 = +√ +5 +√ +12, λ2 = +1 +√ +3, and λ3 = 1 +2 in Eq. (47), then E(3) +g′′,2(|ζ⟩ABC) = 1/4, +but E(2) +g′′,2(|ζ⟩A|BC) = 5/12, E(2) +g′′,2(|ζ⟩AB|C) = 1/3. In +general, for the state +|ω⟩ABC = λ0|000⟩ + λ2|101⟩ + λ3|110⟩ + λ4|111⟩ +with λ0λ4 > 0, max{λ2, λ3} > 0 and min{λ2, λ3} = 0, +then (i) ρAC and ρBC are separable while ρAB is entan- +gled whenever λ3 > 0, and (ii) ρAB and ρBC are separable +while ρAC is entangled whenever λ2 > 0. From this we +can arrive at (i) if λ4 is small enough, then +Cgme(|ω⟩ABC) = C(|ω⟩AB|C) < C(|ω⟩A|BC), +C(|ω⟩AB|C) < C(|ω⟩B|AC), +Cgme(|ω⟩ABC) < C(ρAB), + +12 +and (ii) if λ4 is small enough, then +Cgme(|ω⟩ABC) = C(|ω⟩B|AC) < C(|ω⟩C|AB), +C(|ω⟩B|AC) < C(|ω⟩A|BC), +Cgme(|ω⟩ABC) < C(ρAC). +For example, when taking λ2 +0 = 7/9, λ3 = λ4 = 1/3, we +get +Cgme(|ω⟩ABC) ≈ 0.5879, +C(|ω⟩A|BC) = C(|ω⟩B|AC) ≈ 0.8315, +C(ρAB) ≈ 0.8090; +when taking λ2 +0 = 7/9, λ2 = λ4 = 1/3, we get +Cgme(|ω⟩ABC) ≈ 0.5879, +C(|ω⟩A|BC) = C(|ω⟩C|AB) ≈ 0.8315, +C(ρAC) ≈ 0.8090. +For the generalized GHZ state +|GHZ⟩ = λ0|0⟩⊗n + λ1|1⟩⊗n + · · · λd−1|d − 1⟩⊗n, (48) +E(n) +g′′ +and E(n) +g′′ +are complete monogamous and tightly +complete monogamous. For this state, E(n) +g′′ += E(n) +g′ += +E(n) +g′′ = E(n) +g′ , and nE(n) +g′′ = nE(n) +g′ += 2E(n) +g +. Moreover, for +such a state, all the entanglement are shared between all +of the particles. We thus regard this state as the maxi- +mal genuinely entangled state, and it reaches the maxi- +mal value whenever λ0 = λ1 = · · · = λd−1 = 1/ +√ +d for +the multi-qudit case. +Comparing E(3) +g′′ with E(3) +g′ +and E(3) +g , E(3) +g′ +seems the +best one since (i) it is complete and completely monoga- +mous whenever the reduced function is strictly concave, +(ii) it can be easily calculated, and (iii) it is monogamous +iff it is completely monogamous. For the case of n ≥ 4, +E(n), E(n), E(n) +g +, and E(n) +g +seems better the other cases as +a MEM/GMEM as these measures admit the postulates +of a complete MEM/GMEM. +At last, we calculate these GMEMs for the following +examples, +|Ψ⟩ = +√ +t|000⟩ + +√ +1 − t|111⟩, +|Φ⟩ = √p|100⟩ + √q|010⟩ + +� +1 − p − q|001⟩. +For the GHZ class state |Ψ⟩, Eg′ coincides with Eg′′ and +Eg′ is equivalent to Eg (see Fig. 1 for detail). For |Φ⟩, +Eg, Eg′ and Eg′′ reflect roughly the same tendency (see +Fig. 2 for detail). +VI. +CONCLUSION +We developed a grained scenario of investigating the +MEM and GMEM based on its reduced functions and +then explored these measures in light of the framework of +the complete MEM and complete monogamy relation re- +spectively. We provided criteria that can verify whether +(a) Cg +(b) Eg,F′ +(c) Eg,2 +FIG. 2. (color online). Comparing (a) C(3) +g , C(3) +g′ +and C(3) +g′′ , +(b) E(3) +g,F′, E(3) +g′,F′ and E(3) +g′′,F′, (c) E(3) +g,2, E(3) +g′,2 and E(3) +g′′,2 for |Φ⟩ +with p ≥ q ≥ 1 − p − q > 0, respectively. +a MEM/GMEM is good or not. +By comparision, for +tripartite case, the MEM and GMEM via the maximal +reduced function seems finer than that of the minimal +reduced function as it not only can be easily calculated +but also is complete and completely monogamous. And +for the n-partite case with n ≥ 4, the MEM and GMEM +via the sum of the reduced function sound better than +the other one in the framework of complete MEM and +complete monogamy relation. +In addition, our findings show that, whether the re- +duced function is strictly concave and whether it is +subadditive is of crucial important. +We can also con- + +9,2 +E +q/.2 +E +gll,20.4 +0.6 +p0.5 +0 +0 +0.2 +q +0.4 +0.8 +0.6 +1a.F +E +g'.F +E +gll,F0.4 +0.6 +p0.5 +0 +0 +0.2 +b +0.4 +0.8 +0.6 +1.9 +gr0.4 +0.6 +p0.5 +0 +0 +0.2 +b +0.4 +0.8 +0.6 +113 +clude that the monogamy is stronger than the complete +monogamy in general, they are equivalent to each other +for some case such as the MEM and GMEM via the max- +imal reduced function for the tripartite case, and the +tightly complete monogamy is stronger than the com- +plete monogamy in general. We also find that, in the +framework of complete MEM, the hierarchy condition is +stronger than the unification condition in general but it +is not true for some case such as the MEM and GMEM +via the maximal bipartite entanglement. +ACKNOWLEDGMENTS +This work is supported by the National Natural Sci- +ence Foundation of China under Grant No. 11971277, the +Fund Program for the Scientific Activities of Selected Re- +turned Overseas Professionals in Shanxi Province under +Grant No. 20220031, and the Scientific Innovation Foun- +dation of the Higher Education Institutions of Shanxi +Province under Grant No. 2019KJ034. +[1] M. A. Nielsen, I. L. Chuang, Quantum Computatation +and Quantum Information, (Cambridge University Press, +Cambridge, 2000). +[2] Q. Zhang, A. Goebel, C. Wagenknecht, Y.-A. Chen, B. +Zhao, T. Yang, A. Mair, J. Schmiedmayer, and J.-W. +Pan, Experimental quantum teleportation of a two-qubit +composite system, Nat. Phys. 2, 678 (2006). +[3] C. H. Bennett and S. J. 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A: Math. Theor. 55 +(27),275303 (2022). + diff --git a/GNAyT4oBgHgl3EQffPgF/content/tmp_files/load_file.txt b/GNAyT4oBgHgl3EQffPgF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..135e639a2792bb0663063501836b16dd0d58336e --- /dev/null +++ b/GNAyT4oBgHgl3EQffPgF/content/tmp_files/load_file.txt @@ -0,0 +1,992 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf,len=991 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='00334v1 [quant-ph] 1 Jan 2023 Complete Genuine Multipartite Entanglement Monotone Yu Guo∗ Institute of Quantum Information Science, Shanxi Datong University, Datong, Shanxi 037009, China A complete characterization and quantification of entanglement, particularly the multipartite entanglement, remains an unfinished long-term goal in quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' As long as the multipartite system is concerned, the relation between the entanglement contained in different partitions or different subsystems need to take into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' The complete multipartite entanglement measure and the complete monogamy relation is a framework that just deals with such a issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In this paper, we put forward conditions to justify whether the multipartite entanglement monotone (MEM) and genuine multipartite entanglement monotone (GMEM) are complete, completely monogamous, and tightly complete monogamous according to the feature of the reduced function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Especially, we proposed a class of complete MEMs and a class of complete GMEMs via the maximal reduced function for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' By comparison, it is shown that, for the tripartite case, this class of GMEMs is better than the one defined from the minimal bipartite entanglement in literature under the framework of complete MEM and complete monogamy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In addition, the relation between monogamy, complete monogamy, and the tightly complete monogamy are revealed in light of different kinds of MEMs and GMEMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' PACS numbers: 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='Mn, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='Db, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='Ud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' INTRODUCTION Entanglement, as one of the most puzzling features in quantum mechanics, has been widely used as an essen- tial resource for quantum communication [1–3], quantum cryptography [4, 5], and quantum computing [6, 7], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' The utility of an entangled state for these applications is often directly related to the degree or type of entan- glement contained in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Therefore, efficiently quanti- fying and characterizing multipartite entanglement is of paramount importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Especially, the genuine multi- partite entanglement, as one of the important types of entanglement, offers significant advantages in quantum tasks compared with bipartite entanglement [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' The phenomenon becomes much more complex for multipartite entanglement, particularly the genuinely multipartite entanglement, entanglement shared between all of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Over the years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' many multipar- tite entanglement measures have been proposed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' such as the “residual tangle” which reports the genuine three- qubit entanglement [9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' the genuinely multipartite con- currence [10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' the k-ME concurrence [11],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' the m con- currence [12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' the generalization of negativity [13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' the SL-invariant multipartite measure of entanglement [14– 19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' and the α-entanglement entropy [20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' concurrence triangle [21],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' concentratable entanglement [22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' geomet- ric mean of bipartite concurrence [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' concurrence tri- angle induced genuine multipartite entanglement mea- sure [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' and a general way of constructing genuine mul- tipartite entanglement monotone is proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [26], we proposed a framework of complete mul- tipartite entanglement monotone from which the entan- glement between any partitions or subsystems with the coarsening relation could be compared with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' ∗ guoyu3@aliyun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='com In the context of describing multipartite entanglement, another fundamental task is to understand how entan- glement is distributed over many parties since it reveals fundamental insights into the nature of quantum correla- tions [8] and has profound applications in both quantum communication [27, 28] and other area of physics [29– 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' This characteristic trait of distribution is known as the monogamy law of entanglement [27, 34], which means that the more entangled two parties are, the less correlated they can be with other parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Quan- titatively, the monogamy of entanglement is described by an inequality [9, 29, 34–36] or equality [26, 37, 38], involving a bipartite entanglement monotone or multi- partite entanglement monotone (MEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Consequently, considerable research has been undertaken in this direc- tion [9, 26, 29, 34–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Very recently, we discussed when the genuine multi- partite entanglement measure is complete [40] with the same spirit as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Under such a sense, the hierar- chy structure of the entanglement in the system is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Moreover, whether the multipartite entanglement mea- sure is proper or not can be justified together with the framework of complete monogamy relation for the mul- tipartite system established in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' The framework of complete monogamy relation is based on the complete multipartite entanglement measure [26, 40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' With this postulates, the distribution of entanglement appears more explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Multipartite entanglement measure is always defined via the bipartite entanglement measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Let SX be the set of all density matrices acting on the state space HX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Recall that, a function E : SAB → R+ is called a mea- sure of entanglement [42, 43] if (1) E(σAB) = 0 for any separable density matrix σAB ∈ SAB, and (2) E be- haves monotonically decreasing under local operations and classical communication (LOCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Moreover, convex measures of entanglement that do not increase on average 2 under LOCC are called entanglement monotones [42, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' By replacing SAB with SA1A2···An, it is just the mul- tipartite entanglement measure/monotone, and denoted by E(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Any bipartite entanglement monotone corre- sponds to a concave function on the reduced state when it is evaluated for the pure states [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' For any entangle- ment measure E, if h � ρA� = E � |ψ⟩⟨ψ|AB� (1) is concave, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' h[λρ1 + (1 − λ)ρ2] ≥ λh(ρ1) + (1 − λ)h(ρ2) for any states ρ1, ρ2, and any 0 ≤ λ ≤ 1, then the convex roof extension of E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', EF � ρAB� ≡ min �n j=1 pjE � |ψj⟩⟨ψj|AB� , is an entanglement mono- tone, where the minimum is taken over all pure state decompositions of ρAB = �n j=1 pj|ψj⟩⟨ψj|AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We call h the reduced function of E and HA the reduced subsystem throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' An n-partite pure state |ψ⟩ ∈ HA1A2···An is called biseparable if it can be written as |ψ⟩ = |ψ⟩X ⊗ |ψ⟩Y for some bipartition of A1A2 · · · An (for exam- ple, A1A3|A2A4 is a bipartition of A1A2A3A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' An n- partite mixed state ρ is biseparable if it can be writ- ten as a convex combination of biseparable pure states ρ = � i pi|ψi⟩⟨ψi|, wherein the contained {|ψi⟩} can be biseparable with respect to different bipartitions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', a mixed biseparable state does not need to be separable with respect to any particular bipartition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' If ρ is not biseparable, then it is called genuinely entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' A mul- tipartite entanglement measure E(n) is called a genuine multipartite entanglement measure if (i) E(n)(σ) = 0 for any biseparable state σ, (ii) E(n)(ρ) > 0 for any genuine entangled state, and (iii) it is convex [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' A genuine multipartite entanglement measure is called a genuine multipartite entanglement monotone (GMEM) if it does not increase on average under LOCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [25, 26, 40], we present MEMs and GMEMs that are defined by the sum of the reduced function on pure states and then extended to mixed states via the convex-roof structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' The aim of this paper is to give a condition that can justify when the MEMs and the GMEMs defined in this way is complete and completely monogamous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Moreover, we give another way of defining MEMs and the GMEMs from the maximal reduced func- tion and then discuss when these quantities are complete and completely monogamous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' II, we introduce some preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' III discusses the properties of the reduced functions of the entanglement monotones so far in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' IV is divided into two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' A discusses the MEM defined by the sum of reduced functions, and in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' B, we give the MEMs defined by the max- imal reduced function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Both of theses two MEMs are explored under the framework of the complete measure and the complete monogamy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' V, we consider three kinds of GMEMs which are defined by the sum of reduced functions, the maximal reduced function, and the minimal reduced function, respectively, under the framework the complete measure and the complete monogamy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We present a conclusion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' NOTATIONS AND PRELIMINARIES The framework of the complete entanglement mea- sure/monotone is closely related to the coarser relation of multipartite partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We first introduce three kinds of coarser relation in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' A, from which we then re- view the complete MEM, complete GMEM, monogamy relation and complete monogamy relation, respectively, in the latter three subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Coarser relation of multipartite partition Let X1|X2| · · · |Xk and Y1|Y2| · · · |Yl be two partitions of A1A2 · · · An or subsystem of A1A2 · · · An (for instance, partition AB|C|DE is a 3-partition of the 5-particle sys- tem ABCDE with X1 = AB, X2 = C and X3 = DE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We denote by [40] X1|X2| · · · |Xk ≻a Y1|Y2| · · · |Yl, (2) X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl, (3) X1|X2| · · · |Xk ≻c Y1|Y2| · · · |Yl (4) if Y1|Y2| · · · |Yl can be obtained from X1|X2| · · · |Xk by (a) discarding some subsystem(s) of X1|X2| · · · |Xk, (b) combining some subsystems of X1|X2| · · · |Xk, (c) discarding some subsystem(s) of some subsystem(s) Xk provided that Xk = Ak(1)Ak(2) · · · Ak(f(k)) with f(k) ⩾ 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' For example, A|B|C|D ≻a A|B|D ≻a B|D, A|B|C|D ≻b AC|B|D ≻b AC|BD, A|BC ≻c A|B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Furthermore, if X1|X2| · · · |Xk ≻ Y1|Y2| · · · |Yl, we de- note by Ξ(X1|X2| · · · |Xk −Y1|Y2| · · · |Yl) the set of all the partitions that are coarser than X1|X2| · · · |Xk and either exclude any subsystem of Y1|Y2| · · · |Yl or include some but not all subsystems of Y1|Y2| · · · |Yl [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' For exam- ple, Ξ(A|B|CD|E − A|B) = {CD|E, A|CD|E, B|CD|E, A|CD, B|CD, B|C|E, B|D|E, A|D|E, A|C|E, A|E, B|E, A|C, A|D, B|C, B|D, C|E, D|E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Complete MEM A multipartite entanglement measure E(n) is called a unified multipartite entanglement measure if it satisfies the unification condition [26]: (i) (additivity): E(n)(A1A2 · · · Ak ⊗ Ak+1 · · · An) = E(k)(A1A2 · · · Ak) + E(n−k)(Ak+1 · · · An), (5) 3 holds for all ρA1A2···An ∈ SA1A2···An, hereafter E(n)(X) refers to E(n)(ρX);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (ii) (permutation invariance): E(n)(A1A2 · · · An) = E(n)(Aπ(1)Aπ(2) · · · Aπ(n)), for all ρA1A2···An ∈ SA1A2···An and any permutation π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (iii) (coarsening monotone): E(k)(X1|X2| · · · |Xk) ⩾ E(l)(Y1|Y2| · · · |Yl) (6) holds for all ρA1A2···An ∈ SA1A2···An when- ever X1|X2| · · · |Xk ≻a Y1|Y2| · · · |Yl, where X1|X2| · · · |Xk and Y1|Y2| · · · |Yl are two partitions of A1A2 · · · An or subsystem of A1A2 · · · An, the vertical bar indicates the split across which the en- tanglement is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='. E(n) is called a complete multipartite entanglement mea- sure if it satisfies both the conditions above and the hi- erarchy condition [26]: (iv) (tight coarsening monotone): Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (6) holds for all ρ ∈ SA1A2···An whenever X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Complete GMEM Let E(n) g be a genuine multipartite entanglement mea- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' It is defined to be a unified genuine multipartite entanglement measure if it satisfies the unification con- dition [40], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', (i) (permutation invariance): E(n) g (A1A2 · · · An) = E(n) g (Aπ(1)Aπ(2) · · · Aπ(n)), for all ρA1A2···An ∈ SA1A2···An g and any permutation π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (ii) (coarsening monotone): E(k) g (X1|X2| · · · |Xk) > E(l) g (Y1|Y2| · · · |Yl) (7) holds for all ρA1A2···An ∈ SA1A2···An g whenever X1|X2| · · · |Xk ≻a Y1|Y2| · · · |Yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' A unified GMEM E(n) g is call a complete genuine multi- partite entanglement measure if E(n) g admits the hierar- chy condition [40], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', (iii) (tight coarsening monotone): E(k) g (X1|X2| · · · |Xk) ≥ E(l) g (Y1|Y2| · · · |Yl) (8) holds for all ρ ∈ SA1A2···An g whenever X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Monogamy Relation For an bipartite entanglement measure E, E is said to be monogamous if [9, 39] E(A|BC) ⩾ E(AB) + E(AC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (9) However, Equation (9) is not valid for many entangle- ment measures [9, 35, 37] but some power function of Q admits the monogamy relation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', Eα(A|BC) ⩾ Eα(AB) + Eα(AC) for some α > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [37], we improved the definition of monogamy as: A bipartite measure of entanglement E is monogamous if for any ρ ∈ SABC that satisfies the disentangling condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', E(ρA|BC) = E(ρAB), (10) we have that E(ρAC) = 0, where ρAB = TrCρABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' With respect to this definition, a continuous measure E is monogamous according to this definition if and only if there exists 0 < α < ∞ such that Eα(ρA|BC) ⩾ Eα(ρAB) + Eα(ρAC) (11) for all ρ acting on the state space HABC with fixed dim HABC = d < ∞ (see Theorem 1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [26], in order to characterize the distribution of entanglement in a “complete” sense, the term “com- plete monogamy” of the unified multipartite entangle- ment measure is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' For a unified multipartite entanglement measure E(n), it is said to be completely monogamous if for any ρ ∈ SA1A2···An that satisfies [26] E(k)(X1|X2| · · · |Xk) = E(l)(Y1|Y2| · · · |Yl) (12) with X1|X2| · · · |Xk ≻a Y1|Y2| · · · |Yl we have that E(∗) g (Γ) = 0 (13) holds for all Γ ∈ Ξ(X1|X2| · · · |Xk − Y1|Y2| · · · |Yl), here- after the superscript (∗) is associated with the partition Γ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', if Γ is a n-partite partition, then (∗) = (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' For example, E(3) is completely monogamous if for any ρABC that admits E(3)(ABC) = E(2)(AB) we get E(2)(AC) = E(2)(BC) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Let E(n) be a complete multipartite en- tanglement measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' E(n) is defined to be tightly com- plete monogamous if for any ρ ∈ SA1A2···An that satis- fies [26] E(k)(X1|X2| · · · |Xk) = E(l)(Y1|Y2| · · · |Yl) (14) with X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl we have that E(∗) g (Γ) = 0 (15) holds for all Γ ∈ Ξ(X1|X2| · · · |Xk − Y1|Y2| · · · |Yl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' For instance, E(3) is tightly complete monogamous if for any ρABC that admits E(3)(ABC) = E(2)(A|BC) we have E(2)(BC) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Let E(n) g be a genuine multipartite entanglement mea- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We denote by SA1A2···Am g the set of all genuine en- tangled states in SA1A2···Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' E(n) g is completely monog- amous if it obeys Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (7) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' A complete genuine mul- tipartite entanglement measure E(n) g is tightly complete 4 monogamous if it satisfies the genuine disentangling con- dition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', either for any ρ ∈ SA1A2···Am g that satis- fies [40] E(k) g (X1|X2| · · · |Xk) = E(l) g (Y1|Y2| · · · |Yl) (16) with X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl we have that E(∗) g (Γ) = 0 (17) holds for all Γ ∈ Ξ(X1|X2| · · · |Xk − Y1|Y2| · · · |Yl), or E(k) g (X1|X2| · · · |Xk) > E(l) g (Y1|Y2| · · · |Yl) (18) holds for any ρ ∈ SA1A2···Am g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [26], we showed that the tightly complete monogamy is stronger than the complete monogamy for the complete MEMs that defined by the convex-roof ex- tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' One can easily find that it is also true for any complete GMEM defined by the convex-roof extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' STRICT CONCAVITY AND SUBADDITIVITY OF THE REDUCED FUNCTION Any entanglement monotone, when evaluated on pure states, is uniquely determined by its reduced function and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Therefore, the feature of the entangle- ment monotone defined via the convex-roof extension rests with the quality of its reduced function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [38], we proved that the bipartite entanglement monotone is monogamous whenever its reduced function is strictly concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In ths Section, we review all the reduced func- tions of the entanglement monotones in literature so far and then discuss the subadditivity of theses functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' As what we will show in the next two Sections, the subaddi- tivity is affinitive with the completeness of the measures for some kind of MEM/GMEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Strict concavity The reduced functions of the entanglement of for- mation Ef [45, 46], tangle τ [47], concurrence C [48– 50], negativity N [51], the Tsallis q-entropy of entangle- ment Eq [52], and the R´enyi α-entropy of entanglement Eα [44, 53] are h(ρ) = S(ρ), hτ(ρ) = h2 C(ρ) = 2(1 − Trρ2), hN(ρ) = 1 2[(Tr√ρ)2 − 1], hq(ρ) = 1 − Trρq q − 1 , q > 0, hα(ρ) = (1 − α)−1 ln(Trρα), 0 < α < 1, respectively, where S is the von Neumann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' It has been shown that h, hτ, hC, hN, hq, and hα are not only concave but also strictly concave [38, 44, 54] (where the strict concavity of hN is proved very recently in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' The reduced functions of the entanglement monotones induced by the fidelity-based distances EF, EF ′, and EAF are [56] hF(ρ) = 1 − Trρ3, hF ′(ρ) = 1 − � Trρ2�2 , hAF(ρ) = 1 − � Trρ3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' They are strictly concave [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [55], four kinds of partial norm of entangle- ment are investigated: the partial-norm of entanglement E2, the minimal partial-norm of entanglement Emin, the reinforced minimal partial-norm of entanglement Emin′, and the partial negativity ˆN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' The reduced functions of E2, Emin, E′ min, and ˆN are h2(ρ) = 1 − ∥ρ∥, hmin(ρ) = ∥ρ∥min, hmin′(ρ) = r(ρ)∥ρ∥min, ˆh(ρ) = � δ1δ2, where r(ρ) denotes the rank of ρ, ∥ · ∥ is the operator norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', ∥X∥ = sup|ψ⟩ ∥A|ψ⟩∥, ∥ρ∥min = � λ2 min, λmin < 1, 0, λmin = 1, and δ1, δ2 are the two largest eigenvalues of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' All of them are concave but not strictly concave (ˆh is only strictly concave on qubit states), and these entanglement mono- tones are not monogamous [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Subadditivity We summarize the subadditivity of the reduced func- tions in literature as following: (i) S is additive and subadditive [54], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', S(ρ ⊗ σ) = S(ρ) + S(σ) (19) and S(ρAB) ≤ S(ρA) + S(ρB), (20) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (ii) Sq is subadditive iff q > 1, but not additive, and for 0 < q < 1, Sq is neither subadditive nor super- additive [57] (superadditivity refers to Sq(ρAB) ⩾ Sq(ρA) + Sq(ρB)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In addition, Sq(ρA ⊗ ρB) = Sq(ρA) + Sq(ρB) (21) iff ρA or ρB is pure [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' 5 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Comparing of the properties of the reduced func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' C, SC, SA, and A signify the function is concave, strictly concave, subadditive, and additive, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' E h C SC SA A Ef S ✓ ✓ ✓ ✓ C � 2(1 − Trρ2) ✓ ✓ ✓ × τ 2(1 − Trρ2) ✓ ✓ ✓ × Eq 1−Trρq q−1 ✓(q > 0) ✓(q > 1) ✓(q > 1) × Eα ln(Trρα) 1−α , α ∈ (0, 1) ✓ ✓ × ✓ NF (Tr√ρ)2−1 2 ✓ ✓ × × EF 1 − Trρ3 ✓ ✓ ✓ × EF′ 1 − (Trρ2)2 ✓ ✓ ✓a × EAF 1 − � Trρ3 ✓ ✓ ✓a × E2 1 − ∥ρ∥ ✓ × ✓ × Emin ∥ρ∥min ✓ × × × Emin′ r(ρ)∥ρ∥min ✓ × × × ˆ N √ δ1δ2 ✓b × ✓a × a We conjecture that they are subadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' b We conjecture that it is concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (iii) hα is additive but not subadditive [58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (iv) hτ is subadditive [60], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', 1 + Trρ2 AB ≥ Trρ2 A + Trρ2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (22) In particular, the equality holds iff ρA or ρB is pure [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (v) hN is neither subadditive nor supperadditive [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Item (iv) implies hC is subadditive and the equality holds iff ρA or ρB is pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' hF is subadditive since it coincides with Sq/2 (q = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We conjecture that hF ′ and hAF are subadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' h2 is subadditive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', 1 + ∥ρAB∥ ⩾ ∥ρA∥ + ∥ρB∥ (23) holds for any ρAB ∈ SAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In particular, the equality holds iff ρA or ρB is a pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Note that partial trace is a quantum channel and any quantum channel can be regarded as a operator on the space of the trace-class operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' The norm of quan- tum channel in such a sense is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Therefore ∥ρAB∥ ⩾ ∥ρA,B∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Moreover, if 1 + ∥ρAB∥ = ∥ρA∥ + ∥ρB∥, then ∥ρA∥ = 1 or ∥ρB∥ = 1, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Let ρAB = 1 2|ψ⟩⟨ψ| + 1 2|φ⟩⟨φ| with |ψ⟩ = � 4 5|00⟩+ � 1 5|11⟩ and |φ⟩ = � 4 5|22⟩+ � 1 5|33⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' It is clear that ∥ρAB∥min = 1 2 > ∥ρA∥min + ∥ρB∥min = 1/10 + 1/10 = 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' That is, ∥ · ∥min is not subaddi- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Clearly, hmin′ is also not subadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' According to Proposition 1, hmin and hmin′ are subadditive on the states that satisfies r(ρAB) = r(ρA) = r(ρB) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' One can easily verifies that h2, hmin, hmin′, and ˆh are not additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We conjecture that ˆh is subadditive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', ˆh(ρAB) ⩽ ˆh(ρA) + ˆh(ρB) (24) holds for any ρAB ∈ SAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In what follows, we always assume that hF ′, hAF, and ˆh are subadditive, and that ˆh is concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' The reduced functions of parametrized entanglement monotones in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [61] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [62] are hq′(ρ) = 1 − Trρq, q > 1, and hα′(ρ) = Trρα − 1, 0 < α < 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Obviously, the properties of these two func- tions above are the same as that of hq, although they are different from Eq [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We summarize the properties of theses reduced functions in Table I for more conve- nience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' COMPLETE MEM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Complete MEM from sum of the reduced functions In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [26], we put forward several complete MEMs defined by the sum of the reduced functions on all the single subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In fact, this scenario is valid for all en- tanglement monotones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Let |ψ⟩A1A2···An be a pure state in HA1A2···An and h be a non-negative concave function on SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We define E(n)(|ψ⟩A1A2···An) = 1 2 � i h(ρAi) (25) and then extend it to mixed states by the convex-roof structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We denote E(n) by E(n) f , C(n), τ (n), E(n) q , E(n) α , N (n) F , E(n) F , E(n) F ′ , E(n) AF, E(n) 2 , E(n) min, E(n) min′, and ˆN (n) whenever h = S, hC, hτ, hq, hα, hN, hF, hF ′, hAF, h2, hmin, hmin′, and ˆh, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Here, E(n) f , C(n), τ (n), E(n) q , E(n) α , and N (n) F have been discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [26] for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' The coefficient “1/2” is fixed by the unification condition when E(n) is regarded as a unified MEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' One need note here that E(n) F , E(n) F ′ , and E(n) AF are different from E(n) F,F , E(n) F ′,F , and E(n) AF,F respectively in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Let E(n) be a non-negative function defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Then the following statements hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' 6 (i) E(n) is a unified MEM and is completely monoga- mous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (ii) E(n) is a complete MEM iff h is subadditive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (iii) E(n) is tightly complete monogamous iff h is sub- additive with h(ρAB) = h(ρA) + h(ρB) ⇒ ρAB is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (26) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We only need to discuss the case of n = 3 with no loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (i) For any |ψ⟩ABC ∈ HABC, we let E(2)(ρAB) = � i piE(2)(|ψi⟩) = 1 2 � i pi[h(ρA i ) + h(ρB i )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Then E(3)(|ψ⟩ABC) = 1 2 � h(ρA) + h(ρB) + h(ρC) � ≥ 1 2 � h(ρA) + h(ρB) � ≥ 1 2 � i pi[h(ρA i ) + h(ρB i )] = E(2)(ρAB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' That is, E(3) satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (6) for pure states and it is completely monogamous on pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' For any mixed state ρABC, we let E(3)(ρABC) = � j qjE(3)(|ψj⟩) and E(2)(ρAB j ) = � i pi(j)E(2)(|ψi(j)⟩) = 1 2 � i pi(j)[h(ρA i(j)) + h(ρB i(j))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Then E(3)(ρABC) = 1 2 � j qj � h(ρA j ) + h(ρB j ) + h(ρC j ) � ≥ 1 2 � j qj � hj(ρA) + hj(ρB) � ≥ 1 2 � i,j qjpi(j)[h(ρA i(j)) + h(ρB i(j))] ≥ E(2)(ρAB), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', it is a unified MEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' If E(3)(ρABC) = E(2)(ρAB), it yields h(ρC j ) = 0 for any j, and thus |ψj⟩ABC = |ψj⟩AB|ψj⟩C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Therefore it is completely monogamous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (ii) If E(3) is a complete MEM, then E(3)(|ψ⟩ABC) ≥ E(2)(|ψ⟩A|BC) for any |ψ⟩ABC, which implies h(ρBC) ≤ h(ρB) + h(ρC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' That is, h is subadditive since |ψ⟩ABC is arbitrarily given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Conversely, if h is subadditive, then E(3)(|ψ⟩ABC) ≥ E(2)(|ψ⟩A|BC) for any pure state |ψ⟩ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' For any mixed state ρABC, we let E(3)(ρABC) = � j qjE(3)(|ψj⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Then E(3)(ρABC) = 1 2 � j qj � h(ρA j ) + h(ρB j ) + h(ρC j ) � ≥ 1 2 � j qj � hj(ρA) + hj(ρBC) � ≥ E(2)(ρA|BC), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', it is a complete MEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (iii) It can be easily checked using the argument anal- ogous to that of (ii) together with the fact that, if E(n) is tightly complete monogamous, it is automatically a complete MEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Comparing of E(n) with different different reduced functions, and E (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' CM and TCM signify the measure is completely monogamous and tightly completel monogamous, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' MEM Unified Complete CM TCM E(n) f ✓ ✓ ✓ ✓ C(n) ✓ ✓ ✓ ✓ τ (n) ✓ ✓ ✓ ✓ E(n) q ✓ ✓ ✓ ✓a E(n) α ✓ × ✓ × N (n) F ✓ × ✓ × E(n) F ✓ ✓ ✓ ✓a E(n) F′ ✓ ✓b ✓ ✓a E(n) AF ✓ ✓b ✓ ✓a E(n) 2 ✓ ✓ ✓ ✓ E(n) min ✓ × ✓ × E(n) min′ ✓ × ✓ × ˆ N (n) ✓ ✓b ✓ ✓a E (n) (n ≥ 4) ✓ ✓ ✓ ✓ a It is tightly complete monogamous under the assumption that h is subadditive and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (26) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' b It is complete under the assumption that h is subadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' By Theorem 1, we can conclude: (i) E(n) f , C(n), τ (n), E(n) q , E(n) α , N (n) F , E(n) F , E(n) F ′ , E(n) AF, E(n) 2 , E(n) min, E(n) min′, and ˆN (n) are unified MEMs and are completely monog- amous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (ii) E(n) f , C(n), τ (n), E(n) q , E(n) F , E(n) F ′ , E(n) AF, E(n) 2 , and ˆN (n) are complete MEMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (iii) E(n) α , N (n) F , E(n) min, and E(n) min′ are not complete MEMs since the asso- ciated reduced functions are not subadditive which vio- late the hierarchy condition for some states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' (iv) E(n) f , C(n), τ (n), and E(n) 2 are tightly complete monogamous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' However E(n) 2 , E(n) min, E(n) min′, and ˆN (n) are not monoga- mous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='Together with Theorem in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [38], we obtain that, for these MEMs, both monogamy and tightly complete monogamy are stronger than the complete monogamy under the frame work of the complete MEM, and that monogamy is stronger than both complete monogamy and tightly complete monogamy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=', E(n) 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In particular, if h is subadditive with h(ρAB) = h(ρA) + h(ρB) implies ρAB = ρA ⊗ ρB, then E(n) is tightly complete monogamous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' S, hτ, hC, and h2 be- long to such situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We also conjecture that hq, hF, hF ′, hAF, and ˆh belong to such situations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' That is, we conjecture that E(n) q , E(n) F , E(n) F ′ , E(n) AF, and ˆN (n) are tightly complete monogamous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' [25], we put forward several multipartite en- tanglement measures which are defined by the sum of all bipartite entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' Let |ψ⟩A1A2···An be a pure state in HA1A2···An and h be a non-negative concave function 7 on SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQffPgF/content/2301.00334v1.pdf'} +page_content=' We define [25] E(n)(|ψ⟩A1A2···An) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 2 � i1≤···≤is,s 0 for any t ∈ I, it follows that there exists a t ∈ I and a unit vector ν1(t) ∈ S1 at the t ∈ I +satisfying +dλ +dt (t) = −β(t) (µ(t) · ν1(t)) . +On the other hand, since C(γ,λ) is creative, there must exist a smooth unit vector field �ν : I → S1 +along γ : I → R2 such that +dλ +dt (t) = −β(t) (µ(t) · �ν(t)) +for any t ∈ I. Suppose that the parameter t ∈ I is a regular point of γ. Then, β(t) ̸= 0 at the t ∈ I. +Thus, at the t ∈ I, the unit vector ν1(t) must be �ν(t). Therefore, by the proof of (1) of Theorem 1, at +the regular point t ∈ I of γ, it follows +D = E2. +Suppose that the parameter t ∈ I is a singular point of γ. Then, β(t) = 0 at the t ∈ I. Thus, for any +unit vector v ∈ S1, the following holds at the t ∈ I. +dλ +dt (t) = −β(t) (µ(t) · v) . +Hence, at the singular point t ∈ I, we may choose any unit vector v ∈ S1 as the unit vector ν1(x). +Therefore, by the proof of (1) of Theorem 1, at the singular point t ∈ I of γ, it follows +D = E2 ∪ C(γ(t),λ(t)). +□ + +ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE +13 +Acknowledgement +The first author is supported by the National Natural Science Foundation of China (Grant No. +12001079), Fundamental Research Funds for the Central Universities (Grant No. 3132023205) and China +Scholarship Council. +References +[1] J. W. Bruce and P. J. Giblin, Curves and Singularities (second edition), Cambridge University Press, Cambridge, 1992. +https://doi.org/10.1017/CBO9781139172615 +[2] T. Fukunaga and M. Takahashi, Existence and uniqueness for Legendre curves, J. Geom., 104 (2013), 297–307. +https://doi.org/10.1007/s00022-013-0162-6 +[3] E. Hairer and G. Wanner, Analysis by Its History, Undergraduate Texts in Mathematics, Springer New York, NY, +2008. https://doi.org/10.1007/978-0-387-77036-9 +[4] G. Ishikawa, Singularities of frontals, Adv. Stud. Pure Math., 78, 55–106, Math. Soc. Japan, Tokyo, 2018. +https://doi.org/10.2969/aspm/07810055 +[5] S. Janeczko and T. Nishimura, Anti-orthotomics of frontals and their applications, J. Math. Anal. Appl., 487 (2020), +124019. https://doi.org/10.1016/j.jmaa.2020.124019 +[6] T. Nishimura, Hyperplane families creating envelopes, Nonlinearity, 35 (2022), 2588. https://doi.org/10.1088/1361- +6544/ac61a0 +School of Science, Dalian Maritime University, Dalian 116026, P.R. China +Email address: wangyq@dlmu.edu.cn +Research Institute of Environment and Information Sciences, Yokohama National University, Yokohama +240-8501, Japan +Email address: nishimura-takashi-yx@ynu.ac.jp + diff --git a/JdE3T4oBgHgl3EQfXQrW/content/tmp_files/load_file.txt b/JdE3T4oBgHgl3EQfXQrW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ede3b510fe94d86234108bfd767fdd2590acbbb5 --- /dev/null +++ b/JdE3T4oBgHgl3EQfXQrW/content/tmp_files/load_file.txt @@ -0,0 +1,466 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf,len=465 +page_content='ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE YONGQIAO WANG AND TAKASHI NISHIMURA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In this paper, on envelopes created by circle families in the plane, answers to all four basic problems (existence problem, representation problem, problem on the number of envelopes, problem on relationships of definitions) are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Introduction Throughout this paper, I is an open interval and all functions, mappings are of class C∞ unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Envelopes of planar regular curve families have fascinated many pioneers since the dawn of differential analysis (for instance, see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In most typical cases, straight line families have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In [6], by solving four basic problems on envelopes created by straight line families in the plane (existence problem, representation problem, uniqueness problem and equivalence problem of definitions), the second author constructs a general theory for envelopes created by straight line families in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' On the other hand, circle families in the plane are non-negligible families because the envelopes of them have already had an important application, namely, an application to Seismic Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Following 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='14(9) of [1], a brief explanation of Seismic Survey is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In the Eucledian plane R2, consider the “ground level curve” C parametrized by γ : I → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Suppose that there is a stratum of granite below the top layer of sandstone and that the dividing curve, denoted by M, is parametrized by �f : I → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Seismic Survey is the following method to obtain an approximation of �f as precisely as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Take one fixed point A of C and consider an explosion at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Assume that the sound waves travel in straight lines and are reflected from M, arriving back at points γ(t) of C where their times of arrival are exactly recorded by sensors located along C (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' It is known that there exists a curve W parametrized by f : I → R2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Reflection of sound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' with well-defined normals such that each broken line of a reflected ray starting at A and finishing on C 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 57R45, 58C25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Circle family, Envelope, Frontal, Creative, Creator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='04478v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='DG] 11 Jan 2023 A (ti)(t2)(t3)(t4)(ts) C M2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' NISHIMURA can be replaced by a straight line which is normal to W and of the same total length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The curve W is called the orthotomic of M relative to A and conversely the curve M is called the anti-orthotomic of W relative to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, an envelope created by the circle family � (x, y) ∈ R2 �� ||(x, y) − γ(t)|| = ||f(t) − γ(t)|| � t∈I recovers W (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' After obtaining the parametrization f of W, the parametrization �f of M Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' An envelope created by the circle family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' can be easily obtained by using the anti-orthotomic technique developed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, in order to investigate the parametrization of W as precisely as possible, construction of general theory on envelopes created by circle families is very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In this paper, we construct a general theory on envelopes created by circle families in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' For a point P of R2 and a positive number λ, the circle C(P,λ) centered at P with radius λ is naturally defined as follows, where the dot in the center stands for the standard scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' C(P,λ) = � (x, y) ∈ R2 �� ((x, y) − P) · ((x, y) − P) = λ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' For a curve γ : I → R2 and a positive function λ : I → R+, the circle family C(γ,λ) is naturally defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Here, R+ stands for the set consisting of positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' C(γ,λ) = � C(γ(t),λ(t)) � t∈I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' It is reasonable to assume that at each point γ(t) the normal vector to the curve γ is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, we easily reach the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' A curve γ : I → R2 is called a frontal if there exists a mapping ν : I → S1 such that the following identity holds for each t ∈ I, where S1 is the unit circle in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' dγ dt (t) · ν(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' For a frontal γ, the mapping ν : I → S1 given above is called the Gauss mapping of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By definition, a frontal is a solution of the first order linear differential equation defined by Gauss mapping ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, for a fixed mapping ν : I → S1 the set consisting of frontals with a given Gauss mapping ν : I → S1 is a linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' For frontals, [4] is recommended as an excellent reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hereafter in this paper, the curve γ : I → R2 for a circle family C(γ,λ) is assumed to be a frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In this paper, the following is adopted as the definition of an envelope created by a circle family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let C(γ,λ) be a circle family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' A mapping f : I → R2 is called an envelope created by C(γ,λ) if there exists a mapping �ν : I → S1 such that the following two hold for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (1) df dt(t) · �ν(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' A (ti) (t2) (t3) (t4) (ts) f(ts) f(t4) f(t2)f(t3) f(ti)ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE 3 (2) f(t) ∈ C(γ(t),λ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By definition, as same as an envelope created by a hyperplane family (see [6]), an envelope created by a circle family is a solution of a first order linear differential equation with one constraint condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Moreover, again by definition, an envelope created by a circle family is a frontal with Gauss mapping �ν : I → S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' On the other hand, since there is one constraint condition, again as same as an envelope created by a hyperplane family, the set of envelopes created by a given circle family is in general not a linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (1) Given a circle family C(γ,λ), find a necessary and sufficient codition for the family to create an envelope in terms of γ, ν and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (2) Suppose that a circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, find a parametrization of the envelope in terms of γ, ν and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (3) Suppose that a circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, find a criterion for the number of distinct envelopes created by C(γ,λ) in terms of γ, ν and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Note 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (1) (1) of Problem 1 is a problem to seek the integrability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' There are various cases, for instance the concentric circle family {{(x, y) ∈ R2 | x2 + y2 = t2}}t∈R+ does not create an envelope while the parallel-translated circle family {{(x, y) ∈ R2 | (x − t)2 + y2 = 1}}t∈R does create two envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, (1) of Problem 1 is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (2) The following Example 1 shows that the apparently well-known method to obtain the envelope seems to be useless in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, (2) of Problem 1 is important and the positive answer to it is much desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (3) The following Example 2 shows that there are at least three cases: the case having a unique envelope, the case having exactly two envelopes and the case having uncountably many envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, (3) of Problem 1 is meaningful and interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let γ : R → R2 be the mapping defined by γ(t) = � t3, t6� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Set ν(t) = 1 √ 4t6+1 � −2t3, 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' It is clear that the mapping γ is a frontal with Gauss mapping ν : R → S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let λ : R → R+ be the constant function defined by λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, it seems that the circle family C(γ,λ) creates envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, we can expect that the created envelopes can be obtained by the well-known method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Set F(x, y, t) = � x − t3�2 + � y − t6�2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' D = � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' y) ∈ R2 ���� ∃t such that F(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' t) = ∂F ∂t (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' t) = 0 � = � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' y) ∈ R2 ��� ∃t such that � x − t3�2 + � y − t6�2 − 1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' −6t2 � x − t3� − 12t5 � y − t6� = 0 � = � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' y) ∈ R2 ��� ∃t such that � x − t3�2 + � y − t6�2 − 1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' t2 �� x − t3� + 2t3 � y − t6�� = 0 � = � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' y) ∈ R2 �� x2 + y2 = 1 � � � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' y) ∈ R2 ��� � x − t3�2 + � y − t6�2 − 1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' x = t3 − 2t3 � y − t6�� = � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' y) ∈ R2 �� x2 + y2 = 1 � � � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' y) ∈ R2 ��� � −2t3 � y − t6��2 + � y − t6�2 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' x = t3 � 1 − 2y + 2t6�� = � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' y) ∈ R2 �� x2 + y2 = 1 � � �� t3 ∓ 2t3 √ 4t6 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' t6 ± 1 √ 4t6 + 1 � ∈ R2 ���� t ∈ R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In Example 3 of Section 3, it turns out that the set D calculated here is actually larger than the set of envelopes created by C(γ,λ), namely the unit circle � (x, y) ∈ R2 �� x2 + y2 = 1 � is redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, unfortunately, the apparently well-known method to obtain the envelopes does not work well in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The circle family C(γ,λ) and the candidate of its envelope are depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (1) Let γ : R+ → R2 be the mapping defined by γ(t) = (0, 1 + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, it is clear that γ is a frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let λ : R+ → R+ be the positive function defined by λ(t) = 1+t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, it is easily seen that the origin (0, 0) of the plane R2 itself is a created envelope by the circle family C(γ,λ) and that there are no other envelopes created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence, the number of created envelopes is one in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (2) The parallel-translated circle family {{(x, y) ∈ R2 | (x − t)2 + y2 = 1}}t∈R creates exactly two envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (3) Let γ : R → R2 be the constant mapping defined by γ(t) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, it is clear that γ is a frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let λ : R → R+ be the constant function defined by λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, for any function 4 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' NISHIMURA Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The circle family C(γ,λ) and the candidate of its envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' θ : R → R, the mapping f : R → R2 defined by f(t) = (cos θ(t), sin θ(t)) is an envelope created by the circle family C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence, there are uncountably many created envelopes in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In order to solve Problem 1, we prepare several terminologies that can be derived from a frontal γ : I → R2 with Gauss mapping ν : I → S1 and a positive function λ : I → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' For a frontal γ : I → R2 with Gauss mapping ν : I → S1, following [2], we set µ(t) = J(ν(t)), where J is the anti-clockwise rotation by π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then we have a moving frame {µ(t), ν(t)}t∈I along the frontal γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Set ℓ(t) = dν dt (t) · µ(t), β(t) = dγ dt (t) · µ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The pair of functions (ℓ, β) is called the curvature of the frontal γ with Gauss mapping ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' We want to focus the ratio of dλ dt (t) and β(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The following definition is the key of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let γ : I → R2, λ : I → R+ be a frontal with Gauss mapping ν : I → S1 and a positive function respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, the circle family C(γ,λ) is said to be creative if there exists a mapping �ν : I → S1 such that the following identity holds for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' dλ dt (t) = −β(t) (�ν(t) · µ(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Set cos θ(t) = −�ν(t) · µ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, the creative condition is equivalent to say that there exists a function θ : I → R satisfying the following identity for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' dλ dt (t) = β(t) cos θ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By definition, any family of concentric circles with expanding radius is not creative, and it is clear that such the circle family does not create an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Under the above preparation, Problem 1 is solved as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let γ : I → R2 be a frontal with Gauss mapping ν : I → S1 and let λ : I → R+ be a positive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, the following three holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (1) The circle family C(γ,λ) creates an envelope if and only if C(γ,λ) is creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (2) Suppose that the circle family C(γ,λ) creates an envelope f : I → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, the created envelope f is represented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' f(t) = γ(t) + λ(t)�ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' where �ν : I → S1 is the mapping defined in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE 5 (3) Suppose that the circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, the number of envelopes created by C(γ,λ) is characterized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (3-i) The circle family C(γ,λ) creates a uinique envelope if and only if the set consisting of t ∈ I satisfying β(t) ̸= 0 and dλ dt (t) = ±β(t) is dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (3-ii) There are exactly two distinct envelopes created by C(γ,λ) if and only if the set of t ∈ I satisfying β(t) ̸= 0 is dense in I and there exists at least one t0 ∈ I such that the strict inequality | dλ dt (t0)| < |β(t0)| holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (3-∞) There are uncountably many distinct envelopes created by C(γ,λ) if and only if the set of t ∈ I satisfying β(t) ̸= 0 is not dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By the assertion (2) of Theorem 1, it is reasonable to call �ν the creator for an envelope f created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Theorem 1 is proved in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In Section 3, several examples to which Theorem 1 is effectively applicable are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Finally, in Section 4, relations of several definitions of an envelope created by a circle family are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Proof of Theorem 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Proof of the assertion (1) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Suppose that C(γ,λ) is creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By definition, there exists a mapping �ν : I → S1 such that the equality dλ dt (t) = −β(t) (�ν(t) · µ(t)) holds for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Set f(t) = γ(t) + λ(t)�ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, since (f(t) − γ(t)) · (f(t) − γ(t)) = λ2(t), it follows f(t) ∈ C(γ(t),λ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Morever, since df dt (t) = dγ dt (t) + dλ dt (t)�ν(t) + λ(t)d�ν dt (t), we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' df dt (t) · (f(t) − γ(t)) = �dγ dt (t) + dλ dt (t)�ν(t) + λ(t)d�ν dt (t) � (λ(t)�ν(t)) = dγ dt (t) · (λ(t)�ν(t)) + dλ dt (t)λ(t) = (β(t)µ(t)) · (λ(t)�ν(t)) + (−β(t) (�ν(t) · µ(t))) λ(t) = β(t)λ(t) (µ(t) · �ν(t)) − β(t)λ(t) (�ν(t) · µ(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence, f is an envelope created by the circle family C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Conversely, suppose that the circle family C(γ,λ) creates an envelope f : I → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, by definition, it follows that f(t) ∈ C(γ(t),λ(t)) and df dt(t) · (f(t) − γ(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The condition f(t) ∈ C(γ(t),λ(t)) implies that there exists a mapping �ν : I → S1 such that the following equality holds for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' f(t) = γ(t) + λ(t)�ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, since df dt (t) = dγ dt (t) + dλ dt (t)�ν(t) + λ(t)d�ν dt (t), we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 0 = df dt (t) · (f(t) − γ(t)) = �dγ dt (t) + dλ dt (t)�ν(t) + λ(t)d�ν dt (t) � (λ(t)�ν(t)) = (β(t)µ(t)) · (λ(t)�ν(t)) + dλ dt (t)λ(t) = λ(t) � β(t) (µ(t) · �ν(t)) + dλ dt (t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 6 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' NISHIMURA Since λ(t) is positive for any t ∈ I, it follows β(t) (µ(t) · �ν(t)) + dλ dt (t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, the circle family C(γ,λ) is creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Proof of the assertion (2) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The proof of the assertion (1) given in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='1 proves the assertion (2) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Proof of the assertion (3) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Proof of (3-i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Suppose that the circle family C(γ,λ) creates a unique envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, for any t ∈ I the unit vector �ν(t) satisfying dλ dt (t) = −β(t) (�ν(t) · µ(t)) must be uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence, under considering continuity of two functions dλ dt and β, it follows that the set consisting of t ∈ I satisfying dλ dt (t) = ±β(t) ̸= 0 must be dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Conversely, suppose that the set consisting of t ∈ I satisfying dλ dt (t) = ±β(t) ̸= 0 is dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, under considering continuity of the function t �→ �ν(t) · µ(t), it follows that �ν(t) · µ(t) = ±1 for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, the created envelope f(t) = γ(t) + λ(t)�ν(t) must be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Proof of (3-ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Suppose that there are exactly two distinct envelopes created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, by the equality dλ dt (t) = −β(t) (�ν(t) · µ(t)) , the set consisting of t ∈ I satisfying β(t) ̸= 0 must be dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Suppose moreover that the set of t ∈ I satisfying the equality dλ dt (t) = ±β(t) holds for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, it follows that the set consisting of t ∈ I satisfying dλ dt (t) = ±β(t) ̸= 0 is dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, by the assertion (3-i), the given circle family must create a unique envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' This contradicts the assumption that there are exactly two distinct envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence, there must exist at least one t0 ∈ I such that the strict inequality | dλ dt (t0)| < |β(t0)| holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Conversely, suppose that the set of t ∈ I satisfying β(t) ̸= 0 is dense in I and there exists at least one t0 ∈ I such that the strict inequality | dλ dt (t0)| < |β(t0)| holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, it follows that there must exist an open interval �I in I such that the absolute value |�ν(t) · µ(t)| = | cos θ(t)| is less than 1 for any t ∈ �I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, it follows θ(t) ̸= −θ(t) for any t ∈ �I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence, for any t ∈ �I, there exist exactly two distinct unit vectors �ν+(t), �ν−(t) corresponding �ν+(t) · µ(t) = − cos θ(t) and �ν−(t) · µ(t) = − cos (−θ(t)) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, the circle family must create exactly two distinct envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Proof of (3-∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Suppose that there are uncountably many distinct envelopes created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Suppose moreover that the set of t ∈ I such that β(t) ̸= 0 is dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, from (3-i) and (3-ii), it follows that the circle family C(γ,λ) must create a unique envelope or two distinct envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' This contradicts the assumption that there are uncountably many distinct envelopes created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence, the set of t ∈ I such that β(t) ̸= 0 is never dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Conversely, suppose that the set of t ∈ I such that β(t) ̸= 0 is not dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' This assumption implies that there exists an open interval �I in I such that β(t) = 0 for any t ∈ �I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' On the other hand, since C(γ,λ) creates an envelope f0, the equality dλ dt (t) = −β(t) (�ν(t) · µ(t)) holds for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, there are no restrictions for the value �ν(t) · µ(t) for any t ∈ �I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Take one point t0 of �I and denote the �ν for the envelope f0 by �ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, by using the standard technique on bump functions, we may construct uncountably many distinct creators �νa : I → S1 (a ∈ A) such that the following (a), (b), (c) and (d) hold, where A is a set consisting uncountably many elements such that 0 ̸∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (a) The equality dλ dt (t) = −β(t) (�νa(t) · µ(t)) holds for any t ∈ I and any a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (b) For any t ∈ I − �I and any a ∈ A, the equality �νa(t) = �ν0(t) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (c) For any a ∈ A, the property �νa(t0) ̸= �ν0(t0) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (d) For any wo distinct a1, a2 ∈ A, the property �νa1(t0) ̸= �νa2(t0) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, the circle family C(γ,λ) creates uncountably many distinct envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 2 ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Examples Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' We examine Example 1 by applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In Example 1, γ : R → R2 is given by γ(t) = � t3, t6� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, we can say that ν : R → S1 and µ : R → S1 are given by ν(t) = 1 √ 4t6+1 � −2t3, 1 � and µ(t) = 1 √ 4t6+1 � −1, −2t3� respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Moreover, the radius function λ : R → R is the constant function defined by λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, dλ dt (t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By calculation, we have β(t) = dγ dt (t) · µ(t) = −3t2(1 + 4t6) √ 4t6 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, the unit vector �ν(t) ∈ S1 satisfying dλ dt (t) = −β(t) (�ν(t) · µ(t)) exsists and it must have the form �ν(t) = ±ν(t) = ±1 √ 4t6 + 1 � −2t3, 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence, by (1) of Theorem 1, the circle family C(γ,λ) creates an envelope f : R → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By (2) of Theorem 1, f is parametrized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' f(t) = γ(t) + λ(t)�ν(t) = � t3, t6� ± 1 √ 4t6 + 1 � −2t3, 1 � = � t3 ∓ 2t3 √ 4t6 + 1 , t6 ± 1 √ 4t6 + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Finally, by (3-ii) of Theorem 1, the number of distinct envelopes created by the circle family C(γ,λ) is exactly two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, Theorem 1 reveals that the set D calculated in Example 1 is certainly the union of the unit circle and the set of two envelopes of C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' We examine (1) of Example 2 by applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In (1) of Example 2, γ : R+ → R2 is given by γ(t) = (0, 1 + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, if we define the unit vector ν(t) = (1, 0), ν : R+ → S1 gives the Gauss mapping of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By definition, µ(t) = (0, 1) and thus we have β(t) = dγ dt (t) · µ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' On the other hand, the radius function λ : R+ → R+ has the form λ(t) = 1 + t in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, the created condition dλ dt (t) = −β(t) (�ν(t) · µ(t)) becomes simply (∗) 1 = − (�ν(t) · (0, 1)) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' If we take �ν(t) = (0, −1), then the above equality holds for any t ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, by (1) of Theorem 1, the circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By (2) of Theorem 1, the parametrization of the created envelope is f(t) = γ(t) = λ(t)�ν(t) = (0, 1 + t) + (1 + t) (0, −1) = (0, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Finally, notice that for any t ∈ R+ the creative condition (*) in this case holds if and only if �ν(t) = (0, −1) = −µ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, by (3-i) of Theorem 1, the origin (0, 0) is the unique envelope created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Theorem 1 can be applied also to (2) of Example 2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In this example, γ(t) = (t, 0) and λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, we may take ν(t) = (0, −1), µ(t) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' We have β(t) = dγ dt (t) · µ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Since the radius function λ is a constant function, the created condition dλ dt (t) = −β(t) (�ν(t) · µ(t)) becomes simply 0 = − (�ν(t) · (0, 1)) 8 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' NISHIMURA in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, for any t ∈ R, the created condition is satisfied if and only if �ν(t) = ±(1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence, by (1) of Theorem 1, the circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By (2) of Theorem 1, the parametrization of the created envelope is f(t) = γ(t) = λ(t)�ν(t) = (t, 0) ± (0, −1) = (t, ∓1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Finally, by (3-ii) of Theorem 1, the number of envelope created by C(γ,λ) is exactly two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Theorem 1 can be applied even to (3) of Example 2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In this example, γ(t) = (0, 0) and λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, every mapping ν : R → S1 can be taken as Gauss mapping of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In particular, γ is a frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' We have β(t) = dγ dt (t) · µ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Since the radius function λ is a constant function λ(t) = 1, the created condition dλ dt (t) = −β(t) (�ν(t) · µ(t)) becomes simply 0 = 0 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, for any �ν : R → S1, the created condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence, by (1) of Theorem 1, the circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By (2) of Theorem 1, the parametrization of the created envelope is f(t) = γ(t) = λ(t)�ν(t) = (0, 0) + �ν(t) = �ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Finally, by (3-∞) of Theorem 1, there are uncountably many distinct envelope created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let γ : R+ → R2 be the mapping defined by γ(t) = (t, 0) and let λ : R+ → R+ be the positive function defined by λ(t) = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The circle family C(γ,λ) and the candidate of its envelope is depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Defining the mapping ν : R+ → S1 by ν(t) = (0, −1) clarifies that the mapping γ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The circle family C(γ,λ) and the candidate of its envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' is a frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, µ(t) = J(ν(t)) = (1, 0) and β(t) = dγ dt (t) · µ(t) = (1, 0) · (1, 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' We want to seek a mapping �ν : R+ → S1 satisfying dλ dt (t) = −β(t) (�ν(t) · µ(t)) , namely, a mapping �ν : R+ → S1 satisfying 2t = −((�ν(t) · (1, 0))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Since �ν(t) ∈ S1, from the above expression, it follows that such �ν(t) does not exist if 1 2 < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, the circle family C(γ,λ) is not creative and it creates no envelopes by (1) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='5ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE 9 Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' This example is almost the same as Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The difference from Example 7 is only the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In Example 8, the parameter space I is � 0, 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' That is to say, in this example, R+ in Example 7 is replaced by � 0, 1 2 � and all other settings in Example 7 remain without change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, from calculations in Example 7, it follows that the given circle family C(γ,λ) is creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, by (1) of Theorem 1, C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' It is easily seen that the expression of �ν(t) must be as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' �ν(t) = � −2t, ± � 1 − 4t2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, by (2) of Theorem 1, an envelope f created by C(γ,λ) is parametrized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' f(t) = γ(t) + λ(t)�ν(t) = (t, 0) + t2 � −2t, ± � 1 − 4t2 � = � t − 2t3, ±t2� 1 − 4t2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Finally, by (3-ii) of Theorem 1, it follows that the number of distinct envelopes created by the circle family C(γ,λ) is exactly two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let γ : R → R2 be the mapping defined by γ(t) = (t3, t2) and let λ : R → R+ be the constant function defined by λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The circle family C(γ,λ) and the candidate of its envelope is depicted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' It is easily seen that the mapping ν : R → S1 defined by ν(t) = 1 √ 4+9t2 (2, −3t) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The circle family C(γ,λ) and the candidate of its envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' gives the Gauss mapping for γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, γ is a frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By definition, the mapping µ : R → S1 has the form µ(t) = 1 √ 4+9t2 (3t, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By calculation, we have β(t) = dγ dt (t) · µ(t) = t � 4 + 9t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Since the radius function λ is constant, it follows dλ dt (t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, for any t ∈ R, the unit vector �ν(t) satisfying dλ dt (t) = −β(t) (�ν(t) · µ(t)) , always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Namely we have �ν(t) = ±ν(t) = ±1 √ 4 + 9t2 (2, −3t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 4 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='4 2 2 4 210 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' NISHIMURA Thus, by (1) of Theorem 1, C(γ,λ) creates an envelope, and the created envelope f : R → R2 has the following form by (2) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' f(t) = γ(t) + λ(t)�ν(t) = � t3, t2� ± 1 √ 4 + 9t2 (2, −3t) = � t3 ± 2 √ 4 + 9t2 , t2 ∓ 3t √ 4 + 9t2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Finally, by (3-ii) of Theorem 1, there are no other envelopes created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Alternative definitions In Definition 2 of Section 1, the definition of envelope created by the circle family is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In [1], the set consisting of the images of envelopes defined in Definition 2 is called E2 envelope (denoted by E2) and two alternative definitions (called E1 envelope and D envelope) are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Definition 4 (E1 envelope [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let γ : I → R2, λ : I → R+ be a frontal and a positive function respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let t0 be a parameter of I and fix it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Assume that lim ε→0 C(γ(t0),λ(t0)) ∩ C(γ(t0+ε),λ(t0+ε)) is not the empty set and denote the set by I(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Take one point e1(t0) = (x(t0), y(t0)) of I(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, the set consisting of the images of smooth mappings e1 : I → R2, if exists, is called an E1 envelope created by the circle family C(γ,λ) and is denoted by E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Definition 5 (D envelope [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let γ : I → R2, λ : I → R+ be a frontal and a positive function respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Set F(x, y, t) = ||(x, y) − γ(t)||2 − (λ(t))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, the following set is called the D envelope created by the circle family C(γ,λ) and is denoted by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' � (x, y) ∈ R2 | ∃t ∈ I such that F(x, y, t) = ∂F ∂t (x, y, t) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Concerning the relationships among E1, E2 and D for a given circle family C(γ,λ), the following is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Fact 1 ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' E1 ⊂ D and E2 ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In this section, we study more precise relationships among E1, E2 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The relationship between E1 and E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' We first establish the relationship between E1 and E2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' E1 = E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' We first show E1 ⊂ E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let t0 be a parameter of I and let {ti}i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' be a sequence of I conversing to t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Take a point (x(t0), y(t0)) of E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, we may assume that a point (x(ti), y(ti)) is taken from the intersection of two circles C(γ(ti), λ(ti)) ∩ C(γ(t0), λ(t0)) and satisfies lim ti→t0(x(ti), y(ti)) = (x(t0), y(t0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' ||(x(ti), y(ti)) − γ(ti)||2 = (λ(ti))2 (1) ||(x(ti), y(ti)) − γ(t0)||2 = (λ(t0))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (2) For j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=', set γ(tj) = (γx(tj), γy(tj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Subtracting (2) from (1) yields the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' −2 (x(ti) (γx(ti) − γx(t0)) + y(ti) (γy(ti) − γy(t0))) + (γx(ti))2 − (γx(t0))2 + (γy(ti))2 − (γy(t0))2 = (λ(ti))2 − (λ(t0))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Since limi→∞ ti = t0 and limti→t0(x(ti), y(ti)) = (x(t0), y(t0)), this equality implies −2 � x(t0)dγx dt (t0) + y(t0)dγy dt (t0) � + 2 � γx(t0)dγx dt (t0) + γy(t0)dγy dt (t0) � = 2λ(t0)dλ dt (t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence we have − 1 λ(t0) (x(t0) − γx(t0), y(t0) − γy(t0)) · �dγx dt (t0), dγy dt (t0) � = dλ dt (t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE 11 Notice that the vector 1 λ(t0) (x(t0) − γx(t0), y(t0) − γy(t0)) = 1 λ(t0) ((x(t0), y(t0)) − γ(t0)) is a unit vector and � dγx dt (t0), dγy dt (t0) � = β(t0)µ(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus the creative condtion is satisfied at t = t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, by the proof of (1) of Theorem 1, the point (x(t0), y(t0)) must belong to E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Conversely, suppose that the circle family C(γ,λ) creates an E2 envelope f : I → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' By (2) of Theorem 1, f has the following representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' f(t) = γ(t) + λ(t)�ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' For a point P ∈ R2 and a unit vector v ∈ S1, the straight line L(P, v) is naturally defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' L(P,v) = � (x, y) ∈ R2 | ((x, y) − P) · v = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, since df dt (t)·�ν(t) = �dγ dt (t) + dλ dt (t) · �ν(t) + λ(t)d�ν dt (t) � �ν(t) = dγ dt (t)·�ν(t)+dλ dt (t) = β(t) (µ(t) · �ν(t))+dλ dt (t) = 0, f is an E2 envelope created by the straight line family L(f,�ν) = � L(f(t),�ν(t)) � t∈R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Take one parameter t0 ∈ I and let {ti}i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' ⊂ I be a sequence converging to t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Since for the straight line family L(f,�ν) the image of E2 envelope is the same as E1 emvelope (see (c) of Theorem 1 in [6]), for any sufficiently large i ∈ N there exists a point (x(ti), y(ti)) ∈ L(f(t0),�ν(t0)) ∩ L(f(ti),�ν(ti)) such that limi→∞ (x(ti), y(ti)) = f(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence for any sufficiently large i ∈ N there must exist a point (�x(ti), �y(ti)) ∈ C(γ(t0),λ(t0)) ∩ C(γ(ti),λ(ti)) such that limi→∞ (�x(ti), �y(ti)) = f(t0) (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, the point f(t0) ∈ R2 belongs to E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Existence of (�x(ti), �y(ti)) ∈ C(γ(t0),λ(t0)) ∩ C(γ(ti),λ(ti)) satisfying limi→∞ (�x(ti), �y(ti)) = f(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Since f is an arbitrary envelop created by C(γ,λ) and t0 is an arbitrary parameter in I, it follows that E2 ⊂ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' □ L((ti),入ti) (α(ti),y(ti)) f(to) L((to),入(to) f(ti) (α(ti), y(ti)) C((to),入(to)) ((ti),入(ti))12 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' NISHIMURA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' A relationship between E2 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' In this subsection, we prove the following theorem which asserts that D = E2 if and only if γ : I → R2 is non-singular, and D contains not only E2 but also the circle C(γ(t),λ(t)) at a singular point t of γ when γ is singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let γ : I → R2, λ : I → R+ be a frontal and a positive function respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Suppose that the circle family C(γ,λ) is creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' D = E2 ∪ � � � t∈Σ(γ) C(γ(t),λ(t)) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Here, Σ(γ) stands for the set consisting of singular points of γ : I → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Recall that D = � (x, y) ∈ R2 | ∃t ∈ I such that F(x, y, t) = ∂F ∂t (x, y, t) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Let (x0, y0) be a point of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Since F(x, y, t) = ||(x, y) − γ(t)||2 − |λ(t)|2, it follows the following (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (a) There exists a t ∈ I such that ((x0, y0) − γ(t)) · ((x0, y0) − γ(t)) − (λ(t))2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' (b) d(((x0,y0)−γ(t))·((x0,y0)−γ(t))−(λ(t))2) dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The condition (a) implies that there exists a t ∈ I and a unit vector ν1(t) ∈ S1 at the t ∈ I such that (x0, y0) = γ(t) − λ(t)ν1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' The condition (b) implies that there exists a t ∈ I such that dγ dt (t) · ((x0, y0) − γ(t)) − dλ dt (t)λ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Since dγ dt (t) = β(t)µ(t), just by substituting, we have that there exists a t ∈ I and a unit vector ν1(t) ∈ S1 at the t ∈ I satisfying λ(t) � β(t) (µ(t) · ν1(t)) + dλ dt (t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Since λ(t) > 0 for any t ∈ I, it follows that there exists a t ∈ I and a unit vector ν1(t) ∈ S1 at the t ∈ I satisfying dλ dt (t) = −β(t) (µ(t) · ν1(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' On the other hand, since C(γ,λ) is creative, there must exist a smooth unit vector field �ν : I → S1 along γ : I → R2 such that dλ dt (t) = −β(t) (µ(t) · �ν(t)) for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Suppose that the parameter t ∈ I is a regular point of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, β(t) ̸= 0 at the t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, at the t ∈ I, the unit vector ν1(t) must be �ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, by the proof of (1) of Theorem 1, at the regular point t ∈ I of γ, it follows D = E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Suppose that the parameter t ∈ I is a singular point of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Then, β(t) = 0 at the t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Thus, for any unit vector v ∈ S1, the following holds at the t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' dλ dt (t) = −β(t) (µ(t) · v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Hence, at the singular point t ∈ I, we may choose any unit vector v ∈ S1 as the unit vector ν1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' Therefore, by the proof of (1) of Theorem 1, at the singular point t ∈ I of γ, it follows D = E2 ∪ C(γ(t),λ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' □ ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE 13 Acknowledgement The first author is supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': 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creating envelopes, Nonlinearity, 35 (2022), 2588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='1088/1361- 6544/ac61a0 School of Science, Dalian Maritime University, Dalian 116026, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content=' China Email address: wangyq@dlmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='cn Research Institute of Environment and Information Sciences, Yokohama National University, Yokohama 240-8501, Japan Email address: nishimura-takashi-yx@ynu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} +page_content='jp' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'} diff --git a/KtFOT4oBgHgl3EQfzTRj/content/tmp_files/2301.12931v1.pdf.txt b/KtFOT4oBgHgl3EQfzTRj/content/tmp_files/2301.12931v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..386ff3abdb12542a376d717693feff0a2460a3e0 --- /dev/null +++ b/KtFOT4oBgHgl3EQfzTRj/content/tmp_files/2301.12931v1.pdf.txt @@ -0,0 +1,897 @@ +Human Cognition Surpasses the Nonlocality Tsirelson Bound: Is Mind Outside of +Spacetime? +Stuart Kauffman1, Emeritus Professor of Biochemistry and Biophysics, University of +Pennsylvania, Philadelphia, Pennsylvania 19104, USA +Sudip Patra2, Associate Professor OP Jindal Global University, Founding member CEASP. +India. +Dec 26, 2022 +Abstract +Recent experimental studies on human cognition, particularly where non-separable or +entangled cognitive states have been found, show that in many such cases Bell or CHSH +in-equalities have been maximally violated. The implications are that greater non-local +correlations than allowed in quantum mechanics (often known as the Tsirelson bound), +are found in human cognition. We propose in the current paper that a non-local theory +of mind is needed in order to account for the empirical �indings. This requires a +foundationally different approach than the extant ‘quantum-like’ approach to human +mind. Our account is novel, but still founded on a Hilbert space set up with the physical +constraint of no-signalling. To account for the surpassing of the Tsirelson bound we +propose abandoning the constraint of no-signalling that depends upon spacetime. Thus +we ask; ‘Is mind outside spacetime?’ We discuss a candidate theory of quantum gravity +based on non-locality as fundamental that may accord with our proposal. We are led to +suggest a new 6 part ontological framework linking Mind, Matter, and Cosmos. +Key words: non-locality, no-signalling, Bell inequalities, Tsirelson/Cirelson bound, PR +boxes, Cognition, quantum gravity, six-part framework +Introduction: Is mind outside physical spacetime? +Non-locality has baf�led us since the birth of modern science. For example in Newton’s +gravity framework, we have action at a distance, and Newton himself did not want to +forward any ‘explanation’ of the same by stating, “hypothesis non-�ingo”. Later with the +advent of Special Theory of relativity (SR), and then the General Theory of Relativity +(GR), Einstein nearly singlehandedly challenged the age old concepts of space and time, +proposing the bold and beautiful concept of spacetime, where continuity of action (COA) +plays a central role. COA holds that if a spacetime event A has to in�luence another +spacetime event B then it also has to in�luence any closed 3 surface between them. +Hence no-signalling, or that there is an ultimate limit of signalling between spacetime +1 + +events, which happens to be the speed of light in vacuum, became the foundational +physical constraint for any sound theory of Physics. The orthodox quantum mechanics +(QM) which emerged from intense discussions in Solvay conferences (for example, in +Pylkkanen, 2019), and later known as Copenhagen version, was, however, still based on +a Newtonian space and time background. Later, with the emergence of quantum �ield +theory (QFT) there has been an uncomfortable coexistence of SR and QM. The holy grail +of modern Physics has been to construct uni�ied �ield theories, and particularly quantum +gravity (QG), as the cherished uni�ication of GR with QM. However, in all of these +numerous attempts, non-locality has been a recurrent problem. Even different +interpretations of QM, starting from Collapse of the wave function, to different +alternative theories like Bohmian mechanics (Walleczek et al. 2019), or spontaneous +collapse of wave function (for example in Tumulka, 2006), have been riddled with +different forms of non-localities. Very recently different approaches to QG (Kauffman, +2022) would presume to hold non-locality as fundamental, which is radical. +Spontaneous collapse models or dynamic collapse models have attempted to resolve the +measurement problem by introducing a collapse operator in the Schrödinger equation, +for example in GRW (for example in Wallace, 2014) where a probability of such a +stochastic collapse is small in case of single particles, but grows exponentially in case of +many body systems. Hence, the attempt has been to resolve the incompleteness or +inconsistency problems in orthodox QM, for example, how in the same framework both +deterministic and unitary Schrödinger evolution and random collapse of wave function +due to ‘measurement’ can be accommodated. However, very recent work, (for example +see ‘consciousness and quantum mechanics’ edited by Shan Gao, 2022, and Ball 2022), +now says that any “physically causal” theory for measurement is almost ruled out. +There are also physically “acausal” accounts of measurement. Here we refer to the +recent consciousness induced collapse framework of Chalmers and McQueen (2021), +where phenomenal consciousness plays the role of a superposition resistant, hence +de�inite consciousness state that result in “collapse”. More recently, Kauffman and Roli +(2021), and Kauffman and Radin (2022) have utilized Heisenberg’s interpretation of +quantum mechanics in terms of ontologically real Potentia, Res potentia, and +ontologically real Actuals, Res extensa, where actualization converts Possible to Actuals. +This interpretation does not inherit the mind-body problem because Potentia are not +substances. +In turn this approach suggests a natural role for “mind” in actualizing +quantum potentia, hence “collapsing the wave function. At this point, data supporting +this hypothesis with respect to work using the two slit experiment are strongly +supportive at 6.49 Sigma, or 4 x 10 ^ -11. +We shall base our own discussion on +Heisenberg’s interpretation. +In addition to Heisenberg’s “potentia” interpretation, workers have studied several +other non-realist frameworks, where the wave function is not ontological, but rather a +tool for computing probabilities for epistemological updates of knowledge state of +observers. Two alternative strongly emerging interpretations of QM are relational QM +and QBism. Relational QM holds that QM, or reality for that matter, is not described by +quantum states, but rather by relations among observables. This is a fact ontology (for +more details about “relative” and “stable” facts, we can refer to seminal literature, +2 + +(Pienaar, 2021)). QBism agrees on placing a central role on +phenomenology or +subjective experiences, where QM is the navigation tool for any user (rather than +de�ining who is the user) to make optimal decisions. In QBism relations between the +elements of the framework are objective, such that every agent would agree. We differ +from these frameworks in that these frameworks are largely based on the locality of +physical spacetime, but then they face non-locality problems. +One approach to surpassing the Tsirelson bound is found in PR box worlds (Popescu, +2014, a modern review of Tsirelson bound can be found in Stuckey et al., 2019) that +allow for greater than QM non-local correlation limit the Tsirelson bound. However, the +PR box worlds correspond to no physical model of a universe, (Popescu, 2014 op. cit.). +Hence a related question raised earlier was whether QM is the only theory where there +is a co-existence of non-locality in the sense of Bell inequalities violation and relativistic +causality. +We propose in this paper that if we need to include mind or cognitive aspects in the +foundational frameworks of nature, then we need to have non-locality as the central +feature. In this paper then, we explore our framework of non-local mind or cognition, +and are led to our proposal that “mind” is not in spacetime. By proposing that mind is +not in spacetime, we can naturally eliminate the requirement for Continuity of Action, +hence non-signaling, that makes sense only within a framework of spacetime. By +proposing that mind is not in spacetime, mental events that are in spacetime but +surpass the Tsirelson bound can be explained. +In order to begin to make sense of the concept that “mind is not in spacetime, but +conscious events are in spacetime”, we are led to propose a novel 6 part ontological +framework linking Mind, Matter, and the Cosmos. The grounds for this novel framework +are tentative, but we hope worthy of consideration and are testable in part. +The current paper is organized in sections. Section 1 provide the background of +cognitive frameworks with experimental work, and its recent reformulations in terms of +‘quantum-like’ features, for example entangled cognitive states which +violates Bell +inequalities strongly. Section 2 discusses alternative ways to surpass the Tsirelson +bound. Section 3 presents our novel 6 part framework and the current grounds to +consider it. Section 4 summarizes our results and suggestions for further experiments +and work. +Section 1 Cognition beyond the Tsirelson Bound +Aerts et. al (2013, 2021) have pioneered the study of non-separable states in individual +minds or cognition. This includes how different concepts are combined. Such concept +combinations in individual minds can be re-formulated as non-separable states. In +technical language these are intra state entanglements, which would mean coupling of +different degrees of freedoms of a single system. Here these are individual minds, and +the data can be expressed through inequalities such as CHSH as we explain in section 2. +The statistical values or values obtained from ensembles of ‘minds’ of participants in +such cognitive experiments can be inputted as inequalities. The results have +demonstrated a clean violation of the ‘non-local’ correlation bound which occurs in QM. +3 + +Such a tight bound is, as noted, called a Tsirelson bound, that is maximal and +characteristic for QM. The same authors also provide the statistical signi�icance of their +results. Aerts and Arguelles (2022) have claimed a statistical signi�icance of p values +ranging 0.001-0.005, which is strong enough to suggest, but not yet prove, the viability +of their results. +Aerts et. al (op. cit.) also adopts a Hilbert space framework, but their strategy is of +‘reverse engineering’, i.e. to start with the empirical results, and then describe such +results by a suitable state space modelling, where the state space can be either Hilbert +space or a larger Fock space. Thus, the usage of CHSH inequalities is statistical in nature, +since such inequalities are general. Maximal algebraic violation of such inequalities can +be beyond the Tsirelson bound, but when Hilbert space is the state space then a tight +upper bound comes up as a constraint. +Given the above points, Aerts et al.’s claim of greater than Tsirelson bound violation in +cognitive experiments raises several questions. For example, can any or all Hilbert space +formulations account for such super quantum correlations? Aerts et al. have responded +by suggesting that an entanglement that they consider is of a more complex nature, i.e. +entanglement both in states as well as measure, might account for super violations. We +assess this approach below. +We stress again that any Hilbert space formulation of quantum mechanics implies a +tight Tsirelson bound. And we stress again that the Hilbert space formulation is stated in +a background spacetime with “no signaling” and continuity of action, hence “locality”. +Section 2. Non-locality : Implications for QM +2.1 Non-locality in QM +Here we remind ourselves of the seminal contribution of John Bell (1964, 1966), and +state the basic requirements for Bell factorization conditions, upon which the celebrated +Bell inequalities or later CHSH inequalities are based. Based on continuity of action, the +following three assumptions are required for establishing Bell factorization. +a. +Statistical Independence: +, where +denotes the local hidden +ρ 𝑀 +( +) = ρ µ +( ) +µ +variable, and M stands for measurement settings of apparatuses for different +space-like separated agents. +b. Output independence: +, subscripts a and +ρ𝑎𝑏 𝑎, 𝑏, µ +( +) = ρ𝑎(𝑥𝑎|𝑎, 𝑏, µ)ρ𝑏(𝑥𝑏|𝑎, 𝑏, µ) +b stands for different agents, namely, Alice and Bob, x’s are outcomes at their +ends and a, and b are inputs at their ends respectively. +c. +Parameter independence: ρ𝑎 𝑎, 𝑏, µ +( +) = ρ𝑎 𝑎, µ +( +), 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑙𝑦 ρ𝑏 𝑎, 𝑏, µ +( +) = ρ𝑏 𝑏, µ +( +) +Hence, in conjunction of the three assumptions we have the Bell factorization +(1) +ρ𝑎𝑏 𝑎, 𝑏, µ +( +) = ρ𝑎 𝑎, µ +( +). ρ𝑏 𝑏, µ +( +) +Bell factorization is a general condition based on the local realism assumptions (COA to +be precise), which is violated by different theories in different ways. For example, QM +violates Bell factorization by violating output independence but keeping statistical +4 + +independence and parameter independence. Super Deterministic theory violates the +same by violating Statistical independence, while keeping the other assumptions. And +Cavalcanti and Wiseman (2012) have showed how Bell factorization can be derived +from conjunction of local ‘signalism’ and predictability. +In the form of CHSH, we have two space-like separated agents, Alice and Bob, where say +the measurement settings in Alice’s end are {a, a’} and Bob’s end are {b, b’}, and all +results are dichotomous (say, +/- 1). Here we de�ine the correlation function as +(2) +𝑐 𝑎, 𝑏 +( +) = 𝑃𝑎,𝑏 1, 1 +( +) + 𝑃𝑎,𝑏 − 1, − 1 +( +) − 𝑃𝑎,𝑏 1, − 1 +( +) − 𝑃𝑎,𝑏(− 1, 1) +Hence, we have the CHSH inequality as 𝐶𝐻𝑆𝐻 = 𝑐 𝑎, 𝑏 +( +) + 𝑐 𝑎, 𝑏 +' +( +) + 𝑐 𝑎 +', 𝑏 +( +) − 𝑐(𝑎 +', 𝑏 +') +(3). +Hence CHSH has different upper bounds for different underlying theories. For example +for a local deterministic theory (COA is the requisite here) we would always have as +For QM the maximum violation of the above limit would take place when +𝐶𝐻𝑆𝐻 +| +|≤2. +, hence this gives the Tsirelson (T bound +𝑎, 𝑏 +( +) = 𝑐 𝑎, 𝑏 +' +( +) = 𝑐 𝑎 +', 𝑏 +( +) =− 𝑐 𝑎 +', 𝑏 +' +( +) = +2/2 +from now) bound of +. However algebraically it is possible that we have +|𝐶𝐻𝑆𝐻|≤2√2 +, hence making the maximal upper bound as +𝑐 𝑎, 𝑏 +( +) = 𝑐 𝑎, 𝑏 +' +( +) = 𝑐 𝑎 +', 𝑏 +( +) =− 𝑐 𝑎 +', 𝑏 +' +( +) = 1 +4. +2.2. Different forms of non-separable states: QM and beyond +We mention here that generally composite systems in QM can be represented as tensor +products of states belonging to different Hilbert spaces, such that the total Hilbert space +of the composite system is a tensor product of such Hilbert spaces. This context is called +product states. In addition, we recall that a Tensor product space is strictly larger that +space of direct sums, hence this context also captures ‘quantum-holism’. Now the typical +de�inition of an intersystem entanglement is when the composite system state cannot +be de�ined as simple tensor products of subsystem states. Intersystem entanglement is +most discussed in QM literature, since that is what generates non-local correlations. In +an entangled state the whole is always in a pure state, whereas parts are not in pure +states, this is the classical Schrödinger way of denoting entanglement. Again, as we have +stated earlier, maximally entangled states (often called as Bell states) can violate CHSH +maximally until the T bound. +However, intra-system entanglement, de�ined as coupling between multiple degrees of +freedom of the same system, is also discussed widely. Particularly in the classical +electromagnetism literature authors (Ghose and Mukherjee, 2014 for example) have +observed widely that intra system entanglement, for example coupling between path +and polarization states of a vortex beam, can produce such non-separable states (at +times called Shimony-Wolf states) which can generate violations of CHSH inequalities. +Authors, for example, Khrennikov (2020) has suggested that intra and inter system +entanglements is the main difference between quantum and so-called ‘classical’ +entanglement. +5 + +Multipartite non-locality: +Traditionally Bell tests or CHSH tests are bi-partite +non-locality tests, there have been several modi�ications though, for example GHZ states +or W states, which extends frameworks for many body entanglement. In our previous +framework (Kauffman and Patra, 2022) we start with a multipartite entanglement state. +However, its only recently (Bancal et al. in 2013 for example) that a suitable +mathematical framework is being built. Here we refer to the basic tenets of such a +framework, since this might be harnessed in the framework we suggest here. +We mention here that non-locality is a recurrent feature for many-body systems too (see +for example in Bancal et al. 2013), for example if we consider a tripartite system, with +say each subsystem possessed by Alice, Bob and Charlie who are spatially separated. Say +Alice, Bob and Charlie’s experimental set ups are X, Y and Z respectively and outcomes +of experiments are a, b and c respectively (binary outcomes for simplicity). +If the joint probability (if de�ined) +where, q’s are +𝑃 𝑋𝑌𝑍 +( +) = +𝑙 +∑ 𝑞𝑙𝑃 𝑋 +( )𝑃 𝑌 +( )𝑃 𝑍 +( )(4), +bounded by 0 and 1, and sum to unity, then the sum represents local correlations, where +the subscript l is for underlying hidden variables. However, if the above joint +distribution cannot be written in the above format, then some degree of non-local +correlations +exist. +One +example +of +non-locality +(technically +S2 +non-local): +, where separately q’s +𝑃 𝑋𝑌𝑍 +( +) = +𝑙 +∑ 𝑞𝑙𝑃 𝑋 +( )𝑃 𝑌𝑍 +( +) + +𝑚 +∑ 𝑞𝑚𝑃 𝑌 +( )𝑃 𝑋𝑍 +( +) + +𝑛 +∑ 𝑞𝑛𝑃 𝑍 +( )𝑃 𝑌𝑍 +( +)(5) +sum up to unity. Here we see that in individual sums full factorization is not achieved. At +times, such contexts are also called hybrid non-locality. In another related literature +(Bennet et al., 1999 as one seminal work in this direction) non-locality without +entanglement is theoretically proposed, and later experimentally veri�ied. We refer to +these studies to seek further support for our assertion that non-locality is a more +universal and genuine feature of reality. We also are aware of studies differentiating +between genuine non-locality and direct in�luences (see for example Atmanspacher et +al., 2019). +2.3. Attempts to �it the evidence for non-locality withing the framework of a background +spacetime. +In the last century intense debate on non-locality, or more precisely what non-locality +should mean given relativistic spacetime, was a major debate, and is still continuing. The +non-locality debate has also thrown deeper light on the foundational thinking on QM. +We observe here that the axioms of special theory of relativity (COA) or consequently +fundamental limit for speed of signalling between spacetime events, and the equivalence +of inertial reference frames) seems to be elegant and physically based. However, the +axioms of QM seem to be mathematical with no clear physical basis. +Aharonov and Bohm (1961), and later Popescu, and Rohrlich (1994), and independently +Shimony (1993) have proposed that QM has to be compatible with relativistic causality, +hence with Continuity of Action, COA. The efforts of the authors mentioned showed that +non-local correlations, for example in an EPR set up, can be compatible with relativistic +causality if and only uncertainty of outcomes of measurements is fundamental. Or in +6 + +other words the effect of a cause here is uncertain. (Thus, counterfactuals are required). +Aharonov was the �irst to propose ‘modular’ quantum variables, that are non-local in +spacetime due to non-local relativistic phases, and they have optimal uncertainty for no +signalling. Shimony amusingly observed the whole affair is ‘passion at a distance’. +2.4. Attempts to surpass the Tsirelson bound in formal models. +Based on the dense PR box literature, there have been many attempts to make super +quantum correlations (violating T bound) compatible with relativistic causality, or COA +in general,(for example, Popescu, 2014). Related questions have been whether QM is the +only possible theory where non-local correlation and no signalling co-exists?, (Popescu +2014). +Or why QM does not exhibit greater non-locality?, (Linden et al. 2007 for +example). We further observe that there have been efforts in the line of including +communication complexity, and or, information causality to eradicate super quantum +correlations, (for example, in Jaeger, 2007). We also note that super correlations or +greater than T bound violations are possible in con�iguration spaces with very +particular properties. Overall, there has been an attempt to make violations of Bell +inequalities (not super correlations) compatible with relativistic causality, but it is far +from clear what would be the implications for super correlations for a locality criterion. +As we explore below, how violations of Bell / CHSH or even super correlation results +have been observed in cognitive experiments. +We also note that some authors +(Khrennikov, 2022) have observed that if the observables in a particular theory cannot +be represented by Hermitian operators, there might not be any T bound constraint. +Section 3. Is Spacetime Fundamental? +3.1. Zeilinger and Information: It is important to stress that several authors are +exploring the idea that spacetime is not fundamental. In particular, Zeilinger has +proposed that “information” is fundamental and somehow spacetime emerges from +“information”(see for example Zeilinger’s seminal works since 1998). We note a central +issue, “information” itself implies “possibilities” that are not either true or false. +Consider Shannon information and the source. A given bit string, say (11111) can carry +no information unless one of the bits can, counterfactually, be 0. That is, it must be +possible that one of the bits is 0 not 1. Thus the very concept of “information” requires +more than one simultaneously possible state of the universe. +3.2. Res potentia and Res extensa linked by measurement: In the current article, we base +our approach on Heisenberg’s interpretation of the quantum state as “potentia standing +ghost – like between an idea and reality”. One of us, ( Kastner et al. 2018) has developed +Heisenberg’s interpretation as “Res potentia” ontologically real Possibles, and Res +extensa, ontologically real Actuals. Possibles do not obey Aristotle’s law of the excluded +middle and law of noncontradiction, so are neither ‘true’ nor ‘false’. This allows +“Potentia” to explain quantum superpositions: “Schrödinger’s cat simultaneously is +possibly alive and possibly dead.” This is not a contradiction. +Potentia are non-spatial in nature but ontologically real. By contrast Actuals do obey +Aristotle’s two laws, so are either true or false. All of Classical physics is based on such +true false Boolean variables. Given the concept of Res potentia, one of us, (Kauffman) +7 + +has explored a new approach to quantum gravity that takes non-locality to be +fundamental. Non locality taken as fundamental implies that spacetime is not itself +fundamental, but must somehow arise from the behaviors of entangled coherent, hence +non local, quantum variables. Then non-local entangled coherent quantum variables, +“Res potentia” are not in spacetime. They are Potentia not in spacetime. +3.3. Mind and the Quantum Vacuum: One natural interpretation of the line of thought +above is that the quantum vacuum consists precisely in non-local entangled quantum +coherent variables. Given the above, a natural proposal is that ‘mind’ – non-spatial, is +identical or related to the quantum vacuum. We here both propose this identity and +explore its potential validity. +A �irst implication of the proposed identity of mind and the quantum vacuum is that +both are outside of spacetime. This is a possible step to explaining Aert’s results. To do +so, we need to show that surpassing the Tsirelson bound is straight forward if mind is +outside of spacetime. In this case we can abandon no signalling and continuity of action. +We show this next. But there is a further issue, Aerts et. al data concern experiences of +humans and those experiences are in +spacetime. Powerful recent arguments now +strongly suggest that conscious experiences (phenomenal nature) arise upon collapse of +the wave function, hence, qualia are in spacetime. And further remarkable evidence now +clearly shows that we can purposefully actualize the wave function. A responsible free +will is not ruled out. We address all this below. These recent results and claims will be +part of our proposed 6 part framework introduced below. +Section 4. Surpassing the Tsirelson Bound if Mind Is Outside of Spacetime +Aerts et al.(op. cit.) themselves have attempted to justify the super quantum correlation +values obtained in their ‘concept-combination’ experiments based on complex +entanglement nature in their experimental settings, given that the con�iguration space +of mind is a high dimensional Hilbert space. However the standard belief (going back to +Popescu and Rohrlich) has been that the maximum limit of ‘non-locality’ allowed in a +Hilbert space is the bound. +Our perspective is not to justify the super violations based on the complexity of +entanglement (both in states and in measurements), since there have been critiques of +this line of argument by suggesting that if the ‘marginal selectivity’ rule is also violated +along with Bell inequalities (which Aerts et al. observes) then there can be +contaminations in testing for Bell violations. Hence we propose the six part framework, +where our de�inition of mind need not be constrained by any physical locality condition. +Section 5. Mind and Qualia – Collapse of the Wave Function +Recently Chalmers and McQueen (2022), who have been very sceptical about mind +collapsing wave function, or a relation between QM and phenomenal consciousness in +general, have designed a framework in which phenomenal consciousness might collapse +wave function and thus a de�initive ‘classical’ world emerges. The framework suggested +is based on IIT (Tononi et al. 2016 for example) or integrated information theory, and +also where phenomenal consciousness – qualia – is considered as ‘superposition +resistant’. Here we observe that Chalmers and McQueen (2022) have proposed a partial +8 + +quantum Zeno effect for completing their consciousness induced collapse model. Our +previous framework for the emergence of the classical world naturally includes a partial +Zeno effect, with trade-offs between Zeno effect and atmospheric de-coherence. We +didn’t have non-local mind explicitly in the previous framework. +In addition to Chalmers and McQueen, (op. cit.), Kauffman and Roli (op. cit.) have +recently proposed that the human mind cannot be algorithmic, and that the capacity to +�ind novel affordances requires a quantum mind and qualia associated with the collapse +of the wave function to a single state. The next section presents evidence that humans +can, in fact, collapse the wave function. +Section 6. We Can Collapse the Wavefunction +An old idea in quantum mechanics is that mind might have something to do with +“collapse of the wave function”. +Von Neumann proposed this, (1955/1932). Wigner +suggested the same idea at one point( see for example Wigner, 1995). +Following Heisenberg, as noted, we propose Res potentia, ontologically real Possibles, +and Res extensa, ontologically real Actuals. Here “actualization” converts Possibles to +Actuals. This assertion is fully consistent with recent results, (Gao, op. cit., Bell, op. cit..), +that seem to rule out physical causes of actualization. A physical cause cannot convert a +possible to an actual. +Res potentia and Res extensa plus actualization is the �irst new idea about mind and +body since Descartes’ substance dualism, +Spinoza’s neutral monism, Berkeley’s +Idealism, and pure materialism. +Res potentia and Res extensa is not a substance +dualism. Potentia are not substances. Thus this view does not inherit the mind – body +problem. +Instead Res potentia and Res extensia suggest a natural role for mind. Mind “actualizes” +Possibles to Actuals. +Strong evidence now supports this scienti�ically testable hypothesis. Radin and his +colleagues (for example see Radin 2019) have tested the capacity of humans paying +attention to modify the intensities of the adjacent central bands in the famous +interference pattern of the two slit experiment. The effect is weak, but has been tested in +30 independent experiments. At present the positive results are very strongly +statistically signi�icant at 6.49 Sigma. The probability, “p”, that this arises by chance now +stands a less that 4 x 100,000,000,000, (Kauffman and Radin, op. cit.). +This is strong enough to take very seriously as yet further data are sought. If accepted, +the results alter the foundations of Quantum Mechanics with a fundamental role for +mind. Indeed, even a responsible free will is not ruled out, (Kauffman Radin, op. cit.). +For the purposes of this article, we will accept these results as true. +Section 7. Quantum Gravity if Non locality Is Fundamental +One of us has recently published a work on quantum gravity (Kauffman op. cit.).The +starting point is to take nonlocality as fundamental. Nonlocality arises in the presence of +entangled coherent quantum variables. If one starts with nonlocality it is not necessary +9 + +to explain nonlocality, but necessary to explain locality. Somehow locality – spacetime– +is to emerge from the behaviors of the quantum variables. This immediately �latly +contradicts General Relativity which is local, and in which there is no emergence of +spacetime. Further, General Relativity can be formulated in the absence of matter so +matter cannot be necessary for the very existence of spacetime. But if one starts with +nonlocality, the emergence of spacetime must depend on the matter – the entangled +coherent quantum variables. A further note is that there is no apriori reason not to start +with nonlocality as fundamental. +The steps in building this new theory of quantum gravity start with N entangled +variables in Hilbert space, then constructs a metric distance between each entangled +pair of variables as the sub-additive von Neumann Entropy between that pair. +Sub-additive von Neumann Entropy, therefore, �its the triangle inequality. The next step +notes that quantum variables can be in superposition and interpreted as potentia, +neither true nor false. All the variables of classical physics are true or false. Hence the +next step in the development of the theory maps distances in Hilbert space to classical +spacetime distances between a succession of true actualization events. In this mapping +entangled near-neighbours in Hilbert space construct themselves into nearby points in +classical spacetime. The hypothesis that actualization events construct spacetime is +probably testable using the Casimir effect. +Section 8. Emergence of the Classical World +Here we refer to the ontological framework developed by Kauffman and Patra (2022), +which also forms one reference for the current framework, though we didn’t include +non-local mind in our previous work. We based our previous work on the premise that +measurement and actualization, which creates the de�initive classical world (this +coincides with the contextuality-complementarity philosophy of Bohr1) can happen only +in a speci�ic basis. However we still do not have a comprehensive theory for the +emergence of a speci�ic basis, except the recent attempts from Quantum Darwinism +perspectives as proposed by Zurek (2022) in terms of de-coherence theory. We note +that decoherence does not yield a speci�ic basis. +We have proposed the following steps for the emergence of classical world, in which +testable experiments can be performed. +(i) +We start with sets of N entangled quantum variables, which need not be +maximally entangled. Variables can mutually actualize each other, which is +approximated by the quantum-Zeno effect. +(ii) +Such actualization occurs in one of the 2Nbases. +(iii) +Mutual actualization breaks symmetry among these 2N bases. +(iv) +An amplitude for a speci�ic basis can emerge and increase with further +measurement in the same particular basis, it can also decay between +measurements. +1 Here one can also refer to recent works of Kastrup (), where if we claim that only actualization creates the +definitive world, which would mean no pre-existing values, we should also accept that the world as a whole is +beyond only physical, or the typical physical closure principle would not work. +10 + +(v) + As the number of variables, N, in the system increases, the number of +Quantum Zeno mediated measurements among the N variables increases. +(vi) +Now for experimental purposes, quantum ordered, quantum critical, and +quantum chaotic peptides that decohere at nanosecond versus femtosecond +time scales can be used as test objects. +(vii) +By varying the number of amino acids, N, and the use of quantum ordered, +critical, or chaotic peptides, the ratio of decoherence to Quantum Zeno effects +can be tuned. This enables new means to probe the emergence of one among +a set of initially entangled bases via weak measurements after preparing the +system in a mixed basis condition. +(viii) +Use of the three stable isotopes of carbon, oxygen, and nitrogen and the �ive +stable isotopes of sulfur allows any ten atoms in the test peptide or protein to +be discriminably labelled and the basis of emergence for those labelled atoms +can be detected by weak measurements. We present an initial mathematical +framework for this theory, and we propose experiments. +Section 9. If Mind is Outside of spacetime, What is “My” Mind? +If we are to make sense of Aerts et. al data (op. cit.), and do so by proposing that mind is +outside of spacetime but that the cognitive experience is in spacetime, we must claim +that qualia emerge upon actualization events, as discussed above. But in addition, it +becomes fundamental to address the issue: What maps the quantum variables in Hilbert +space and the vacuum to “My Mind”? +The theory of quantum gravity based on nonlocality as fundamental almost +automatically affords a possible answer to this issue. +Compare the relatively simple +quantum behaviors of a quantum variables in a quartz crystal and the presumably far +more complex behaviors of the quantum variables in the diverse proteins in a speci�ic +human brain with its unique genetic background and life experiences. The proposal is +that when these quantum variables become coherent, they are not in spacetime but part +of the quantum vacuum. The behaviors of these variables in the vacuum must exhibit +and re�lect the complexities the quantum behaviors in that speci�ic brain. Because +entangled neighbors in Hilbert space map to spatial neighbors in classical spacetime and +the matter in it, actualization events with qualia will typically map to and occur in the +same brain. Thus, “What is My Mind” seems naturally answered. +These proposals claim to answer Aerts et. al (op. cit.). My mind is not in spacetime, so +not bound by continuity of action and nonsignaling. The Tsirelson bound can be +surpassed. But actualization occurs in my brain so are my qualia. +Section 10. The Quantum Vacuum and the Matter in the Universe +Our proposal to start with nonlocality as fundamental drives a different conception of +the quantum vacuum. This vacuum is normally conceived in the absence of any matter +and as a coupling of all the fundamental �ields. The same can be considered as coupled +quantum harmonic oscillators whose zero point energy can be studied. As so conceived, +the spectrum of the quantum vacuum must be stationary. +11 + +By contrast, if nonlocality is taken as fundamental, spacetime is not fundamental and +can only arise due to the behaviors of the quantum variables when coherent and also +when not coherent. In the latter case, the Schrödinger equation no longer applies. The +quantum behaviors of quarks, protons, neutrons, and electrons in complex proteins +must differ from those in a simple crystal. With this seemingly necessary inference, the +behaviors of the quantum vacuum – coherent entangled quantum variables, cannot be +stationary over the history of the universe as more and more complex classical systems, +stable for long periods, come into existence. We can propose, quantum vacuum must +also re�lect the history of the behaviours of the ever more complex matter than has +come to exist and vanished. +Section 11. The Six Part Ontological Framework: Mind Matter and Cosmos +The above considerations lead us to propose that: i. The quantum vacuum is composed +of entangled coherent quantum variables that are ontologically real “Possibles”; ii +“Mind” is identical to the Possibles of the quantum vacuum. Hence this is the de�inition +of mind in our framework; iii. The vacuum is outside of spacetime; iv. Mind can mediate +actualization of potentia ,(Kauffman and Radin, op. cit.); actualization of potentia then +constructs classical spacetime, where a metric exists in the quantum vacuum Hilbert +space via non-additive von Neumann Entropies between pairs of entangled variables, +that is then mapped to events at speci�ic classical spacetime locations, (Kauffman +quantum gravity); v. We experience such actualized quantum variables as “qualia”, +(Chalmers and McQueen, Kauffman and Roli, 2021); vi. In the last, sixth, part we +propose the emergence of classical world, which is based on our previously proposed +framework (Kauffman and Patra, 2022). We suggest a mutual actualization process of +quantum variables. Through a trade-off between the quantum Zeno effect and +atmospheric de-coherence, such de-cohering and re-cohering variables creates the +observable classical variables. In this framework we suggest veri�iable experiments with +peptides whose entangled variables decohere exponentially fast versus peptides whose +entangled variables decohere power law slowly as a possible ground of test. +We hope to show that the six part proposal above allows us to account for Aert’s et. al +results that surpass the Tsirelson bound. Far more, this new six part framework may +help organize our emerging ideas about “Mind, Matter, and Cosmos”. +Section 12. Discussion and Further Work +Non-locality has always baf�led us. The non-local and non-deterministic collapse of wave +function in QM worried Einstein throughout his working life, since the fear was such +non-locality would mean action at a distance and thus break- down of the causality +structure of spacetime. The latter is fundamental to any Physical theory. Certainly, a +huge literature has demonstrated that non-local collapse may not mean any +superluminal signaling. Later since Bell’s seminal contribution, there have been many +versions of such frameworks (CHSH being the most popular), which have suggested that +local hidden variable theories cannot reproduce QM faithfully. Loophole free Bell +inequality/ CHSH inequality violations have demonstrated that one or more of the basic +underlying assumptions, of localism, realism or non-contextuality, or statistical +independence have to be relaxed to describe the empirical results of QM. +12 + +Many workers have shown that Bell inequalities violations are considered to be +evidence of non-local correlations between subsystems. The canonical example is +entangled pairs of particles (EPR set up for example) with agents measuring on each +half of the pair who are space like separated. +In the presence of an assumed background spacetime, the only way a no signaling +theorem is going to be preserved is by introducing inherent quantum uncertainty in +outcomes. In the words of Shimony there can be a happy co-existence between Special +Relativity and quantum fundamental uncertainty. However, the Hilbert space structure, +assumed to be the state space in such frameworks, inherently does set up an upper +bound for violations of inequalities, the celebrated Tsirelson (or Cirelson) bound. Thus, +the question arises what if in any empirical observation exist where such a limit is +violated? +Over the last decade there has been strong evidence of violations of CHSH inequalities, +pertaining to cognitive experiments (Aerts et. al). The data are now con�irmed at p = +.001 to .005. Further work is needed to con�irm these results more strongly. However, +they are already strong enough to warrant consideration of the implications. +Aerts et. al (op. cit.) have tried to preserve a background spacetime and “no signaling” +by assuming more complex entanglement, i.e. both states and measurements. The same +authors have also claimed that quantum entities might be conceptual or cognitive +entities, hence non-spatial. +We propose here a novel, yet unexplored framework based on non-localism, where +spacetime need not be fundamental to existence. Non-locality is not mysterious in our +framework. Our attempt is to start from non-locality, and derive locality from �irst +principles. Then in such a constructed local spacetime we have standard QM and SR +operate with restricted non-locality which is no signaling also. +Our proposal is related to that of Aerts et. al in an unexpected way. As just noted, these +authors propose quantum entities might be conceptual or cognitive entities, hence +non-spatial. Almost in parallel, we propose that the quantum vacuum consists in +ontologically real Possibles, that Possibles are non-spatial, i.e. not in spacetime, that +Mind is identical to these Possibles, that Mind can actualize these potentia, and we can +experience these as qualia. +What should we make of this extensive new six part framework? A �irst point is that +other attempts such as PR boxes correspond to no know physical reality. +Our proposal is not too distant from Aharanov’s non local proposal . But as Shimony +notes, this is “passion at a distance” in spacetime. In our six part framework, the +correlations are among ontologically real possibles that are not in spacetime, but “mind” +is/are part of the quantum vacuum of possibles. These possibles then constitute the +information Zeilinger hopes is the basis, somehow, of spacetime. However Zeilinger +offers no account of what information is, other than a “bit”, nor any idea of how these +might be related to spacetime. +13 + +In our account, spacetime is constructed by the sequential actualization of quantum +variables in Hilbert space with a metric via non-additive von Neumann Entropies that +then map to Actual events whose mutual distance relations re�lect the metric in Hilbert +space to constitute spacetime, (Kauffman quantum gravity). This claim underlies our +�irst part, “i”, and “iii”. There are data at 6.49 sigma to support “iv” and “v” +above,(Kauffman, op. cit.). +The second part, ii “mind” is identical to the possibles of the quantum vacuum, is an +entirely new proposal. Oddly, this proposal just might afford a highly speculative answer +to the point raised in a recent article on Biocosmology, (Cortes et al., 2022), about a link +between the emergence of life 4 billion years ago and the recent dominance of dark +energy whose tight temporal coincidence in Cosmology is strange. If living organisms +actualize quantum variables far more often than the quantum variables of the abiotic +universe, then life, via mind, can have played a role in the emerging dominance of dark +energy in the past four billion years. +Our vi. part concerns the emergence of the classical world from the quantum world. Our +own proposal, (referring to Kauffman and Patra, 2022), has the virtue of being testable. +In addition it automatically supplies the incomplete Quantum Zeno Effect desired by +Chalmers and McQueen, (op. cit.). Our speci�ic proposal for the emergence of the +classical world is consistent with our general framework i. to vi., and is consistent with +efforts to study how an increase in the mass of molecules such as the Buckyball may +increase decoherence . +The proposal that quantum gravity is a quantum construction of spacetime is not yet +united with General Relativity, but may be a new pathway to do so . Such a union with +our proposals in the present article might be fundamentally new. Such a union would +embrace Mind, Matter and Cosmos. +The most important lines of further work are: i. Experiments to test and extend the +Aerts et. al results, (op. cit.). A ‘p’ value of 0.001 is of interest, but hardly persuasive. ii. +Our six part framework rests heavily on taking non-locality as fundamental. +Experiments testing the hypothesis that actualization constructs spacetime are needed. +The Casimir effect may prove useful, (Kauffman et al., 2021 for example). iii. Further +testing of the capacity of the human mind to ‘collapse the wave function’ are needed. +The current data at a Sigma of 6.49 are strong. But this is a truly major claim that must +pass muster with critics. iv. Were it possible to demonstrate that actualization events +constructed spacetime and with good grounds established that mind can mediate +actualization, it might become possible someday to test if mind by mediating +actualization can construct spacetime. v. The current article is at best a conceptual +framework. A far more formal and integrated mathematical theory must be constructed +and ultimately tested. +Section 13. Summary and Conclusion +The dream of physics since the discoveries of General Relativity and Quantum +Mechanics nearly a century ago has been their union in Quantum Gravity. Yet since +Newton, a role for mind in the becoming of the Cosmos has seemed precluded. In 2022 +14 + +NASA launched a rocket that nudged a distant asteroid in the Solar System into a slightly +different orbit altering the orbital dynamics of the solar system. Mind has cosmic +consequences. +In the current article we take the results of Aerts et. al as if they were �irmly established. +With a p value of .001 the results are at most grounds for consideration. More +experiments are needed. However, assuming such �irm results, human cognitive events +surpass the Tsirelson bound. Attempts to explain such a result within a backgound +spacetime that demands Continuity of Action and non-signaling have severe dif�iculties. +Were mind not in spacetime the requirement for Continuity of Action and nonsignaling +would not arise. The possibility of cognitive events surpassing the Tsirelson bound +would be arise. But this would require that mind correspond to something “real” that is +not in spacetime, and also that cognitive events themselves exist in the actual experience +of humans, hence in spacetime. +We approach quantum gravity by taking nonlocality as fundamental. If nonlocality is +fundamental, spacetime is not fundamental. Non locality arises with two or more +entangled coherent variables. Thus, we are forced to the conclusion that spacetime +somehow emerges from the behaviors of coherent entangled variables. This �latly +contradicts General Relativity which is local, spacetime does not emerge in General +relativity, and GR can be formulated without matter �ields so the very existence of +spacetime cannot depend upon matter. +Based on the above it is straightforward to de�ine a metric distance between each pair of +entangled variables in Hilbert space as the subadditive von Neumann Entropy of that +entangled pair. But this set of distances is in Hilbert space whose variables can be in +superposition. Classical events in spacetime cannot be in superposition. Hence it +becomes natural to �ind a map between the metric in Hilbert space and classical +spacetime by successive actualization events in which the distance between actual +events correlates with those of the corresponding variables in Hilbert space. Nearby +entangled variables in Hilbert space construct themselves into nearby points in classical +spacetime. +In the present article we hope to account for evidence that cognitive events do surpass +the Tsirelson bound by identifying mind with coherent entangled quantum variables +that constitute the quantum vacuum and are not in spacetime. These are to map to +cognitive events within spacetime by actualization events that constitute qualia. +Increasingly strong grounds exist to support the view that conscious events – qualia – +accord with actualization events. Further, evidence now stands at 6.49 sigma, or 4 in +100,000,000,000, in support the claim that mind acausally mediates actualization. +The union of the above issues then constitute a vision of quantum gravity in which mind +is identical to the entangled coherent quantum variables of the quantum vacuum and +mind itself mediates the actualization events that construct classical spacetime. +15 + +Such a vision is not yet united with General Relativity. A new union may be possible in +which quantum gravity constructs the classical spacetime in which General Relativity +operates. +General Relativity requires a world of classical objects. Among these, some are very +simple, some like the evolved proteins in the human brain are very complex. The +quantum behaviors of very complex molecules and groups of molecules will be far +richer than those of a simple small quartz crystal. Therefore, the mind of a brain can be +far more complex that a mind of a crystal. And the quantum behaviors of one brain will +be partially unique to that brain and its ontogenetic and experiential history. But the +quantum behaviors of entangled variables in brains, when coherent, are not in +spacetime and are part of the quantum vacuum. Upon actualization, these entangled +variables that are neighbors in Hilbert space construct themselves to nearby points in +the matter in classical spacetime, thus typically to events located in the same brain. “My +memories and thoughts are mine, not yours.” Yet by entanglement between brains, +telepathy is possible, and precognition is possible. By entanglement between variables +in a brain and other physical objects, psychokinesis is possible The data for all these are +now abundant at high Sigma values. +The concepts and data we have discussed do not yet warrant such enormous +conclusions. Far more would be required. Yet, perhaps for the �irst time since Newton, +they may constitute the start of a conceptual framework uniting Mind, Matter and +Cosmos. +References +Atmanspacher, H. and Filk, T., 2019. Contextuality revisited: Signaling may differ from +communicating. In Quanta and Mind (pp. 117-127). Springer, Cam. +Aharonov, Y., & Bohm, D. (1961). Further considerations on electromagnetic potentials +in the quantum theory. Physical Review, 123(4), 1511. +Aerts, +D., +Gabora, +L. +and +Sozzo, +S., +2013. +Concepts and their dynamics: A +quantum-theoretic modelling of human thought. Topics in Cognitive science, 5(4), +pp.737-772. +Aerts, D., Sassoli de Bianchi, M., Sozzo, S. and Veloz, T., 2021. Modeling human +decision-making: An overview of the Brussels quantum approach. Foundations of +Science, 26(1), pp.27-54. +Aerts, D. and Arguëlles, J.A., 2022. 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Life and death in the tails of the GRW wave function. arXiv preprint +arXiv:1407.4746. +Zeilinger, A., 1998. Fundamentals of quantum information. Physics World, 11(3), p.35. +Zurek, W.H., 2022. Quantum Theory of the Classical: Einselection, Envariance, Quantum +Darwinism and Extantons. Entropy, 24(11), p.1520. +18 + diff --git a/KtFOT4oBgHgl3EQfzTRj/content/tmp_files/load_file.txt b/KtFOT4oBgHgl3EQfzTRj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88b5b8eb72072c8e37dabb420aae5dcc0ff7a9fb --- /dev/null +++ b/KtFOT4oBgHgl3EQfzTRj/content/tmp_files/load_file.txt @@ -0,0 +1,739 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf,len=738 +page_content='Human Cognition Surpasses the Nonlocality Tsirelson Bound: Is Mind Outside of Spacetime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Stuart Kauffman1, Emeritus Professor of Biochemistry and Biophysics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA Sudip Patra2, Associate Professor OP Jindal Global University, Founding member CEASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Dec 26, 2022 Abstract Recent experimental studies on human cognition, particularly where non-separable or entangled cognitive states have been found, show that in many such cases Bell or CHSH in-equalities have been maximally violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The implications are that greater non-local correlations than allowed in quantum mechanics (often known as the Tsirelson bound), are found in human cognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We propose in the current paper that a non-local theory of mind is needed in order to account for the empirical �indings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This requires a foundationally different approach than the extant ‘quantum-like’ approach to human mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Our account is novel, but still founded on a Hilbert space set up with the physical constraint of no-signalling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' To account for the surpassing of the Tsirelson bound we propose abandoning the constraint of no-signalling that depends upon spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Thus we ask;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' ‘Is mind outside spacetime?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We discuss a candidate theory of quantum gravity based on non-locality as fundamental that may accord with our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We are led to suggest a new 6 part ontological framework linking Mind, Matter, and Cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Key words: non-locality, no-signalling, Bell inequalities, Tsirelson/Cirelson bound, PR boxes, Cognition, quantum gravity, six-part framework Introduction: Is mind outside physical spacetime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Non-locality has baf�led us since the birth of modern science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' For example in Newton’s gravity framework, we have action at a distance, and Newton himself did not want to forward any ‘explanation’ of the same by stating, “hypothesis non-�ingo”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Later with the advent of Special Theory of relativity (SR), and then the General Theory of Relativity (GR), Einstein nearly singlehandedly challenged the age old concepts of space and time, proposing the bold and beautiful concept of spacetime, where continuity of action (COA) plays a central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' COA holds that if a spacetime event A has to in�luence another spacetime event B then it also has to in�luence any closed 3 surface between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Hence no-signalling, or that there is an ultimate limit of signalling between spacetime 1 events, which happens to be the speed of light in vacuum, became the foundational physical constraint for any sound theory of Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The orthodox quantum mechanics (QM) which emerged from intense discussions in Solvay conferences (for example, in Pylkkanen, 2019), and later known as Copenhagen version, was, however, still based on a Newtonian space and time background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Later, with the emergence of quantum �ield theory (QFT) there has been an uncomfortable coexistence of SR and QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The holy grail of modern Physics has been to construct uni�ied �ield theories, and particularly quantum gravity (QG), as the cherished uni�ication of GR with QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However, in all of these numerous attempts, non-locality has been a recurrent problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Even different interpretations of QM, starting from Collapse of the wave function, to different alternative theories like Bohmian mechanics (Walleczek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 2019), or spontaneous collapse of wave function (for example in Tumulka, 2006), have been riddled with different forms of non-localities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Very recently different approaches to QG (Kauffman, 2022) would presume to hold non-locality as fundamental, which is radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Spontaneous collapse models or dynamic collapse models have attempted to resolve the measurement problem by introducing a collapse operator in the Schrödinger equation, for example in GRW (for example in Wallace, 2014) where a probability of such a stochastic collapse is small in case of single particles, but grows exponentially in case of many body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Hence, the attempt has been to resolve the incompleteness or inconsistency problems in orthodox QM, for example, how in the same framework both deterministic and unitary Schrödinger evolution and random collapse of wave function due to ‘measurement’ can be accommodated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However, very recent work, (for example see ‘consciousness and quantum mechanics’ edited by Shan Gao, 2022, and Ball 2022), now says that any “physically causal” theory for measurement is almost ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' There are also physically “acausal” accounts of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Here we refer to the recent consciousness induced collapse framework of Chalmers and McQueen (2021), where phenomenal consciousness plays the role of a superposition resistant, hence de�inite consciousness state that result in “collapse”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' More recently, Kauffman and Roli (2021), and Kauffman and Radin (2022) have utilized Heisenberg’s interpretation of quantum mechanics in terms of ontologically real Potentia, Res potentia, and ontologically real Actuals, Res extensa, where actualization converts Possible to Actuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This interpretation does not inherit the mind-body problem because Potentia are not substances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In turn this approach suggests a natural role for “mind” in actualizing quantum potentia, hence “collapsing the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' At this point, data supporting this hypothesis with respect to work using the two slit experiment are strongly supportive at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='49 Sigma, or 4 x 10 ^ -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We shall base our own discussion on Heisenberg’s interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In addition to Heisenberg’s “potentia” interpretation, workers have studied several other non-realist frameworks, where the wave function is not ontological, but rather a tool for computing probabilities for epistemological updates of knowledge state of observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Two alternative strongly emerging interpretations of QM are relational QM and QBism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Relational QM holds that QM, or reality for that matter, is not described by quantum states, but rather by relations among observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This is a fact ontology (for more details about “relative” and “stable” facts, we can refer to seminal literature, 2 (Pienaar, 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' QBism agrees on placing a central role on phenomenology or subjective experiences, where QM is the navigation tool for any user (rather than de�ining who is the user) to make optimal decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In QBism relations between the elements of the framework are objective, such that every agent would agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We differ from these frameworks in that these frameworks are largely based on the locality of physical spacetime, but then they face non-locality problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' One approach to surpassing the Tsirelson bound is found in PR box worlds (Popescu, 2014, a modern review of Tsirelson bound can be found in Stuckey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', 2019) that allow for greater than QM non-local correlation limit the Tsirelson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However, the PR box worlds correspond to no physical model of a universe, (Popescu, 2014 op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Hence a related question raised earlier was whether QM is the only theory where there is a co-existence of non-locality in the sense of Bell inequalities violation and relativistic causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We propose in this paper that if we need to include mind or cognitive aspects in the foundational frameworks of nature, then we need to have non-locality as the central feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In this paper then, we explore our framework of non-local mind or cognition, and are led to our proposal that “mind” is not in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' By proposing that mind is not in spacetime, we can naturally eliminate the requirement for Continuity of Action, hence non-signaling, that makes sense only within a framework of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' By proposing that mind is not in spacetime, mental events that are in spacetime but surpass the Tsirelson bound can be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In order to begin to make sense of the concept that “mind is not in spacetime, but conscious events are in spacetime”, we are led to propose a novel 6 part ontological framework linking Mind, Matter, and the Cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The grounds for this novel framework are tentative, but we hope worthy of consideration and are testable in part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The current paper is organized in sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 1 provide the background of cognitive frameworks with experimental work, and its recent reformulations in terms of ‘quantum-like’ features, for example entangled cognitive states which violates Bell inequalities strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 2 discusses alternative ways to surpass the Tsirelson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 3 presents our novel 6 part framework and the current grounds to consider it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 4 summarizes our results and suggestions for further experiments and work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 1 Cognition beyond the Tsirelson Bound Aerts et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' al (2013, 2021) have pioneered the study of non-separable states in individual minds or cognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This includes how different concepts are combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Such concept combinations in individual minds can be re-formulated as non-separable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In technical language these are intra state entanglements, which would mean coupling of different degrees of freedoms of a single system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Here these are individual minds, and the data can be expressed through inequalities such as CHSH as we explain in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The statistical values or values obtained from ensembles of ‘minds’ of participants in such cognitive experiments can be inputted as inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The results have demonstrated a clean violation of the ‘non-local’ correlation bound which occurs in QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 3 Such a tight bound is, as noted, called a Tsirelson bound, that is maximal and characteristic for QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The same authors also provide the statistical signi�icance of their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Aerts and Arguelles (2022) have claimed a statistical signi�icance of p values ranging 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='001-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='005, which is strong enough to suggest, but not yet prove, the viability of their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Aerts et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' al (op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=') also adopts a Hilbert space framework, but their strategy is of ‘reverse engineering’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' to start with the empirical results, and then describe such results by a suitable state space modelling, where the state space can be either Hilbert space or a larger Fock space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Thus, the usage of CHSH inequalities is statistical in nature, since such inequalities are general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Maximal algebraic violation of such inequalities can be beyond the Tsirelson bound, but when Hilbert space is the state space then a tight upper bound comes up as a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Given the above points, Aerts et al.’s claim of greater than Tsirelson bound violation in cognitive experiments raises several questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' For example, can any or all Hilbert space formulations account for such super quantum correlations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Aerts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' have responded by suggesting that an entanglement that they consider is of a more complex nature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' entanglement both in states as well as measure, might account for super violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We assess this approach below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We stress again that any Hilbert space formulation of quantum mechanics implies a tight Tsirelson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' And we stress again that the Hilbert space formulation is stated in a background spacetime with “no signaling” and continuity of action, hence “locality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Non-locality : Implications for QM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='1 Non-locality in QM Here we remind ourselves of the seminal contribution of John Bell (1964, 1966), and state the basic requirements for Bell factorization conditions, upon which the celebrated Bell inequalities or later CHSH inequalities are based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Based on continuity of action, the following three assumptions are required for establishing Bell factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Statistical Independence: , where denotes the local hidden ρ 𝑀 ( ) = ρ µ ( ) µ variable, and M stands for measurement settings of apparatuses for different space-like separated agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Output independence: , subscripts a and ρ𝑎𝑏 𝑎, 𝑏, µ ( ) = ρ𝑎(𝑥𝑎|𝑎, 𝑏, µ)ρ𝑏(𝑥𝑏|𝑎, 𝑏, µ) b stands for different agents, namely, Alice and Bob, x’s are outcomes at their ends and a, and b are inputs at their ends respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Parameter independence: ρ𝑎 𝑎, 𝑏, µ ( ) = ρ𝑎 𝑎, µ ( ), 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑙𝑦 ρ𝑏 𝑎, 𝑏, µ ( ) = ρ𝑏 𝑏, µ ( ) Hence, in conjunction of the three assumptions we have the Bell factorization (1) ρ𝑎𝑏 𝑎, 𝑏, µ ( ) = ρ𝑎 𝑎, µ ( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' ρ𝑏 𝑏, µ ( ) Bell factorization is a general condition based on the local realism assumptions (COA to be precise), which is violated by different theories in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' For example, QM violates Bell factorization by violating output independence but keeping statistical 4 independence and parameter independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Super Deterministic theory violates the same by violating Statistical independence, while keeping the other assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' And Cavalcanti and Wiseman (2012) have showed how Bell factorization can be derived from conjunction of local ‘signalism’ and predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In the form of CHSH, we have two space-like separated agents, Alice and Bob, where say the measurement settings in Alice’s end are {a, a’} and Bob’s end are {b, b’}, and all results are dichotomous (say, +/- 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=" Here we de�ine the correlation function as (2) 𝑐 𝑎, 𝑏 ( ) = 𝑃𝑎,𝑏 1, 1 ( ) + 𝑃𝑎,𝑏 − 1, − 1 ( ) − 𝑃𝑎,𝑏 1, − 1 ( ) − 𝑃𝑎,𝑏(− 1, 1) Hence, we have the CHSH inequality as 𝐶𝐻𝑆𝐻 = 𝑐 𝑎, 𝑏 ( ) + 𝑐 𝑎, 𝑏 ' ( ) + 𝑐 𝑎 ', 𝑏 ( ) − 𝑐(𝑎 ', 𝑏 ') (3)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Hence CHSH has different upper bounds for different underlying theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' For example for a local deterministic theory (COA is the requisite here) we would always have as For QM the maximum violation of the above limit would take place when 𝐶𝐻𝑆𝐻 | |≤2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=" , hence this gives the Tsirelson (T bound 𝑎, 𝑏 ( ) = 𝑐 𝑎, 𝑏 ' ( ) = 𝑐 𝑎 ', 𝑏 ( ) =− 𝑐 𝑎 ', 𝑏 ' ( ) = 2/2 from now) bound of ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=" However algebraically it is possible that we have |𝐶𝐻𝑆𝐻|≤2√2 , hence making the maximal upper bound as 𝑐 𝑎, 𝑏 ( ) = 𝑐 𝑎, 𝑏 ' ( ) = 𝑐 𝑎 ', 𝑏 ( ) =− 𝑐 𝑎 ', 𝑏 ' ( ) = 1 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Different forms of non-separable states: QM and beyond We mention here that generally composite systems in QM can be represented as tensor products of states belonging to different Hilbert spaces, such that the total Hilbert space of the composite system is a tensor product of such Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This context is called product states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In addition, we recall that a Tensor product space is strictly larger that space of direct sums, hence this context also captures ‘quantum-holism’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Now the typical de�inition of an intersystem entanglement is when the composite system state cannot be de�ined as simple tensor products of subsystem states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Intersystem entanglement is most discussed in QM literature, since that is what generates non-local correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In an entangled state the whole is always in a pure state, whereas parts are not in pure states, this is the classical Schrödinger way of denoting entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Again, as we have stated earlier, maximally entangled states (often called as Bell states) can violate CHSH maximally until the T bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However, intra-system entanglement, de�ined as coupling between multiple degrees of freedom of the same system, is also discussed widely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Particularly in the classical electromagnetism literature authors (Ghose and Mukherjee, 2014 for example) have observed widely that intra system entanglement, for example coupling between path and polarization states of a vortex beam, can produce such non-separable states (at times called Shimony-Wolf states) which can generate violations of CHSH inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Authors, for example, Khrennikov (2020) has suggested that intra and inter system entanglements is the main difference between quantum and so-called ‘classical’ entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 5 Multipartite non-locality: Traditionally Bell tests or CHSH tests are bi-partite non-locality tests, there have been several modi�ications though, for example GHZ states or W states, which extends frameworks for many body entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In our previous framework (Kauffman and Patra, 2022) we start with a multipartite entanglement state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However, its only recently (Bancal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' in 2013 for example) that a suitable mathematical framework is being built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Here we refer to the basic tenets of such a framework, since this might be harnessed in the framework we suggest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We mention here that non-locality is a recurrent feature for many-body systems too (see for example in Bancal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 2013), for example if we consider a tripartite system, with say each subsystem possessed by Alice, Bob and Charlie who are spatially separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Say Alice, Bob and Charlie’s experimental set ups are X, Y and Z respectively and outcomes of experiments are a, b and c respectively (binary outcomes for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' If the joint probability (if de�ined) where, q’s are 𝑃 𝑋𝑌𝑍 ( ) = 𝑙 ∑ 𝑞𝑙𝑃 𝑋 ( )𝑃 𝑌 ( )𝑃 𝑍 ( )(4), bounded by 0 and 1, and sum to unity, then the sum represents local correlations, where the subscript l is for underlying hidden variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However, if the above joint distribution cannot be written in the above format, then some degree of non-local correlations exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' One example of non-locality (technically S2 non-local): , where separately q’s 𝑃 𝑋𝑌𝑍 ( ) = 𝑙 ∑ 𝑞𝑙𝑃 𝑋 ( )𝑃 𝑌𝑍 ( ) + 𝑚 ∑ 𝑞𝑚𝑃 𝑌 ( )𝑃 𝑋𝑍 ( ) + 𝑛 ∑ 𝑞𝑛𝑃 𝑍 ( )𝑃 𝑌𝑍 ( )(5) sum up to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Here we see that in individual sums full factorization is not achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' At times, such contexts are also called hybrid non-locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In another related literature (Bennet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', 1999 as one seminal work in this direction) non-locality without entanglement is theoretically proposed, and later experimentally veri�ied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We refer to these studies to seek further support for our assertion that non-locality is a more universal and genuine feature of reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We also are aware of studies differentiating between genuine non-locality and direct in�luences (see for example Atmanspacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Attempts to �it the evidence for non-locality withing the framework of a background spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In the last century intense debate on non-locality, or more precisely what non-locality should mean given relativistic spacetime, was a major debate, and is still continuing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The non-locality debate has also thrown deeper light on the foundational thinking on QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We observe here that the axioms of special theory of relativity (COA) or consequently fundamental limit for speed of signalling between spacetime events, and the equivalence of inertial reference frames) seems to be elegant and physically based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However, the axioms of QM seem to be mathematical with no clear physical basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Aharonov and Bohm (1961), and later Popescu, and Rohrlich (1994), and independently Shimony (1993) have proposed that QM has to be compatible with relativistic causality, hence with Continuity of Action, COA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The efforts of the authors mentioned showed that non-local correlations, for example in an EPR set up, can be compatible with relativistic causality if and only uncertainty of outcomes of measurements is fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Or in 6 other words the effect of a cause here is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' (Thus, counterfactuals are required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Aharonov was the �irst to propose ‘modular’ quantum variables, that are non-local in spacetime due to non-local relativistic phases, and they have optimal uncertainty for no signalling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Shimony amusingly observed the whole affair is ‘passion at a distance’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Attempts to surpass the Tsirelson bound in formal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Based on the dense PR box literature, there have been many attempts to make super quantum correlations (violating T bound) compatible with relativistic causality, or COA in general,(for example, Popescu, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Related questions have been whether QM is the only possible theory where non-local correlation and no signalling co-exists?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', (Popescu 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Or why QM does not exhibit greater non-locality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', (Linden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 2007 for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We further observe that there have been efforts in the line of including communication complexity, and or, information causality to eradicate super quantum correlations, (for example, in Jaeger, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We also note that super correlations or greater than T bound violations are possible in con�iguration spaces with very particular properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Overall, there has been an attempt to make violations of Bell inequalities (not super correlations) compatible with relativistic causality, but it is far from clear what would be the implications for super correlations for a locality criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' As we explore below, how violations of Bell / CHSH or even super correlation results have been observed in cognitive experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We also note that some authors (Khrennikov, 2022) have observed that if the observables in a particular theory cannot be represented by Hermitian operators, there might not be any T bound constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Is Spacetime Fundamental?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Zeilinger and Information: It is important to stress that several authors are exploring the idea that spacetime is not fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In particular, Zeilinger has proposed that “information” is fundamental and somehow spacetime emerges from “information”(see for example Zeilinger’s seminal works since 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We note a central issue, “information” itself implies “possibilities” that are not either true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Consider Shannon information and the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' A given bit string, say (11111) can carry no information unless one of the bits can, counterfactually, be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' That is, it must be possible that one of the bits is 0 not 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Thus the very concept of “information” requires more than one simultaneously possible state of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Res potentia and Res extensa linked by measurement: In the current article, we base our approach on Heisenberg’s interpretation of the quantum state as “potentia standing ghost – like between an idea and reality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' One of us, ( Kastner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 2018) has developed Heisenberg’s interpretation as “Res potentia” ontologically real Possibles, and Res extensa, ontologically real Actuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Possibles do not obey Aristotle’s law of the excluded middle and law of noncontradiction, so are neither ‘true’ nor ‘false’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This allows “Potentia” to explain quantum superpositions: “Schrödinger’s cat simultaneously is possibly alive and possibly dead.” This is not a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Potentia are non-spatial in nature but ontologically real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' By contrast Actuals do obey Aristotle’s two laws, so are either true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' All of Classical physics is based on such true false Boolean variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Given the concept of Res potentia, one of us, (Kauffman) 7 has explored a new approach to quantum gravity that takes non-locality to be fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Non locality taken as fundamental implies that spacetime is not itself fundamental, but must somehow arise from the behaviors of entangled coherent, hence non local, quantum variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Then non-local entangled coherent quantum variables, “Res potentia” are not in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' They are Potentia not in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Mind and the Quantum Vacuum: One natural interpretation of the line of thought above is that the quantum vacuum consists precisely in non-local entangled quantum coherent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Given the above, a natural proposal is that ‘mind’ – non-spatial, is identical or related to the quantum vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We here both propose this identity and explore its potential validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' A �irst implication of the proposed identity of mind and the quantum vacuum is that both are outside of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This is a possible step to explaining Aert’s results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' To do so, we need to show that surpassing the Tsirelson bound is straight forward if mind is outside of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In this case we can abandon no signalling and continuity of action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We show this next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' But there is a further issue, Aerts et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' al data concern experiences of humans and those experiences are in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Powerful recent arguments now strongly suggest that conscious experiences (phenomenal nature) arise upon collapse of the wave function, hence, qualia are in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' And further remarkable evidence now clearly shows that we can purposefully actualize the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' A responsible free will is not ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We address all this below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' These recent results and claims will be part of our proposed 6 part framework introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Surpassing the Tsirelson Bound if Mind Is Outside of Spacetime Aerts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='(op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=') themselves have attempted to justify the super quantum correlation values obtained in their ‘concept-combination’ experiments based on complex entanglement nature in their experimental settings, given that the con�iguration space of mind is a high dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However the standard belief (going back to Popescu and Rohrlich) has been that the maximum limit of ‘non-locality’ allowed in a Hilbert space is the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Our perspective is not to justify the super violations based on the complexity of entanglement (both in states and in measurements), since there have been critiques of this line of argument by suggesting that if the ‘marginal selectivity’ rule is also violated along with Bell inequalities (which Aerts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' observes) then there can be contaminations in testing for Bell violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Hence we propose the six part framework, where our de�inition of mind need not be constrained by any physical locality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Mind and Qualia – Collapse of the Wave Function Recently Chalmers and McQueen (2022), who have been very sceptical about mind collapsing wave function, or a relation between QM and phenomenal consciousness in general, have designed a framework in which phenomenal consciousness might collapse wave function and thus a de�initive ‘classical’ world emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The framework suggested is based on IIT (Tononi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 2016 for example) or integrated information theory, and also where phenomenal consciousness – qualia – is considered as ‘superposition resistant’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Here we observe that Chalmers and McQueen (2022) have proposed a partial 8 quantum Zeno effect for completing their consciousness induced collapse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Our previous framework for the emergence of the classical world naturally includes a partial Zeno effect, with trade-offs between Zeno effect and atmospheric de-coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We didn’t have non-local mind explicitly in the previous framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In addition to Chalmers and McQueen, (op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' ), Kauffman and Roli (op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=') have recently proposed that the human mind cannot be algorithmic, and that the capacity to �ind novel affordances requires a quantum mind and qualia associated with the collapse of the wave function to a single state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The next section presents evidence that humans can, in fact, collapse the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We Can Collapse the Wavefunction An old idea in quantum mechanics is that mind might have something to do with “collapse of the wave function”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Von Neumann proposed this, (1955/1932).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Wigner suggested the same idea at one point( see for example Wigner, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Following Heisenberg, as noted, we propose Res potentia, ontologically real Possibles, and Res extensa, ontologically real Actuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Here “actualization” converts Possibles to Actuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This assertion is fully consistent with recent results, (Gao, op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', Bell, op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='.), that seem to rule out physical causes of actualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' A physical cause cannot convert a possible to an actual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Res potentia and Res extensa plus actualization is the �irst new idea about mind and body since Descartes’ substance dualism, Spinoza’s neutral monism, Berkeley’s Idealism, and pure materialism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Res potentia and Res extensa is not a substance dualism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Potentia are not substances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Thus this view does not inherit the mind – body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Instead Res potentia and Res extensia suggest a natural role for mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Mind “actualizes” Possibles to Actuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Strong evidence now supports this scienti�ically testable hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Radin and his colleagues (for example see Radin 2019) have tested the capacity of humans paying attention to modify the intensities of the adjacent central bands in the famous interference pattern of the two slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The effect is weak, but has been tested in 30 independent experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' At present the positive results are very strongly statistically signi�icant at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='49 Sigma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The probability, “p”, that this arises by chance now stands a less that 4 x 100,000,000,000, (Kauffman and Radin, op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This is strong enough to take very seriously as yet further data are sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' If accepted, the results alter the foundations of Quantum Mechanics with a fundamental role for mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Indeed, even a responsible free will is not ruled out, (Kauffman Radin, op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' For the purposes of this article, we will accept these results as true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Quantum Gravity if Non locality Is Fundamental One of us has recently published a work on quantum gravity (Kauffman op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='The starting point is to take nonlocality as fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Nonlocality arises in the presence of entangled coherent quantum variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' If one starts with nonlocality it is not necessary 9 to explain nonlocality, but necessary to explain locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Somehow locality – spacetime– is to emerge from the behaviors of the quantum variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This immediately �latly contradicts General Relativity which is local, and in which there is no emergence of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Further, General Relativity can be formulated in the absence of matter so matter cannot be necessary for the very existence of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' But if one starts with nonlocality, the emergence of spacetime must depend on the matter – the entangled coherent quantum variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' A further note is that there is no apriori reason not to start with nonlocality as fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The steps in building this new theory of quantum gravity start with N entangled variables in Hilbert space, then constructs a metric distance between each entangled pair of variables as the sub-additive von Neumann Entropy between that pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Sub-additive von Neumann Entropy, therefore, �its the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The next step notes that quantum variables can be in superposition and interpreted as potentia, neither true nor false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' All the variables of classical physics are true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Hence the next step in the development of the theory maps distances in Hilbert space to classical spacetime distances between a succession of true actualization events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In this mapping entangled near-neighbours in Hilbert space construct themselves into nearby points in classical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The hypothesis that actualization events construct spacetime is probably testable using the Casimir effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Emergence of the Classical World Here we refer to the ontological framework developed by Kauffman and Patra (2022), which also forms one reference for the current framework, though we didn’t include non-local mind in our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We based our previous work on the premise that measurement and actualization, which creates the de�initive classical world (this coincides with the contextuality-complementarity philosophy of Bohr1) can happen only in a speci�ic basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However we still do not have a comprehensive theory for the emergence of a speci�ic basis, except the recent attempts from Quantum Darwinism perspectives as proposed by Zurek (2022) in terms of de-coherence theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We note that decoherence does not yield a speci�ic basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We have proposed the following steps for the emergence of classical world, in which testable experiments can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' (i) We start with sets of N entangled quantum variables, which need not be maximally entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Variables can mutually actualize each other, which is approximated by the quantum-Zeno effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' (ii) Such actualization occurs in one of the 2Nbases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' (iii) Mutual actualization breaks symmetry among these 2N bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' (iv) An amplitude for a speci�ic basis can emerge and increase with further measurement in the same particular basis, it can also decay between measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 1 Here one can also refer to recent works of Kastrup (), where if we claim that only actualization creates the definitive world, which would mean no pre-existing values, we should also accept that the world as a whole is beyond only physical, or the typical physical closure principle would not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 10 (v) As the number of variables, N, in the system increases, the number of Quantum Zeno mediated measurements among the N variables increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' (vi) Now for experimental purposes, quantum ordered, quantum critical, and quantum chaotic peptides that decohere at nanosecond versus femtosecond time scales can be used as test objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' (vii) By varying the number of amino acids, N, and the use of quantum ordered, critical, or chaotic peptides, the ratio of decoherence to Quantum Zeno effects can be tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This enables new means to probe the emergence of one among a set of initially entangled bases via weak measurements after preparing the system in a mixed basis condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' (viii) Use of the three stable isotopes of carbon, oxygen, and nitrogen and the �ive stable isotopes of sulfur allows any ten atoms in the test peptide or protein to be discriminably labelled and the basis of emergence for those labelled atoms can be detected by weak measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We present an initial mathematical framework for this theory, and we propose experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' If Mind is Outside of spacetime, What is “My” Mind?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' If we are to make sense of Aerts et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' al data (op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' ), and do so by proposing that mind is outside of spacetime but that the cognitive experience is in spacetime, we must claim that qualia emerge upon actualization events, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' But in addition, it becomes fundamental to address the issue: What maps the quantum variables in Hilbert space and the vacuum to “My Mind”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The theory of quantum gravity based on nonlocality as fundamental almost automatically affords a possible answer to this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Compare the relatively simple quantum behaviors of a quantum variables in a quartz crystal and the presumably far more complex behaviors of the quantum variables in the diverse proteins in a speci�ic human brain with its unique genetic background and life experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The proposal is that when these quantum variables become coherent, they are not in spacetime but part of the quantum vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The behaviors of these variables in the vacuum must exhibit and re�lect the complexities the quantum behaviors in that speci�ic brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Because entangled neighbors in Hilbert space map to spatial neighbors in classical spacetime and the matter in it, actualization events with qualia will typically map to and occur in the same brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Thus, “What is My Mind” seems naturally answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' These proposals claim to answer Aerts et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' al (op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' My mind is not in spacetime, so not bound by continuity of action and nonsignaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The Tsirelson bound can be surpassed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' But actualization occurs in my brain so are my qualia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The Quantum Vacuum and the Matter in the Universe Our proposal to start with nonlocality as fundamental drives a different conception of the quantum vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This vacuum is normally conceived in the absence of any matter and as a coupling of all the fundamental �ields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The same can be considered as coupled quantum harmonic oscillators whose zero point energy can be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' As so conceived, the spectrum of the quantum vacuum must be stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 11 By contrast, if nonlocality is taken as fundamental, spacetime is not fundamental and can only arise due to the behaviors of the quantum variables when coherent and also when not coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In the latter case, the Schrödinger equation no longer applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The quantum behaviors of quarks, protons, neutrons, and electrons in complex proteins must differ from those in a simple crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' With this seemingly necessary inference, the behaviors of the quantum vacuum – coherent entangled quantum variables, cannot be stationary over the history of the universe as more and more complex classical systems, stable for long periods, come into existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We can propose, quantum vacuum must also re�lect the history of the behaviours of the ever more complex matter than has come to exist and vanished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The Six Part Ontological Framework: Mind Matter and Cosmos The above considerations lead us to propose that: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The quantum vacuum is composed of entangled coherent quantum variables that are ontologically real “Possibles”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' ii “Mind” is identical to the Possibles of the quantum vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Hence this is the de�inition of mind in our framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The vacuum is outside of spacetime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Mind can mediate actualization of potentia ,(Kauffman and Radin, op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' actualization of potentia then constructs classical spacetime, where a metric exists in the quantum vacuum Hilbert space via non-additive von Neumann Entropies between pairs of entangled variables, that is then mapped to events at speci�ic classical spacetime locations, (Kauffman quantum gravity);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We experience such actualized quantum variables as “qualia”, (Chalmers and McQueen, Kauffman and Roli, 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In the last, sixth, part we propose the emergence of classical world, which is based on our previously proposed framework (Kauffman and Patra, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We suggest a mutual actualization process of quantum variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Through a trade-off between the quantum Zeno effect and atmospheric de-coherence, such de-cohering and re-cohering variables creates the observable classical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In this framework we suggest veri�iable experiments with peptides whose entangled variables decohere exponentially fast versus peptides whose entangled variables decohere power law slowly as a possible ground of test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We hope to show that the six part proposal above allows us to account for Aert’s et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' al results that surpass the Tsirelson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Far more, this new six part framework may help organize our emerging ideas about “Mind, Matter, and Cosmos”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Discussion and Further Work Non-locality has always baf�led us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The non-local and non-deterministic collapse of wave function in QM worried Einstein throughout his working life, since the fear was such non-locality would mean action at a distance and thus break- down of the causality structure of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The latter is fundamental to any Physical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Certainly, a huge literature has demonstrated that non-local collapse may not mean any superluminal signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Later since Bell’s seminal contribution, there have been many versions of such frameworks (CHSH being the most popular), which have suggested that local hidden variable theories cannot reproduce QM faithfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Loophole free Bell inequality/ CHSH inequality violations have demonstrated that one or more of the basic underlying assumptions, of localism, realism or non-contextuality, or statistical independence have to be relaxed to describe the empirical results of QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 12 Many workers have shown that Bell inequalities violations are considered to be evidence of non-local correlations between subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The canonical example is entangled pairs of particles (EPR set up for example) with agents measuring on each half of the pair who are space like separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In the presence of an assumed background spacetime, the only way a no signaling theorem is going to be preserved is by introducing inherent quantum uncertainty in outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In the words of Shimony there can be a happy co-existence between Special Relativity and quantum fundamental uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However, the Hilbert space structure, assumed to be the state space in such frameworks, inherently does set up an upper bound for violations of inequalities, the celebrated Tsirelson (or Cirelson) bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Thus, the question arises what if in any empirical observation exist where such a limit is violated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Over the last decade there has been strong evidence of violations of CHSH inequalities, pertaining to cognitive experiments (Aerts et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' al).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The data are now con�irmed at p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='001 to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Further work is needed to con�irm these results more strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However, they are already strong enough to warrant consideration of the implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Aerts et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' al (op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=') have tried to preserve a background spacetime and “no signaling” by assuming more complex entanglement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' both states and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The same authors have also claimed that quantum entities might be conceptual or cognitive entities, hence non-spatial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We propose here a novel, yet unexplored framework based on non-localism, where spacetime need not be fundamental to existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Non-locality is not mysterious in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Our attempt is to start from non-locality, and derive locality from �irst principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Then in such a constructed local spacetime we have standard QM and SR operate with restricted non-locality which is no signaling also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Our proposal is related to that of Aerts et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' al in an unexpected way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' As just noted, these authors propose quantum entities might be conceptual or cognitive entities, hence non-spatial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Almost in parallel, we propose that the quantum vacuum consists in ontologically real Possibles, that Possibles are non-spatial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' not in spacetime, that Mind is identical to these Possibles, that Mind can actualize these potentia, and we can experience these as qualia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' What should we make of this extensive new six part framework?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' A �irst point is that other attempts such as PR boxes correspond to no know physical reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Our proposal is not too distant from Aharanov’s non local proposal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' But as Shimony notes, this is “passion at a distance” in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In our six part framework, the correlations are among ontologically real possibles that are not in spacetime, but “mind” is/are part of the quantum vacuum of possibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' These possibles then constitute the information Zeilinger hopes is the basis, somehow, of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However Zeilinger offers no account of what information is, other than a “bit”, nor any idea of how these might be related to spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 13 In our account, spacetime is constructed by the sequential actualization of quantum variables in Hilbert space with a metric via non-additive von Neumann Entropies that then map to Actual events whose mutual distance relations re�lect the metric in Hilbert space to constitute spacetime, (Kauffman quantum gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This claim underlies our �irst part, “i”, and “iii”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' There are data at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='49 sigma to support “iv” and “v” above,(Kauffman, op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The second part, ii “mind” is identical to the possibles of the quantum vacuum, is an entirely new proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Oddly, this proposal just might afford a highly speculative answer to the point raised in a recent article on Biocosmology, (Cortes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', 2022), about a link between the emergence of life 4 billion years ago and the recent dominance of dark energy whose tight temporal coincidence in Cosmology is strange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' If living organisms actualize quantum variables far more often than the quantum variables of the abiotic universe, then life, via mind, can have played a role in the emerging dominance of dark energy in the past four billion years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Our vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' part concerns the emergence of the classical world from the quantum world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Our own proposal, (referring to Kauffman and Patra, 2022), has the virtue of being testable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In addition it automatically supplies the incomplete Quantum Zeno Effect desired by Chalmers and McQueen, (op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Our speci�ic proposal for the emergence of the classical world is consistent with our general framework i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' to vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', and is consistent with efforts to study how an increase in the mass of molecules such as the Buckyball may increase decoherence .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The proposal that quantum gravity is a quantum construction of spacetime is not yet united with General Relativity, but may be a new pathway to do so .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Such a union with our proposals in the present article might be fundamentally new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Such a union would embrace Mind, Matter and Cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The most important lines of further work are: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Experiments to test and extend the Aerts et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' al results, (op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' A ‘p’ value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='001 is of interest, but hardly persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Our six part framework rests heavily on taking non-locality as fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Experiments testing the hypothesis that actualization constructs spacetime are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The Casimir effect may prove useful, (Kauffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', 2021 for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Further testing of the capacity of the human mind to ‘collapse the wave function’ are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The current data at a Sigma of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='49 are strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' But this is a truly major claim that must pass muster with critics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Were it possible to demonstrate that actualization events constructed spacetime and with good grounds established that mind can mediate actualization, it might become possible someday to test if mind by mediating actualization can construct spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The current article is at best a conceptual framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' A far more formal and integrated mathematical theory must be constructed and ultimately tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Summary and Conclusion The dream of physics since the discoveries of General Relativity and Quantum Mechanics nearly a century ago has been their union in Quantum Gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Yet since Newton, a role for mind in the becoming of the Cosmos has seemed precluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In 2022 14 NASA launched a rocket that nudged a distant asteroid in the Solar System into a slightly different orbit altering the orbital dynamics of the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Mind has cosmic consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In the current article we take the results of Aerts et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' al as if they were �irmly established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' With a p value of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='001 the results are at most grounds for consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' More experiments are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' However, assuming such �irm results, human cognitive events surpass the Tsirelson bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Attempts to explain such a result within a backgound spacetime that demands Continuity of Action and non-signaling have severe dif�iculties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Were mind not in spacetime the requirement for Continuity of Action and nonsignaling would not arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The possibility of cognitive events surpassing the Tsirelson bound would be arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' But this would require that mind correspond to something “real” that is not in spacetime, and also that cognitive events themselves exist in the actual experience of humans, hence in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' We approach quantum gravity by taking nonlocality as fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' If nonlocality is fundamental, spacetime is not fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Non locality arises with two or more entangled coherent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Thus, we are forced to the conclusion that spacetime somehow emerges from the behaviors of coherent entangled variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' This �latly contradicts General Relativity which is local, spacetime does not emerge in General relativity, and GR can be formulated without matter �ields so the very existence of spacetime cannot depend upon matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Based on the above it is straightforward to de�ine a metric distance between each pair of entangled variables in Hilbert space as the subadditive von Neumann Entropy of that entangled pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' But this set of distances is in Hilbert space whose variables can be in superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Classical events in spacetime cannot be in superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Hence it becomes natural to �ind a map between the metric in Hilbert space and classical spacetime by successive actualization events in which the distance between actual events correlates with those of the corresponding variables in Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Nearby entangled variables in Hilbert space construct themselves into nearby points in classical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In the present article we hope to account for evidence that cognitive events do surpass the Tsirelson bound by identifying mind with coherent entangled quantum variables that constitute the quantum vacuum and are not in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' These are to map to cognitive events within spacetime by actualization events that constitute qualia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Increasingly strong grounds exist to support the view that conscious events – qualia – accord with actualization events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Further, evidence now stands at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='49 sigma, or 4 in 100,000,000,000, in support the claim that mind acausally mediates actualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The union of the above issues then constitute a vision of quantum gravity in which mind is identical to the entangled coherent quantum variables of the quantum vacuum and mind itself mediates the actualization events that construct classical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 15 Such a vision is not yet united with General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' A new union may be possible in which quantum gravity constructs the classical spacetime in which General Relativity operates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' General Relativity requires a world of classical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Among these, some are very simple, some like the evolved proteins in the human brain are very complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The quantum behaviors of very complex molecules and groups of molecules will be far richer than those of a simple small quartz crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Therefore, the mind of a brain can be far more complex that a mind of a crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' And the quantum behaviors of one brain will be partially unique to that brain and its ontogenetic and experiential history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' But the quantum behaviors of entangled variables in brains, when coherent, are not in spacetime and are part of the quantum vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Upon actualization, these entangled variables that are neighbors in Hilbert space construct themselves to nearby points in the matter in classical spacetime, thus typically to events located in the same brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' “My memories and thoughts are mine, not yours.” Yet by entanglement between brains, telepathy is possible, and precognition is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' By entanglement between variables in a brain and other physical objects, psychokinesis is possible The data for all these are now abundant at high Sigma values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' The concepts and data we have discussed do not yet warrant such enormous conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Far more would be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Yet, perhaps for the �irst time since Newton, they may constitute the start of a conceptual framework uniting Mind, Matter and Cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' References Atmanspacher, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' and Filk, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Contextuality revisited: Signaling may differ from communicating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' In Quanta and Mind (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 117-127).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Springer, Cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Aharonov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', & Bohm, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Further considerations on electromagnetic potentials in the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Physical Review, 123(4), 1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Aerts, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', Gabora, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' and Sozzo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Concepts and their dynamics: A 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='4746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Zeilinger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Fundamentals of quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Physics World, 11(3), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Zurek, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Quantum Theory of the Classical: Einselection, Envariance, Quantum Darwinism and Extantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' Entropy, 24(11), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content='1520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFOT4oBgHgl3EQfzTRj/content/2301.12931v1.pdf'} diff --git a/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf b/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cb3936564b63862af2c34157fea35414a8873645 --- /dev/null +++ b/MNAzT4oBgHgl3EQfV_xV/content/2301.01293v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9fb0865baa923e43f00964e10350ee04e11efbe3b62ece7e55c227e84f23f1b8 +size 689688 diff --git a/N9E0T4oBgHgl3EQfjgF4/content/tmp_files/2301.02460v1.pdf.txt b/N9E0T4oBgHgl3EQfjgF4/content/tmp_files/2301.02460v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c4d37fe1d47c715b158484fb8df1b7785f81d4e3 --- /dev/null +++ b/N9E0T4oBgHgl3EQfjgF4/content/tmp_files/2301.02460v1.pdf.txt @@ -0,0 +1,854 @@ +arXiv:2301.02460v1 [cond-mat.mes-hall] 6 Jan 2023 +Half-metal and other fractional metal phases in doped AB bilayer graphene +A.L. Rakhmanov,1 A.V. Rozhkov,1 A.O. Sboychakov,1 and Franco Nori2, 3 +1Institute for Theoretical and Applied Electrodynamics, +Russian Academy of Sciences, 125412 Moscow, Russia +2Center for Quantum Computing and Cluster for Pioneering Research, RIKEN, Wako-shi, Saitama, 351-0198, Japan +3Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA +(Dated: January 9, 2023) +We theoretically argue that, in doped AB bilayer graphene, the electron-electron coupling can give +rise to the spontaneous formation of fractional metal phases. These states, being generalizations of +a more common half-metal, have a Fermi surface that is perfectly polarized not only in terms of +a spin-related quantum number, but also in terms of the valley index. The proposed mechanism +assumes that the ground state of undoped bilayer graphene is a spin density wave insulator, with a +finite gap in the single-electron spectrum. Upon doping, the insulator is destroyed, and replaced by +a fractional metal phase. As doping increases, transitions between various types of fractional metal +(half-metal, quarter-metal, etc.) are triggered. Our findings are consistent with recent experiments +on doped AB bilayer graphene, in which a cascade of phase transitions between different isospin +states was observed. +PACS numbers: 73.22.Pr, 73.22.Gk +Introduction.— A usual metal demonstrates perfect +symmetry with regard to the carriers’ spin projection. +This symmetry manifests itself in the vanishing total +spin magnetization and the Fermi-surface spin degener- +acy. Yet the symmetry can be spontaneously destroyed +by sufficiently strong electron-electron interaction, which +may result, for example, in the formation of two non- +identical Fermi surfaces for the two spin projections. In +the extreme case of the so-called half-metals (HM), one +of these projections is completely absent from the Fermi +surface, while all states at the Fermi energy have identi- +cal spin quantum number [1–3]. Various rather dissim- +ilar materials with transition-metal atoms are found to +be half-metals [4–7]. Several papers [8–12] predicted the +half-metallicity in carbon-based systems as well. The ex- +istence of spin-polarized currents in such systems makes +them promising materials for applications in spintron- +ics [3, 13]. +Graphene-based bilayer and multi-layer systems posses +additional quantum number, +the valley index. +In +these materials, besides the spin-related polarization, a +many-body state may demonstrate a valley polarization. +Therefore, for graphene-based materials, the notion of +a HM can be generalized to include the possibility of +a Fermi surface with perfect valley polarization as well. +Such a proposal was put forward in Ref. 14, where the +concept of a quarter-metal (QM) was formulated. +A +Fermi surface of a QM state is perfectly polarized both in +valley and in spin-related indices. Furthermore, the lat- +ter paper explained that both an HM and a QM should +be viewed as specific instances of a more general notion, +‘a fractional metal’ (FraM). This many-body phase may +be realized in materials with degenerate Fermi surface. +The higher the degeneracy, the stronger fractionalization +of the Fermi surface can be achieved. +Since our publication [14] the experimental observation +of a QM state in graphene trilayer has been claimed [15]. +The experimental data of Ref. 16 suggest that a QM and +FraM states can be stabilized in a sample of AB bilayer +graphene (AB-BLG). Given these experimental successes +it appears important to develop a microscopic theoretical +framework that can explain the existence of the FraM +in the AB-BLG. In this letter, a suitable mechanism is +proposed and discussed. +Model.— An elementary unit cell of the AB-BLG con- +sists of four atoms (sublattices A and B, and layers 1 and +2) with the distance between neighboring carbon atoms +a0 ≈ 0.142 nm and interlayer distance c0 ≈ 0.335 nm. +The hoping amplitude t connecting the nearest A and B +sites in the layer is 2.5 eV ≲ t ≲ 3 eV. The hopping be- +tween the nearest sites in different layers can be estimated +as 0.3 eV ≲ t0 ≲ 0.4 eV. It is possible to introduce addi- +tional, longer-range, hopping amplitudes into the model. +We assume, however, that the effect of these amplitudes +is weak, and they are neglected. +The AB-BLG Brillouin zone is a regular hexagon, +with +two +non-equivalent +Dirac +points +at +K1 += +2π( +√ +3, 1)/3 +√ +3a0 and K2 = 2π( +√ +3, −1)/3 +√ +3a0. +It is +convenient to measure momentum relative to the Dirac +points. Thus, we introduce q = k − K1,2. +The energy spectrum of undoped AB-BLG consists of +four bands, two electron and two hole ones. Since we are +interested in the low-energy spectrum of AB-BLG, q ≪ +2t0/3ta0, we restrict our consideration to the effective +two-band model. It has one electron and one hole band, +both bands have quadratic dispersion. The bands touch +at the Fermi energy. In such a model, the Hamiltonian +for a single-electron wave function reads [17] +H0 = −ℏ2v2 +F +t0 +� +0 +(iqx + ξqy)2 +(iqx − ξqy)2 +0 +� +, +(1) +where the graphene Fermi velocity is vF = 3a0t/2ℏ and + +2 +ξ is the valley index. The value ξ = 1 corresponds to +K1 and ξ = −1 corresponds to K2. +In the second- +quantization formalism we can write +H0 = +� +qσξl +εqlγ† +qlσξγqlσξ, +(2) +where the spin projection is denoted by σ, the index l +labels the electron (l = 1) or hole (l = 2) band, and γqlσξ +is the corresponding second quantization operator. The +eigenenergies εql of the Hamiltonian (1) are +εql = (−1)l ℏ2v2 +F +t0 +q2. +(3) +Next we include the electron-electron repulsion into the +model. The latter is a highly non-trivial task. Clearly, +the low-energy two-band effective model (1) is incom- +patible with the bare Coulomb repulsion. +Instead, an +effective interaction Hamiltonian must be derived. Un- +fortunately, a compact description of such an effective +interaction remains an elusive theoretical goal. Indeed, +due to multiple factors affecting the many-body physics +in graphene and graphene-based systems, an effective +interaction term is quite complex, with multiple cou- +pling constants, whose non-universal values are poorly +known [18–22]. +In this situation we prefer to adopt a +semi-phenomenological approach, keeping only the terms +that directly contribute to the spin-density wave (SDW) +ordering. It is possible to identify three types of such +terms. The first term arises due to the forward-scattering +Hf +int = VC +Nc +� +kk′,ll′ +σσ′,ξξ′ +γ† +klσξγk′lσξγ† +k′l′σ′ξ′γkl′σ′ξ′, +(4) +where Nc is the number of unit cells in the sample, and +VC is an effective interaction constant whose value can be +potentially extracted from the low-temperature data [23– +32] on spontaneous symmetry breaking in AB-BLG. The +forward scattering is characterized by a small momentum +transfer |k − k′| ≪ |K1 − K2|, and preserves the band +indices l and l′ of the two participating electrons. Next, +one can define the backscattering term +Hb +int = V b +C +Nc +� +kk′,ll′ +σσ′,ξ +γ† +klσξγk′lσ¯ξγ† +k′l′σ′ ¯ξγkl′σ′ξ, +(5) +where a bar on top of a binary-valued index implies the +inversion of the index value (for example, if ξ = 1 then +¯ξ = −1). For Hb +int the transferred momentum is large +|k−k′| ∼ |K1 −K2|, thus we can assume that V b +C ≪ VC. +Finally, the umklapp-type interaction +Hu +int = V u +C +Nc +� +kk′, +σσ′,ξξ′ +γ† +k1σξγk′2σξγ† +k′1σ′ξ′γk2σ′ξ′ + h.c., +(6) +represents scattering events in which both electrons +change their bands. It accounts for the coupling between +inter-layer dipole moments, which is also weaker than the +coupling between charge densities represented by Hf +int. In +principle, there is backscattering umklapp, which we do +not consider due to it being even weaker than Hu +int. +Mean-field approximation.— +We +consider a +zero- +temperature SDW instability of the AB-BLG. This is +characterized by the spontaneous generation of staggered +spin magnetization violating the spin-rotation symmetry. +The direction of this magnetization is not fixed and there +are several equivalent choices for an SDW order parame- +ter that differ by the spin-magnetization direction. It is +convenient to assume that ⟨γ† +k1σξγk2¯σξ⟩ ̸= 0. This choice +corresponds to the magnetization in the xy-plane. Note +also that the introduced order parameter accounts for the +coupling of single-electron states in the same valley ξ. +Now, assuming that the backscattering (5) and the +umklapp (6) are weak, we apply the mean-field approxi- +mation to Hf +int +HMF +int = − +� +kσξ +∆σξγ† +k2σξγk1¯σξ + h.c. + B, +(7) +where the order parameter ∆σξ and c-number B are +∆σξ = VC +Nc +� +q +⟨γ† +q1σξγq2¯σξ⟩Θ(qC − q), +(8) +B = +� +qσξ +∆σξ⟨γ† +q2σξγq1¯σξ⟩Θ(qC − q) = Nc +VC +� +σξ +|∆σξ|2. +(9) +In these expressions, the momentum cutoff for the inter- +action qC satisfies qC ≪ |K1 − K2|. +The mean-field Hamiltonian (7) does not conserve spin +(spin-rotation symmetry is spontaneously broken for non- +zero ∆σξ). However, quasi-momentum q is conserved. In +addition to q, one can introduce valley and spin-flavor +operators +Sf +q = +� +σξl +σ(−1)lγ† +qlσξγqlσξ, +Sv +q = +� +σξl +ξγ† +qlσξγqlσξ, +(10) +which commute with the Hamiltonian H0 +HMF +int and are +good quantum numbers. Thus, in this approximation all +fermionic degrees of freedom can be grouped into four +uncoupled sectors, each sector having its own values of +spin-flavor index (−1)lσ and valley index ξ. +A sector +is characterized by its own order parameter ∆σξ, and +single-particle spectrum +E1,2 +qσξ = ± +� +∆2 +σξ + +�ℏ2v2 +F +t0 +�2 +q4. +(11) +The thermodynamic grand potential Ω can be expressed +as a sum Ω = � +σξ Ωσξ + B, where Ωσξ are four partial +grand potentials corresponding to specific sectors. +At + +3 +zero temperature, these are +Ωσξ = +� +ql +� +El +qσξ − µ +� +Θ +� +µ − El +qσξ +� +, +(12) +where µ is the chemical potential. +Minimization of Ω over the order parameters allows +us to derive the following independent self-consistency +equations for the order parameters in the four sectors +1 = VC +Nc +� +|q| 0 case only. For positive chem- +ical potential: Θ(µ + E1 +qσ) − Θ(µ − E1 +qσ) = Θ(E1 +qσ − µ). +Introducing dimensionless variables +g = VCt0 +√ +3πt2 , m = 4t0µ +9t2 , δσξ = 4t0∆σξ +9t2 +, +(14) +we obtain from Eq. (13) +1 = 2g +� QC +Qm +σξ +QdQ +� +δ2 +σξ + Q4 , +(15) +where +QC = a0qC, +Qm +σξ = (m2 − δ2 +σξ)1/4. +(16) +It is evident that the gap in the spectrum of electrons in +the sector (σ, ξ) arises only if QC > Qm +σξ, that is, if the +number of the doped charge carriers in this sector is not +too large. One can perform the integration in Eq. (15) +and obtain that +1 = g ln + + +Q2 +C + +� +δ2 +σξ + Q4 +C +m + +� +m2 − δ2 +σξ + + . +(17) +In the weak coupling limit, g ≪ 1, we have δσξ ≪ Q2 +C. +Consequently +∆σξ = +� +∆0(2µ − ∆0), +(18) +where +∆0 = 9t2 +4t0 +q2 +Ca2 +0e−1/g +(19) +is the mean-field gap of undoped AB-BLG. Introducing +δ0 = 4t0∆0/(9t2) we can express Eq. (18) in dimension- +less form +δσξ = +� +δ0(2m − δ0). +(20) +Since experiments are performed at fixed doping, we need +to connect the values of ∆σξ with doping. It is conve- +nient to introduce partial doping, that is, the number of +electrons with specific values of σ(−1)l and ξ: +xσξ = −∂Ωσξ +∂µ += 2π +VBZ +� +σξ +� +kdkΘ(µ − E1 +kσξ). (21) +The total doping x is equal to +x = +� +σξ +xσξ. +(22) +If µ > ∆σξ, we obtain the relation between the partial +doping and the chemical potential in the form +xσξ = 3 +√ +3 +8π +� +m2 − δ2 +σξ. +(23) +Otherwise, xσξ = 0. As a result, we derive in the case of +non-zero xσξ +m = δ0 − 8π +3 +√ +3xσξ = δ0 +� +1 − 2xσξ +x0 +� +, +(24) +δσξ = δ0 +� +1 − 4xσξ +x0 +, +(25) +where +x0 = t0∆0 +√ +3πt2 . +(26) +Equation (25) indicates that for xσξ = x0/4 the order +parameter in the sector vanishes. That is, for xσξ > x0/4 +one has +∆σξ(xσξ) ≡ 0, +m = 8π +3 +√ +3xσξ = 2δ0 +x0 +xσξ. +(27) +Note that the chemical potential, as given by Eqs. (24) +and (27), demonstrates non-monotonic behavior as a +function of xσξ. Of particular importance is the fact that, +for low doping, µ = µ(xσξ) is a decreasing function. This +means that the compressibility of the homogeneous phase +is negative and points to a possibility of the phase sepa- +ration of the electronic liquid. We will assume below that +the long-range Coulomb interaction is sufficiently strong +to arrest the phase separation, restoring the stability of +homogeneous states. +Quarter metal state of doped AB-BLG.— Disregarding +the possibility of the phase separation, we use Eqs. (24) +and (25) to characterize the thermodynamics of the sys- +tem. To describe the doped state of the electronic liquid +for a specific x, one must determine partial dopings in +all four sectors. To achieve this goal, we should calculate +the free energy F(x) = F(0) + +� +µ(x)dx. In so doing, we +obtain +F(x) = F(0) + ∆0 + +x − +� +σξ +x2 +σξ +x0 + + , +(28) +when 0 < xσξ < x0/4, and +F(x) = F(0) + ∆0 + +x0 +8 + +� +σξ +x2 +σξ +x0 + + , +(29) + +4 +if xσξ > x0/4. This free energy must be minimized over +xσξ under the constraint (22). For small x, simple calcu- +lations demonstrate that F is smallest when all charges +are placed into a single sector +xσξ = x, +xσ′ξ′ = 0, for σ′ ̸= σ or ξ′ ̸= ξ. +(30) +The free energy corresponding to distribution (30) equals +to FQM = ∆0(x−x2/x0). It is smaller, for example, than +the free energy Feq = ∆0x−∆0x2/(4x0), that represents +an equal distribution of doping between all four sectors +(xσξ = x/4 for all σ and ξ). +The state described by Eq. (30) is metallic, with (al- +most) circular Fermi surface whose radius kF = kF(x) is +set by the equation +a2 +0k2 +F = 8πx +3 +√ +3. +(31) +This Fermi surface, however, is quite unique: all single- +electronic states reaching the Fermi energy are perfectly +polarized in terms of Sf and Sv. In other words, they +have an identical value of (−1)lσ, and the Fermi surface +is located within a single valley Kξ. Since among four +possible Fermi surface sheets of the non-interacting the- +ory, only one sheet emerges in the system, it is natural +to designate such a conducting state as a QM. +Cascade +of +phase +transition +between +different +symmetry-broken phases.— The QM state described +above remains stable only for sufficiently low x: +one +sector cannot accommodate too much doping. Indeed, +when x = x0/2, Eq. (27) implies that µ = ∆0. Doping +a single sector beyond this point is impossible: adding +more charge to this sector increases the chemical po- +tential beyond ∆0, unavoidably placing charges into +the remaining sectors as well. As a result, a cascade of +doping-driven phase transitions emerges. +The transi- +tions connect different metallic states, each state being +characterized by a number of doped sectors: 1, 2, 3, or +4 [paramagnetic (PM) state] sectors. +Let us briefly describe this cascade of transitions. At +zero doping the system is gapped with the gap equal to +∆0 in all sectors. For small x, the system absorbes all ex- +tra charge carriers into a single sector [say, sector (σ =↑, +ξ = +1)]. This is a QM state. At x = x0/4, a second +order phase transition inside the QM state takes place. +Beyond this doping, ∆↑+1 vanishes. However, the QM +state remains stable for x < x0/2. At higher doping the +QM energy becomes higher than the HM energy, and a +first order phase transition from QM to HM state occurs. +In the HM state, the gap is zero in two sectors [for defi- +niteness, we assign these to be (σ =↑, ξ = +1) and (σ =↑, +ξ = −1); however, other configurations are equiproba- +bly possible], and extra charge carriers are equally dis- +tributed between these two sectors. +As x increases further, one reaches the point where the +HM energy becomes equal to that of a 3/4 metal ( 3 +4M) +state. In such a state, three sectors [say, (σ =↑, ξ = +1), +x +1st +1st +1st +2nd +� +� � +� +� � +�� +�� +���� � �� +���� � �� +���� � �� +���� � �� +���� � � +���� � � +���� � � +���� � �� +���� � � +���� � � +���� � �� +���� � �� +���� � � +���� � �� +���� � �� +���� � �� +� +� �� +� +� �� +� +� �� +� +� �� +FIG. 1. Cascade of the doping-driven phase transitions be- +tween different FraM states with different valley and/or spin- +flavor (isospin) polarizations. Only the region of electron dop- +ing is shown. For hole doping the picture is identical up to a +replacement x → −x. Vertical solid (dashed) lines represent +first (second) order transitions. +(σ =↑, ξ = −1), and (σ =↓, ξ = +1)] are doped, and +the fourth sector, (σ =↓, ξ = −1), is gapped, with the +extra charge carriers being equally distributed among the +three doped sectors. For our simple model, the transition +into the 3 +4M happens at x = +� +3/4x0. The transition is +first-order. +If doping is continued even further, the +3 +4M state is +replaced by the PM state. This is yet another first-order +transition, and the last one in the transition cascade. It +occurs at x = +� +3/2x0. The phase diagram of the system +is shown in Fig. 1. In this figure only the electron doping +is shown. Due to electron-hole symmetry of our model, +the phase diagram at hole doping is equivalent to that +shown in Fig. 1 up to the replacement x → −x. +Discussion.— We would like to stress here several im- +portant points. One must remember that the HM state +realized in our model upon sufficiently strong doping +is not the conventional HM [1, 2] whose Fermi surface +demonstrates perfect spin polarization. Instead, we now +have a spin-flavor HM [33–36], with perfect spin-flavor +polarization of the Fermi surface. This means that the +electron (hole) single-particle states reaching the Fermi +energy have their spin projection being equal to σ (to ¯σ). +(The related feature of the QM state was already men- +tioned above.) In a model with electron-hole symmetry +a spin-flavor-polarized FraM state does not accumulate +net spin polarization. +However, a finite spin polariza- +tion may accompany a finite spin-flavor polarization [33] +when such a symmetry is absent. The spin polarization +was indeed observed in Ref. 16. +We argued above that the relative stability of vari- +ous metallic states is affected by doping, triggering the +transitions between them. Doping is not, however, the +only factor that influence the competition between the +FraM phases. +Particular model’s ingredients favoring +HM states are the umklapp and backscattering interac- +tion terms. Specifically, the umklapp couples two sectors + +5 +with unequal (−1)lσ, the backscattering, on the other +hand, connect the sectors with non-identical values of the +ξ index. Thus, in the presence of either strong Hum +int or +strong Hb +int only two (not four) decoupled sectors of the +mean-field Hamiltonian can be defined, promoting the +HM phase over other FraM’s. Therefore, in more realistic +models, the critical doping values are no longer propor- +tional to x0, with universal proportionality coefficients. +Instead, they become functions of the backscattering and +umklapp coupling constants. +The qualitative agreement between the remarkable re- +cent experiments reported in Ref. 16 and our formalism +is very encouraging. The proposed theory can account +for such experimentally observed features as the cascade +of phase transitions, magnetization, and valley polariza- +tions. Yet one must keep in mind that the experiments +were performed at finite electric field applied transverse +to a sample. In our formalism, this field is assumed to be +zero. Further research is needed to understand the role +of this field. +To conclude, we proposed a mechanism responsible for +the formation of the FraM states in doped AB-BLG. 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B 101, 235141 (2020). + diff --git a/N9E0T4oBgHgl3EQfjgF4/content/tmp_files/load_file.txt b/N9E0T4oBgHgl3EQfjgF4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a42176f83ab54ec068ba03ece1b377ee8db6c4c3 --- /dev/null +++ b/N9E0T4oBgHgl3EQfjgF4/content/tmp_files/load_file.txt @@ -0,0 +1,578 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf,len=577 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='02460v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='mes-hall] 6 Jan 2023 Half-metal and other fractional metal phases in doped AB bilayer graphene A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' 3 1Institute for Theoretical and Applied Electrodynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Russian Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' 125412 Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Russia 2Center for Quantum Computing and Cluster for Pioneering Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' RIKEN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Wako-shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Saitama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' 351-0198,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Japan 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' MI 48109-1040,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' USA (Dated: January 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' 2023) We theoretically argue that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' in doped AB bilayer graphene,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' the electron-electron coupling can give rise to the spontaneous formation of fractional metal phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' These states, being generalizations of a more common half-metal, have a Fermi surface that is perfectly polarized not only in terms of a spin-related quantum number, but also in terms of the valley index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The proposed mechanism assumes that the ground state of undoped bilayer graphene is a spin density wave insulator, with a finite gap in the single-electron spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Upon doping, the insulator is destroyed, and replaced by a fractional metal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' As doping increases, transitions between various types of fractional metal (half-metal, quarter-metal, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=') are triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Our findings are consistent with recent experiments on doped AB bilayer graphene, in which a cascade of phase transitions between different isospin states was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' PACS numbers: 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='Pr, 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='Gk Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='— A usual metal demonstrates perfect symmetry with regard to the carriers’ spin projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' This symmetry manifests itself in the vanishing total spin magnetization and the Fermi-surface spin degener- acy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Yet the symmetry can be spontaneously destroyed by sufficiently strong electron-electron interaction, which may result, for example, in the formation of two non- identical Fermi surfaces for the two spin projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' In the extreme case of the so-called half-metals (HM), one of these projections is completely absent from the Fermi surface, while all states at the Fermi energy have identi- cal spin quantum number [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Various rather dissim- ilar materials with transition-metal atoms are found to be half-metals [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Several papers [8–12] predicted the half-metallicity in carbon-based systems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The ex- istence of spin-polarized currents in such systems makes them promising materials for applications in spintron- ics [3, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Graphene-based bilayer and multi-layer systems posses additional quantum number, the valley index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' In these materials, besides the spin-related polarization, a many-body state may demonstrate a valley polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Therefore, for graphene-based materials, the notion of a HM can be generalized to include the possibility of a Fermi surface with perfect valley polarization as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Such a proposal was put forward in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' 14, where the concept of a quarter-metal (QM) was formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' A Fermi surface of a QM state is perfectly polarized both in valley and in spin-related indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Furthermore, the lat- ter paper explained that both an HM and a QM should be viewed as specific instances of a more general notion, ‘a fractional metal’ (FraM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' This many-body phase may be realized in materials with degenerate Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The higher the degeneracy, the stronger fractionalization of the Fermi surface can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Since our publication [14] the experimental observation of a QM state in graphene trilayer has been claimed [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The experimental data of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' 16 suggest that a QM and FraM states can be stabilized in a sample of AB bilayer graphene (AB-BLG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Given these experimental successes it appears important to develop a microscopic theoretical framework that can explain the existence of the FraM in the AB-BLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' In this letter, a suitable mechanism is proposed and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='— An elementary unit cell of the AB-BLG con- sists of four atoms (sublattices A and B, and layers 1 and 2) with the distance between neighboring carbon atoms a0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='142 nm and interlayer distance c0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='335 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The hoping amplitude t connecting the nearest A and B sites in the layer is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='5 eV ≲ t ≲ 3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The hopping be- tween the nearest sites in different layers can be estimated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='3 eV ≲ t0 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='4 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' It is possible to introduce addi- tional, longer-range, hopping amplitudes into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' We assume, however, that the effect of these amplitudes is weak, and they are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The AB-BLG Brillouin zone is a regular hexagon, with two non-equivalent Dirac points at K1 = 2π( √ 3, 1)/3 √ 3a0 and K2 = 2π( √ 3, −1)/3 √ 3a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' It is convenient to measure momentum relative to the Dirac points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Thus, we introduce q = k − K1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The energy spectrum of undoped AB-BLG consists of four bands, two electron and two hole ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Since we are interested in the low-energy spectrum of AB-BLG, q ≪ 2t0/3ta0, we restrict our consideration to the effective two-band model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' It has one electron and one hole band, both bands have quadratic dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The bands touch at the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' In such a model, the Hamiltonian for a single-electron wave function reads [17] H0 = −ℏ2v2 F t0 � 0 (iqx + ξqy)2 (iqx − ξqy)2 0 � , (1) where the graphene Fermi velocity is vF = 3a0t/2ℏ and 2 ξ is the valley index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The value ξ = 1 corresponds to K1 and ξ = −1 corresponds to K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' In the second- quantization formalism we can write H0 = � qσξl εqlγ† qlσξγqlσξ, (2) where the spin projection is denoted by σ, the index l labels the electron (l = 1) or hole (l = 2) band, and γqlσξ is the corresponding second quantization operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The eigenenergies εql of the Hamiltonian (1) are εql = (−1)l ℏ2v2 F t0 q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' (3) Next we include the electron-electron repulsion into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The latter is a highly non-trivial task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Clearly, the low-energy two-band effective model (1) is incom- patible with the bare Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Instead, an effective interaction Hamiltonian must be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Un- fortunately, a compact description of such an effective interaction remains an elusive theoretical goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Indeed, due to multiple factors affecting the many-body physics in graphene and graphene-based systems, an effective interaction term is quite complex, with multiple cou- pling constants, whose non-universal values are poorly known [18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' In this situation we prefer to adopt a semi-phenomenological approach, keeping only the terms that directly contribute to the spin-density wave (SDW) ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' It is possible to identify three types of such terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The first term arises due to the forward-scattering Hf int = VC Nc � kk′,ll′ σσ′,ξξ′ γ† klσξγk′lσξγ† k′l′σ′ξ′γkl′σ′ξ′, (4) where Nc is the number of unit cells in the sample, and VC is an effective interaction constant whose value can be potentially extracted from the low-temperature data [23– 32] on spontaneous symmetry breaking in AB-BLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The forward scattering is characterized by a small momentum transfer |k − k′| ≪ |K1 − K2|, and preserves the band indices l and l′ of the two participating electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Next, one can define the backscattering term Hb int = V b C Nc � kk′,ll′ σσ′,ξ γ† klσξγk′lσ¯ξγ† k′l′σ′ ¯ξγkl′σ′ξ, (5) where a bar on top of a binary-valued index implies the inversion of the index value (for example, if ξ = 1 then ¯ξ = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' For Hb int the transferred momentum is large |k−k′| ∼ |K1 −K2|, thus we can assume that V b C ≪ VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Finally, the umklapp-type interaction Hu int = V u C Nc � kk′, σσ′,ξξ′ γ† k1σξγk′2σξγ† k′1σ′ξ′γk2σ′ξ′ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=', (6) represents scattering events in which both electrons change their bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' It accounts for the coupling between inter-layer dipole moments, which is also weaker than the coupling between charge densities represented by Hf int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' In principle, there is backscattering umklapp, which we do not consider due to it being even weaker than Hu int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='— We consider a zero- temperature SDW instability of the AB-BLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' This is characterized by the spontaneous generation of staggered spin magnetization violating the spin-rotation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The direction of this magnetization is not fixed and there are several equivalent choices for an SDW order parame- ter that differ by the spin-magnetization direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' It is convenient to assume that ⟨γ† k1σξγk2¯σξ⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' This choice corresponds to the magnetization in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Note also that the introduced order parameter accounts for the coupling of single-electron states in the same valley ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Now, assuming that the backscattering (5) and the umklapp (6) are weak, we apply the mean-field approxi- mation to Hf int HMF int = − � kσξ ∆σξγ† k2σξγk1¯σξ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' + B, (7) where the order parameter ∆σξ and c-number B are ∆σξ = VC Nc � q ⟨γ† q1σξγq2¯σξ⟩Θ(qC − q), (8) B = � qσξ ∆σξ⟨γ† q2σξγq1¯σξ⟩Θ(qC − q) = Nc VC � σξ |∆σξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' (9) In these expressions, the momentum cutoff for the inter- action qC satisfies qC ≪ |K1 − K2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' The mean-field Hamiltonian (7) does not conserve spin (spin-rotation symmetry is spontaneously broken for non- zero ∆σξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' However, quasi-momentum q is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' In addition to q, one can introduce valley and spin-flavor operators Sf q = � σξl σ(−1)lγ† qlσξγqlσξ, Sv q = � σξl ξγ† qlσξγqlσξ, (10) which commute with the Hamiltonian H0 +HMF int and are good quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Thus, in this approximation all fermionic degrees of freedom can be grouped into four uncoupled sectors, each sector having its own values of spin-flavor index (−1)lσ and valley index ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' A sector is characterized by its own order parameter ∆σξ, and single-particle spectrum E1,2 qσξ = ± � ∆2 σξ + �ℏ2v2 F t0 �2 q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' (11) The thermodynamic grand potential Ω can be expressed as a sum Ω = � σξ Ωσξ + B, where Ωσξ are four partial grand potentials corresponding to specific sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' At 3 zero temperature, these are Ωσξ = � ql � El qσξ − µ � Θ � µ − El qσξ � , (12) where µ is the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf'} +page_content=' Minimization of Ω over the order parameters allows us to derive the following independent self-consistency equations for the order parameters in the four sectors 1 = VC Nc � |q|231) particles, dark matter +halos and subhalos are detected in real space with the local maxima +of dark matter particle density field. Their edge is defined as the point +© 2022 RAS, MNRAS 000, 1–10 + +Velocity waves +3 +Figure 2. Schema of the cylinder used to select (sub)halos whose radial pecu- +liar velocities, derived as a function of the synthetic observer at the simulated +box center, are used to study the velocity wave arisen from the massive halo in +its center. While open circles stand for selected halos, dashed circles represent +excluded ones. +where the overdensity of dark matter mass drops below 80 times the +background density. We further apply a lower threshold of a minimum +of 100 dark matter particles. Fig. 1 shows the ∼40 Mpc thick XY super- +galactic slice of the CLONE. Black (red) dots stand for the dark matter +halos (galaxies from the 2MASS Galaxy Redshift Catalog - XSCz3). +Note that XSCz galaxies are used for sole comparison purposes. XSCz +is indeed far more complete than the peculiar velocity catalog used to +constrain the simulation (∼2.5% of the redshift catalog is used to de- +rive the peculiar velocity). In fact, it shows the constraining power of +the peculiar velocities that are correlated on large scales. Namely, the +simulation is constrained also in regions where no peculiar velocity +measurements were available and thus used as constraints. It confirms +once more that peculiar velocity catalogs fed to our technique, to re- +construct/constrain the local density and velocity fields, do not need to +be complete (Sorce et al. 2017). +3 +VELOCITY WAVE +3.1 +In simulated data +Positioning a synthetic observer at the simulation box center, we de- +rive radial peculiar velocities for all the dark matter halos and subha- +los in the z=0 catalog. We then draw lines-of-sight in the direction of +each dark matter halo more massive than 5 1014M⊙. All the (sub)halos +within 10 Mpc from the line-of-sight and within 74 Mpc along the +line-of-sight from a given massive dark matter halo (with the center +and edge of the box as upper limits) are selected to plot the latter cor- +responding velocity wave. Namely, as shown on Fig. 2, radial peculiar +velocities, with respect to the synthetic observer, of (sub)halos within +a cylinder at maximum 148 Mpc long and 20 Mpc wide are used to vi- +sualize the velocity wave caused by the massive dark matter halo in the +cylinder center. Note that because the simulation is constrained to re- +produce the local Universe, we choose not to use the periodic boundary +conditions to wrap around the box edges. It will indeed not be repre- +sentative of local structures. A 10 Mpc radius cylinder corresponds to +about three times the virial radius of the massive clusters under study +here (M>5 1014M⊙). Since the goal is to study the link between veloc- +ity wave properties and cluster masses, exact masses cannot be used to +define the cylinder shape. Finally, such large volumes permit probing +the infall region around the massive halos. Note that a cylinder shape +is preferable to a cone shape to get an unbiased wave signal. A cone +would indeed result in a distorted signal as it would probe a larger and +larger region around a massive halo with the distance. +3.2 +In observational data +Observational data are taken from the raw second and third catalogs +of the Cosmicflows project (Tully et al. 2013, 2016). Note that the +3 https://wise2.ipac.caltech.edu/staff/jarrett/2mass/XSCz/specz.html +second catalog containing ∼8000 galaxies, with a mean distance of +∼90 Mpc, serves as the basis to build the constraint-catalog of ∼5000 +bias-minimized radial peculiar velocities of galaxies and groups with +a mean distance of ∼60 Mpc. By contrast, the third catalog contains +∼17,000 galaxies with a mean distance of ∼120 Mpc. The third +catalog is not used to constrained our CLONE initial conditions and +thus constitute partly an independent dataset for consistency check. +More precisely, it serves the two-fold goal of extending the number +of observational datapoints to be compared with the simulation and +highlighting again the constraining power of peculiar velocities. The +latter can indeed permit recovering structures that are not directly +probed and that are at the limit of the non-linear threshold. In the +sense that there is no direct measurement in a given region but, +because the latter influences the velocities of other regions (large scale +correlations), it can still be reconstructed. +Uncertainties on distances and radial peculiar velocities in these +catalogs depend on the distance indicator used to derive the distance +moduli. Error bar sizes need to be limited to see clearly velocity waves. +Thus, to be able to compare with the simulated data, only galaxies with +uncertainties on distance moduli smaller than 0.2 dex are retained. +There remain 338 and 424 galaxies respectively from the second and +third catalogs with a mean distance of ∼50 Mpc. These galaxies are +mostly hosts of supernovae, especially those the furthest from us (dis- +tance indicator with a small uncertainty even as the distance increases). +To derive the radial peculiar velocities of these galaxies, we use +both galaxy distance moduli (µ) and observational redshifts (zobs) +following Davis & Scrimgeour (2014). We add supergalactic longitude +and latitude coordinates to derive galaxy cartesian supergalactic +coordinates. A cosmological model is then required to determine +peculiar velocities. While we use ΛCDM, as cosmicflows catalog zero +points are calibrated through a long process on WMAP (rather than +Planck) values (Ωm=0.27, ΩΛ=0.73, H0=74 +km s−1 Mpc−1, Tully +et al. 2013, 2016), we have to use the same parameter values. We +indeed showed that when applying the bias minimization technique +to the peculiar velocity catalog of constraints, we drastically reduce +the dependence on ΛCDM cosmological parameter values (Sorce +& Tempel 2017). However, in order to be able to probe the whole +velocity wave for the comparisons, we have to use the raw catalog i.e. +with neither galaxy grouping nor bias minimization. Consequently, +if were to take Planck values to derive galaxy peculiar velocities, +the WMAP calibration would translate into a residual Hubble flow +visible in the background-expansion-subtracted Hubble diagram. +Subsequently, using WMAP values for the observations: +Luminosity distances, dlum, are derived from distance modulus mea- +surements, µ, obtained via distance indicators: +µ = 5log10(dlum (Mpc)) + 25 +(1) +Cosmological redshifts, zcos, are then obtained through the equation: +dlum = (1 + zcos) +� zcos +0 +cdz +H0 +� +(1 + z)3Ωm + ΩΛ +(2) +Galaxy radial peculiar velocity, vpec, are finally estimated, using the +observational zobs and cosmological zcos redshifts with the following +formula: +vpec = czobs − zcos +1 + zcos +(3) +where vpec will always refer to the radial peculiar velocity in this paper +and c is the speed of light. +© 2022 RAS, MNRAS 000, 1–10 + +4 +Sorce et al. +Virgo +0 +20 +40 +60 +80 +d (Mpc) +0 +2000 +4000 +6000 +v (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Centaurus +0 +20 +40 +60 +80 +100 +d (Mpc) +0 +2000 +4000 +6000 +v (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Virgo +0 +20 +40 +60 +80 +d (Mpc) +-2000 +-1000 +0 +1000 +2000 +3000 +vpec (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +Centaurus +0 +20 +40 +60 +80 +100 +d (Mpc) +-2000 +-1000 +0 +1000 +2000 +3000 +vpec (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +Figure 3. Radial velocities of simulated dark matter (sub)halos (black and grey scale) and observed galaxies (orange, blue and red) as a function of the distance +from the synthetic observer and us respectively. Error bars stand for uncertainties on observational distance and velocity estimates. Orange and light blue (red and +dark blue) filled squares and diamonds show observed galaxies assuming H0=74 (67.77) km s−1 Mpc−1 for scaling positions. CF2 (CF3) corresponds to the second +(third) catalog of the Cosmicflows project. Larger symbols are used for galaxies, with a peculiar velocity higher than 1000 km s−1, identified as the closest to the +simulated massive halos assuming the synthetic observer at the box center and the same Supergalactic coordinate system and orientation as the local Universe. The +arrow indicates the position of the massive dark matter halo in the simulation. Names of corresponding observed clusters are given at the top of each panel. Velocity +waves stand out in the different lines-of-sight and there is a good agreement with observational datapoints for those two best-constrained clusters the closest to us. +Top: Hubble diagram. Bottom: Hubble flow subtracted. The solid and dashed yellow lines are respectively the simulated positive-half velocity wave envelope and its +Gaussian-plus-continuum fit. The color scale filling the black circles stands for their distance from the line-of-sight. From black to light grey, objects are less than +2.5, 5, 7.5 and 10 Mpc away from the line-of-sight. The dark matter halo virial masses in the simulation are M=9.8×1014M⊙ and M=9.0×1014M⊙ for the Virgo and +Centaurus cluster counterparts respectively. +3.3 +Simulated vs. observed data +Assuming the synthetic observer at the box center and the simulated +volume oriented similarly to the local volume, observed and simulated +positions and lines-of-sight can be matched. We can only compare +velocity waves born from local galaxy clusters for which infalling +galaxy peculiar velocities, with uncertainties on corresponding +distance moduli smaller than 0.2 dex, are available in the observed +cluster surroundings. We thus select these clusters. For each simulated +massive dark matter halo, the quickest way is then to search for the +closest observed galaxy, in our selected above samples, with a radial +peculiar velocity greater than 1000 km s−1 (∼2σ above the average). +This is indeed a signature that it has most probably an observed cluster +with a mass of at least a few 1014M⊙ as a neighbor. Whenever a +simulated massive dark matter halo is within the 2σ uncertainty of +the observed galaxy distance, we select all the observed galaxies in +the cylinder corresponding to the line-of-sight. For every case, there +is indeed a massive observed cluster in the vicinity of the galaxies. +More to the point, given the Supergalactic coordinates of the observed +clusters and those of the simulated ones in the box, they indeed match. +Fig. 3 superimposes observed and simulated lines-of-sight with +the velocity waves born from the two closest most massive local +clusters. Observational data is of sufficient quality in their respec- +tive infall region to warrant adequate comparisons. From left to right, +galaxy clusters (dark matter halos) are at increasing distance from +us (the synthetic observer). The name of the clusters is indicated at +the top of each panel. Filled black and grey circles stand for simu- +lated (sub)halos while filled light blue and orange squares and dia- +monds represent observed galaxies. Because the simulation was run +with H0 = 67.77 km s−1 Mpc−1, filled dark blue and red squares and +diamonds are observed galaxies at positions rescaled with this latter +value. Position differences are always within about the 1σ uncertainty +© 2022 RAS, MNRAS 000, 1–10 + +Velocity waves +5 +Cluster +CLONE/CF2 +CLONE/CF2 +CLONE/CF3 +CLONE/CF3 +Cylinder radius +10 Mpc +2.5 Mpc +10 Mpc +2.5 Mpc +Virgo +0.0098 +0.011 +0.0058 +0.0071 +Centaurus +0.010 +0.011 +0.006 +0.0073 +Abell 569 +0.25 +0.25 +0.084 +0.085 +Coma +0.17 +0.17 +0.25 +0.25 +Abell 85 +0.25 +0.25 +0.25 +0.25 +Abell 2256 +0.50 +0.50 +0.50 +0.50 +PGC 765572 +0.050 +0.051 +0.10 +0.10 +PGC 999654 +0.50 +0.50 +0.50 +0.50 +PGC 340526 +0.25 +0.25 +0.50 +0.50 +PGC 46604 +0.50 +0.50 +0.50 +0.50 +Table 1. Kolmogorov-Smirnov statistic or highest distance between the cumula- +tive distribution functions of the observed and simulated lines-of-sight including +the velocity waves. +on the distance. Arrows indicate the position of the most massive halos +in the lines-of-sight of interest. +In the top panels, the Hubble diagrams are clearly distorted by +the presence of massive halos. Their corresponding velocity wave or +triple-value region signatures show up. Bottom panels with the Hub- +ble flow subtracted equally confirms the waves. The simulated velocity +waves stand out in the peculiar velocity of (sub)halos plotted as a func- +tion of the distance from the synthetic observer diagrams for the two +massive dark matter halos. The agreement with the observational data +points is qualitatively good. All the more since only sparse peculiar ve- +locities of today field galaxies and groups are used to constrained the +linear initial density and velocity fields, at the positions of the latter +progenitors, using solely linear theory and a power spectrum assuming +a given cosmology. Then the full non-linear theory is used to evolved +these initial conditions from the initial redshift down to z=0 within a +ΛCDM framework. +The signatures of Virgo West and the group around NGC4709 +that are respectively beyond Virgo and Centaurus in the lines-of-sight +can also be identified as secondary waves. These smaller waves follow +the highest ones representing the main clusters in both the observations +and the simulation. Additionally, a void between us and Centaurus +in the line-of-sight shows equally well in both the simulation and +the observations. The accuracy with which the CLONE reproduces +the lines-of-sight dynamical state of Virgo and Centaurus is visually +excellent. +To quantify the agreement between simulated and observed +lines-of-sight, we use a 2D-Kolmogorov-Smirnov statistic test applied +to the simulated and observed galaxy velocity and position samples +following Peacock (1983); Fasano & Franceschini (1987). p-values +obtained for Virgo and Centaurus are above 0.20. They are actually +close to 1.0 but values above 0.20 have no particular significance. They +only confirm that the observed and simulated distributions along the +line-of-sight are not significantly different. Additionally, Table 1 gives +the 2D-Kolmogorov-Smirnov (KS) statistic or the highest distance +between the cumulative distribution functions of the observed and +simulated lines-of-sight including the velocity waves. A single 2D-KS +statistic value has no particular meaning but several together permit +ordering the simulated lines-of-sight from those that match the most +their observational counterpart to those that match it the less (smallest +to largest values). Virgo and Centaurus lines-of-sight happen to be +equally well reproduced by the simulation. 2D-KS statistic values are +barely different when considering all the subhalos/galaxies within +a 10 Mpc radius or solely those within a 2.5 Mpc radius from the +line-of-sight. The agreement is slightly better with galaxies from the +third catalog (CF3) of the Cosmicflows project than with those of the +Cluster +CLONE/CF2 +CLONE/CF2 +CLONE/CF3 +CLONE/CF3 +Cylinder radius +10 Mpc +2.5 Mpc +10 Mpc +2.5 Mpc +Virgo +6 +10 +9 +12 +Centaurus +21 +37 +25 +36 +Abell 569 +14 +23 +11 +22 +Coma +27 +40 +205 +225 +Abell 85 +184 +286 +299 +400 +Abell 2256 +152 +152 +364 +364 +PGC 765572 +39 +53 +56 +70 +PGC 999654 +687 +687 +662 +662 +PGC 340526 +92 +99 +16 +41 +PGC 46604 +544 +544 +544 +544 +Table 2. ζ-metric in km s−1. It measures the difference between the simulated +and observed lines-of-sight. The higher ζ is the more different the lines-of-sight +are. See the text for a detailed explanation. +second one, although the second one is the starting point to build the +constrained initial conditions. However, given that the third catalog +has more points and smaller uncertainties, it is encouraging that the +simulation matches more the third catalog than the second one. The +2D-KS statistic test cannot indeed take into account uncertainties. +Finally, 2D-KS statistic values do not differ when using H0 = 67.77 +rather than 74 km s−1 Mpc−1. +The 2D-KS statistic test cannot take into account the real distance +of galaxies. It compares only the cumulative distributions of galaxies +along the lines-of-sight using four directions (smallest to largest dis- +tances to the y-axis and vice versa, smallest to largest distances to the +x-axis - in that case velocities because they are centered on zero - and +vice versa). Consequently, we also define our own ζ-metric to compare +simulated and observed lines-of-sight as follows: +ζ = 1 +n +n +� +i=1 +� +(min[vobs[i] − vsim])2 + [(min[dobs[i] − dsim]) × H0]2 +(4) +where n is the number of observed galaxies in the line-of-sight. vX are +the galaxy/subhalo observed and simulated peculiar velocities and dX +are their distances. +Table 2 gives the values of ζ for the different lines-of-sight. +Because ζ-values are only modified by a few percent when changing +H0 value, their mean is reported in the table. Like for the 2D-KS +statistic values, ζ-values permit ordering the simulated lines-of-sight +(including waves) that are the best reproduction of the observed ones +to those that reproduce them the less. Since our ζ-metric results in +similar conclusions as the 2D-KS statistic does, it seems appropriate. +Moreover, contrary to the 2D-KS statistic, it is sensitive to the real +distance of the cluster, not solely to its position on the fraction of +the line-of-sight that is studied. It thus includes both differences due +to a difference in height and to a shift in position along the entire +line-of-sight. It is easily checked by randomly shuffling observed +and simulated lines-of-sights and comparing them. The ζ-metric then +gives values on average between a 100 and up to 1000 km s−1. The +ζ-metric though, like the 2D-KS statistic, does not take into account +uncertainties on observational distance and velocity estimates. +In the rest of the paper, we work solely with the background ex- +pansion subtracted since it does not affect our conclusion and ease the +comparisons, studies and analyses. +Given the above mentioned success, although the simulation +matches best the local large-scale structure by construction in the inner +part, where most of the constraints are, Fig. 4 shows an additional four +massive halos that are more distant. These halos are still matching +nicely observational clusters that are further away. Tables 1 and 2 +© 2022 RAS, MNRAS 000, 1–10 + +6 +Sorce et al. +Abell 569 +20 +40 +60 +80 +100 +120 +140 +160 +d (Mpc) +-2000 +-1000 +0 +1000 +2000 +3000 +vpec (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +Coma +60 +80 +100 +120 +140 +160 +180 +d (Mpc) +-2000 +-1000 +0 +1000 +2000 +3000 +vpec (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +Abell 85 +160 +180 +200 +220 +240 +260 +280 +300 +d (Mpc) +-2000 +-1000 +0 +1000 +2000 +3000 +vpec (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +Abell 2256 +180 +200 +220 +240 +260 +280 +300 +320 +d (Mpc) +-2000 +-1000 +0 +1000 +2000 +3000 +vpec (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +Figure 4. Same as Figure 3 bottom panels for four clusters at increasing distance from us from left to right, top to bottom. Although these clusters are less con- +strained, the agreement between observed and simulated waves is still visually good especially for the first two. The dark matter halo masses in the simulation are +M=9.0×1014M⊙, M=12.6×1014M⊙, M=6.6×1014M⊙ and M=11.7×1014M⊙ for Abell 569, Coma, Abell 85 and Abell 2256 cluster counterparts respectively. +confirm the visual impression. The different values also show the +limitation of both metrics and confirm their complementarity. On +the one hand, the ζ-metric is more robust to small samples than the +2D-KS statistic: e.g. Abell 569 has a smaller observational sample in +the second catalog of the Cosmicflows project than in the third one. +However, the ζ-values when comparing both observational samples to +the simulated one differ by only a few percent. On the contrary, the +2D-KS statistic values grandly differ. One the other hand, the 2D-KS +statistic is more robust to observational uncertainties: peculiar velocity +values of galaxies in Coma, Abell 85 and Abell 2256 surroundings are +compatible, given their uncertainties, between the second and third +catalogs of the Cosmicflows project. They are higher though in the +third catalog. Consequently, the ζ-metric gives higher values when +comparing lines-of-sight from this third catalog to the simulated ones +rather than lines-of-sight from the second catalog to the simulated one. +Note though that it is not completely unexpected that the simulated +lines-of-sight match better those from the second catalog than the +third one. Indeed, the second catalog is the starting point to build the +constrained initial conditions. +Additionally, since observed galaxies with low distance uncertain- +ties are usually not exactly along the line-of-sight of the massive clus- +ters, their velocity constitutes a lower limit for the mass estimate of +the observed clusters. Indeed, galaxies perfectly aligned with the ob- +server and the cluster would have the highest possible velocity but such +galaxies are difficult to distinguish from those belonging to the cluster. +Consequently, for Virgo, Centaurus and Abell 569, the maximum pe- +culiar velocity in the simulation is slightly higher than that in the ob- +servations: it confirms that the simulated cluster have reached the low +mass limit set by the observations. Moreover, the difference between +the observed and simulated wave maxima is small enough that masses +are within the same mass range according to the Least Action modeling +(see for instance Mohayaee & Tully 2005; Tully & Mohayaee 2004). +This agreement is confirmed by observational data that follow the wave +shape so as to reproduce its width. The next section expands on the link +between wave properties and cluster masses. Note that the adequacy +between simulated and observed velocity wave shapes is really good +for Abell 569 given that even small uncertainty peculiar velocities, not +used to constrain this wave progenitor in the initial conditions’ linear +regime, follow also the simulated wave contour. There are indeed two +orange/red datapoints from the third catalog that have no blue counter- +part in the second catalog. The 2D-KS statistic small value confirms +the adequacy. +For Coma, Abell 85 and Abell 2256, given their hosted galaxy +peculiar velocity uncertainties, masses are also in good agreement and +the lower mass limit is reached. This is not fully expected given that +these clusters are at the edge of the constrained region (50%, 90% and +99% of the constraints are in ∼75-80, 150-160 and 275-290 Mpc). +Additional precise observational data are however required to probe +the wave slopes and check their width to tighten the constraint on the +masses. +Fig. 5 shows four additional velocity waves born from massive +dark matter halos to which we can associate observed galaxies. The +© 2022 RAS, MNRAS 000, 1–10 + +Velocity waves +7 +PGC765572 +100 +120 +140 +160 +180 +200 +220 +240 +d (Mpc) +-2000 +-1000 +0 +1000 +2000 +3000 +vpec (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +PGC999654 +120 +140 +160 +180 +200 +220 +240 +260 +d (Mpc) +-2000 +-1000 +0 +1000 +2000 +3000 +vpec (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +PGC340526 +160 +180 +200 +220 +240 +260 +280 +d (Mpc) +-2000 +-1000 +0 +1000 +2000 +3000 +vpec (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +PGC46604 +160 +180 +200 +220 +240 +260 +280 +d (Mpc) +-2000 +-1000 +0 +1000 +2000 +3000 +vpec (km s-1) +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +CLONE +CF2-67 +CF3-67 +CF2-74 +CF3-74 +Envelope +Fit +Figure 5. Same as Figure 3 bottom panels for four additional clusters. Names at the top of each panel are PGC (Principal Galaxy Catalog) numbers of the galaxies +with the highest velocity in the observational catalog at the given locations. +galaxies with the largest peculiar velocities are identified by their PGC +(Principal Galaxy Catalog) number at the top of each panel. Here again, +given the distance of these clusters and the sparsity and limit of our +constraint-catalog, the agreement is quite good. Tables 1 and 2 confirm +again the visual impression. They also highlight again the limitations of +both metrics. Both values must be given together to conclude on how +much the observed and simulated lines-of-sight match. Note that we +identify other simulated velocity waves corresponding to local clusters +(e.g. in the Perseus-Pisces region) but observational data is not of suf- +ficient quality or absent in the infall region for comparisons. Nonethe- +less, all the halos and associated waves are used for the next section +studies. The mass range is actually extended down to 2 1014M⊙. +4 +WAVE PROPERTIES VS. CLUSTER MASSES +4.1 +The amplitude +The wave amplitude is the first obvious property to check against halo +mass. Indeed, the deeper the gravitational potential well, the faster +should galaxies fall onto it. The amplitude is thus defined as the dif- +ference between the maximum and minimum peculiar velocities of +galaxies falling onto the cluster either from the front or from behind +with respect to the synthetic observer. Fig. 6 thus shows the amplitude +of the simulated velocity waves as a function of the dark matter halo +masses. Each black and red filled circle corresponds to a halo. Red +ones stand for clusters identified in Fig. 3 to 5. While it is immediate to +notice that there is a clear correlation between the wave amplitude and +the halo mass, one can also point out that the amplitude is extremely +difficult to measure in observational data and that there is a residual +scatter. Indeed, measuring the amplitude in observational data implies +getting exquisite distance (peculiar velocity) estimates of galaxies ex- +actly in the line-of-sight of the cluster with respect to us. It supposes +first that there are actually galaxies exactly aligned. Then, identifying +these galaxies and measuring their distances with great accuracy, while +they fall onto the cluster from the front is already quite a challenge, let +alone when they fall from behind. +In any case, the residual scatter suggests that the amplitude, be it +measurable, alone cannot be used as a precise proxy for cluster mass +estimates. Part of this scatter is probably due to the fact the galaxies are +not perfectly aligned with us and the cluster. The gravitational potential +well shape might also be responsible for another part of this scatter. To +a lesser extent, the large-scale structure environment might also play a +role. +4.2 +The height +While the wave height is not expected to be a better proxy than the +wave amplitude, it is interesting to check whether there still is a tight +Figure 6. Amplitude of the simulated velocity waves as a function of the dark +matter halo masses. Halos shown in Fig. 3 to 5 are identified in red. +enough correlation. Indeed, while it is challenging to have precise dis- +tance measurements for both galaxies falling from the front and from +behind a cluster in the line-of-sight with respect to us, it might be fea- +sible especially for galaxies falling from the front. The height is thus +defined as the maximum (minimum) peculiar velocities of galaxies +falling onto the cluster from the front (behind) with respect to the syn- +thetic observer. In Fig. 7, each black and red (blue and orange) filled +circles stand for the height of a dark matter halo positive-(negative- +)half wave as a function of its mass. Red and orange are used for dark +matter halos from Fig. 3 to 5. +A similar correlation as with the amplitude is found although +with a somewhat larger scatter. Interestingly it also shows that velocity +waves are not symmetric: their maximum differs from their minimum. +Both the potential well shape and the non-perfect alignement observer- +galaxy-halo or observer-halo-galaxy might be the reason for this asym- +metry. Nonetheless because there still is a correlation and because in +observational data it is easier to get accurate datapoints at the wave +front than in its wake, it is legitimate to focus on the positive-half ve- +locity wave shape to study more thoroughly the relation with the halo +mass. +© 2022 RAS, MNRAS 000, 1–10 + +8 +Sorce et al. +Figure +7. +Dark +(blue) +filled +circles +are +heights +of +the +simulated +positive(negative)-half velocity waves as a function of the dark matter halo +masses. Halos shown in Fig. 3 to 5 are identified in red (orange). +4.3 +Height, width and continuum +After deriving the positive-half wave envelope of every dark matter +halo, we choose to fit the simplest model possible, a Gaussian-plus- +continuum model, to each one of them as follows: +vpec = Afit × e +−(d−d0)2 +2σ2 +fit ++ C fit +(5) +where Afit, σ fit and C fit are respectively the Gaussian amplitude, +its standard deviation and a continuum. d0 depends on the halo +distance and has no other purpose than centering the Gaussian on +zero. Its sole physical meaning is to be the actual distance of the +halo. The amplitude is related to the positive-half wave envelope +height while the standard deviation is linked to its width. Finally, the +continuum gives the positive-half wave offset from a zero average +velocity. For visualization, envelopes and their fits for halos presented +in Fig. 3 to 5 are shown as solid and dashed lines on these same figures. +Fig. 8 gathers the three parameters of the fits and halo masses +for a concomitant study to highlight an eventual multi-parameter +correlation. The Gaussian amplitude is represented as a function of +the Gaussian standard deviation while the color scale stands for the +continuum. From black-violet to red, the continuum decreases from +positive values to negative ones. The model uncertainty is shown as +error bars for the amplitude and standard deviation. The color scale +smoothness includes the continuum uncertainty. The Gaussian-plus- +continuum model choice proves to be robust given the tiny error bars +that it results in. The filled circle sizes are proportional to the dark +matter halo masses. Finally, an additional small red filled circle is used +to identify each halo analyzed in Fig. 3 to 5. +The previous subsection (4.2) showed that there is a correlation +between the wave height and the halo mass. It is thus not surprising to +find back that the more massive the halo is (larger circle), the larger +500 +1000 +1500 +2000 +Afit (km s-1) +2 +4 +6 +8 +10 +12 +14 +σfit (Mpc) +574 +407 +239 +71 +-96 +-264 -432 +Cfit (km s-1) +Virgo +Centaurus +Abell 569 +Coma +Abell 85 +Abell 2256 +PGC765572 +PGC999654 +PGC340526 +PGC46604 +Figure 8. Parameters of the Gaussian-plus-continuum fit to the simulated +positive-half velocity waves. σfit stands for the Gaussian standard deviation, +Afit for its amplitude and C fit for the continuum. The filled circle sizes are pro- +portional to dark matter halo masses. Tiny error bars on the standard deviation +and amplitude resulting from fitting the envelopes highlight the adequacy of the +model choice. Halos shown in Fig. 3 to 5 are identified with red nametags and +additional small red filled circles. +the Gaussian amplitude is (larger value). As stated above, the Gaussian +amplitude is indeed the counterpart of the positive-half wave height. +In addition, there is a small correlation between the amplitude +and standard deviation thus halo mass. More massive halos seem to +give birth to wider waves. The scatter is however quite large. It cer- +tainly depends greatly on the halo triaxiality and thus on its orientation +with respect to us. A similar conclusion is valid for the continuum, the +smaller the continuum but for extreme values is, the more massive the +halo is on average. Anyhow, the scatter is quite large in that case. A +strong dependence on the global environment of the dark matter halo +in addition to the halo mass might be in cause here. +Interestingly a general pattern emerges quite clearly though: +• the most massive halos (≳ 6 1014 M⊙) tend to give birth to positive- +half waves that have a continuum compatible with zero or slightly +negative/positive in addition to high amplitude and standard deviation +values. +• the less massive halos ( 2 1014 M⊙≲M≲ 4 1014 M⊙) tend to give birth +to positive-half waves that have a continuum compatible with zero or +slightly negative/positive in addition to low amplitude and standard +deviation values. +• intermediate mass halos (4 1014 M⊙≲M≲ 6 1014 M⊙) give rise to +positive-half waves that have high continuum absolute values. Such +values permit distinguishing them from the most massive halos with +which they share high amplitude and possibly standard deviation +values, especially in the negative continuum case. +It is highly probable that the global environment or cosmic web is +responsible for such a finding. We will investigate this link in more +details in future studies. +The halo segregation in different continuum value classes is an- +other quite inspiring source. There seems to be a different correlation +for each continuum value class: +• Halos with fits resulting in a high (close to zero) continuum value +© 2022 RAS, MNRAS 000, 1–10 + +Velocity waves +9 +seems to have masses correlated with the Gaussian amplitudes but not +so much with the Gaussian standard deviations that appear to have low +values (present a large scatter). +• Halos with fits resulting in a very low continuum value have both +amplitudes and standard deviations correlated together as well as with +the masses. +• Halos with fits resulting in either positive or negative intermediate +continuum values present masses correlated with amplitudes and up +to a certain point with standard deviations. Consequently, although to +a lesser extent than for halos whose continuum values are quite low, +amplitudes and standard deviations are slightly correlated. +To summarize, since the fit parameters are interdependent, a +global fit to the velocity wave seems the best approach to obtain clus- +ter rough mass estimates rather than single and independent measure- +ments of amplitude, height and width. Because different categories +appear among halos, in future studies, a machine learning approach +might become handy to actually get accurate enough mass estimates +from sparse observations. In a first approach, the simple Gaussian-plus- +continuum fit presented here could be used as a model reduction. +5 +CONCLUSIONS +Galaxy clusters are excellent cosmological probes provided their +mass estimates are accurately determined. Fueled with large imaging +surveys, stacked weak lensing is the most promising mass estimate +method though it provides estimates within relatively small radii. +Given the large amount of accompanying redshift and spectroscopic +data overlapping the imaging surveys, we must take the opportunity to +calibrate also with a reasonable accuracy a method based on galaxy +dynamics. Two independent measures hold indeed better constraints +on the cosmological model. Infall zones of galaxy clusters are proba- +bly the less sensitive to baryonic physics, thus mostly shielded from +challenging systematics, and probe large radii. These manifestations +of a tug of war between gravity and dark energy provide a unique +avenue to test modified gravity theories when comparing resulting +mass estimates to those from stacked weak lensing measurements. +Combined with stacked weak lensing results, they might even yield +evidence that departure from General Relativity on cosmological +scales is responsible for the expansion acceleration. +The accurate calibration of the relation between infall zones +properties and cluster masses starts with careful comparisons between +cosmological simulations and observations. In this paper, we thus +present our largest and highest resolution Constrained Local & +Nesting Environment Simulation (CLONE) built so far to reproduce +numerically our cosmic environment. This simulation stems from +initial conditions constrained by peculiar velocities of local galax- +ies. By introducing this cosmological dark matter CLONE of the +local large-scale structure with a particle mass of ∼109M⊙ within a +∼738 Mpc box, we have sufficient resolution to study the effect of +the gravitational potential of massive local halos onto the velocity +of (sub)halos. We can also compare with that of their observational +cluster counterparts. +Velocity waves stand out in radial peculiar velocity - distance to +a box-centered synthetic observer diagram. The agreement between +lines-of-sight including velocity waves, caused by the most massive +dark matter halos of the CLONE and those born from their observa- +tional local cluster counterparts, is visually good especially for the +clusters the closest to us that are the best constrained (e.g. Virgo, +Centaurus). Secondary waves due to smaller groups in (quasi) the +same line-of-sight as the most massive clusters stand out equally even +though they are further into the non-linear regime. Indeed, prior to +full non-linear evolution to the z=0 state, assuming ΛCDM, CLONE +initial conditions are constrained with solely the linear theory, a power +spectrum and highly uncertain and sparse local peculiar velocities. The +visual matching between the simulated and observed lines-of-sight +is confirmed with 2D-Kolmogorov Smirnov (KS) statistic values and +tests as well as with our own ζ-metric. Contrary to the 2D-KS statistic, +the ζ-metric takes into account the real distance of galaxies along the +entire lines-of-sight (not only the studied fractions). The ζ-metric is +however more sensitive to the fact that observational uncertainties are +not taken into account in these metrics. The two metrics appear to +be complementary. They show that the closest clusters have the best +reproduced lines-of-sight. The lines-of-sight of clusters at the edges of +the constrained region and even slightly beyond are also reproduced +by the simulation although to a smaller extent. +Additionally, a Gaussian-plus-continuum fit to the envelope of +the positive-half of all the velocity waves born from dark matter +halos more massive than 2 1014M⊙ in the simulation reveals both the +variety and complexity of the potential wells as well as the correlation +of the fit parameters with the halo masses. Overall, the Gaussian +amplitude is mostly linked to the halo mass, but for a few exceptions, +with a residual scatter. Although the Gaussian standard deviation +is not always correlated with the mass, it can be slightly correlated +with the Gaussian amplitude thus with the mass. The continuum is +certainly an interesting parameter to consider as it permits splitting +the halos into different classes. Each continuum value seems to drive a +given correlation between the Gaussian amplitude and the halo mass +and, to a smaller extent, with the Gaussian standard deviation. To +summarize, parameter fits are completely interdependent, a global fit +to the velocity wave is then the best approach to obtain a first rough +cluster mass estimate. +First and foremost, this work confirms the potential of the +velocity wave technique to get massive cluster mass estimates and +test gravity and cosmological models. Our CLONES, with the first +shown reproduction of observed lines-of-sight including velocity +waves, could in the near future provide the zero point of galaxy +infall kinematic technique calibrations (Zu & Weinberg 2013). A +bayesian inference model or/and a machine learning technique built +and trained on random simulated galaxy surveys that is then applied to +both constrained simulated and observed galaxy surveys must recover +the same local velocity waves and corresponding mass estimates to +be validated. Our CLONES will moreover allow minimizing obser- +vational biases as any real environmental and cluster property will +be reproduced for perfect one-to-one comparisons. Local kinematic +mass estimates can then become accurate. Once compared with other +techniques of local galaxy cluster mass estimates, they will permit +calibrating the zero-point of these other techniques to be applied to +further-and-further away clusters. +DATA AVAILABILITY +Synthetic catalogs are available upon reasonable request to the authors. +ACKNOWLEDGEMENTS +The authors acknowledge the Gauss Centre for Supercomputing e.V. +(www.gauss-centre.eu) and GENCI (https://www.genci.fr/) for funding +this project by providing computing time on the GCS Supercomputer +SuperMUC-NG at Leibniz Supercomputing Centre (www.lrz.de) and +Joliot-Curie at TGCC (http://www-hpc.cea.fr), grants ID: 22307/22736 +© 2022 RAS, MNRAS 000, 1–10 + +10 +Sorce et al. +and A0080411510 respectively. This work was supported by the grant +agreements ANR-21-CE31-0019 / 490702358 from the French Agence +Nationale de la Recherche / DFG for the LOCALIZATION project and +ERC-2015-AdG 695561 from the European Research Council (ERC) +under the European Union’s Horizon 2020 research and innovation +program for the ByoPiC project (https://byopic.eu). KD acknowledges +support by the COMPLEX project from the ERC under the European +Union’s Horizon 2020 research and innovation program grant agree- +ment ERC-2019-AdG 882679. The authors thank the referee for their +comments. JS thanks Marian Douspis for useful comments, the By- +oPiC team and her CLUES collaborators for continuous discussions. +REFERENCES +Aubert D., Pichon C., Colombi S., 2004, MNRAS, 352, 376 +Burke D., 2006, in APS April Meeting Abstracts +Carlberg R. 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H., Jennings E., Li B., Wyman M., 2014, MN- +RAS, 445, 1885 +© 2022 RAS, MNRAS 000, 1–10 + diff --git a/PNAzT4oBgHgl3EQfWvwu/content/tmp_files/load_file.txt b/PNAzT4oBgHgl3EQfWvwu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..665e8dcc6962354eeb21b573e3284507761234f6 --- /dev/null +++ b/PNAzT4oBgHgl3EQfWvwu/content/tmp_files/load_file.txt @@ -0,0 +1,1301 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf,len=1300 +page_content='Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 000, 1–10 (2022) Printed 5 January 2023 (MN LATEX style file v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='2) Velocity waves in the Hubble diagram: signature of local galaxy clusters Jenny G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Sorce1,2,3,4⋆, Roya Mohayaee5,6, Nabila Aghanim1, Klaus Dolag7,8, Nicola Malavasi7,1 1 Universit´e Paris-Saclay, CNRS, Institut d’Astrophysique Spatiale, 91405, Orsay, France 2 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Lyon, ENS de Lyon, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Lyon1, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69007, Lyon, France 3Leibniz-Institut f¨ur Astrophysik (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany 4Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France 5CNRS, UPMC, Institut d’Astrophysique de Paris, 98 bis Bld Arago, Paris, France 6Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom 7University Observatory Munich, Scheinerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 1, 81679 M¨unchen, Germany 8Max-Planck Institut f¨ur Astrophysik, Karl-Schwarzschild Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 1, D-85741 Garching, Germany ABSTRACT The Universe expansion rate is modulated around local inhomogeneities due to their gravita- tional potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Velocity waves are then observed around galaxy clusters in the Hubble diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' This paper studies them in a ∼738 Mpc wide, with 20483 particles, cosmological simulation of our cosmic environment (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' CLONE: Constrained LOcal & Nesting Environment Simulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' For the first time, the simulation shows that velocity waves that arise in the lines-of-sight of the most massive dark matter halos agree with those observed in local galaxy velocity catalogs in the lines-of- sight of Coma and several other local (Abell) clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' For the best-constrained clusters such as Virgo and Centaurus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' those closest to us, secondary waves caused by galaxy groups, further into the non-linear regime, also stand out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' This match is not utterly expected given that before being evolved into a fully non-linear z=0 state, assuming ΛCDM, CLONE initial conditions are constrained with solely linear theory, power spectrum and highly uncertain and sparse local peculiar velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Addi- tionally, Gaussian fits to velocity wave envelopes show that wave properties are tightly tangled with cluster masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' This link is complex though and involves the environment and formation history of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Using machine learning techniques to grasp more thoroughly the complex wave-mass relation, velocity waves could in the near future be used to provide additional and independent mass estimates from galaxy dynamics within large cluster radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Key words: galaxies: clusters: individual – waves – methods: numerical – methods: analytical – techniques: radial velocities – gravitation 1 INTRODUCTION As the largest gravitationally bound structures in the Universe, galaxy clusters bear imprints of the cosmic growth visible through the distribution and motion of galaxies in their surrounding environment (see Kravtsov & Borgani 2012, for a review and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' They constitute therefore powerful complementary probes to super- novae and baryon acoustic oscillations in testing theories explaining cosmic acceleration origin (see Weinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2013, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Relations between halo masses and observables (optical galaxy richness, Sunyaev-Zel’dovich effect, X-ray luminosity) must however be calibrated beforehand to study the evolution of the cluster mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Our capacity to discriminate among cosmological models is thus tightly linked to the accuracy of cluster mass estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' However, most of the cluster matter content is not directly visible making their mass estimates a particularly challenging task (see for a review Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' With future imaging surveys to come (LSST, Burke 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Euclid, ⋆ E-mail: jenny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='sorce@universite-paris-saclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='fr / jsorce@aip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='de Peacock 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' WFIRST, Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2012), stacked weak lensing mea- surements will certainly provide the best cluster mass estimates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' with the 1% accuracy required (Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2006) but limited to small radii around clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Independent virial mass estimators (Heisler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 1985), hydrostatic estimators for galaxy population (Carlberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 1997) or velocity caustics (boundaries between galaxies bound to and escaping from the cluster potential, Diaferio 1999) constitute com- plementary tools once calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Their calibration suffers though from the influence of baryonic physics and galaxy bias on velocity fields and dispersion profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Perhaps velocity caustics are less prone to such sys- tematics (Diaferio 1999) explaining their recent increased popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Galaxy clusters can indeed be seen as disrupters of the expansion, thus creating a velocity wave first mentioned by Tonry & Davis (1981) as a triple-value region1 whose properties (mostly height and width) de- pend on the cluster mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Combined with infall models (Mohayaee & Tully 2005), velocities of galaxies in the infall zones constitute thus 1 Such an appellation derives directly from the fact that in a disrupted Hubble diagram, galaxies at three distinct distances, d, share a similar velocity value whereas in an unperturbed diagram, these galaxy velocities would differ pre- cisely because of the expansion proportional to H0 × d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' © 2022 RAS arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='01305v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='CO] 3 Jan 2023 2 Sorce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' good mass proxies for galaxy clusters shown to be in good agreement with virial mass estimates (Tully 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' They have been used in dif- ferent studies to retrieve the total amount of dark matter in groups and clusters as well as to detect groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Karachentsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Karachentsev & Nasonova 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Moreover, Zu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' (2014) showed that the wave shape is an excellent complementary probe: for instance, f(R) modified gravity models enhance the wave height (infall veloc- ity) and broaden its width (velocity dispersions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' This translates into a higher mass when considering a ΛCDM framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Subsequently, it would lead to cosmological tensions between S 8 values measured with the cosmic microwave background and with the galaxy cluster counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Furthermore, velocity waves probe a cluster mass within radii larger than those reached with weak lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Subsequently, combined together, stacked weak lensing and velocity wave mass measurements hold tighter constraints on dark energy than each of them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Indeed, velocity waves are signatures of a tug of war between gravity and dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Differences between these two independent mass esti- mates, one dynamic and one static, permit measuring the gravitational slip between the Newtonian and curvature potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' This constitutes an excellent test of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Given future galaxy redshift and large spectroscopic follow-up surveys (with Euclid, Peacock 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 4MOST, de Jong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' MOONS, Cirasuolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2014) of imaging ones, studying galaxy infall kinematics to derive better cluster dynamic mass estimates is surely the next priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Cosmological simulations constitute critical tools to test, understand and eventually calibrate this mass estimate method applied to galaxy cluster observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Ideally these simula- tions must be constrained simulations2 to properly set the zero point of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Namely, simulations must be designed to ensure that the simulated and observed waves match in every aspect but if the theoretical model somewhere fails and not because of, for instance, different formation histories and/or environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' We are now able to produce such simulations valid down to the cluster scale including the formation history of the clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Sorce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2016a, 2019, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Sorce 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' These simulations are thus faithful reproduction of our local environment including its clusters such as Virgo, Coma, Centaurus, Perseus and several Abell clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' This paper thus starts with the first comparison between line-of- sight velocity waves due to several observed local clusters and their counterparts from a Constrained LOcal & Nesting Environment Sim- ulation (CLONE) built within a ΛCDM framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' First, we present the numerical CLONE used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Next, we compare the ob- served and simulated lines-of-sight that host velocity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' To facil- itate the comparisons, the background expansion is subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Before concluding, wave envelopes are fitted to study relations between wave properties and cluster masses in a ΛCDM cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2 THE CLONE SIMULATION Constrained simulations are designed to match the local large-scale structure around the Local Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Several techniques have been developed to build the initial conditions of such simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Gottl¨ober et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Kitaura 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Jasche & Wandelt 2013) with density, velocity or both constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Here we use the technique whose details (algorithms and steps) are described in Sorce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' (2016a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Sorce (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Local observational data used to constrain the initial conditions are distances of galaxies and groups (Tully et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Sorce & Tempel 2017) converted to peculiar velocities (Sorce 2 The initial conditions of such simulations stem from observational constraints applied to the density and velocity fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 222 148 74 0 74 148 222 SGX (Mpc) 222 148 74 0 74 148 222 SGY (Mpc) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' ∼40 Mpc thick XY supergalactic slice of the CLONE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Black dots stand for the dark matter halos (subhalos are excluded for clarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Red dots are galaxies from the 2MASS Galaxy Redshift Catalog (XSCz) for comparison pur- poses only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Indeed, only a small fraction of local galaxy observational redshifts have been used to derive peculiar velocities that were used as constraints (about ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='5% of the XSCz catalog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Sorce & Tempel 2018) that are bias-minimized (Sorce 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' We showed that constrained simulations obtained from this particular technique, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' the CLONES (Sorce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2021), are currently the sole replicas of the local large-scale structure that include the largest local clusters using only galaxy peculiar velocities as constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Namely, the cosmic variance is effectively reduced within a 200 Mpc radius centered on the Local Group down to the cluster scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3-4 Mpc, (Sorce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Galaxy clusters (such as Virgo, Centaurus, Coma) have masses in agreement with observational estimates (Sorce 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Several ensuing studies focused in particular on the Virgo galaxy cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' These studies confirmed the necessity of using CLONES to get a high-fidelity Virgo-like cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Additionally, they confirmed observationally-based formation scenarios of the latter (Olchanski & Sorce 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Sorce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2019, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' To actually probe a large range of velocities in the infall zones, the CLONE for the present study needs to have a sufficient resolution to simulate, with a hundred particles at z=0, halos of intermediate mass (∼1011-1012 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Its constrained initial conditions contain thus 20483 particles in a ∼738 Mpc comoving box (particle mass ∼109 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' It ran on more than 10,000 cores from z=120 to z=0 in the Planck cosmology framework (Ωm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='307 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' ΩΛ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='693 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' H0=67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='77 km s−1 Mpc−1 and σ8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='829, Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2014) using the adaptive mesh refinement Ramses code (Teyssier 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The mesh is dynamically (de-)refined from level 11 up to 18 according to a pseudo-Lagrangian criterion, namely when the total density in a cell is larger (smaller) than the density of a cell containing 8 dark matter particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The initial coarse grid is thus adaptively refined up to a best-achieved spatial resolution of ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='8 kpc roughly constant in proper length (a new level is added at expansion factors a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='8 up to level 18 after a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Using the halo finder, described in Aubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' (2004) and Tweed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' (2009), modified to work with 20483 (>231) particles, dark matter halos and subhalos are detected in real space with the local maxima of dark matter particle density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Their edge is defined as the point © 2022 RAS, MNRAS 000, 1–10 Velocity waves 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Schema of the cylinder used to select (sub)halos whose radial pecu- liar velocities, derived as a function of the synthetic observer at the simulated box center, are used to study the velocity wave arisen from the massive halo in its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' While open circles stand for selected halos, dashed circles represent excluded ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' where the overdensity of dark matter mass drops below 80 times the background density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' We further apply a lower threshold of a minimum of 100 dark matter particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 1 shows the ∼40 Mpc thick XY super- galactic slice of the CLONE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Black (red) dots stand for the dark matter halos (galaxies from the 2MASS Galaxy Redshift Catalog - XSCz3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Note that XSCz galaxies are used for sole comparison purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' XSCz is indeed far more complete than the peculiar velocity catalog used to constrain the simulation (∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='5% of the redshift catalog is used to de- rive the peculiar velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' In fact, it shows the constraining power of the peculiar velocities that are correlated on large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Namely, the simulation is constrained also in regions where no peculiar velocity measurements were available and thus used as constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' It confirms once more that peculiar velocity catalogs fed to our technique, to re- construct/constrain the local density and velocity fields, do not need to be complete (Sorce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3 VELOCITY WAVE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='1 In simulated data Positioning a synthetic observer at the simulation box center, we de- rive radial peculiar velocities for all the dark matter halos and subha- los in the z=0 catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' We then draw lines-of-sight in the direction of each dark matter halo more massive than 5 1014M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' All the (sub)halos within 10 Mpc from the line-of-sight and within 74 Mpc along the line-of-sight from a given massive dark matter halo (with the center and edge of the box as upper limits) are selected to plot the latter cor- responding velocity wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Namely, as shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2, radial peculiar velocities, with respect to the synthetic observer, of (sub)halos within a cylinder at maximum 148 Mpc long and 20 Mpc wide are used to vi- sualize the velocity wave caused by the massive dark matter halo in the cylinder center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Note that because the simulation is constrained to re- produce the local Universe, we choose not to use the periodic boundary conditions to wrap around the box edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' It will indeed not be repre- sentative of local structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' A 10 Mpc radius cylinder corresponds to about three times the virial radius of the massive clusters under study here (M>5 1014M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Since the goal is to study the link between veloc- ity wave properties and cluster masses, exact masses cannot be used to define the cylinder shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Finally, such large volumes permit probing the infall region around the massive halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Note that a cylinder shape is preferable to a cone shape to get an unbiased wave signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' A cone would indeed result in a distorted signal as it would probe a larger and larger region around a massive halo with the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='2 In observational data Observational data are taken from the raw second and third catalogs of the Cosmicflows project (Tully et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2013, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Note that the 3 https://wise2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='edu/staff/jarrett/2mass/XSCz/specz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='html second catalog containing ∼8000 galaxies, with a mean distance of ∼90 Mpc, serves as the basis to build the constraint-catalog of ∼5000 bias-minimized radial peculiar velocities of galaxies and groups with a mean distance of ∼60 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' By contrast, the third catalog contains ∼17,000 galaxies with a mean distance of ∼120 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The third catalog is not used to constrained our CLONE initial conditions and thus constitute partly an independent dataset for consistency check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' More precisely, it serves the two-fold goal of extending the number of observational datapoints to be compared with the simulation and highlighting again the constraining power of peculiar velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The latter can indeed permit recovering structures that are not directly probed and that are at the limit of the non-linear threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' In the sense that there is no direct measurement in a given region but, because the latter influences the velocities of other regions (large scale correlations), it can still be reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Uncertainties on distances and radial peculiar velocities in these catalogs depend on the distance indicator used to derive the distance moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Error bar sizes need to be limited to see clearly velocity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Thus, to be able to compare with the simulated data, only galaxies with uncertainties on distance moduli smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='2 dex are retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' There remain 338 and 424 galaxies respectively from the second and third catalogs with a mean distance of ∼50 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' These galaxies are mostly hosts of supernovae, especially those the furthest from us (dis- tance indicator with a small uncertainty even as the distance increases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' To derive the radial peculiar velocities of these galaxies, we use both galaxy distance moduli (µ) and observational redshifts (zobs) following Davis & Scrimgeour (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' We add supergalactic longitude and latitude coordinates to derive galaxy cartesian supergalactic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' A cosmological model is then required to determine peculiar velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' While we use ΛCDM, as cosmicflows catalog zero points are calibrated through a long process on WMAP (rather than Planck) values (Ωm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='27, ΩΛ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='73, H0=74 km s−1 Mpc−1, Tully et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2013, 2016), we have to use the same parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' We indeed showed that when applying the bias minimization technique to the peculiar velocity catalog of constraints, we drastically reduce the dependence on ΛCDM cosmological parameter values (Sorce & Tempel 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' However, in order to be able to probe the whole velocity wave for the comparisons, we have to use the raw catalog i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' with neither galaxy grouping nor bias minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Consequently, if were to take Planck values to derive galaxy peculiar velocities, the WMAP calibration would translate into a residual Hubble flow visible in the background-expansion-subtracted Hubble diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Subsequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' using WMAP values for the observations: Luminosity distances,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' dlum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' are derived from distance modulus mea- surements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' obtained via distance indicators: µ = 5log10(dlum (Mpc)) + 25 (1) Cosmological redshifts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' zcos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' are then obtained through the equation: dlum = (1 + zcos) � zcos 0 cdz H0 � (1 + z)3Ωm + ΩΛ (2) Galaxy radial peculiar velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' vpec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' are finally estimated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' using the observational zobs and cosmological zcos redshifts with the following formula: vpec = czobs − zcos 1 + zcos (3) where vpec will always refer to the radial peculiar velocity in this paper and c is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' © 2022 RAS, MNRAS 000, 1–10 4 Sorce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='Virgo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='20 ' metadata={'source': 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and us respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Error bars stand for uncertainties on observational distance and velocity estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Orange and light blue (red and dark blue) filled squares and diamonds show observed galaxies assuming H0=74 (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='77) km s−1 Mpc−1 for scaling positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' CF2 (CF3) corresponds to the second (third) catalog of the Cosmicflows project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Larger symbols are used for galaxies, with a peculiar velocity higher than 1000 km s−1, identified as the closest to the simulated massive halos assuming the synthetic observer at the box center and the same Supergalactic coordinate system and orientation as the local Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The arrow indicates the position of the massive dark matter halo in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Names of corresponding observed clusters are given at the top of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Velocity waves stand out in the different lines-of-sight and there is a good agreement with observational datapoints for those two best-constrained clusters the closest to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Top: Hubble diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Bottom: Hubble flow subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The solid and dashed yellow lines are respectively the simulated positive-half velocity wave envelope and its Gaussian-plus-continuum fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The color scale filling the black circles stands for their distance from the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' From black to light grey, objects are less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='5, 5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='5 and 10 Mpc away from the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The dark matter halo virial masses in the simulation are M=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='8×1014M⊙ and M=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='0×1014M⊙ for the Virgo and Centaurus cluster counterparts respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='3 Simulated vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' observed data Assuming the synthetic observer at the box center and the simulated volume oriented similarly to the local volume, observed and simulated positions and lines-of-sight can be matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' We can only compare velocity waves born from local galaxy clusters for which infalling galaxy peculiar velocities, with uncertainties on corresponding distance moduli smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='2 dex, are available in the observed cluster surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' We thus select these clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' For each simulated massive dark matter halo, the quickest way is then to search for the closest observed galaxy, in our selected above samples, with a radial peculiar velocity greater than 1000 km s−1 (∼2σ above the average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' This is indeed a signature that it has most probably an observed cluster with a mass of at least a few 1014M⊙ as a neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Whenever a simulated massive dark matter halo is within the 2σ uncertainty of the observed galaxy distance, we select all the observed galaxies in the cylinder corresponding to the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' For every case, there is indeed a massive observed cluster in the vicinity of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' More to the point, given the Supergalactic coordinates of the observed clusters and those of the simulated ones in the box, they indeed match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3 superimposes observed and simulated lines-of-sight with the velocity waves born from the two closest most massive local clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Observational data is of sufficient quality in their respec- tive infall region to warrant adequate comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' From left to right, galaxy clusters (dark matter halos) are at increasing distance from us (the synthetic observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The name of the clusters is indicated at the top of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Filled black and grey circles stand for simu- lated (sub)halos while filled light blue and orange squares and dia- monds represent observed galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Because the simulation was run with H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='77 km s−1 Mpc−1, filled dark blue and red squares and diamonds are observed galaxies at positions rescaled with this latter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Position differences are always within about the 1σ uncertainty © 2022 RAS, MNRAS 000, 1–10 Velocity waves 5 Cluster CLONE/CF2 CLONE/CF2 CLONE/CF3 CLONE/CF3 Cylinder radius 10 Mpc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='5 Mpc 10 Mpc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='5 Mpc Virgo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='0098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='0058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='0071 Centaurus 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='50 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Kolmogorov-Smirnov statistic or highest distance between the cumula- tive distribution functions of the observed and simulated lines-of-sight including the velocity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' on the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Arrows indicate the position of the most massive halos in the lines-of-sight of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' In the top panels, the Hubble diagrams are clearly distorted by the presence of massive halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Their corresponding velocity wave or triple-value region signatures show up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Bottom panels with the Hub- ble flow subtracted equally confirms the waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The simulated velocity waves stand out in the peculiar velocity of (sub)halos plotted as a func- tion of the distance from the synthetic observer diagrams for the two massive dark matter halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The agreement with the observational data points is qualitatively good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' All the more since only sparse peculiar ve- locities of today field galaxies and groups are used to constrained the linear initial density and velocity fields, at the positions of the latter progenitors, using solely linear theory and a power spectrum assuming a given cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Then the full non-linear theory is used to evolved these initial conditions from the initial redshift down to z=0 within a ΛCDM framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The signatures of Virgo West and the group around NGC4709 that are respectively beyond Virgo and Centaurus in the lines-of-sight can also be identified as secondary waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' These smaller waves follow the highest ones representing the main clusters in both the observations and the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Additionally, a void between us and Centaurus in the line-of-sight shows equally well in both the simulation and the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The accuracy with which the CLONE reproduces the lines-of-sight dynamical state of Virgo and Centaurus is visually excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' To quantify the agreement between simulated and observed lines-of-sight, we use a 2D-Kolmogorov-Smirnov statistic test applied to the simulated and observed galaxy velocity and position samples following Peacock (1983);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Fasano & Franceschini (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' p-values obtained for Virgo and Centaurus are above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' They are actually close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='0 but values above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='20 have no particular significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' They only confirm that the observed and simulated distributions along the line-of-sight are not significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Additionally, Table 1 gives the 2D-Kolmogorov-Smirnov (KS) statistic or the highest distance between the cumulative distribution functions of the observed and simulated lines-of-sight including the velocity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' A single 2D-KS statistic value has no particular meaning but several together permit ordering the simulated lines-of-sight from those that match the most their observational counterpart to those that match it the less (smallest to largest values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Virgo and Centaurus lines-of-sight happen to be equally well reproduced by the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 2D-KS statistic values are barely different when considering all the subhalos/galaxies within a 10 Mpc radius or solely those within a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='5 Mpc radius from the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The agreement is slightly better with galaxies from the third catalog (CF3) of the Cosmicflows project than with those of the Cluster CLONE/CF2 CLONE/CF2 CLONE/CF3 CLONE/CF3 Cylinder radius 10 Mpc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='5 Mpc 10 Mpc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='5 Mpc Virgo 6 10 9 12 Centaurus 21 37 25 36 Abell 569 14 23 11 22 Coma 27 40 205 225 Abell 85 184 286 299 400 Abell 2256 152 152 364 364 PGC 765572 39 53 56 70 PGC 999654 687 687 662 662 PGC 340526 92 99 16 41 PGC 46604 544 544 544 544 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' ζ-metric in km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' It measures the difference between the simulated and observed lines-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The higher ζ is the more different the lines-of-sight are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' See the text for a detailed explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' second one, although the second one is the starting point to build the constrained initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' However, given that the third catalog has more points and smaller uncertainties, it is encouraging that the simulation matches more the third catalog than the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The 2D-KS statistic test cannot indeed take into account uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Finally, 2D-KS statistic values do not differ when using H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='77 rather than 74 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The 2D-KS statistic test cannot take into account the real distance of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' It compares only the cumulative distributions of galaxies along the lines-of-sight using four directions (smallest to largest dis- tances to the y-axis and vice versa, smallest to largest distances to the x-axis - in that case velocities because they are centered on zero - and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Consequently, we also define our own ζ-metric to compare simulated and observed lines-of-sight as follows: ζ = 1 n n � i=1 � (min[vobs[i] − vsim])2 + [(min[dobs[i] − dsim]) × H0]2 (4) where n is the number of observed galaxies in the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' vX are the galaxy/subhalo observed and simulated peculiar velocities and dX are their distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Table 2 gives the values of ζ for the different lines-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Because ζ-values are only modified by a few percent when changing H0 value, their mean is reported in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Like for the 2D-KS statistic values, ζ-values permit ordering the simulated lines-of-sight (including waves) that are the best reproduction of the observed ones to those that reproduce them the less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Since our ζ-metric results in similar conclusions as the 2D-KS statistic does, it seems appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Moreover, contrary to the 2D-KS statistic, it is sensitive to the real distance of the cluster, not solely to its position on the fraction of the line-of-sight that is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' It thus includes both differences due to a difference in height and to a shift in position along the entire line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' It is easily checked by randomly shuffling observed and simulated lines-of-sights and comparing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The ζ-metric then gives values on average between a 100 and up to 1000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The ζ-metric though, like the 2D-KS statistic, does not take into account uncertainties on observational distance and velocity estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' In the rest of the paper, we work solely with the background ex- pansion subtracted since it does not affect our conclusion and ease the comparisons, studies and analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Given the above mentioned success, although the simulation matches best the local large-scale structure by construction in the inner part, where most of the constraints are, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 4 shows an additional four massive halos that are more distant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' These halos are still matching nicely observational clusters that are further away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Tables 1 and 2 © 2022 RAS, MNRAS 000, 1–10 6 Sorce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='Abell 569 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='40 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='Fit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Same as Figure 3 bottom panels for four clusters at increasing distance from us from left to right, top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Although these clusters are less con- strained, the agreement between observed and simulated waves is still visually good especially for the first two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The dark matter halo masses in the simulation are M=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='0×1014M⊙, M=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='6×1014M⊙, M=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='6×1014M⊙ and M=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='7×1014M⊙ for Abell 569, Coma, Abell 85 and Abell 2256 cluster counterparts respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' confirm the visual impression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The different values also show the limitation of both metrics and confirm their complementarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' On the one hand, the ζ-metric is more robust to small samples than the 2D-KS statistic: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Abell 569 has a smaller observational sample in the second catalog of the Cosmicflows project than in the third one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' However, the ζ-values when comparing both observational samples to the simulated one differ by only a few percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' On the contrary, the 2D-KS statistic values grandly differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' One the other hand, the 2D-KS statistic is more robust to observational uncertainties: peculiar velocity values of galaxies in Coma, Abell 85 and Abell 2256 surroundings are compatible, given their uncertainties, between the second and third catalogs of the Cosmicflows project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' They are higher though in the third catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Consequently, the ζ-metric gives higher values when comparing lines-of-sight from this third catalog to the simulated ones rather than lines-of-sight from the second catalog to the simulated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Note though that it is not completely unexpected that the simulated lines-of-sight match better those from the second catalog than the third one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Indeed, the second catalog is the starting point to build the constrained initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Additionally, since observed galaxies with low distance uncertain- ties are usually not exactly along the line-of-sight of the massive clus- ters, their velocity constitutes a lower limit for the mass estimate of the observed clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Indeed, galaxies perfectly aligned with the ob- server and the cluster would have the highest possible velocity but such galaxies are difficult to distinguish from those belonging to the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Consequently, for Virgo, Centaurus and Abell 569, the maximum pe- culiar velocity in the simulation is slightly higher than that in the ob- servations: it confirms that the simulated cluster have reached the low mass limit set by the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Moreover, the difference between the observed and simulated wave maxima is small enough that masses are within the same mass range according to the Least Action modeling (see for instance Mohayaee & Tully 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Tully & Mohayaee 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' This agreement is confirmed by observational data that follow the wave shape so as to reproduce its width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The next section expands on the link between wave properties and cluster masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Note that the adequacy between simulated and observed velocity wave shapes is really good for Abell 569 given that even small uncertainty peculiar velocities, not used to constrain this wave progenitor in the initial conditions’ linear regime, follow also the simulated wave contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' There are indeed two orange/red datapoints from the third catalog that have no blue counter- part in the second catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The 2D-KS statistic small value confirms the adequacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' For Coma, Abell 85 and Abell 2256, given their hosted galaxy peculiar velocity uncertainties, masses are also in good agreement and the lower mass limit is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' This is not fully expected given that these clusters are at the edge of the constrained region (50%, 90% and 99% of the constraints are in ∼75-80, 150-160 and 275-290 Mpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Additional precise observational data are however required to probe the wave slopes and check their width to tighten the constraint on the masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 5 shows four additional velocity waves born from massive dark matter halos to which we can associate observed galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The © 2022 RAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 1–10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='Velocity waves ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='PGC765572 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='120 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Same as Figure 3 bottom panels for four additional clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Names at the top of each panel are PGC (Principal Galaxy Catalog) numbers of the galaxies with the highest velocity in the observational catalog at the given locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' galaxies with the largest peculiar velocities are identified by their PGC (Principal Galaxy Catalog) number at the top of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Here again, given the distance of these clusters and the sparsity and limit of our constraint-catalog, the agreement is quite good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Tables 1 and 2 confirm again the visual impression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' They also highlight again the limitations of both metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Both values must be given together to conclude on how much the observed and simulated lines-of-sight match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Note that we identify other simulated velocity waves corresponding to local clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' in the Perseus-Pisces region) but observational data is not of suf- ficient quality or absent in the infall region for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Nonethe- less, all the halos and associated waves are used for the next section studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The mass range is actually extended down to 2 1014M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 4 WAVE PROPERTIES VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' CLUSTER MASSES 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='1 The amplitude The wave amplitude is the first obvious property to check against halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Indeed, the deeper the gravitational potential well, the faster should galaxies fall onto it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The amplitude is thus defined as the dif- ference between the maximum and minimum peculiar velocities of galaxies falling onto the cluster either from the front or from behind with respect to the synthetic observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 6 thus shows the amplitude of the simulated velocity waves as a function of the dark matter halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Each black and red filled circle corresponds to a halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Red ones stand for clusters identified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' While it is immediate to notice that there is a clear correlation between the wave amplitude and the halo mass, one can also point out that the amplitude is extremely difficult to measure in observational data and that there is a residual scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Indeed, measuring the amplitude in observational data implies getting exquisite distance (peculiar velocity) estimates of galaxies ex- actly in the line-of-sight of the cluster with respect to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' It supposes first that there are actually galaxies exactly aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Then, identifying these galaxies and measuring their distances with great accuracy, while they fall onto the cluster from the front is already quite a challenge, let alone when they fall from behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' In any case, the residual scatter suggests that the amplitude, be it measurable, alone cannot be used as a precise proxy for cluster mass estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Part of this scatter is probably due to the fact the galaxies are not perfectly aligned with us and the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The gravitational potential well shape might also be responsible for another part of this scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' To a lesser extent, the large-scale structure environment might also play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='2 The height While the wave height is not expected to be a better proxy than the wave amplitude, it is interesting to check whether there still is a tight Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Amplitude of the simulated velocity waves as a function of the dark matter halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Halos shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3 to 5 are identified in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' enough correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Indeed, while it is challenging to have precise dis- tance measurements for both galaxies falling from the front and from behind a cluster in the line-of-sight with respect to us, it might be fea- sible especially for galaxies falling from the front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The height is thus defined as the maximum (minimum) peculiar velocities of galaxies falling onto the cluster from the front (behind) with respect to the syn- thetic observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 7, each black and red (blue and orange) filled circles stand for the height of a dark matter halo positive-(negative- )half wave as a function of its mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Red and orange are used for dark matter halos from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' A similar correlation as with the amplitude is found although with a somewhat larger scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Interestingly it also shows that velocity waves are not symmetric: their maximum differs from their minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Both the potential well shape and the non-perfect alignement observer- galaxy-halo or observer-halo-galaxy might be the reason for this asym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Nonetheless because there still is a correlation and because in observational data it is easier to get accurate datapoints at the wave front than in its wake, it is legitimate to focus on the positive-half ve- locity wave shape to study more thoroughly the relation with the halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' © 2022 RAS, MNRAS 000, 1–10 8 Sorce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Dark (blue) filled circles are heights of the simulated positive(negative)-half velocity waves as a function of the dark matter halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Halos shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3 to 5 are identified in red (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='3 Height, width and continuum After deriving the positive-half wave envelope of every dark matter halo, we choose to fit the simplest model possible, a Gaussian-plus- continuum model, to each one of them as follows: vpec = Afit × e −(d−d0)2 2σ2 fit + C fit (5) where Afit, σ fit and C fit are respectively the Gaussian amplitude, its standard deviation and a continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' d0 depends on the halo distance and has no other purpose than centering the Gaussian on zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Its sole physical meaning is to be the actual distance of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The amplitude is related to the positive-half wave envelope height while the standard deviation is linked to its width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Finally, the continuum gives the positive-half wave offset from a zero average velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' For visualization, envelopes and their fits for halos presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3 to 5 are shown as solid and dashed lines on these same figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 8 gathers the three parameters of the fits and halo masses for a concomitant study to highlight an eventual multi-parameter correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The Gaussian amplitude is represented as a function of the Gaussian standard deviation while the color scale stands for the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' From black-violet to red, the continuum decreases from positive values to negative ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The model uncertainty is shown as error bars for the amplitude and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The color scale smoothness includes the continuum uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The Gaussian-plus- continuum model choice proves to be robust given the tiny error bars that it results in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The filled circle sizes are proportional to the dark matter halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Finally, an additional small red filled circle is used to identify each halo analyzed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The previous subsection (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='2) showed that there is a correlation between the wave height and the halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' It is thus not surprising to find back that the more massive the halo is (larger circle), the larger 500 1000 1500 2000 Afit (km s-1) 2 4 6 8 10 12 14 σfit (Mpc) 574 407 239 71 96 264 -432 Cfit (km s-1) Virgo Centaurus Abell 569 Coma Abell 85 Abell 2256 PGC765572 PGC999654 PGC340526 PGC46604 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Parameters of the Gaussian-plus-continuum fit to the simulated positive-half velocity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' σfit stands for the Gaussian standard deviation, Afit for its amplitude and C fit for the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The filled circle sizes are pro- portional to dark matter halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Tiny error bars on the standard deviation and amplitude resulting from fitting the envelopes highlight the adequacy of the model choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Halos shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 3 to 5 are identified with red nametags and additional small red filled circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' the Gaussian amplitude is (larger value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' As stated above, the Gaussian amplitude is indeed the counterpart of the positive-half wave height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' In addition, there is a small correlation between the amplitude and standard deviation thus halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' More massive halos seem to give birth to wider waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The scatter is however quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' It cer- tainly depends greatly on the halo triaxiality and thus on its orientation with respect to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' A similar conclusion is valid for the continuum, the smaller the continuum but for extreme values is, the more massive the halo is on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Anyhow, the scatter is quite large in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' A strong dependence on the global environment of the dark matter halo in addition to the halo mass might be in cause here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Interestingly a general pattern emerges quite clearly though: the most massive halos (≳ 6 1014 M⊙) tend to give birth to positive- half waves that have a continuum compatible with zero or slightly negative/positive in addition to high amplitude and standard deviation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' the less massive halos ( 2 1014 M⊙≲M≲ 4 1014 M⊙) tend to give birth to positive-half waves that have a continuum compatible with zero or slightly negative/positive in addition to low amplitude and standard deviation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' intermediate mass halos (4 1014 M⊙≲M≲ 6 1014 M⊙) give rise to positive-half waves that have high continuum absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Such values permit distinguishing them from the most massive halos with which they share high amplitude and possibly standard deviation values, especially in the negative continuum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' It is highly probable that the global environment or cosmic web is responsible for such a finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' We will investigate this link in more details in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The halo segregation in different continuum value classes is an- other quite inspiring source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' There seems to be a different correlation for each continuum value class: Halos with fits resulting in a high (close to zero) continuum value © 2022 RAS, MNRAS 000, 1–10 Velocity waves 9 seems to have masses correlated with the Gaussian amplitudes but not so much with the Gaussian standard deviations that appear to have low values (present a large scatter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Halos with fits resulting in a very low continuum value have both amplitudes and standard deviations correlated together as well as with the masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Halos with fits resulting in either positive or negative intermediate continuum values present masses correlated with amplitudes and up to a certain point with standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Consequently, although to a lesser extent than for halos whose continuum values are quite low, amplitudes and standard deviations are slightly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' To summarize, since the fit parameters are interdependent, a global fit to the velocity wave seems the best approach to obtain clus- ter rough mass estimates rather than single and independent measure- ments of amplitude, height and width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Because different categories appear among halos, in future studies, a machine learning approach might become handy to actually get accurate enough mass estimates from sparse observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' In a first approach, the simple Gaussian-plus- continuum fit presented here could be used as a model reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' 5 CONCLUSIONS Galaxy clusters are excellent cosmological probes provided their mass estimates are accurately determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Fueled with large imaging surveys, stacked weak lensing is the most promising mass estimate method though it provides estimates within relatively small radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Given the large amount of accompanying redshift and spectroscopic data overlapping the imaging surveys, we must take the opportunity to calibrate also with a reasonable accuracy a method based on galaxy dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Two independent measures hold indeed better constraints on the cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Infall zones of galaxy clusters are proba- bly the less sensitive to baryonic physics, thus mostly shielded from challenging systematics, and probe large radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' These manifestations of a tug of war between gravity and dark energy provide a unique avenue to test modified gravity theories when comparing resulting mass estimates to those from stacked weak lensing measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Combined with stacked weak lensing results, they might even yield evidence that departure from General Relativity on cosmological scales is responsible for the expansion acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The accurate calibration of the relation between infall zones properties and cluster masses starts with careful comparisons between cosmological simulations and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' In this paper, we thus present our largest and highest resolution Constrained Local & Nesting Environment Simulation (CLONE) built so far to reproduce numerically our cosmic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' This simulation stems from initial conditions constrained by peculiar velocities of local galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' By introducing this cosmological dark matter CLONE of the local large-scale structure with a particle mass of ∼109M⊙ within a ∼738 Mpc box, we have sufficient resolution to study the effect of the gravitational potential of massive local halos onto the velocity of (sub)halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' We can also compare with that of their observational cluster counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Velocity waves stand out in radial peculiar velocity - distance to a box-centered synthetic observer diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The agreement between lines-of-sight including velocity waves, caused by the most massive dark matter halos of the CLONE and those born from their observa- tional local cluster counterparts, is visually good especially for the clusters the closest to us that are the best constrained (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Virgo, Centaurus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Secondary waves due to smaller groups in (quasi) the same line-of-sight as the most massive clusters stand out equally even though they are further into the non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Indeed, prior to full non-linear evolution to the z=0 state, assuming ΛCDM, CLONE initial conditions are constrained with solely the linear theory, a power spectrum and highly uncertain and sparse local peculiar velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The visual matching between the simulated and observed lines-of-sight is confirmed with 2D-Kolmogorov Smirnov (KS) statistic values and tests as well as with our own ζ-metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Contrary to the 2D-KS statistic, the ζ-metric takes into account the real distance of galaxies along the entire lines-of-sight (not only the studied fractions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The ζ-metric is however more sensitive to the fact that observational uncertainties are not taken into account in these metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The two metrics appear to be complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' They show that the closest clusters have the best reproduced lines-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The lines-of-sight of clusters at the edges of the constrained region and even slightly beyond are also reproduced by the simulation although to a smaller extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Additionally, a Gaussian-plus-continuum fit to the envelope of the positive-half of all the velocity waves born from dark matter halos more massive than 2 1014M⊙ in the simulation reveals both the variety and complexity of the potential wells as well as the correlation of the fit parameters with the halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Overall, the Gaussian amplitude is mostly linked to the halo mass, but for a few exceptions, with a residual scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Although the Gaussian standard deviation is not always correlated with the mass, it can be slightly correlated with the Gaussian amplitude thus with the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The continuum is certainly an interesting parameter to consider as it permits splitting the halos into different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Each continuum value seems to drive a given correlation between the Gaussian amplitude and the halo mass and, to a smaller extent, with the Gaussian standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' To summarize, parameter fits are completely interdependent, a global fit to the velocity wave is then the best approach to obtain a first rough cluster mass estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' First and foremost, this work confirms the potential of the velocity wave technique to get massive cluster mass estimates and test gravity and cosmological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Our CLONES, with the first shown reproduction of observed lines-of-sight including velocity waves, could in the near future provide the zero point of galaxy infall kinematic technique calibrations (Zu & Weinberg 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' A bayesian inference model or/and a machine learning technique built and trained on random simulated galaxy surveys that is then applied to both constrained simulated and observed galaxy surveys must recover the same local velocity waves and corresponding mass estimates to be validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Our CLONES will moreover allow minimizing obser- vational biases as any real environmental and cluster property will be reproduced for perfect one-to-one comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Local kinematic mass estimates can then become accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' Once compared with other techniques of local galaxy cluster mass estimates, they will permit calibrating the zero-point of these other techniques to be applied to further-and-further away clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' DATA AVAILABILITY Synthetic catalogs are available upon reasonable request to the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors acknowledge the Gauss Centre for Supercomputing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='gauss-centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='eu) and GENCI (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='genci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='fr/) for funding this project by providing computing time on the GCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='lrz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='de) and Joliot-Curie at TGCC (http://www-hpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='cea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='fr), grants ID: 22307/22736 © 2022 RAS, MNRAS 000, 1–10 10 Sorce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' and A0080411510 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' This work was supported by the grant agreements ANR-21-CE31-0019 / 490702358 from the French Agence Nationale de la Recherche / DFG for the LOCALIZATION project and ERC-2015-AdG 695561 from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program for the ByoPiC project (https://byopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content='eu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' KD acknowledges support by the COMPLEX project from the ERC under the European Union’s Horizon 2020 research and innovation program grant agree- ment ERC-2019-AdG 882679.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfWvwu/content/2301.01305v1.pdf'} +page_content=' The authors thank the referee for their comments.' metadata={'source': 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mode 100644 index 0000000000000000000000000000000000000000..7e7eca3fb0c7cd86ea825c7741c73c61a3c15c52 --- /dev/null +++ b/PdAyT4oBgHgl3EQf7fpb/content/tmp_files/2301.00839v1.pdf.txt @@ -0,0 +1,5967 @@ +arXiv:2301.00839v1 [math-ph] 2 Jan 2023 +Integrable and superintegrable 3d Newtonian potentials using +quadratic first integrals: A review +Antonios Mitsopoulos1,a) and Michael Tsamparlis2,3,b) +1Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics, +University of Athens, Panepistemiopolis, Athens 157 83, Greece +2NITheCS, National Institute for Theoretical and Computational Sciences, +Pietermaritzburg 3201, KwaZulu-Natal, South Africa +3TCCMMP, Theoretical and Computational Condensed Matter and Materials Physics Group, +School of Chemistry and Physics, University of KwaZulu-Natal, +Pietermaritzburg 3201, KwaZulu-Natal, South Africa +a)Author to whom correspondence should be addressed: antmits@phys.uoa.gr +b)Email: mtsampa@phys.uoa.gr +Abstract +The determination of the first integrals (FIs) of a dynamical system and the subsequent assessment of +their integrability or superintegrability in a systematic way is still an open subject. One method which has +been developed along these lines for holonomic autonomous dynamical systems with dynamical equations +¨qa = −Γa +bc(q) ˙qb ˙qc − Qa(q), where Γa +bc(q) are the coefficients of the Riemannian connection defined by the +kinetic metric of the system and −Qa(q) are the generalized forces, is the so-called direct method. According +to this method, one assumes a general functional form for the FI I and requires the condition dI +dt = 0 along +the dynamical equations. This results to a system of partial differential equations (PDEs) to which one +adds the necessary integrability conditions of the involved scalar quantities. It is found that the final system +of PDEs breaks into two sets: a. One set containing geometric elements only and b. A second set with +geometric and dynamical quantities. Then, provided the geometric quantities are known or can be found, +one uses the second set to compute the FIs and, accordingly, assess on the integrability of the dynamical +system. The ‘solution’ of the system of PDEs for quadratic FIs (QFIs) has been given in a recent paper (M. +Tsamparlis and A. Mitsopoulos, J. Math. Phys. 61, 122701 (2020) ). In the present work, we consider the +application of this ‘solution’ to Newtonian autonomous conservative dynamical systems with three degrees of +freedom, and compute integrable and superintegrable potentials V (x, y, z) whose integrability is determined +via autonomous and/or time-dependent QFIs. The geometric elements of these systems are the ones of the +Euclidean space E3 which are known. Setting various values for the parameters determining the geometric +elements, we determine in a systematic way all known integrable and superintegrable potentials in E3 +together with new ones obtained in this work. +For easy reference, the results are collected in tables so +that the present work may act as an updated review of the QFIs of Newtonian autonomous conservative +dynamical systems with three degrees of freedom. It is emphasized that by assuming different values for the +parameters, other authors may find more integrable potentials of this type of systems. +Keywords: Integrable potentials; superintegrable potentials; 3d Newtonian potentials; quadratic first inte- +grals; time-dependent first integrals; autonomous conservative dynamical systems; Killing tensors. +1 +Introduction +According to Liouville integrability theorem [1], a three-dimensional (3d) Newtonian autonomous conservative +system is (Liouville) integrable if it admits three (functionally) independent first integrals (FIs) in involution. +1 + +Integrable systems that admit five independent FIs are called maximally superintegrable, while if they admit +four independent FIs they are called minimally superintegrable. A superintegrable potential is always integrable; +however, some authors [2, 3, 4, 5] define superintegrability without the requirement of integrability, that is, they +look only for sets of independent FIs whose number exceeds the degrees of freedom of the system. +For 3d Newtonian autonomous conservative systems one quadratic FI (QFI) is the Hamiltonian H; therefore, +one needs two additional independent autonomous1 FIs in involution in order to establish integrability. If in +addition to these FIs there exist one/two more independent autonomous or time-dependent FIs, then the +system is minimally/maximally superintegrable. Besides establishing superintegrability, time-dependent FIs +can be used also to establish the integrability of a dynamical system provided they are in involution (see e.g. +[6, 7]). +The maximum number of independent autonomous FIs of a Hamiltonian dynamical system of n degrees of +freedom is 2n − 1. However, if time-dependent FIs are considered, this maximum limit can be exceeded. For +example, the 3d potential V = −kr2, where r = +� +x2 + y2 + z2 and k is an arbitrary constant, admits the six +(five are enough) time-dependent linear FIs (LFIs) I3a±, a = 1, 2, 3 (see Table V in [8]): +k > 0 +: +I3a± = e± +√ +2kt � +˙qa ∓ +√ +2kqa +� +k < 0 +: +I3a± = e±i√−2kt � +˙qa ∓ i +√ +−2kqa +� +which are functionally independent. Since the three LFIs I3a+ (or I3a−) are also in involution, the considered +3d potential is superinetgarble. +Concerning the number of the free parameters that define a 3d superintegrable potential, the following +terminology is used (see e.g. [5]): +a. The degenerate (or three-parameter) potentials, and +b. The non-degenerate (or four-parameter) potentials. +In many works [3, 4, 5, 9], the term second order superintegrable potentials is used for potentials that are +superintegrable due to QFIs only. Such potentials have the following special properties [3, 4]: +1) Multi-integrability. They are integrable in multiple ways and the comparison of ways of integration leads to +new facts about the system. +2) They are multi-separable. +3) The second order symmetries expressed by second order Killing tensors (KTs) generate a closed quadratic +algebra. In the quantum case, the representation of this algebra yields results concerning the spectral resolution +of the Schr¨odinger operator and the other symmetry operators. +There are two types of integrable potentials in E3. The decomposable potentials (or 2+1 separable integrable +potentials) generated from integrable potentials in E2 and the non-decomposable ones. +Let V (x, y) be a 2d integrable potential in E2 which admits an additional autonomous FI I1. Then, the +3d Newtonian z-separable potential ¯V (x, y, z) = V (x, y) + F(z), where F is an arbitrary smooth function of +z, is a 2 + 1 separable integrable potential in E3. The integrability of these potentials is due to the three +independent FIs H, I1 and I2 = 1 +2 ˙z2 + F(z) which are in involution. If V (x, y) is superintegrable with respect +to (wrt) two additional FIs, say J1 and J2, then ¯V (x, y, z) is minimally superintegrable because of the four +independent FIs H, J1, J2, and I2. If in addition to J1 and J2 the 2d superintegrable potential V (x, y) admits +also a time-dependent FI J3, then ¯V (x, y, z) is maximally superintegrable. For example, the second potential +of Table II in [2] is not minimally superintegrable but maximally superintegrable because it admits in addition +the time-dependent FIs I73a and I73b from the last Table of [10]. +The non-decomposable (i.e. non-separable) 3d Newtonian integrable potentials V (x, y, z) cannot be written +in the form ¯V (x, y, z) = V (x, y) + F(z) where V (x, y) is a 2d Newtonian integrable potential. In general, their +determination is more difficult and various methods of escalating complexity have been proposed. Furthermore, +the existing results concern autonomous FIs only and are limited in number. The purpose of the present work is +to provide a systematic (i.e. algorithmic) method which enables one to determine integrable and superintegrable +potentials in E3 using autonomous and time-dependent QFIs. The method relies on Theorem 1 [11] (see section +3) which relates the QFIs of the dynamical system with the dynamical elements (i.e. the potential) and the +geometry defined by the kinetic energy of the system. The structure of the paper is as follows. +1These additional FIs must be autonomous because the Poisson bracket (PB) of the Hamiltonian with an arbitrary time- +dependent FI J(t, q, ˙q) does not vanish. Indeed, we have {H, J} = ∂J +∂t ̸= 0. +2 + +In section 2, we determine the 3d integrable/superintegrable 2+1 decomposable potentials directly from the +well-known 2d integrable/superintegrable potentials listed in the reference works [10] and [12]. The results are +presented in tables where the known potentials with the corresponding reference are listed together with the +new ones determined in this work. In section 3, we state Theorem 1 from which follows that there are three +types of QFIs to consider, denoted as I(1,ℓ), I(2,ℓ), I(3), which are expressed in terms of the geometric elements +of the kinetic metric and the potential function. In section 4, we state the geometric quantities of E3 which are +required for the application of Theorem 1. It is seen that the number of parameters introduced from the KT +components is large. This remark and the fact that the associated system of PDEs is overdetermined have the +result that one will find special solutions only by assuming particular values of the geometric parameters. In +section 5, we consider the QFI I(1,1) (ℓ = 1) and the relevant PDEs for this case. We consider various values +for the parameters and recover all the existing results together with new ones. For easy reference, the various +potentials are grouped in Tables 4 - 7. In section 6, we consider the potentials admitting QFIs of the type I(2,0) +(ℓ = 0). These results are presented in Tables 8 - 10. In section 7, we consider time-dependent LFIs/QFIs of +the type I(3) and the results are collected in Tables 11 - 13. In section 8, we compare and discuss the results +listed in the tables with the existing results of the literature. Finally, in section 9, we draw our conclusions. +List of abbreviations and notations/conventions +For the convenience of the reader, we give a list of abbreviations and notations used throughout the text. +Abbreviations: +• FI = first integral +• HV = homothetic vector +• KT = Killing tensor +• KV = Killing vector +• LFI = linear first integral +• Nd = N-dimensional +• ODE = ordinary differential equation +• PB = Poisson bracket +• PDE = partial differential equation +• QFI = quadratic first integral +Mathematical notations/conventions: +• En = n-dimensional Euclidean space +• r = +� +x2 + y2 + z2, R = +� +x2 + y2, tan θ = y +x, and w = x + iy = Reiθ. +• The angular momentum M ≡ Mi = (M1, M2, M3) = (y ˙z − z ˙y, z ˙x − x ˙z, x ˙y − y ˙x) with square magnitude +M2 = M 2 +1 + M 2 +2 + M 2 +3 . +• The kinetic metric γab(q) of the dynamical system is used for lowering and raising the indices. +• A comma indicates partial derivative and a semicolon Riemannian covariant derivative. +Coordinate systems of E3: +• Cartesian coordinates: (x, y, z). +• Spherical coordinates: (r, θ, φ) with x = r sin θ cos φ, y = r sin θ sin φ and z = r cos θ. +• Parabolic cylindrical coordinates: (λ′, µ′, z) with λ′ = R + y and µ′ = R − y. +• Rotational parabolic coordinates: (ζ, η, φ) with ζ = r + z, η = r − z, φ = tan−1 � y +x +� +or, equivalently, +x = √ζη cos φ, y = √ζη sin φ, z = 1 +2 (ζ − η). +3 + +2 +Integrable/superintegrable 2+1 separable potentials +As it has been remarked, the 2+1 separable integrable/superintegrable potentials in E3 are given in terms of the +integrable/superintegrable potentials Φ(x, y) in E2. From the latter potentials, the ones that admit LFIs/QFIs +are collected in the review papers [10] and [12]. Using these results, the 2 + 1 separable potentials in E3 +V (x, y, z) = Φ(x, y) + F(z) +(1) +where F(z) is an arbitrary smooth function, are integrable/superintegrable due to the additional QFI I = +1 +2 ˙z2 + F(z) which is in involution with the FIs of Φ(x, y). +Applying the above procedure to the results of [10, 12], we find the integrable and superintegrable potentials +in E3 listed in Tables 1 - 3. The QFI of the Hamiltonian H is not included in the tables. In Tables 2 and +3, we compare with the results of [2]. A similar comparison cannot be done in Table 1 because in [2] only +superintegrable potentials are considered. Concerning the notation, we set r = +� +x2 + y2 + z2, R = +� +x2 + y2 +and the angular momentum Mi = (y ˙z − z ˙y, z ˙x − x ˙z, x ˙y − y ˙x). +4 + +Integrable 2 + 1 separable potentials +Potential +LFIs and QFIs +V = F1 +� +R2 +2 + b1y − b2x +� ++ F2(z) +I1 = M3 − b1 ˙x − b2 ˙y, I2 = 1 +2 ˙z2 + F2(z) +V = +F1( y +x) +R2 ++ F2(R) + F3(z) +I1 = M 2 +3 + 2F1 +� y +x +� +, I2 = 1 +2 ˙z2 + F3(z) +V = +k +x2+ℓy2 + F1(R) + F2(z) +I1 = M 2 +3 + 2k(1−ℓ)y2 +x2+ℓy2 , I2 = 1 +2 ˙z2 + F2(z) +V = F1(u)−F2(v) +u2−v2 ++ F3(z) +u2 = R2 + A + +� +(R2 + A)2 − 4Ax2�1/2 and +v2 = R2 + A − +� +(R2 + A)2 − 4Ax2�1/2 +I1 = M 2 +3 + A ˙x2 + v2F1(u)−u2F2(v) +u2−v2 +I2 = 1 +2 ˙z2 + F3(z) +V = F1(u)−F2(v) +u2−v2 ++ F3(z) +u2 = R2 + +� +R4 − 4A(x ± iy)2�1/2 and +v2 = R2 − +� +R4 − 4A(x ± iy)2�1/2 +I1 = M 2 +3 + A( ˙x ± i ˙y)2 + v2F1(u)−u2F2(v) +u2−v2 +I2 = 1 +2 ˙z2 + F3(z) +V = F1(R+y)+F2(R−y) +R ++ F3(z) +I1 = −M3 ˙x + (R+y)F2(R−y)−(R−y)F1(R+y) +R +I2 = 1 +2 ˙z2 + F3(z) +V = ¯w−1/2 � +F1(w + √ ¯w) + F2(w − √ ¯w) +� ++ F3(z) +w = x + iy and ¯w = x − iy +I1 = −M3( ˙x + i ˙y) + i +8( ˙x − i ˙y)2+ ++i +� +1 − +w +√ ¯ +w +� +F1(w + √ ¯w)+ ++i +� +−1 − +w +√ ¯w +� +F2(w − √ ¯w) +I2 = 1 +2 ˙z2 + F3(z) +V = F1(w) +r ++ F ′ +2(w) + F3(z) +F ′ +2 = dF2 +dw and w = x ± iy +I1 = −M3( ˙x ± i ˙y) − iwV + iF2(w) +I2 = 1 +2 ˙z2 + F3(z) +V = F1(x) + F2(y) + F3(z) +I1 = 1 +2 ˙x2 + F1, I2 = 1 +2 ˙y2 + F2, I3 = 1 +2 ˙z2 + F3 +V = F1 +� +y + b0x + +� +b2 +0 + 1x +� ++ ++F2 +� +y + b0x − +� +b2 +0 + 1x +� ++ F3(z) +where b0 ≡ A−B +2C +I1 = A ˙x2 + B ˙y2 + 2C ˙x ˙y + (A + B)(F1 + F2)+ ++2C +� +b2 +0 + 1(F1 − F2) +I2 = 1 +2 ˙z2 + F3(z) +V (b0 = 0) = F1(y + x) + F2(y − x) + F3(z) +I1 = ˙x ˙y + F1 − F2, I2 = 1 +2 ˙z2 + F3(z) +Table 1: Integrable potentials V (x, y, z) = Φ(x, y) + F(z) in E3, +where Φ(x, y) are integrable potentials in E2. +5 + +Minimally superintegrable 2 + 1 separable potentials +Potential +Ref [2] +LFIs and QFIs +V = cx + F1(y − bx) + F2(z) +c ̸= 0, d2F1 +dw2 ̸= 0 and w ≡ y − bx +New +I1 = ˙x + b ˙y + ct +I2 = ( ˙x + b ˙y)2 + 2c(x + by) +I3 = 1 +2 ˙z2 + F2(z) +V = F1(y − bx) + F2(z) +d2F1 +dw2 ̸= 0 and w ≡ y − bx +New +I1 = ˙x + b ˙y +I2 = t( ˙x + b ˙y) − (x + by) +I3 = 1 +2 ˙z2 + F2(z) +V = k1 +2 (x2 + 4y2) + k2 +x2 + k3y + F(z) +Table II +k3 = 0 +x ↔ y +I1 = M3 ˙x + k1yx2 − 2k2y +x2 + k3 +2 x2 +I2 = 1 +2 ˙x2 + k1 +2 x2 + k2 +x2 +I3 = 1 +2 ˙y2 + 2k1y2 + k3y +I4 = 1 +2 ˙z2 + F(z) +V = k1 +x2 + k2 +R + k3y +Rx2 + F(z) +Table II +x ↔ y +I1 = M 2 +3 + 2k1 +y2 +x2 + 2k3 +Ry +x2 +I2 = M3 ˙x − 2k1 +y +x2 − k2 +y +R − k3 +x2+2y2 +Rx2 +I3 = 1 +2 ˙z2 + F(z) +V = k1 +R + k2 +√R+y +R ++ k3 +√R−y +R ++ F(z) +Table II +I1 = M3 ˙x − k1y +R − k3(R+y)√R−y−k2(R−y)√R+y +R +I2 = M3 ˙y + G(x, y) +I3 = 1 +2 ˙z2 + F(z) +G,x = −yV,y and G,y = 2xV,y − yV,x +V = F1(x) + +k +(y+c)2 + F2(z) +New +I1 = 1 +2 ˙x2 + F1 +I2 = 1 +2 ˙y2 + +k +(y+c)2 +I3 = 1 +2 ˙z2 + F2 +I4 = − t2 +2 ˙y2 + t(y + c) ˙y − t2 +k +(y+c)2 − 1 +2y2 − cy +V = λ +2 R2 + b1y − b2x + F(z) +λ ̸= 0 +New +I1 = λM3 − b1 ˙x − b2 ˙y +I2 = 1 +2 ˙x2 + 1 +2λx2 − b2x +I3 = 1 +2 ˙y2 + 1 +2λy2 + b1y +I4 = ˙x ˙y + λxy + b1x − b2y +I5 = 1 +2 ˙z2 + F(z) +Table 2: +Minimally superintegrable potentials V (x, y, z) += +Φ(x, y) + F(z) in E3, where Φ(x, y) are superintegrable potentials +in E2. +6 + +Maximally superintegrable 2 + 1 separable potentials +Potential +Ref [2] +LFIs and QFIs +V = cx + λy + F(z) +New +I1 = ˙x + ct, I2 = ˙y + λt, I3 = 1 +2 ˙x2 + cx, +I4 = 1 +2 ˙y2 + λy, I5 = 1 +2 ˙z2 + F(z) +V = cx − 1 +2λ2y2 + F(z) +λ ̸= 0 +New +I1 = ˙x + ct, I2 = eλt( ˙y − λy), I3 = 1 +2 ˙x2 + cx, +I4 = 1 +2 ˙y2 − 1 +2λ2y2, I5 = 1 +2 ˙z2 + F(z) +V = − k2 +2 R2 + F(z) +k ̸= 0 +New +I1 = M3, I2 = 1 +2 ˙x2 − 1 +2k2x2, I3 = 1 +2 ˙y2 − 1 +2k2y2, +I4 = ˙x ˙y − k2xy, I5 = 1 +2 ˙z2 + F(z), +I6± = e±kt( ˙x ∓ kx), I7± = e±kt( ˙y ∓ ky), +4I2I3 = I2 +4 − k2M 2 +3 +V = k +2R2 + +b +x2 + +c +y2 + F(z) +Table II +I1 = M 2 +3 + 2b y2 +x2 + 2c x2 +y2 , I2 = 1 +2 ˙z2 + F(z) +I3 = 1 +2 ˙x2 + k +2x2 + +b +x2 , I4 = 1 +2 ˙y2 + k +2y2 + +c +y2 +- For k = 0: +I5 = − t2 +2 ˙y2 + ty ˙y − t2 c +y2 − 1 +2y2 +I6 = − t2 +2 ˙x2 + tx ˙x − t2 b +x2 − 1 +2x2 +- For k = − λ2 +4 ̸= 0: +I5 = eλt � +− ˙x2 + λx ˙x − λ2 +4 x2 − 2b +x2 +� +I6 = eλt � +− ˙y2 + λy ˙y − λ2 +4 y2 − 2c +y2 +� +V = +k1 +(x+c1)2 + +k2 +(y+c2)2 + F(z) +New +I1 = 1 +2 ˙x2 + +k1 +(x+c1)2 +I2 = 1 +2 ˙y2 + +k2 +(y+c2)2 +I3 = 1 +2 ˙z2 + F(z) +I4 = − t2 +2 ˙y2 + t(y + c2) ˙y − t2 +k2 +(y+c2)2 − 1 +2y2 − c2y +I5 = − t2 +2 ˙x2 + t(x + c1) ˙x − t2 +k1 +(x+c1)2 − 1 +2x2 − c1x +V = − λ2 +8 R2 − λ2 +4 (c1x + c2y) − +− +k1 +(x+c1)2 − +k2 +(y+c2)2 + F(z) +λ ̸= 0 +New +I1 = 1 +2 ˙x2 − λ2 +8 x2 − c1λ2 +4 x − +k1 +(x+c1)2 +I2 = 1 +2 ˙y2 − λ2 +8 y2 − c2λ2 +4 y − +k2 +(y+c2)2 +I3 = 1 +2 ˙z2 + F(z) +I4 = eλt � +− ˙x2 + λ(x + c1) ˙x − λ2 +4 (x + c1)2 + +2k1 +(x+c1)2 +� +I5 = eλt � +− ˙y2 + λ(y + c2) ˙y − λ2 +4 (y + c2)2 + +2k2 +(y+c2)2 +� +Table 3: +Maximally superintegrable potentials V (x, y, z) += +Φ(x, y) + F(z) in E3, where Φ(x, y) are superintegrable potentials +in E2. +Note 1: The results indicated as ‘New’ in Tables 2 and 3 do not appear in [2] where only autonomous QFIs +are considered. +Note 2: In Table II of [2], the potential (see Table 3) +V = k +2 R2 + b +x2 + c +y2 + F(z) +(2) +where k, b, c are arbitrary constants and F(z) is an arbitrary smooth function, is said to be minimally superin- +tegrable because of the four independent autonomous QFIs: +I1 = M 2 +3 + 2by2 +x2 + 2cx2 +y2 , I2 = 1 +2 ˙z2 + F(z), I3 = 1 +2 ˙x2 + k +2 x2 + b +x2 , I4 = 1 +2 ˙y2 + k +2 y2 + c +y2 . +However, using in addition the time-dependent QFIs: +For k = 0: +I5 = −t2 +2 ˙y2 + ty ˙y − t2 c +y2 − 1 +2y2, I6 = −t2 +2 ˙x2 + tx ˙x − t2 b +x2 − 1 +2x2 +7 + +and +For k = −λ2 +4 ̸= 0: +I5 = eλt +� +− ˙x2 + λx ˙x − λ2 +4 x2 − 2b +x2 +� +, I6 = eλt +� +− ˙y2 + λy ˙y − λ2 +4 y2 − 2c +y2 +� +it is seen that the potential (2) for these values of k is maximally superintegrable. +Moreover, if we assume the canonical transformation x → x+c1 and y → y+c2 where c1 and c2 are arbitrary +constants, it is shown that the potential (2) is transformed canonically into the last two potentials of Table 3. +Indeed, for k = 0, b = k1 and c = k2, we get the potential +V = +k1 +(x + c1)2 + +k2 +(y + c2)2 + F(z) +while for k = − λ2 +4 , b = −k1 and c = −k2, we get the potential +V = −λ2 +8 R2 − λ2 +4 (c1x + c2y) − +k1 +(x + c1)2 − +k2 +(y + c2)2 − λ2 +8 (c2 +1 + c2 +2) + F(z). +The constant term − λ2 +8 (c2 +1 + c2 +2) is overlooked because it does not contribute to the dynamical equations. +Note 3: From Table 2, we observe that the minimally superintegrable potential +V = k1 +R + k2 +√R + y +R ++ k3 +√R − y +R ++ F(z) +(3) +where k1, k2, k3 are arbitrary constants and F(z) is an arbitrary smooth function, admits the two autonomous +QFIs: +I1 += +M3 ˙x − k1y +R + k2(R − y)√R + y +R +− k3(R + y)√R − y +R +(4) +I2 += +M3 ˙y + G(x, y). +(5) +The function G(x, y) must satisfy the system of PDEs: +G,x + yV,y += +0 +(6) +G,y + yV,x − 2xV,y += +0. +(7) +Using the parabolic cylindrical coordinates (λ′, µ′, z) (see eqs. (3.19) and (3.51) in [2]) with λ′ = R + y and +µ′ = R − y, the QFI (4) becomes2 +I1 = M3 ˙x − +2 +λ′ + µ′ +�k1 +2 (λ′ − µ′) − k2µ′√ +λ′ + k3λ′� +µ′ +� +. +(8) +The QFI I2 in eq. (3.57) of [2] is not correct and should be replaced by the QFI (8). +In the parabolic cylindrical coordinates (u, v, z) with u = R + x, v = R − x and3 x, y > 0, the system of +PDEs (6) - (7) becomes G,v = uV,v and G,u = −vV,u. The solution of this system is +G(u, v) = +2 +u + v +�k1 +2 (u − v) − (k2 + k3)v +�u +2 + (k2 − k3)u +�v +2 +� +or, equivalently, in Cartesian coordinates +G(x, y) = 1 +R +� +k1x − (k2 + k3)(R − x) +� +R + x +2 ++ (k2 − k3)(R + x) +� +R − x +2 +� +. +Then, the QFI (5) is +I2 = M3 ˙y + +2 +u + v +�k1 +2 (u − v) − (k2 + k3)v +�u +2 + (k2 − k3)u +�v +2 +� +. +(9) +2We recall that the coordinates λ′, µ′ are either positive or zero because λ′ + µ′ = 2R, λ′ − µ′ = 2y, and λ′µ′ = x2. +3For x, y > 0 we have: √R + x + √R − x = +√ +2√R + y and √R + x − √R − x = +√ +2√R − y. +8 + +There is a misprint in the QFI I3 of eq. (3.57) in [2]; the correct answer is the QFI (9). +Note 4: The two superintegrable potentials given in eq. (17) of [4] are subcases of the potential (see Table +2) +V = λ +2 R2 + b1y − b2x + F(z) +(10) +for F(z) = λ +2 z2 + b3z and F(z) = λ +8 z2 + b3 +z2 , where b3 is an arbitrary constant. +Note 5: The potential (see Table 2) +V1 = cx + F1(y − bx) + F2(z) +(11) +where c is an arbitrary non-zero constant, w ≡ y − bx and d2F1 +dw2 ̸= 0, admits the following LFIs/QFIs (apart +from the Hamiltonian H): +I1 = ˙x + b ˙y + ct, I2 = t( ˙x + b ˙y) − (x + by) + c +2t2, I3 = ( ˙x + b ˙y)2 + 2c(x + by), I4 = 1 +2 ˙z2 + F2(z). +We compute the PBs: +{H, I1} = c, {H, I2} = I1, {I1, I2} = 1 + b2, {I1, I3} = −2c(1 + b2), {I2, I3} = −2(1 + b2)I1. +The three FIs H, I3, I4 are (functionally) independent and in involution; therefore, the potential (11) is in- +tegrable. The five FIs H, I1, I2, I3, I4 are not independent because I2 +1 = I3 + 2cI2. +However, the four FIs +H, I3, I4, I1, or the H, I3, I4, I2, are independent and, therefore, the potential (11) is minimally superintegrable. +3 +The Theorem for QFIs +In order to compute in a systematic way the QFIs of non-decomposable potentials, we need to recall a theorem +which is proved in [11]. +Theorem 1 The independent QFIs of the n-dimensional autonomous holonomic dynamical system +¨qa = −Γa +bc(q) ˙qb ˙qc − Qa(q) +(12) +where qa are the coordinates of the configuration space, ˙qa = dqa +dt , t is the time variable, Γa +bc(q) are the Rieman- +nian connection coefficients of the kinetic metric γab(q) defined by the kinetic energy of the system and −Qa(q) +are the generalized forces, are the following: +Integral 1. +I(1,ℓ) += +� +−t2ℓ +2ℓ L(2ℓ−1)(a;b) − ... − t4 +4 L(3)(a;b) − t2 +2 L(1)(a;b) + Cab +� +˙qa ˙qb + t2ℓ−1L(2ℓ−1)a ˙qa + ... + t3L(3)a ˙qa + ++tL(1)a ˙qa + t2ℓ +2ℓ L(2ℓ−1)aQa + ... + t4 +4 L(3)aQa + t2 +2 L(1)aQa + G(q) +where4 Cab(q) and L(M)(a;b)(q) for M = 1, 3, ..., 2ℓ−1 are KTs, +� +L(2ℓ−1)bQb� +,a = −2L(2ℓ−1)(a;b)Qb, +� +L(k−1)bQb� +,a = +−2L(k−1)(a;b)Qb − k(k + 1)L(k+1)a for k = 2, 4, ..., 2ℓ − 2, and G,a = 2CabQb − L(1)a. +Integral 2. +I(2,ℓ) += +� +− t2ℓ+1 +2ℓ + 1L(2ℓ)(a;b) − ... − t3 +3 L(2)(a;b) − tL(0)(a;b) +� +˙qa ˙qb + t2ℓL(2ℓ)a ˙qa + ... + t2L(2)a ˙qa + ++L(0)a ˙qa + t2ℓ+1 +2ℓ + 1L(2ℓ)aQa + ... + t3 +3 L(2)aQa + tL(0)aQa +4We note that for ℓ = 0 the conditions for the QFI I(1,0) are given by nullifying all the vectors L(M)a. +9 + +where LM(a;b)(q) for M = 0, 2, ..., 2ℓ are KTs, +� +L(2ℓ)bQb� +,a = −2L(2ℓ)(a;b)Qb, and +� +L(k−1)bQb� +,a = −2L(k−1)(a;b)Qb − k(k + 1)L(k+1)a for k = 1, 3, ..., 2ℓ − 1. +Integral 3. +I(3) = eλt � +−L(a;b) ˙qa ˙qb + λLa ˙qa + LaQa� +where the vector La(q) is such that L(a;b) is a KT and +� +LbQb� +,a = −2L(a;b)Qb − λ2La. +Notation: The Einstein summation convention is used, round (square) brackets indicate symmetrization +(antisymmetrization) of the enclosed indices, indices enclosed between vertical lines are overlooked by anti- +symmetrization or symmetrization symbols, a comma indicates partial derivative and a semicolon Riemannian +covariant derivative. +Before we proceed, we recall the geometric quantities of the Euclidean space E3 required by Theorem 1. +4 +The geometric quantities of E3 +- E3 admits three gradient Killing vectors (KVs) ∂x, ∂y, ∂z whose generating functions are x, y, z, respectively, +and three non-gradient KVs y∂x − x∂y, z∂y − y∂z, z∂x − x∂z. These vectors are written collectively as +La = + + +b1 − b4y + b5z +b2 + b4x − b6z +b3 − b5x + b6y + + +(13) +where b1, b2, ..., b6 are arbitrary constants. +- The general second order KT in E3 has independent components: +C11 += +a6 +2 y2 + a1 +2 z2 + a4yz + a5y + a2z + a3 +C12 += +a10 +2 z2 − a6 +2 xy − a4 +2 xz − a14 +2 yz − a5 +2 x − a15 +2 y + a16z + a17 +C13 += +a14 +2 y2 − a4 +2 xy − a1 +2 xz − a10 +2 yz − a2 +2 x + a18y − a11 +2 z + a19 +(14) +C22 += +a6 +2 x2 + a7 +2 z2 + a14xz + a15x + a12z + a13 +C23 += +a4 +2 x2 − a14 +2 xy − a10 +2 xz − a7 +2 yz − (a16 + a18)x − a12 +2 y − a8 +2 z + a20 +C33 += +a1 +2 x2 + a7 +2 y2 + a10xy + a11x + a8y + a9 +where aK with K = 1, 2, ..., 20 are arbitrary constants. +- The vector La generating the reducible KT Cab = L(a;b) is +La = + + +−a15y2 − a11z2 + a5xy + a2xz + 2(a16 + a18)yz + a3x + 2a4y + 2a1z + a6 +−a5x2 − a8z2 + a15xy − 2a18xz + a12yz + 2(a17 − a4)x + a13y + 2a7z + a14 +−a2x2 − a12y2 − 2a16xy + a11xz + a8yz + 2(a19 − a1)x + 2(a20 − a7)y + a9z + a10 + + +(15) +and the generated KT is +Cab = + + +a5y + a2z + a3 +− a5 +2 x − a15 +2 y + a16z + a17 +− a2 +2 x + a18y − a11 +2 z + a19 +− a5 +2 x − a15 +2 y + a16z + a17 +a15x + a12z + a13 +−(a16 + a18)x − a12 +2 y − a8 +2 z + a20 +− a2 +2 x + a18y − a11 +2 z + a19 +−(a16 + a18)x − a12 +2 y − a8 +2 z + a20 +a11x + a8y + a9 + + +(16) +which is a subcase of the general KT (14) for a1 = a4 = a6 = a7 = a10 = a14 = 0. +5 +The QFI I(1,1) where ℓ = 1 +We set L(1)a = La and the QFI I(1,ℓ) for ℓ = 1 becomes +I(1,1) = +� +−t2 +2 L(a;b) + Cab +� +˙qa ˙qb + tLa ˙qa + t2 +2 LaV ,a + G(x, y, z) +(17) +10 + +where Cab is a second order KT given by (14), the vector La is given by (15), the generated KT L(a;b) is the +(16) and the following conditions must be satisfied: +� +LbV ,b� +,a += +−2L(a;b)V ,b +(18) +G,a += +2CabV ,b − La. +(19) +Equations (18) and (19) must be supplemented with the three integrability conditions for the function G and +the three integrability conditions for the function LaV ,a. +Finally, we have an overdetermined system of twelve PDEs with unknowns the two functions G(x, y, z) and +V (x, y, z), and forty free parameters. Obviously, the general solution is not possible, and we have to look for +special solutions which are achieved by introducing simplifying assumptions. +5.1 +Case La = 0 +In this case, the QFI (17) is the well-known autonomous QFI +I(1,1)(La = 0) = Cab ˙qa ˙qb + G(x, y, z) +(20) +where the second order KT Cab has independent components +C11 += +a6 +2 y2 + a1 +2 z2 + a4yz + a5y + a2z + a3 +C12 += +a10 +2 z2 − a6 +2 xy − a4 +2 xz − a14 +2 yz − a5 +2 x − a15 +2 y + a16z + a17 +C13 += +a14 +2 y2 − a4 +2 xy − a1 +2 xz − a10 +2 yz − a2 +2 x + a18y − a11 +2 z + a19 +(21) +C22 += +a6 +2 x2 + a7 +2 z2 + a14xz + a15x + a12z + a13 +C23 += +a4 +2 x2 − a14 +2 xy − a10 +2 xz − a7 +2 yz − (a16 + a18)x − a12 +2 y − a8 +2 z + a20 +C33 += +a1 +2 x2 + a7 +2 y2 + a10xy + a11x + a8y + a9 +the parameters a1, ..., a20 are arbitrary constants and the function G(x, y, z) satisfies the condition +G,a = 2CabV ,b. +(22) +The integrability condition G,[ab] = 0 gives: +0 += +C12 (V,yy − V,xx) + +�a6(y2 − x2) +2 ++ (a1 − a7)z2 +2 +− (a14x − a4y)z − a15x + a5y + (a2 − a12)z+ ++a3 − a13] V,xy + C13V,yz − C23V,xz + 3 +2(a6y + a4z + a5)V,x − 3 +2(a6x + a14z + a15)V,y + ++ +�3a14 +2 +y − 3a4 +2 x + 2a18 + a16 +� +V,z +(23) +0 += +C13 (V,zz − V,xx) + +�a1(z2 − x2) +2 ++ (a6 − a7)y2 +2 +− (a10x − a4z)y − a11x + (a5 − a8)y + a2z+ ++a3 − a9] V,xz + C12V,yz − C23V,xy + 3 +2(a4y + a1z + a2)V,x + +�3a10 +2 +z − 3a4 +2 x + 2a16 + a18 +� +V,y − +−3 +2(a1x + a10y + a11)V,z +(24) +0 += +C23 (V,zz − V,yy) + +�a7(z2 − y2) +2 ++ (a6 − a1)x2 +2 +− (a10y − a14z)x + (a15 − a11)x − a8y + a12z+ ++a13 − a9] V,yz + C12V,xz − C13V,xy + +�3a10 +2 +z − 3a14 +2 +y + a16 − a18 +� +V,x + 3 +2(a14x + a7z + a12)V,y − +11 + +−3 +2(a10x + a7y + a8)V,z. +(25) +We note that in the case of 2d Newtonian potentials the integrability condition G,[ab] = 0 leads to just one +equation, which is the well-known Bertrand-Darboux PDE (see e.g. eq. (28) of [10] and eq. (3.2.5) of [12]). +The system of PDEs (23) - (25) has to be solved in order to find potentials V (x, y, z) that admit autonomous +QFIs of the form (20). Replacing these potentials in the remaining condition (22), we find the functions G(x, y, x) +which determine the associated QFIs (20). Since the general solution V (x, y, z) is not possible, we consider again +several cases for various values of the twenty free parameters a1, a2, ..., a20. +5.1.1 +The components of the KT Cab are constants +In this case, the possibly non-zero parameters are the a3, a9, a13, a17, a19, and a20. A detailed study leads to +the following five cases (only non-vanishing parameters are listed). +1) a3 = a, a17 = b +2, and a19 = c +2, where a, b, c are arbitrary constants5. +The potential is +V (x, y, z) = F1 +� +cz + by + ( +� +a2 + b2 + c2 + a)x +� ++ F2 +� +cz + by − ( +� +a2 + b2 + c2 − a)x +� ++ F3(bz − cy) (26) +where F1, F2, and F3 are arbitrary smooth functions of their arguments. +The associated QFI (20) is +I(1,1) = (a ˙x + b ˙y + c ˙z) ˙x + a(F1 + F2) + +� +a2 + b2 + c2(F1 − F2). +(27) +We note that the constants a, b, c are parameters of the potential (26); therefore, they cannot generate three +distinct QFIs. +2)a3 = a, a13 = −a, and a17 = ia, where a is an arbitrary constant. +The potential is +V (x, y, z) = F1(w, z) + F2(w) ¯w +(28) +where w = x + iy, ¯w = x − iy and F1, F2 are arbitrary smooth functions of their arguments. +The associated autonomous QFI (20) is +I(1,1) = ( ˙x + i ˙y)2 + 4 +� +F2(w)dw. +(29) +If F1(w, z) = F3(w) + F4(z), then +V (x, y, z) = F2(w) ¯w + F3(w) + F4(z) +(30) +is a new integrable potential due to the additional QFI I = 1 +2 ˙z2 + F4(z). +3) a3 = a, a13 = −a, and a17 = −ia, where a is an arbitrary constant. +The potential +V (x, y, z) = F1( ¯w, z) + F2( ¯w)w +(31) +and the associated QFI +I(1,1) = ( ˙x − i ˙y)2 + 4 +� +F2( ¯w)d ¯w. +(32) +If F1( ¯w, z) = F3( ¯w) + F4(z), then +V (x, y, z) = F2( ¯w)w + F3( ¯w) + F4(z) +(33) +is a new integrable potential due to the additional QFI I = 1 +2 ˙z2 + F4(z). +4) a19 ̸= 0 and a20 = ia19. +The potential is +V (x, y, z) = F ′ +2z2 + F3(w)z + F4(w) + F2(w) ¯w +(34) +5If instead of the triplet a3, a17, a19 we take as non-vanishing parameters the triplets a13, a17, a20 or a9, a19, a20, the resulting +potentials are symmetric up to a cyclic permutation of the coordinates x, y, z. +12 + +where w = x + iy, ¯w = x − iy, F2, F3, F4 are arbitrary smooth functions of their arguments, and F ′ +2 ≡ dF2 +dw . +The associated autonomous QFI (20) is +I(1,1) = 1 +2 ˙z ( ˙x + i ˙y) + F2(w)z + 1 +2 +� +F3(w)dw. +(35) +Because the potential (34) is of the general form (28) for F1(w, z) = F ′ +2z2 + F3(w)z + F4(w), it admits +the additional QFI (29). Therefore, it is integrable because the independent QFIs H, (29) and (35) are in +involution6. +For F2 = k1w + k2 and F3 = k3 where k1, k2, k3 are arbitrary constants, the potential (34) becomes +V (x, y, z) = k1r2 + k2 ¯w + k3z + F4(w). +(36) +This is a new minimally superintegrable potential because it is separable in the coordinate z. +5) a19 ̸= 0 and a20 = −ia19. +The potential is +V (x, y, z) = F ′ +2z2 + F3( ¯w)z + F4( ¯w) + F2( ¯w)w. +(37) +where w = x + iy and F ′ +2 ≡ dF2 +d ¯ +w . +The associated autonomous QFI (20) is +I(1,1) = 1 +2 ˙z ( ˙x − i ˙y) + F2( ¯w)z + 1 +2 +� +F3( ¯w)d ¯w. +(38) +The potential (37) is of the general form (31) for F1( ¯w, z) = F ′ +2z2+F3( ¯w)z+F4( ¯w); therefore, it is integrable +due to the additional QFI (32). +Moreover, for F2 = k1 ¯w + k2 and F3 = k3, the potential (37) becomes +V (x, y, z) = k1r2 + k2w + k3z + F4( ¯w). +(39) +This is a new minimally superintegrable potential because it is separable in the coordinate z. +5.1.2 +The components of the KT Cab are linear functions of x, y, z +The possibly non-zero parameters are the a2, a5, a8, a11, a12, a15, a16, and a18. In this case, there are six different +combinations which lead to new results. +1) a2 = a and a5 = b, where a, b are arbitrary constants. +The potential is +V (x, y, z) = (a2 + b2)x2 + 4(az + by)2 + k1 +x2 + k2(az + by) + F(ay − bz) +(40) +where F is an arbitrary smooth function of its argument and k1, k2 are arbitrary constants. For a = 0, the +potential (40) reduces to the minimally superintegrable potential of the form (see Table 2) +V (x, y, z) = k1 +2 (x2 + 4y2) + k2 +x2 + k3y + F(z). +(41) +The associated QFI (20) is +I(1,1) = (aM2 − bM3) ˙x − k2 +2 (a2 + b2)x2 − 2(a2 + b2)(az + by)x2 + 2k1(az + by) +x2 +(42) +where Mi = (y ˙z − z ˙y, z ˙x − x ˙z, x ˙y − y ˙x) is the angular momentum. +Since the potential (40) is separable in the coordinate x, it admits the additional QFI +I = 1 +2 ˙x2 + (a2 + b2)x2 + k1 +x2 . +6The PB of the QFIs (29) and (35) vanishes because for an integrable function of the form M(w) = +� +F (w)dw with w = x + iy, +it holds that: M′ ≡ dM +dw = F , M,x = F and M,y = iF . +13 + +However, it is not integrable because the PB {I(1,1), I} ̸= 0. +2) a2 = a and a12 = b, where a, b are arbitrary constants. +We find the fully separable potential +V (x, y, z) = k1(x2 + y2 + 4z2) + k2 +x2 + k3 +y2 + k4z +(43) +where k1, k2, k3, and k4 are arbitrary constants. We note that the potential in Table I of [2] is a subcase of the +potential (43) for k4 = 0. +The associated QFI (20) consists of the following two independent QFIs: +J1 += +M2 ˙x + 2z +�k2 +x2 − k1x2 +� +− k4 +2 x2 +(44) +J2 += +−M1 ˙y + 2z +�k3 +y2 − k1y2 +� +− k4 +2 y2. +(45) +Moreover, the potential (43) is of the integrable form V = +F1( y +x) +R2 ++ F2(R) + F3(z) (see Table 1) for +F1 +�y +x +� += k2 +� +1 + +�y +x +�2� ++ k3 +� +1 + +�x +y +�2� +, F2(R) = k1(x2 + y2), F3(z) = 4k1z2 + k4z. +Therefore, it admits the additional QFI +J3 = 1 +2M 2 +3 + k2 +�y +x +�2 ++ k3 +�x +y +�2 +. +(46) +In order to compare the QFIs (44), (45), (46) with the QFIs I3, I4 of eq. (3.43) in [2], we set k4 = 0 and we +use the rotational parabolic coordinates (see eqs. (3.8) and (3.9) in [2]): +ζ = r + z, η = r − z, φ = tan−1 �y +x +� +or equivalently +x = +� +ζη cos φ, y = +� +ζη sin φ, z = 1 +2 (ζ − η) . +We compute (for k4 = 0): +J1 += +M2 ˙x + (ζ − η) +� +k2 +ζη cos2 φ − k1ζη cos2 φ +� +(47) +J2 += +−M1 ˙y + (ζ − η) +� +k3 +ζη sin2 φ − k1ζη sin2 φ +� +(48) +J3 += +1 +2M 2 +3 + +k2 +cos2 φ + +k3 +sin2 φ − k2 − k3. +(49) +Then, J3 = I3 − k2 − k3 and +J1 + J2 = M2 ˙x − M1 ˙y − (ζ − η) +� +k1ζη − +k2 +ζη cos2 φ − +k3 +ζη sin2 φ +� += I4. +There is a misprint in eq. (3.43) of [2] concerning the leading term of the QFI I4. It must be L2P1 − L1P2. +We conclude that the potential (43) is maximally superintegrable. However, from the seven QFIs (the QFIs +H, J1, J2, J3 plus the three QFIs arising from the separability of x, y, z) only five are functionally independent. +3) a2 = a and a8 = b, where a, b are arbitrary constants. +We find the separable potential +V (x, y, z) = k1 +4 (x2 + 16y2 + 4z2) + k2 +x2 + k3y +(50) +14 + +where k1, k2, and k3 are arbitrary constants. +The associated QFI (20) consists of the following two independent QFIs: +J1 += +M2 ˙x + z +�2k2 +x2 − k1 +2 x2 +� +(51) +J2 += +M1 ˙z − z2 +� +2k1y + k3 +2 +� +. +(52) +Therefore, the separable potential (50) is a new maximally superintegrable potential. +4) a2 = a and a16 = −a18 = b +2, where a, b are arbitrary constants. +The potential is +V (x, y, z) = +k +� +(ax + by)2 + (a2 + b2)z2 +(53) +where k is an arbitrary constant. +The associated QFI (20) is +I(1,1) = aM2 ˙x − bM1 ˙x + azV. +(54) +In order to show that the potential (53) is integrable, we need one more independent FI in involution. +5) a2 = a12 ̸= 0. +In this case, we find the following three potentials7: +V1(x, y, z) += +F1 +� y +x +� +R2 ++ k(R2 + 4z2) +(55) +V2(x, y, z) += +F1 +� y +x +� +R2 ++ k1z +rR2 − k2 +r +(56) +V3(x, y, z) += +F1 +� y +x +� +R2 ++ kz +(57) +where k, k1, k2 are arbitrary constants, R = +� +x2 + y2 and r = +� +x2 + y2 + z2. The first two potentials, i.e. V1 +and V2, are included in Table II of [2], whereas the third potential V3 is not included. +The associated QFIs (20) are: +- For the potential8 (55): +I(1,1) += +M2 ˙x − M1 ˙y + 2zF1 +� y +x +� +x2 + y2 − 2kz(x2 + y2) += +M2 ˙x − M1 ˙y + (ζ − η) +�F1(tan φ) +ζη +− kζη +� +. +(58) +- For the potential (56): +I(1,1) += +M2 ˙x − M1 ˙y + 2zF1 +� y +x +� +x2 + y2 + +2k1z2 +r(x2 + y2) + k1 +r − k2z +r += +M2 ˙x − M1 ˙y + (ζ − η)F1(tan φ) +ζη ++ k1(ζ2 + η2) +ζη(ζ + η) − k2(ζ − η) +ζ + η +. +(59) +- For the potential (57): +I(1,1) += +M2 ˙x − M1 ˙y + 2zF1 +� y +x +� +x2 + y2 − k R2 +2 . +(60) +We note that both the potentials (55) and (57) are minimally superintegrable, because they are of the form +V = +F1( y +x) +R2 ++ F2(R) + F3(z) (see Table 1). +7We note that any linear combination of these potentials is a solution of the system of PDEs (23) - (25) for a2 = a12 ̸= 0. +8In Table II of [2], there is a misprint in the QFIs I3 associated with the potentials (55) and (56). The leading part of I3 should +be L2P1 − P2L1. +15 + +Moreover, from cases 2) and 3) of the following subsection 5.1.3, the potential (56) becomes minimally +superintegrable because it admits the additional QFIs (75) and (81) which are also in involution. +6) a2 ̸= 0, a12 = −a2, a16 = ia2 and a18 = −i a2 +2 . +The potential is +V (x, y, z) = k1(R2 + 4z2) + k2z + k3 +w2 + k4 +¯w +w3 +(61) +where k1, k2, k3, k4 are arbitrary constants, w = x + iy, and ¯w = x − iy. This result coincides with the potential +given in eq. (14) of [4] if we apply the canonical transformation x → y, y → z and z → x. +The associated autonomous QFI (20) is +I1 = 1 +2( ˙x + i ˙y) (M2 − iM1) − k1zw2 − k2 +4 w2 − k4 +z +w2 . +(62) +Moreover, the potential (61) admits the additional QFIs: +I2 += +1 +2 ( ˙x + i ˙y)2 + k1w2 − k4 +w2 +(63) +I3 += +1 +2 ˙z2 + 4k1z2 + k2z +(64) +I4 += +1 +2M 2 +3 + k3e−2iθ + k4e−4iθ +(65) +I5 += +1 +2 (M2 ˙x − M1 ˙y) + k3 +z +w2 + k4 +z ¯w +w3 − k1zR2 − k2 +R2 +4 +(66) +because it is of the form (28) for F1 = 4k1z2 + k2z + k3 +w2 and F2 = k1w + k4 +w3 , and of the form (see subsection +5.1.2 and Table 6) +V (x, y, z) = F1 +� y +x +� +R2 ++ k1(R2 + 4z2) + k2z +(67) +for F1 +� y +x +� += k3e−2iθ + k4e−4iθ, where tan θ = y +x. Therefore, the potential (61) is maximally superintegrable. +5.1.3 +The components of the KT Cab depend on xy, xz, yz, x2, y2, z2 +In this case, the possibly non-zero parameters are the a1, a4, a6, a7, a10, and a14. New results are produced for +the following six cases. +1) a1 = a, a6 = b, and a7 = c, where a, b, c are arbitrary constants. +The potential is (see Table II in [2]) +V (x, y, z) = k1 +x2 + k2 +y2 + k3 +z2 + F(r) +(68) +where k1, k2, k3 are arbitrary constants, r = +� +x2 + y2 + z2 and F is an arbitrary smooth function of r. +The associated QFI (20) consists of the following three independent FIs (one for each parameter a, b, c): +I1 += +1 +2M 2 +1 + k2 +z2 +y2 + k3 +y2 +z2 +(69) +I2 += +1 +2M 2 +2 + k1 +z2 +x2 + k3 +x2 +z2 +(70) +I3 += +1 +2M 2 +3 + k1 +y2 +x2 + k2 +x2 +y2 . +(71) +Using spherical coordinates x = r sin θ cos φ, y = r sin θ sin φ and z = r cos θ, the QFIs (69) - (71) coincide +with those found in Table II of [2]. Moreover, by adding the above QFIs, we find the QFI +I4 = 1 +2M2 + +k1 +sin2 θ cos2 φ + +k2 +sin2 θ sin2 φ + +k3 +cos2 θ +(72) +where M2 = M 2 +1 + M 2 +2 + M 2 +3 is the square magnitude of the angular momentum. +16 + +Even though the potential (68) admits the four independent QFIs H, I1, I2, and I3, it is not integrable +because the PBs {Ii, Ij} ̸= 0. +For F(r) = kr2, where k is an arbitrary constant, the potential (68) becomes (see Table I in [2]) +V (x, y, z) = k +� +x2 + y2 + z2� ++ k1 +x2 + k2 +y2 + k3 +z2 +(73) +which is maximally superintegrable (see Tables 1 and 3). +2) a6 ̸= 0. +The potential is +V (x, y, z) = F1 +� y +x +� +R2 ++ F2(R, z) +(74) +where F1 and F2 are arbitrary smooth functions of their arguments. +The associated QFI (20) is +I(1,1) = 1 +2M 2 +3 + F1 +�y +x +� +. +(75) +If F2(R, z) = F3(R) + F4(z) where F3 and F4 are arbitrary smooth functions, then the potential (74) is +integrable (see Table 1). +3) a1 = a6 = a7 ̸= 0. +The potential is +V (x, y, z; m) = +m +� +j=1 +Fj +� y +x +� +R2 +Nj +� z +R +� ++ F(r) +(76) +where Fj, Nj and F with j = 1, 2, ..., m are smooth functions of their arguments. +The associated QFI (20) is +I(1,1) = 1 +2M2 + +m +� +j=1 +r2Fj +� y +x +� +R2 +Nj +� z +R +� +. +(77) +We note that for +a. m = 2, N1 = F2 = 1, N2 = k2 R2 +z2 , F(r) = k1r2; and +b. m = 2, N1 = F2 = 1, N2 = +k1 z +R +� +1+ z2 +R2 +, F(r) = − k2 +r +the potential (76) reduces, respectively, to the following potentials (see Table II in [2]): +V1(x, y, z) += +F1 +� y +x +� +x2 + y2 + k1(x2 + y2 + z2) + k2 +z2 +(78) +V2(x, y, z) += +F1 +� y +x +� +x2 + y2 + +k1z +r(x2 + y2) − k2 +r +(79) +where k1 and k2 are arbitrary constants. Both the above potentials are also of the general form (74). +The associated QFIs (77) are as follows: +- For the potential (78): +I(1,1) = 1 +2M2 + r2F1 +� y +x +� +x2 + y2 + k2(x2 + y2) +z2 +. +(80) +-For the potential (79): +I(1,1) = 1 +2M2 + r2F1 +� y +x +� +x2 + y2 + +k1zr +x2 + y2 . +(81) +We note that the potential (78) is minimally superintegrable because it is separable in the coordinate z and +is also of the form (74). +4) a1 ̸= 0, a7 = −a1 and a10 = ia1. +The potential is +V (x, y, z) = F3(w) + z2 +w F2(w) + F4 +� z +w +� +w2 ++ F2(w) ¯w +(82) +17 + +where w = x + iy, ¯w = x − iy, and F2, F3, F4 are arbitrary smooth functions of their arguments. +The associated QFI (20) is +I1 = 1 +2 (M2 − iM1)2 + F4 +� z +w +� +. +(83) +Moreover, the potential (82) admits the additional QFI +I2 = ( ˙x + i ˙y)2 + 4 +� +F2(w)dw +(84) +because it is of the form (28) with F1 = F3(w) + z2 +w F2(w) + +F4( z +w) +w2 +. Since the PB {I1, I2} = 0, the potential +(82) is (Liouville) integrable. +Finally, for F2 = k1w and F4 = k2 w2 +z2 , the potential (82) becomes +V (x, y, z) = k1r2 + k2 +z2 + F3(w) +(85) +which is a new minimally superintegrable potential due to the separability in the z-coordinate. +5) a1 ̸= 0, a7 = −a1 and a10 = −ia1. +Similarly to the previous case 4), we find the integrable potential +V (x, y, z) = F3( ¯w) + z2 +¯w F2( ¯w) + F4 +� z +¯ +w +� +¯w2 ++ F2( ¯w)w +(86) +and the associated QFI +I1 = 1 +2 (M2 + iM1)2 + F4 +� z +¯w +� +. +(87) +Moreover, the potential (86) admits the additional QFI +I2 = ( ˙x − i ˙y)2 + 4 +� +F2( ¯w)d ¯w +(88) +because it is of the form (31) for F1 = F3( ¯w) + z2 +¯ +w F2( ¯w) + +F4( z +¯ +w) +¯ +w2 +. +Finally, for F2 = k1 ¯w and F4 = k2 ¯w2 +z2 , the potential (86) becomes +V (x, y, z) = k1r2 + k2 +z2 + F3( ¯w) +(89) +which is a new minimally superintegrable potential due to the separability in the z-coordinate. +6) a4 ̸= 0 and a14 = −ia4. +The potential is (see eq. (12) of [4]) +V (x, y, z) = k1r2 + k2 +w2 + k3 +z +w3 + k4 +R2 − 3z2 +w4 +(90) +where k1, k2, k3, k4 are arbitrary constants and w = x + iy. +The associated QFI (20) is +I1 = M3 (iM1 − M2) + k3 +y +w − 2ik2 +z +w − 3ik3 +2 +z2 +w2 − 4ik4 +z(x2 + y2 − z2) +w3 +. +(91) +Moreover, the potential (90) admits the additional QFIs: +I2 += +1 +2 (M2 − iM1)2 + k3 +z +w − 4k4 +z2 +w2 +(92) +I3 += +1 +2 ˙z ( ˙x + i ˙y) + k1zw − k3 +4w2 + k4 +z +w3 . +(93) +I4 += +1 +2 ( ˙x + i ˙y)2 + k1w2 − k4 +w2 +(94) +18 + +I5 += +1 +2M2 + k2 +r2 +w2 + k3 +zr2 +w3 + k4 +r2(x2 + y2 − 3z2) +w4 +. +(95) +Specifically, we have the following: +1) It admits the QFI (92) because it is of the form (82) for F2 = k1w + k4 +w3 , F3 = k2 +w2 and F4 = k3 z +w − 4k4 z2 +w2 . +2) It admits the QFI (93) because it is of the form (34) for F2 = k1w + k4 +w3 , F3 = k3 +w3 and F4 = k2 +w2 . +3) It admits the QFI (94) because it is of the form (28) for F1 = k1z2 + k2 +w2 + k3 z +w3 − 3k4 z2 +w4 and F2 = k1w + k4 +w3 . +4) It admits the QFI (95) because it is of the form (76) for m = 4, N1 = 1, F1 = k2e−2iθ, N2 = z +R, F2 = k3e−3iθ, +N3 = 1, F3 = k4e−4iθ, N4 = z2 +R2 , F4 = −3k4e−4iθ and F(r) = k1r2. +We note that the variable θ = tan−1 � y +x +� +; hence, w = x + iy = Reiθ and R2 = w ¯w. +We conclude that the potential (90) is maximally superintegrable. Specifically, it is integrable due to the +triplet H, I2, I4 and superintegrable because it admits the five independent QFIs H, I1, I2, I3, I4. +5.1.4 +The components of the KT Cab depend on products of x, y, z of mixed degree +In this subsection, we continue by considering mixed combinations of the twenty parameters a1, ..., a20 so that +the components of the KT Cab contain products of x, y, z of mixed degree. We note that we do not exhaust +all possible cases; therefore, other authors could consider other cases and determine new non-decomposable +integrable/superintegrable potentials in E3. +1) The only non-vanishing parameters are the a3 = iB +4 , a5 = B, a13 = − iB +4 , a17 = − B +4 and a15 = iB, where +B is an arbitrary constant. +The KT (21) is +Cab = + + +By + iB +4 +− B +2 x − iB +2 y − B +4 +0 +− B +2 x − iB +2 y − B +4 +iBx − iB +4 +0 +0 +0 +0 + + . +(96) +For the KT (96) the system of PDEs (23) - (25) gives the potential +V (x, y, z) = 4k1 +� +R2 − ¯w3 +2 +� ++ k2 +� +2w − 3 ¯w2� ++ k3 ¯w + F(z) +(97) +where k1, k2, k3 are arbitrary constants and F(z) is an arbitrary smooth function. +The associated QFI (20) is +I1 += +1 +4 ( ˙x + i ˙y)2 + iM3 ( ˙x − i ˙y) − 2k1 +�3 +4 ¯w4 − w2 + R2 ¯w +� +− +−2k2 +� +¯w3 + 2R2� ++ k3 +� ¯w2 +2 + w +� +. +(98) +Moreover, the potential (97) admits the additional QFIs: +I2 += +1 +8 ( ˙x − i ˙y)2 + k1 ¯w2 + k2 ¯w +(99) +I3 += +1 +2 ˙z2 + F(z) +(100) +because it is of the form (31) for F1 = −2k1 ¯w3 − 3k2 ¯w2 + k3 ¯w + F(z) and F2 = 4k1 ¯w + 2k2, and it is separable +on the z-coordinate. Therefore, it is minimally superintegrable due to the four independent QFIs H, I1, I2, I3. +We note that the potential given in eq. (15) of [4] is a subcase of (97) for F(z) = k1z2 + k4 +z2 , where k4 is +an arbitrary constant. As it will be shown below, in this special case, the resulting potential admits additional +QFIs which promote it to a maximally superintegrable potential. +2) The only non-vanishing parameters are the a1 = C, a7 = −C, a8 = −iD + 2iC, a10 = −iC and a11 = D, +where C, D are arbitrary constants. +The KT (21) is +Cab = + + +C +2 z2 +− iC +2 z2 +− C +2 xz + iC +2 yz − D +2 z +− iC +2 z2 +− C +2 z2 +iC +2 xz + C +2 yz + i +� D +2 − C +� +z +− C +2 xz + iC +2 yz − D +2 z +iC +2 xz + C +2 yz + i +� D +2 − C +� +z +C +2 (x2 − y2) + iCy(2 − x) + D(x − iy) + + . (101) +19 + +For the KT (101) the system of PDEs (23) - (25) gives the potential (see eq. (15) of [4]) +V (x, y, z) = 4k1 +� +R2 − ¯w3 +2 + z2 +4 +� ++ k2 +� +2w − 3 ¯w2� ++ k3 ¯w + k4 +z2 +(102) +where k1, k2, k3, and k4 are arbitrary constants. +The associated QFI (20) consists of the following independent QFIs: +I1 += +1 +2 (M2 + iM1)2 + 2i ˙zM1 + k1z2 � +3 ¯w2 − 4iy +� ++ 2k2z2 (2 ¯w + 1) − +−k3z2 + k4 +z2 +� +¯w2 + 4iy +� +(103) +I2 += +1 +2 ˙z (M2 + iM1) + k1z2 ¯w + k2z2 − k4 +¯w +z2 . +(104) +Moreover, the potential (102) admits the three additional QFIs (98) - (100) because it is of the form (97) +for F(z) = k1z2 + k4 +z2 . Therefore, it is maximally superintegrable. +3) The only non-vanishing parameters are the: +a1 = −2C, a2 = iB − C, a5 = a8 = iA, a7 = 2C, a9 = iB +4 − C +4 , a10 = −2iC, a11 = a15 = A, +a12 = −iB + 2C, a13 = C +4 , a16 = −B − 3iC +2 , a18 = B +2 + 3iC +2 , a20 = iA +4 +where A, B, and C are arbitrary constants. +The KT (21) has independent components: +C11 += +−Cz2 + iAy + (iB − C)z +C12 += +−iCz2 − iA +2 ¯w − +� +B + 3iC +2 +� +z +C13 += +Cxz − −iCyz − iB − C +2 +x + 1 +2(B + 3iC)y − A +2 z +(105) +C22 += +Cz2 + Ax − (iB − 2C)z − C +4 +C23 += +iCxz − Cyz + B +2 x + iB − 2C +2 +y − iA +2 z + iA +4 +C33 += +−Cx2 + Cy2 − 2iCxy + Aw + iB − C +4 +where w = x + iy. +For the KT (105) the system of PDEs (23) - (25) gives the potential (see eq. (16) of [4]) +V (x, y, z) = k1w + k2 +� +3w2 + z +� ++ k3 +� +4w3 + 3wz + ¯w +4 +� ++ k4 +�5 +2w4 + r2 +2 + 3w2z +� +(106) +where k1, k2, k3, and k4 are arbitrary constants. +The associated QFI (20) consists of the following independent QFIs: +I1 += +M3 ˙w − (M1 + iM2) ˙z − 1 +2 ˙y ˙z + ik1 +2 +� +w2 − z +� ++ ik2 +� +2w3 − zw + i +2y +� +− ik3 +8 +� +w2 − z +� ++ ++ik3 +� +3w4 − z2 + iyw +� +− k4 +2 y +� +w2 + z +� ++ ik4w +� +2w4 + zw2 − z2� +(107) +I2 += +M1 (2 ˙x + i ˙y) + iM2 ˙x + M3 ˙z + i +4 ˙z2 + ik2 +2 +� +z − w2� ++ ik3w +� +z − w2� ++ ++ik4 +4 +� +z2 + 2zw2 − 3w4� +(108) +20 + +I3 += +(M1 + iM2)2 + (2iM1 − M2) ˙w − iM3 ˙z + 1 +4 +� +˙y2 − ˙z2� ++ k1zw + ik1 +2 y + ++2k2w (2zw + iy) + k2 +2 +� +w2 − z +� ++ k3w3(6z + 1) + k3zw +� +2z − 1 +4 +� ++ ++ik3y +� +3w2 − 1 +8 +� +− k3xz + k4w4 +� +4z + 3 +4 +� ++ 2ik4yw3 + ++k4z2 +� +3w2 − 1 +4 +� ++ k4 +2 +�y2 +2 − zR2 +� +. +(109) +We note that the parameter A produces the QFI I1, B the I2, and C the I3. +Moreover, the potential (106) admits the additional QFIs: +I4 += +˙w2 + k3w + k4w2 +(110) +I5 += +˙z ˙w + +� +k4w + k3 +2 +� +z + k4w3 + 3k3 +2 w2 + k2w +(111) +because it is of the form (28) for F1 = k1w + k2(3w2 + z) + k3w(4w2 + 3z) + k4 +� +5 +2w4 + 3w2z + z2 +2 +� +and F2 = +k4 +2 w+ k3 +4 ; and of the form (34) for F2 = k4 +2 w+ k3 +4 , F3 = 3k4w2+3k3w+k2 and F4 = 5k4 +2 w4+4k3w3+3k2w2+k1w. +We compute the PB {I2, I4} = 0; therefore, the potential (106) is maximally superintegrable. +5.1.5 +Special superintegrable potentials +In this subsection, we construct potentials whose form belongs to two or more of the previous general results. +We have the following cases: +1) Consider the potential (see Table I in [2]) +V (x, y, z) = −c1 +r + c2 +x2 + c3 +y2 +(112) +where c1, c2, and c3 are arbitrary constants. This potential is of the general form (68) for F(r) = − c1 +r , k1 = c2, +k2 = c3 and k3 = 0; and of the form (56) for k1 = 0, k2 = c1 and F1 +� y +x +� += c2 +� +1 + +� y +x +�2� ++ c3 +� +1 + +� +x +y +�2� +. +Therefore, it admits the additional QFIs: +I1 += +1 +2M 2 +1 + c3 +z2 +y2 +(113) +I2 += +1 +2M 2 +2 + c2 +z2 +x2 +(114) +I3 += +1 +2M 2 +3 + c2 +y2 +x2 + c3 +x2 +y2 +(115) +I4 += +M2 ˙x − M1 ˙y − 2z +� c1 +2r − c2 +x2 − c3 +y2 +� +. +(116) +We conclude that the potential (112) is maximally superintegrable because the QFIs H, I3, I4 are in involution +and the five QFIs H, I1, I2, I3, I4 are functionally independent. +2) Consider the potential (see Table I in [2]) +V (x, y, z) = c1y +x2R + c2 +x2 + c3 +z2 +(117) +where c1, c2, and c3 are arbitrary constants. This potential is of the form V = +k1 +x2 + k2 +R + k3y +Rx2 + F(z) (see +Table 2) for k1 = c2, k2 = 0, k3 = c1, and F(z) = +c3 +z2 ; and of the form (78) for k1 = 0, k2 = c3, and +F1 +� y +x +� += +� +c1 +� +1+ x2 +y2 ++ c2 +� � +1 + y2 +x2 +� +. Therefore, it admits the additional QFIs: +I1 += +1 +2 ˙z2 + c3 +z2 +(118) +21 + +I2 += +1 +2M 2 +3 + c2 +y2 +x2 + c1 +yR +x2 +(119) +I3 += +M3 ˙x − 2c2 +y +x2 − c1 +x2 + 2y2 +x2R +(120) +I4 += +1 +2M2 + c1 +yr2 +Rx2 + c2 +r2 +x2 + c3 +R2 +z2 +(121) +and it is maximally superintegrable. +3) Another maximally superintegrable potential is the (see Table I in [2]) +V (x, y, z) = c1y +x2R + c2 +x2 + c3z +(122) +where c1, c2, and c3 are arbitrary constants. This potential is of the form V = +k1 +x2 + k2 +R + k3y +Rx2 + F(z) (see +Table 2) for k1 = c2, k2 = 0, k3 = c1, and F(z) = c3z; and of the form (57) for k = c3, and F1 +� y +x +� += +� +c1 +� +1+ x2 +y2 ++ c2 +� � +1 + y2 +x2 +� +. Therefore, it admits the additional QFIs: +I1 += +1 +2 ˙z2 + c3z +(123) +I2 += +1 +2M 2 +3 + c2 +y2 +x2 + c1 +yR +x2 +(124) +I3 += +M3 ˙x − 2c2 +y +x2 − c1 +x2 + 2y2 +x2R +(125) +I4 += +M2 ˙x − M1 ˙y + c1 +2yz +x2R + c2 +2z +x2 − c3 +R2 +2 . +(126) +4) Consider the potential (see eq. (11) of [4]) +V (x, y, z) = k1r2 + k2 +¯w +w3 + k3 +w2 + k4 +z2 +(127) +where k1, k2, k3, k4 are arbitrary constants, w = x + iy and ¯w = x − iy. +This potential admits the additional QFIs: +I1 += +1 +2 (M2 − iM1)2 + k4 +w2 +z2 − k2 +z2 +w2 . +(128) +I2 += +1 +2 ( ˙x + i ˙y)2 + k1w2 − k2 +w2 +(129) +I3 += +1 +2M 2 +3 + k2e−4iθ + k3e−2iθ = 1 +2M 2 +3 + k2 +� ¯w +w +�2 ++ k3 +¯w +w +(130) +I4 += +1 +2 ˙z2 + k1z2 + k4 +z2 +(131) +I5 += +1 +2M2 + k2 +r2 ¯w +w3 + k3 +r2 +w2 + k4 +r2 +z2 +(132) +because it is of the form (82) for F2 = k1w + k2 +w3 , F3 = +k3 +w2 and F4 = −k2 z2 +w2 + k4 w2 +z2 ; of the form (28) for +F1 = k1z2 + k3 +w2 + k4 +z2 and F2 = k1w + k2 +w3 ; of the form (74) for F1 = k2e−4iθ + k3e−2iθ and F2 = k1r2 + k4 +z2 ; +separable on the z-coordinate; and of the form (76) for m = 2, F1 = N2 = 1, N1 = k4 R2 +z2 , F2 = k2e−4iθ +k3e−2iθ +and F(r) = k1r2. +The variable θ = tan−1 � y +x +� +and, hence, w = x + iy = Reiθ. We recall that +w = Reiθ =⇒ einθ = +�w +R +�n +=⇒ einθ = + + +1 + i y +x +� +1 + +� y +x +�2 + + +n +22 + +where n is an arbitrary real constant. If n = 2k ∈ R, then e2ikθ = +� w +¯ +w +�k because R2 = w ¯w. +We conclude that the potential (127) is maximally superintegrable. +We collect the results of this section in Tables 4 - 7. +23 + +Potential +LFIs and QFIs +V = F1 +� +cz + by + ( +√ +a2 + b2 + c2 + a)x +� ++ ++F2 +� +cz + by − ( +√ +a2 + b2 + c2 − a)x +� ++ ++F3(bz − cy) +I1 = (a ˙x + b ˙y + c ˙z) ˙x + a(F1 + F2)+ ++ +√ +a2 + b2 + c2(F1 − F2) +V = (a2 + b2)x2 + 4(az + by)2 + k1 +x2 + ++k2(az + by) + F(ay − bz) +I1 = aM2 ˙x − bM3 ˙x − k2 +2 (a2 + b2)x2− +−2(a2 + b2)(az + by)x2 + 2k1(az+by) +x2 +I2 = 1 +2 ˙x2 + (a2 + b2)x2 + k1 +x2 +V = +k +√ +(ax+by)2+(a2+b2)z2 +I1 = aM2 ˙x − bM1 ˙x + azV +V = k1 +x2 + k2 +y2 + k3 +z2 + F(r) +I1 = 1 +2M 2 +1 + k2 z2 +y2 + k3 +y2 +z2 +I2 = 1 +2M 2 +2 + k1 z2 +x2 + k3 x2 +z2 +I3 = 1 +2M 2 +3 + k1 +y2 +x2 + k2 x2 +y2 +V = +F1( y +x) +x2+y2 + F2(x2 + y2, z) +I1 = 1 +2M 2 +3 + F1 +� y +x +� +V = �m +j=1 +Fj( y +x) +R2 +Nj +� z +R +� ++ F(r) +I = 1 +2M2 + �m +j=1 +r2Fj( y +x) +R2 +Nj +� z +R +� +V = F1(w, z) + F2(w) ¯w +I1 = ( ˙x + i ˙y)2 + 4 +� +F2(w)dw +V = F1( ¯w, z) + F2( ¯w)w +I1 = ( ˙x − i ˙y)2 + 4 +� +F2( ¯w)d ¯w +Table 4: Possibly non-integrable potentials V (x, y, z) in E3 that +admit one or more QFIs of the type I(1,1) which are not in involu- +tion. +Integrable potentials +Potential +LFIs and QFIs +V = F2(w) ¯w + F3(w) + F4(z) +I1 = ( ˙x + i ˙y)2 + 4 +� +F2(w)dw +I2 = 1 +2 ˙z2 + F4(z) +V = F2( ¯w)w + F3( ¯w) + F4(z) +I1 = ( ˙x − i ˙y)2 + 4 +� +F2( ¯w)d ¯w +I2 = 1 +2 ˙z2 + F4(z) +V = F ′ +2z2 + F3(w)z + F4(w) + F2(w) ¯w +I1 = 1 +2 ˙z ( ˙x + i ˙y) + F2(w)z + 1 +2 +� +F3(w)dw +I2 = ( ˙x + i ˙y)2 + 4 � F2(w)dw +V = F ′ +2z2 + F3( ¯w)z + F4( ¯w) + F2( ¯w)w +I1 = 1 +2 ˙z ( ˙x − i ˙y) + F2( ¯w)z + 1 +2 +� +F3( ¯w)d ¯w +I2 = ( ˙x − i ˙y)2 + 4 +� +F2( ¯w)d ¯w +V = F3(w) + z2 +w F2(w) + +F4( z +w) +w2 ++ F2(w) ¯w +I1 = 1 +2 (M2 − iM1)2 + F4 +� z +w +� +I2 = ( ˙x + i ˙y)2 + 4 +� +F2(w)dw +V = F3( ¯w) + z2 +¯ +w F2( ¯w) + +F4( z +¯ +w) +¯ +w2 ++ F2( ¯w)w +I1 = 1 +2 (M2 + iM1)2 + F4 +� z +¯w +� +I2 = ( ˙x − i ˙y)2 + 4 +� +F2( ¯w)d ¯w +Table 5: Integrable potentials V (x, y, z) in E3 that admit QFIs of +the type I(1,1). +24 + +Minimally superintegrable potentials +Potential +Ref [2] +LFIs and QFIs +V = +F1( y +x) +R2 ++ k1(R2 + 4z2) + k2z +Table II +k2 = 0 +I1 = 1 +2 ˙z2 + 4k1z2 + k2z +I2 = 1 +2M 2 +3 + F1 +� y +x +� +I3 = M2 ˙x − M1 ˙y + +2zF1( y +x) +R2 +− 2k1zR2 − k2 R2 +2 +V = +F1( y +x) +R2 ++ k1z +rR2 − k2 +r +Table II +I1 = 1 +2M 2 +3 + F1 +� y +x +� +I2 = 1 +2M2 + +r2F1( y +x) +R2 ++ k1zr +R2 +I3 = M2 ˙x − M1 ˙y + +2zF1( y +x) +R2 ++ 2k1z2 +rR2 + k1 +r − k2z +r +V = +F1( y +x) +R2 ++ k1r2 + k2 +z2 +Table II +I1 = 1 +2 ˙z2 + k1z2 + k2 +z2 +I2 = 1 +2M 2 +3 + F1 +� y +x +� +I3 = 1 +2M2 + +r2F1( y +x) +R2 ++ k2R2 +z2 +V = 4k1 +� +R2 − ¯w3 +2 +� ++ k2 +� +2w − 3 ¯w2� ++ ++k3 ¯w + F(z) +New +I1 = 1 +8 ( ˙x − i ˙y)2 + k1 ¯w2 + k2 ¯w +I2 = 1 +4 ˙w2 + iM3 ( ˙x − i ˙y) − 2k1 +� 3 +4 ¯w4 − w2 + R2 ¯w +� +− +−2k2 +� +¯w3 + 2R2� ++ k3 +� +¯w2 +2 + w +� +I3 = 1 +2 ˙z2 + F(z) +V = k1r2 + k2 ¯w + k3z + F4(w) +New +I1 = 1 +2 ˙z ( ˙x + i ˙y) + k1wz + k2z + k3 +2 w +I2 = ( ˙x + i ˙y)2 + 2k1w2 + 4k2w +I3 = 1 +2 ˙z2 + k1z2 + k3z +V = k1r2 + k2w + k3z + F4( ¯w) +New +I1 = 1 +2 ˙z ( ˙x − i ˙y) + k1 ¯wz + k2z + k3 +2 ¯w +I2 = ( ˙x − i ˙y)2 + 2k1 ¯w2 + 4k2 ¯w +I3 = 1 +2 ˙z2 + k1z2 + k3z +V = k1r2 + k2 +z2 + F3(w) +New +I1 = 1 +2 (M2 − iM1)2 + k2 w2 +z2 +I2 = ( ˙x + i ˙y)2 + 2k1w2 +I3 = 1 +2 ˙z2 + k1z2 + k2 +z2 +V = k1r2 + k2 +z2 + F3( ¯w) +New +I1 = 1 +2 (M2 + iM1)2 + k2 ¯w2 +z2 +I2 = ( ˙x − i ˙y)2 + 2k1 ¯w2 +I3 = 1 +2 ˙z2 + k1z2 + k2 +z2 +Table 6: Minimally superintegrable potentials V (x, y, z) in E3 that +admit QFIs of the type I(1,1). +25 + +Maximally superintegrable potentials +Potential +Ref [2] +Ref [4] +LFIs and QFIs +V = k1(R2 + 4z2) + k2 +x2 + k3 +y2 + k4z +Table I +k4 = 0 +eq. (13) +z ↔ x +I1 = 1 +2 ˙x2 + k1x2 + k2 +x2 +I2 = 1 +2 ˙y2 + k1y2 + k3 +y2 +I3 = 1 +2 ˙z2 + 4k1z2 + k4z +I4 = M2 ˙x + 2z +� k2 +x2 − k1x2� +− k4 +2 x2 +I5 = −M1 ˙y + 2z +� +k3 +y2 − k1y2� +− k4 +2 y2 +I6 = 1 +2M 2 +3 + k2 +� y +x +�2 + k3 +� +x +y +�2 +V = k1 +4 +� +x2 + 16y2 + 4z2� ++ k2 +x2 + ++k3y +New +New +I1 = 1 +2 ˙x2 + k1 +4 x2 + k2 +x2 +I2 = 1 +2 ˙y2 + 4k1y2 + k3y +I3 = 1 +2 ˙z2 + k1z2 +I4 = M2 ˙x + z +� 2k2 +x2 − k1 +2 x2� +I5 = M1 ˙z − z2 � +2k1y + k3 +2 +� +V = kr2 + k1 +x2 + k2 +y2 + k3 +z2 +Table I +eq. (10) +I1 = 1 +2 ˙x2 + kx2 + k1 +x2 +I2 = 1 +2 ˙y2 + ky2 + k2 +y2 +I3 = 1 +2 ˙z2 + kz2 + k3 +z2 +I4 = 1 +2M 2 +1 + k2 z2 +y2 + k3 +y2 +z2 +I5 = 1 +2M 2 +2 + k1 z2 +x2 + k3 x2 +z2 +I6 = 1 +2M 2 +3 + k1 +y2 +x2 + k2 x2 +y2 +V = − c1 +r + c2 +x2 + c3 +y2 +Table I +not +included +I1 = 1 +2M 2 +1 + c3 z2 +y2 +I2 = 1 +2M 2 +2 + c2 z2 +x2 +I3 = 1 +2M 2 +3 + c2 +y2 +x2 + c3 x2 +y2 +I4 = M2 ˙x − M1 ˙y − 2z +� +c1 +2r − c2 +x2 − c3 +y2 +� +V = c1y +x2R + c2 +x2 + c3 +z2 +Table I +x ↔ y +not +included +I1 = 1 +2 ˙z2 + c3 +z2 +I2 = 1 +2M 2 +3 + c2 +y2 +x2 + c1 +yR +x2 +I3 = M3 ˙x − 2c2 +y +x2 − c1 +x2+2y2 +x2R +I4 = 1 +2M2 + c1 +yr2 +Rx2 + c2 r2 +x2 + c3 R2 +z2 +V = c1y +x2R + c2 +x2 + c3z +Table I +x ↔ y +not +included +I1 = 1 +2 ˙z2 + c3z +I2 = 1 +2M 2 +3 + c2 +y2 +x2 + c1 +yR +x2 +I3 = M3 ˙x − 2c2 +y +x2 − c1 +x2+2y2 +x2R +I4 = M2 ˙x − M1 ˙y + c1 +2yz +x2R + c2 2z +x2 − c3 R2 +2 +V = k1r2 + k2 ¯w +w3 + k3 +w2 + k4 +z2 +w = x + iy = Reiθ +¯w = x − iy +not +included +eq. (11) +I1 = 1 +2 (M2 − iM1)2 + k4 w2 +z2 − k2 z2 +w2 +I2 = 1 +2 ( ˙x + i ˙y)2 + k1w2 − k2 +w2 +I3 = 1 +2M 2 +3 + k2e−4iθ + k3e−2iθ += 1 +2M 2 +3 + k2 +� ¯w +w +�2 + k3 ¯ +w +w +I4 = 1 +2 ˙z2 + k1z2 + k4 +z2 +I5 = 1 +2M2 + k2 r2 ¯w +w3 + k3 r2 +w2 + k4 r2 +z2 +V = k1r2 + k2 +w2 + k3 z +w3 + ++k4 R2−3z2 +w4 +not +included +eq. (12) +I1 = 1 +2 (M2 − iM1)2 + k3 z +w − 4k4 z2 +w2 +I2 = M3 (iM1 − M2) + k3 +y +w − 2ik2 z +w− +− 3ik3 +2 +z2 +w2 − 4ik4 +z(x2+y2−z2) +w3 +I3 = 1 +2 ( ˙x + i ˙y)2 + k1w2 − k4 +w2 +I4 = 1 +2 ˙z ( ˙x + i ˙y) + k1zw − +k3 +4w2 + k4 z +w3 +I5 = 1 +2M2 + k2 r2 +w2 + k3 zr2 +w3 + k4 +r2(x2+y2−3z2) +w4 +26 + +V = k1(R2 + 4z2) + k2z + k3 +w2 + ++k4 ¯w +w3 +not +included +eq. (14) +I1 = 1 +2 ( ˙x + i ˙y)2 + k1w2 − k4 +w2 +I2 = 1 +2( ˙x + i ˙y) (M2 − iM1) − k1zw2− +− k2 +4 w2 − k4 z +w2 +I3 = 1 +2 ˙z2 + 4k1z2 + k2z +I4 = 1 +2M 2 +3 + k3e−2iθ + k4e−4iθ +I5 = 1 +2 (M2 ˙x − M1 ˙y) + k3 z +w2 + k4 z ¯ +w +w3 − +−k1zR2 − k2 R2 +4 +V = 4k1 +� +R2 − ¯w3 +2 + z2 +4 +� ++ ++k2 +� +2w − 3 ¯w2� ++ k3 ¯w+ ++ k4 +z2 +not +included +eq. (15) +I1 = 1 +8 ( ˙x − i ˙y)2 + k1 ¯w2 + k2 ¯w +I2 = 1 +4 ˙w2 + iM3 ˙¯w − 2k1 +� 3 +4 ¯w4 − w2 + R2 ¯w +� +− +−2k2 +� +¯w3 + 2R2� ++ k3 +� +¯ +w2 +2 + w +� +I3 = 1 +2 ˙z2 + k1z2 + k4 +z2 +I4 = 1 +2 (M2 + iM1)2 + 2i ˙zM1+ ++k1z2 � +3 ¯w2 − 4iy +� ++ 2k2z2 (2 ¯w + 1) − +−k3z2 + k4 +z2 +� +¯w2 + 4iy +� +I5 = 1 +2 ˙z (M2 + iM1) + k1z2 ¯w + k2z2 − k4 ¯ +w +z2 +V = k1w + k2 +� +3w2 + z +� ++ ++k3 +� +4w3 + 3wz + ¯ +w +4 +� ++ ++k4 +� +5 +2w4 + r2 +2 + 3w2z +� +not +included +eq. (16) +I1 = M3 ˙w − (M1 + iM2) ˙z − 1 +2 ˙y ˙z+ ++ ik1 +2 +� +w2 − z +� ++ ik2 +� +2w3 − zw + i +2y +� +− +− ik3 +8 +� +w2 − z +� ++ ik3 +� +3w4 − z2 + iyw +� +− +− k4 +2 y +� +w2 + z +� ++ ik4w +� +2w4 + zw2 − z2� +I2 = M1 (2 ˙x + i ˙y) + iM2 ˙x + M3 ˙z + i +4 ˙z2+ ++ ik2 +2 +� +z − w2� ++ ik3w +� +z − w2� ++ ++ ik4 +4 +� +z2 + 2zw2 − 3w4� +I3 = (M1 + iM2)2 + (2iM1 − M2) ˙w− +−iM3 ˙z + 1 +4 +� +˙y2 − ˙z2� ++ k1zw+ ++ ik1 +2 y + 2k2w (2zw + iy) + ++ k2 +2 +� +w2 − z +� ++ k3w3(6z + 1)+ ++k3zw +� +2z − 1 +4 +� ++ ik3y +� +3w2 − 1 +8 +� +− +−k3xz + k4w4 � +4z + 3 +4 +� ++ 2ik4yw3+ ++k4z2 � +3w2 − 1 +4 +� ++ k4 +2 +� +y2 +2 − zR2� +I4 = ˙w2 + k3w + k4w2 +I5 = ˙z ˙w + +� +k4w + k3 +2 +� +z + k4w3+ ++ 3k3 +2 w2 + k2w +Table 7: Maximally superintegrable potentials V (x, y, z) in E3 that +admit QFIs of the type I(1,1). +Remark: The potential (68) admits the four independent QFIs H, I1, I2 and I3 (see Table 4); however, +it is not second order integrable because the PBs {Ii, Ij} ̸= 0 for i ̸= j. +In Table II of [2], it is claimed +that this potential is minimally superintegrable because in that paper superintegrability is defined without the +requirement of the integrability (i.e. the vanishing of the PBs). Indeed, we have: +I4 ≡ {I1, I3} = {I2, I1} = {I3, I2} = M1 +� +x2 ˙y ˙z + 2k1 +yz +x2 +� ++M2 +� +y2 ˙x ˙z + 2k2 +xz +y2 +� ++M3 +� +z2 ˙x ˙y + 2k3 +xy +z2 +� +. (133) +The third order (cubic) FI I4 cannot be used for establishing integrability because the PBs {Ii, I4} ̸= 0, where +i, j = 1, 2, 3. +6 +The QFI I(2,0) where ℓ = 0 +We set L(0)a = La and the QFI I(2,ℓ) for ℓ = 0 becomes +I(2,0) = −tL(a;b) ˙qa ˙qb + La ˙qa + tLaV ,a +(134) +27 + +where the vector La is given by (15), the generated KT L(a;b) is given by (16) and the following condition is +satisfied +� +LbV ,b� +,a = −2L(a;b)V ,b. +(135) +Condition (135) is a subcase of the general condition (22) in the case that the function G = −LaV ,a and the +general second order KT Cab = L(a;b) is reducible. In section 5.1, we have computed (not all) pairs of functions +(G, V ) which satisfy the condition (22). Therefore, in order to find potentials V (x, y, z) that admit QFIs of the +form (134), it is sufficient to solve the constraint +G = −LaV ,a +(136) +for all pairs (G, V ) for which the KT Cab is given by the reducible form (16). If the constraint (136) is not +satisfied for some pairs (G, V ), then the corresponding potentials V of these pairs do not admit QFIs of the +type I(2,0). +Moreover, the QFI (134) is written as +I(2,0) = −Jt + La ˙qa +where J is the associated autonomous QFI (20). The PB {H, I(2,0)} = +∂I(2,0) +∂t += −J. Therefore: +The time-dependent QFI I(2,0) generates an autonomous QFI of the type I(1,0). +This is an interesting connection between (first degree) time-dependent and autonomous QFIs. +We consider the following cases. +1) In section 5.1.2, we determined the functions: +V (x, y, z) += +(a2 + b2)x2 + 4(az + by)2 + k1 +x2 + k2(az + by) + F(ay − bz) +(137) +G(x, y, z) += +−k2 +2 (a2 + b2)x2 − 2ab(ay + bz)x2 − 2(a3z + b3y)x2 + 2k1(by + az) +x2 +. +(138) +Then, the vector +La = + + +bxy + axz + 2b2y + 2b1z + b3 +−bx2 − 2b2x + 2b4z + b6 +−ax2 − 2b1x − 2b4y + b5 + + +(139) +where b1, b2, ..., b6 are arbitrary constants. Replacing (137), (138) and (139) in the condition (136), we find: +b1 = b2 = b3 = b4 = 0, a = ±ib, b5 = ±b6. +Therefore, the potential (137) becomes9 (see the potential V = F1(y − bx) + F2(z) in Table 2) +V (x, y, z) = k1 +x2 + 4b(y ± iz)2 + bk2(y ± iz) + F(y ± iz) +� +�� +� +=F1(y±iz) += k1 +x2 + F1(y ± iz) +(140) +and the vector +La = + + +bx(y ± iz) +−bx2 + b6 +±i(−bx2 + b6) + + . +(141) +The associated time-dependent QFI (134) is +I(2,0) += +−bt(y ± iz) ˙x2 + btx ˙x( ˙y ± i ˙z) + b(y ± iz)x ˙x − (bx2 − b6) ˙y ∓ i(bx2 − b6) ˙z − 2k1bt(y ± iz) +x2 += +b6J1 − bJ2 +(142) +which contains the independent FIs: +J1 = ˙y ± i ˙z, J2 = t +� +˙x2 + 2k1 +x2 +� +(y ± iz) − x ˙x(y ± iz) − J1x(t ˙x − x). +9The function F (iz ± y) is either the F (y + iz) or the F (y − iz). Therefore, we can write F (y ± iz). +28 + +From section 5.1.2 we have that the potential (140) admits also the autonomous QFIs: +J3 = (±iM2 − M3) ˙x + 2k1(y ± iz) +x2 +, J4 = 1 +2 ˙x2 + k1 +x2 . +We note that J2 = J3t − x ˙x(y ± iz) + J1x2. +The potential (140) is maximally superintegrable due to the five linearly independent FIs H, J1, J2, J3, and +J4. The autonomous FIs H, J1, J4 are in involution. This is a new result which could not be found in [2] because +of the additional time-dependent QFI J2. +The PBs are: +{H, J2} = ∂J2 +∂t = J3, {J1, J2} = {J1, J3} = {J1, J4} = 0, {J3, J4} = −J1 +� +˙x2 + 2k1 +x2 +� +, +{J2, J3} = −2(M3 ∓ iM2)2 − 4k1 +x2 (y ± iz)2, {J2, J4} = − (J1t + y ± iz) +� +˙x2 + 2k1 +x2 +� ++ 2J1x ˙x. +2) In section 5.1.2, we determined the functions: +V (x, y, z) += +F1 +� y +x +� +R2 ++ k1z +rR2 − k2 +r +(143) +G(x, y, z) += +a2 +2zF1 +� y +x +� +R2 ++ a2 +2k1z2 +rR2 + a2 +k1 +r − a2 +k2z +r . +(144) +Then, the vector +La = + + +axz + 2b2y + 2b1z + b3 +ayz − 2b2x + 2b4z + b6 +−aR2 − 2b1x − 2b4y + b5 + + . +(145) +Replacing (143), (144) and (145) in the condition (136), we get: +b1 = b2 = b3 = b4 = b5 = b6 = 0, k2 = 0. +Therefore, the potential (143) becomes +V (x, y, z) = F1 +� y +x +� +R2 ++ k1z +rR2 = R−2 +� +F1 +�y +x +� ++ k1z +r +� +(146) +and the vector Lb = a +� +xz, yz, −R2� +. +The associated time-dependent QFI (134) is +I(2,0) = −J1t + z(x ˙x + y ˙y) − (x2 + y2) ˙z +(147) +where J1 is the autonomous QFI +J1 = M2 ˙x − M1 ˙y + 2zF1 +� y +x +� +x2 + y2 + +2k1z2 +r(x2 + y2) + k1 +r . +From Table 6 we have that the potential (146) admits the additional autonomous QFIs: +J2 = 1 +2M 2 +3 + F1 +�y +x +� +, J3 = 1 +2M2 + r2F1 +� y +x +� +x2 + y2 + +k1zr +x2 + y2 . +Therefore, (146) is a new maximally superintegrable potential due to the five independent QFIs H, J1, J2, J3, +and (147). We note that this potential was considered to be minimally superintegrable (see e.g. [2]) because +only autonomous QFIs were considered. +The PBs are {H, I(2,0)} = −J1 and {I(2,0), J2} = 0. +3) In section 5.1.1, we determined the functions: +V (x, y, z) += +F1(x) + F2(y) + F3(z) +(148) +29 + +G(x, y, z) += +2a3F1(x) + 2a13F2(y) + 2a9F3(z). +(149) +Then, the vector +La = + + +a3x + 2b2y + 2b1z + b3 +a13y − 2b2x + 2b4z + b6 +a9z − 2b1x − 2b4y + b5 + + . +(150) +Replacing (148), (149) and (150) in the condition (136), we obtain the following ordinary differential equation +(ODE): +0 += +a3 [xF ′ +1 + 2F1(x)] + b3F ′ +1 + a13 [yF ′ +2 + 2F2(y)] + b6F ′ +2 + a9 [zF ′ +3 + 2F3(z)] + b5F ′ +3 + ++2b2 (F ′ +1y − F ′ +2x) + 2b1 (F ′ +1z − F ′ +3x) + 2b4 (F ′ +2z − F ′ +3y) +(151) +where F ′ +1 = dF1 +dx , F ′ +2 = dF2 +dy and F ′ +3 = dF3 +dz . +We consider the following subcases: +3.1. Subcase b1 = b2 = b4 = 0 and the pairs (a3, b3), (a13, b6), (a9, b5) are not the origin (0, 0). +Then, the ODE (151) gives: +a3 [xF ′ +1 + 2F1(x)] + b3F ′ +1 += +λ1 +(152) +a13 [yF ′ +2 + 2F2(y)] + b6F ′ +2 += +λ2 +(153) +a9 [zF ′ +3 + 2F3(z)] + b5F ′ +3 += +−λ1 − λ2 +(154) +where λ1 and λ2 are arbitrary constants, and the vector La = + + +a3x + b3 +a13y + b6 +a9z + b5 + +. +Solving the system of ODEs (152) - (154), we find the functions: +F1(x) = λ1 +� a3 +2 x2 + b3x +� ++ c1 +(a3x + b3)2 +, F2(y) = λ2 +� a13 +2 y2 + b6y +� ++ c2 +(a13y + b6)2 +, F3(z) = −(λ1 + λ2) +� a9 +2 z2 + b5z +� ++ c3 +(a9z + b5)2 +where c1, c2, and c3 are arbitrary constants. +Then, the potential (148) becomes +V (x, y, z) = λ1 +� a3 +2 x2 + b3x +� ++ c1 +(a3x + b3)2 ++ λ2 +� a13 +2 y2 + b6y +� ++ c2 +(a13y + b6)2 +− (λ1 + λ2) +� a9 +2 z2 + b5z +� ++ c3 +(a9z + b5)2 +. +(155) +The associated time-dependent QFI (134) is +I(2,0) = −Jt + (a3x ˙x + a13y ˙y + a9z ˙z) + b3 ˙x + b6 ˙y + b5 ˙z +(156) +where J = 2a3I1 + 2a13I2 + 2a9I3 is the sum of the three separated QFIs: +I1 = 1 +2 ˙x2 + F1(x), I2 = 1 +2 ˙y2 + F2(y), I3 = 1 +2 ˙z2 + F3(z). +(157) +Therefore, (155) is a new minimally superintegrable potential due to the four independent QFIs I1, I2, I3, and +(156). We note that (155) depends on the eleven parameters a3, a9, a13, b3, b5, b6, c1, c2, c3, λ1 and λ2; hence, the +time-dependent QFI (156) is irreducible. +- For λ1 = λ2 = 0 and a3a13a9 ̸= 0 we obtain the potential10 +V (x, y, z) = +k1 +(x + m1)2 + +k2 +(y + m2)2 + +k3 +(z + m3)2 +(158) +where k1, k2, k3, m1, m2, and m3 are new arbitrary constants. +Then, the associated time-dependent QFI (156) consists of the independent QFIs: +I4 = −2I1t + (x + m1) ˙x, I5 = −2I2t + (y + m2) ˙y, I6 = −2I3t + (z + m3) ˙z. +10Since a3a13a9 ̸= 0, we can set b3 = m1a3, b6 = m2a13, b5 = m3a9, k1 = c1 +a2 +3 , k2 = +c2 +a2 +13 and k3 = c3 +a2 +9 . +30 + +Therefore, the potential (158) is maximally superintegrable due to the independent QFIs I1, I2, ..., I6. Because +time-dependent FIs are considered, the maximum number of independent FIs is six (i.e. greater than five). +3.2. Subcase b1 = b2 = b4 = 0, a3 ̸= 0 and a9 = a13 = b5 = b6 = 0 (i.e. two pairs of parameters from subcase +3.1 vanish). +From the system of ODEs (152) - (154), we find that λ1 = λ2 = 0 and the potential (148) becomes +V (x, y, z) = +k1 +(x + m1)2 + F2(y) + F3(z) +(159) +where k1, m1 are arbitrary constants and F2(y), F3(z) are arbitrary smooth functions. +The associated time-dependent QFI (134) is +I(2,0) = −2I1t + (x + m1) ˙x +(160) +where the QFI I1 = 1 +2 ˙x2 + +k1 +(x+m1)2 . Therefore, the potential (159) is minimally superintegrable (see Table 2). +3.3. Subcase b1 = b2 = b4 = 0, a9 = b5 = 0 and the pairs (a3, b3), (a13, b6) are not the origin (0, 0). +From the system of ODEs (152) - (154), we find that λ2 = −λ1 and the potential (148) becomes +V (x, y, z) = λ1 +� a3 +2 x2 + b3x +� ++ c1 +(a3x + b3)2 +− λ1 +� a13 +2 y2 + b6y +� ++ c2 +(a13y + b6)2 ++ F3(z) +(161) +where F3(z) is an arbitrary smooth function. +The associated time-dependent QFI (134) is +I(2,0) = −2(a3I1 + a13I2)t + (a3x + b3) ˙x + (a13y + b6) ˙y +(162) +where the QFIs I1 and I2 are given by (157). We note that the potential (161) is a minimally superintegrable +potential. +- For λ1 = 0 and a3a13 ̸= 0 we obtain the maximally superintegrable potential (see Table 3) +V (x, y, z) = +k1 +(x + m1)2 + +k2 +(y + m2)2 + F3(z) +(163) +which admits the additional time-dependent QFIs: +I4 = −2I1t + (x + m1) ˙x, I5 = −2I2t + (y + m2) ˙y. +3.4. Subcase a3 = a9 = a13 = 0 (autonomous LFIs, L(a;b) = 0 and G = 0). +The ODE (151) becomes +2b2 (F ′ +1y − F ′ +2x) + 2b1 (F ′ +1z − F ′ +3x) + 2b4 (F ′ +2z − F ′ +3y) + b3F ′ +1 + b6F ′ +2 + b5F ′ +3 = 0 +(164) +and the vector +La = + + +2b2y + 2b1z + b3 +−2b2x + 2b4z + b6 +−2b1x − 2b4y + b5 + + . +(165) +The ODE (164) admits solutions of the form: +F1(x) = kx2 + k1x, F2(y) = ky2 + k2y, F3(z) = kz2 + k3z +(166) +where k, k1, k2, and k3 are arbitrary constants. Then, we get the separable potential +V (x, y, z) = kr2 + k1x + k2y + k3z. +(167) +Replacing (166) in (164), we find the following system of equations: +k1b3 + k2b6 + k3b5 += +0 +(168) +kb3 − k2b2 − k3b1 += +0 +(169) +31 + +kb6 + k1b2 − k3b4 += +0 +(170) +kb5 + k1b1 + k2b4 += +0. +(171) +We consider the following cases. +- Case k = 0. +The potential (167) becomes +V (x, y, z) = k1x + k2y + k3z +(172) +where k1k2k3 ̸= 0 in order to have a 3d potential. +Solving the system of equations (168) - (171) for k = 0, we find: +b1 = −k2 +k1 +b4, b2 = k3 +k1 +b4, b3 = −k2 +k1 +b6 − k3 +k1 +b5. +The associated QFI (134) reduces to the LFI +I = La ˙qa = −2b4 +3 +� +i=1 +kiMi − b5(k3 ˙x − k1 ˙z) − b6(k2 ˙x − k1 ˙y) +which consists of the LFIs: +J1 = +3 +� +i=1 +kiMi, J2 = k3 ˙x − k1 ˙z, J3 = k2 ˙x − k1 ˙y. +Therefore, the separable potential (172) is maximally superintegrable. +- Case k ̸= 0. +The system of equations (168) - (171) implies that b3 = k2 +k b2+ k3 +k b1, b5 = − k1 +k b1− k2 +k b4, and b6 = k3 +k b4− k1 +k b2. +Similarly, we find the LFIs: +J1 = 2kM1 + k2 ˙z − k3 ˙y, J2 = 2kM2 + k3 ˙x − k1 ˙z, J3 = 2kM3 + k1 ˙y − k2 ˙x. +Therefore, the separable potential (167) is maximally superintegrable. We note that the k ̸= 0 introduces the +term kr2 which is the oscillator; therefore, the corresponding change in the FIs is the addition of the components +of the angular momentum. +6.1 +Case La is a KV +We consider that La is a KV in E3. Then, L(a;b) = 0 and the time-dependent QFI (134) becomes the time- +dependent LFI +I = La ˙qa + st +(173) +where the arbitrary constant s satisfies the condition +LaV ,a = s. +(174) +Replacing the general KV La given by (13) in (174), we find the PDE +(b1 − b4y + b5z) ∂V +∂x + (b2 + b4x − b6z) ∂V +∂y + (b3 − b5x + b6y) ∂V +∂z = s +(175) +where b1, ..., b6 are arbitrary constants. +We consider the following cases. +1) Case b1 ̸= 0 and b4 = b5 = b6 = 0. +Then, the PDE (175) gives the potential +V = c1x + F(y − c2x, z − c3x) +(176) +where c1 = +s +b1 , c2 = b2 +b1 , c3 = b3 +b1 and F is an arbitrary smooth function of its arguments. +32 + +The associated LFI (173) is +I = ˙x + c2 ˙y + c3 ˙z + c1t. +(177) +2) Case b2 ̸= 0, b1 = 0 and b4 = b5 = b6 = 0. +We find a subcase of the potential (176) for x ↔ y and c2 = 0. +3) Case b3 ̸= 0, b1 = b2 = 0 and b4 = b5 = b6 = 0. +We find a subcase of the potential (176) for c1 = c, c2 = c3 = 0 and x ↔ z. +4) Case b4 ̸= 0 and b5 = b6 = 0. +Then, the PDE (175) gives the potential +V = c0 tan−1 +�x + c2 +y + c1 +� ++ F +�1 +2(x2 + y2) + c2x + c1y, z + c3 tan−1 +�x + c2 +y + c1 +�� +(178) +where c0 = +s +b4 , c1 = − b1 +b4 , c2 = b2 +b4 , c3 = − b3 +b4 and F is an arbitrary function of its arguments. +The associated LFI (173) is +I = M3 − c1 ˙x + c2 ˙y − c3 ˙z + c0t. +(179) +5) Case b4 ̸= 0, b6 = 0 and b2 = b3 = 0. +Then, the PDE (175) gives the potential +V = +c0 +� +1 + c2 +1 +tan−1 +� +y + c1z + c2 +� +1 + c2 +1x +� ++ F +� +z − c1y, x2 + (1 − c2 +1)y2 + 2c2y + 2c1yz +� +(180) +where c0 = +s +b4 , c1 = − b5 +b4 , c2 = − b1 +b4 and F is an arbitrary function of its arguments. +The associated LFI (173) is +I = M3 − c1M2 − c2 ˙x + c0t. +(181) +6) Case b1 = b2 = b3 = s = 0 and b6 ̸= 0. +Then, the PDE (175) gives the potential +V = F(r, x − c1y − c2z) +(182) +where c1 = − b5 +b6 , c2 = − b4 +b6 and F is an arbitrary function of its arguments. +The associated LFI (173) is +I = M1 − c1M2 − c2M3. +(183) +We collect the results of section 6 in Tables 8 - 10. +33 + +Minimally superintegrable potentials +Potential +Ref [2] +LFIs and QFIs +V = +λ1( +a3 +2 x2+b3x)+c1 +(a3x+b3)2 ++ +λ2( +a13 +2 y2+b6y)+c2 +(a13y+b6)2 +− +− +(λ1+λ2)( +a9 +2 z2+b5z)+c3 +(a9z+b5)2 +New +I = −Jt + (a3x ˙x + a13y ˙y + a9z ˙z)+ ++b3 ˙x + b6 ˙y + b5 ˙z +J = 2a3I1 + 2a13I2 + 2a9I3 +I1 = 1 +2 ˙x2 + +λ1( +a3 +2 x2+b3x)+c1 +(a3x+b3)2 +I2 = 1 +2 ˙y2 + +λ2( +a13 +2 y2+b6y)+c2 +(a13y+b6)2 +I3 = 1 +2 ˙z2 − +(λ1+λ2)( +a9 +2 z2+b5z)+c3 +(a9z+b5)2 +V = +k1 +(x+m1)2 + F2(y) + F3(z) +New +I1 = 1 +2 ˙x2 + +k1 +(x+m1)2 +I2 = 1 +2 ˙y2 + F2(y) +I3 = 1 +2 ˙z2 + F3(z) +I4 = −2I1t + (x + m1) ˙x +V = +λ1( +a3 +2 x2+b3x)+c1 +(a3x+b3)2 +− +− +λ1( +a13 +2 y2+b6y)+c2 +(a13y+b6)2 ++ F3(z) +New +I1 = 1 +2 ˙x2 + +λ1( +a3 +2 x2+b3x)+c1 +(a3x+b3)2 +I2 = 1 +2 ˙y2 − +λ1( +a13 +2 y2+b6y)+c2 +(a13y+b6)2 +I3 = 1 +2 ˙z2 + F3(z) +I4 = −2(a3I1 + a13I2)t + (a3x + b3) ˙x+ ++(a13y + b6) ˙y +Table 8: Minimally superintegrable potentials V (x, y, z) in E3 that +admit time-dependent QFIs of the form I(2,0). +34 + +Maximally superintegrable potentials +Potential +Ref [2] +LFIs and QFIs +V = k1 +x2 + F1(y ± iz) +New +J1 = ˙y ± i ˙z +J2 = J3t − x ˙x(y ± iz) + J1x2 +J3 = (±iM2 − M3) ˙x + 2k1(y±iz) +x2 +J4 = 1 +2 ˙x2 + k1 +x2 +V = R−2 � +F1 +� y +x +� ++ k1z +r +� +New +J1 = M2 ˙x − M1 ˙y + +2zF1( y +x) +x2+y2 ++ +2k1z2 +r(x2+y2) + k1 +r +J2 = 1 +2M 2 +3 + F1 +� y +x +� +J3 = 1 +2M2 + +r2F1( y +x) +x2+y2 + +k1zr +x2+y2 +J4 = −J1t + z(x ˙x + y ˙y) − (x2 + y2) ˙z +V = +k1 +(x+m1)2 + +k2 +(y+m2)2 + +k3 +(z+m3)2 +New +I1 = 1 +2 ˙x2 + +k1 +(x+m1)2 +I2 = 1 +2 ˙y2 + +k2 +(y+m2)2 +I3 = 1 +2 ˙z2 + +k3 +(z+m3)2 +I4 = −2I1t + (x + m1) ˙x +I5 = −2I2t + (y + m2) ˙y +I6 = −2I3t + (z + m3) ˙z +V = +k1 +(x+m1)2 + +k2 +(y+m2)2 + F3(z) +New +I1 = 1 +2 ˙x2 + +k1 +(x+m1)2 +I2 = 1 +2 ˙y2 + +k2 +(y+m2)2 +I3 = 1 +2 ˙z2 + F3(z) +I4 = −2I1t + (x + m1) ˙x +I5 = −2I2t + (y + m2) ˙y +V = k1x + k2y + k3z +New +I1 = 1 +2 ˙x2 + k1x +I2 = 1 +2 ˙y2 + k2y +I3 = 1 +2 ˙z2 + k3z +I4 = �3 +i=1 kiMi +I5 = k3 ˙x − k1 ˙z +I6 = k2 ˙x − k1 ˙y +V = kr2 + k1x + k2y + k3z +New +I1 = 1 +2 ˙x2 + kx2 + k1x +I2 = 1 +2 ˙y2 + ky2 + k2y +I3 = 1 +2 ˙z2 + kz2 + k3z +I4 = 2kM1 + k2 ˙z − k3 ˙y +I5 = 2kM2 + k3 ˙x − k1 ˙z +I6 = 2kM3 + k1 ˙y − k2 ˙x +Table 9: Maximally superintegrable potentials V (x, y, z) in E3 that +admit QFIs of the form I(2,0). +Potential +LFIs and QFIs +V = c1x + F(y − c2x, z − c3x) +I = ˙x + c2 ˙y + c3 ˙z + c1t +V = c0 tan−1 � +x+c2 +y+c1 +� ++ ++F +� +1 +2(x2 + y2) + c2x + c1y, z + c3 tan−1 � +x+c2 +y+c1 +�� +I = M3 − c1 ˙x + c2 ˙y − c3 ˙z + c0t +V = +c0 +√ +1+c2 +1 tan−1 +� +y+c1z+c2 +√ +1+c2 +1x +� ++ ++F +� +z − c1y, x2 + (1 − c2 +1)y2 + 2c2y + 2c1yz +� +I = M3 − c1M2 − c2 ˙x + c0t +V = F(r, x − c1y − c2z) +I = M1 − c1M2 − c2M3 +Table 10: Possibly non-integrable potentials V (x, y, z) in E3 that +admit LFIs of the form I = La ˙qa + st. +35 + +7 +The QFI I(3) +In this section, we consider the QFI +I(3) = eλt � +−L(a;b) ˙qa ˙qb + λLa ˙qa + LaV ,a� +where the vector La is given by (15), the generated KT L(a;b) is given by (16) and the following condition is +satisfied +� +LbV ,b� +,a = −2L(a;b)V ,b − λ2La. +(184) +We consider several cases concerning the parameters a1, a2, ..., a20 which define the vector La given in (15). +7.1 +Case containing KVs and the HV: parameters a1, a3, a4, a6, a7, a9, a10, a13, a14 +In this case, the vector La given in (15) has the general form +La = + + +k1x +k2y +k3z + + + + + +b1 − b4y + b5z +b2 + b4x − b6z +b3 − b5x + b6y + + +(185) +where k1, ..., k3, b1, ..., b6 are arbitrary constants and the generated KT L(a;b) = diag(k1, k2, k3). +We assume k1 = k2 = k3 = k is an arbitrary constant. Then, the vector (185) is the linear combination of the +homothetic vector (HV) with the gradient and non-gradient KVs. The KT L(a;b) = kδab and the time-dependent +QFI I(3) becomes +I = eλt (−k ˙qa ˙qa + λLa ˙qa + LaV ,a) . +(186) +The condition (184) is +� +LbV ,b + 2kV +� +,a + λ2La = 0. +(187) +From the integrability condition of (187), we get: +La,b − Lb,a = 0 =⇒ La,b = kδab =⇒ b4 = b5 = b6 = 0. +This implies that only the HV and the gradient KVs survive, that is, the vector (185) becomes +La = + + +kx + b1 +ky + b2 +kz + b3 + + . +(188) +We consider the following special cases. +1) Case k = 0, b3 = 0 and b1 ̸= 0. +The vector La = (b1, b2, 0). Then, equation (187) gives the potential +V (x, y, z) = λ2 +2 +� +c2 +1 − 1 +� +x2 + c2x − c1λ2xy + F(y − c1x, z) +(189) +where c1 ≡ b2 +b1 , c2 are arbitrary constants and F is an arbitrary smooth function of its arguments. +The associated time-dependent LFI is +I = eλt � +λ ˙x + c1λ ˙y − λ2x − c1λ2y + c2 +� +. +(190) +We note that {H, I} = ∂I +∂t = λI. +- For c1 = 0, the potential (189) becomes +V (x, y, z) = −λ2 +2 x2 + c2x + F(y, z) +(191) +and the associated LFI (190) is +I = eλt � +λ ˙x − λ2x + c2 +� +. +(192) +36 + +In the case that F(y, z) = F1(y) + F2(z), the potential (191) is separable; therefore, it is minimally superinte- +grable due to the additional independent LFI (192). +2) Case k = 0 and b1 = b2 = b3. +We have La = (1, 1, 1). +The potential (after the transformation x ↔ z) +V (x, y, z) = λ2 +2 x2 + kx − λ2(y + z)x + F(x − z, y − z) +(193) +where k is an arbitrary constant and F is an arbitrary smooth function of its arguments. +The associated LFI is +I = eλt � +λ( ˙x + ˙y + ˙z) − λ2(x + y + z) + k +� +. +(194) +3) Case k ̸= 0. +We find the potential +V (x, y, z) = −λ2 +8 r2 − λ2 +4 +�b1 +k x + b2 +k y + b3 +k z +� ++ +1 +� +z + b3 +k +�2 F +� +y + b2 +k +x + b1 +k +, z + b3 +k +x + b1 +k +� +(195) +where F is an arbitrary function of its arguments. +The associated QFI is +I += +eλt +�� +˙x − λ +2 x +�2 ++ +� +˙y − λ +2 y +�2 ++ +� +˙z − λ +2 z +�2 +− λ +�b1 +k ˙x + b2 +k ˙y + b3 +k ˙z +� ++ ++λ2 +2 +�b1 +k x + b2 +k y + b3 +k z + b2 +1 +2k2 + b2 +2 +2k2 + b2 +3 +2k2 +� ++ +2F +� +z + b3 +k +�2 +� +. +(196) +3.1. For b1 = b2 = b3 = 0. +The potential +V (x, y, z) = −λ2 +8 r2 + F +� y +x, z +x +� +z2 +(197) +and the associated QFI is +I = eλt +� +˙x2 + ˙y2 + ˙z2 − λ(x ˙x + y ˙y + z ˙z) + λ2 +4 r2 + 2F +� y +x, z +x +� +z2 +� +. +(198) +3.2. For b1 = k and b2 = b3 = 0. +The potential +V (x, y, z) = −λ2 +8 r2 − λ2 +4 x + +F +� +y +x+1, +z +x+1 +� +z2 +(199) +and the associated QFI is +I = eλt +�� +˙x − λ +2 x +�2 ++ +� +˙y − λ +2 y +�2 ++ +� +˙z − λ +2 z +�2 +− λ ˙x + λ2 +2 x + λ2 +4 + 2F +z2 +� +. +(200) +3.3. For b1 = b2 = k and b3 = 0. +The potential +V (x, y, z) = −λ2 +8 r2 − λ2 +4 (x + y) + 1 +z2 F +�y + 1 +x + 1, +z +x + 1 +� +(201) +and the associated QFI is +I = eλt +�� +˙x − λ +2 x +�2 ++ +� +˙y − λ +2 y +�2 ++ +� +˙z − λ +2 z +�2 +− λ( ˙x + ˙y) + λ2 +2 (x + y + 1) + 2F +z2 +� +. +(202) +37 + +3.4. For b1 = b2 = b3 = k. +The potential +V (x, y, z) = −λ2 +8 r2 − λ2 +4 (x + y + z) + +1 +(z + 1)2 F +�y + 1 +x + 1, z + 1 +x + 1 +� +(203) +and the associated QFI is +I += +eλt +�� +˙x − λ +2 x +�2 ++ +� +˙y − λ +2 y +�2 ++ +� +˙z − λ +2 z +�2 +− λ ( ˙x + ˙y + ˙z) + ++λ2 +2 +� +x + y + z + 3 +2 +� ++ +2F +(z + 1)2 +� +. +(204) +3.5. For F +� +y+ b2 +k +x+ b1 +k , z+ b3 +k +x+ b1 +k +� += F1 +� +y+ b2 +k +x+ b1 +k +x+ b1 +k +z+ b3 +k +� ++ +c0 +(x+ b1 +k ) +2 = F1 +� +y+ b2 +k +z+ b3 +k +� ++ +c0 +(x+ b1 +k ) +2 , where c0 is an arbitrary +constant. +The potential +V (x, y, z) = −λ2 +8 r2 − λ2 +4 (c1x + c2y + c3z) + +c0 +(x + c1)2 + +1 +(z + c3)2 F1 +�y + c2 +z + c3 +� +(205) +where ci = bi +k . +The associated QFI consists of the independent QFIs: +I1 += +eλt +�� +˙x − λ +2 x +�2 +− λc1 +� +˙x − λ +2 x − λc1 +4 +� ++ +2c0 +(x + c1)2 +� +(206) +I2 += +eλt +�� +˙y − λ +2 y +�2 ++ +� +˙z − λ +2 z +�2 +− λ (c2 ˙y + c3 ˙z) + λ2 +2 +� +c2y + c3z + c2 +2 +2 + c2 +3 +2 +� ++ +2F1 +(z + c3)2 +� +. (207) +4) Case k1 = k2 = k3 = k and La = V,a. +We find the potential +V (x, y, z) = k +2 r2 + b1x + b2y + b3z. +(208) +Then, equation (187) gives k = − λ2 +4 and the potential (208) becomes +V (x, y, z) = −λ2 +8 r2 + b1x + b2y + b3z. +(209) +The associated QFI is +I = eλt +� +λ2 +4 +3 +� +i=1 +� +˙qi − λ +2 qi +�2 ++ λ(bi ˙qi) − λ2 +2 biqi + +3 +� +i=1 +b2 +i +� +. +(210) +This QFI consists of the independent QFIs: +I1 = eλt +� +λ2 +4 +� +˙x − λ +2 x +�2 ++ λb1 ˙x − λ2 +2 b1x + b2 +1 +� +, I2 = eλt +� +λ2 +4 +� +˙y − λ +2 y +�2 ++ λb2 ˙y − λ2 +2 b2y + b2 +2 +� +, +I3 = eλt +� +λ2 +4 +� +˙z − λ +2 z +�2 ++ λb3 ˙z − λ2 +2 b3z + b2 +3 +� +. +Therefore, the potential (209) is maximally superintegrable (see Table 3). +5) Case k1k2k3 ̸= 0 and b4 = b5 = b6 = 0. +38 + +The potential +V (x, y, z) = −λ2 +8 r2 − λ2 +4 +� b1 +k1 +x + b2 +k2 +y + b3 +k3 +z +� ++ +c1 +� +x + b1 +k1 +�2 + +c2 +� +y + b2 +k2 +�2 + +c3 +� +z + b3 +k3 +�2 +(211) +where c1, c2, and c3 are arbitrary constants. +The associated QFI gives the following three independent QFIs +Ii = eλt + + +� +˙qi − λ +2 qi +�2 +− λ bi +ki +� +˙qi − λ +2 qi − λbi +4ki +� ++ +2ci +� +qi + bi +ki +�2 + + +(212) +where i = 1, 2, 3 and qi = (x, y, z). +Therefore, the separable potential (211) is maximally superintegrable (see Table 3). +We note that, as expected, for k1 = k2 = k3 = k the resulting potential (211) belongs to the family of +potentials (195) if we set +F +� +y + b2 +k +x + b1 +k +, z + b3 +k +x + b1 +k +� += c1 +� +z + b3 +k +x + b1 +k +�2 ++ c2 +� +y + b2 +k +x + b1 +k +�−2 � +z + b3 +k +x + b1 +k +�2 ++ c3. +6) Case k1b2 ̸= 0, k2 = k3 = 0 and b4 = b5 = b6 = 0. +The vector La = (k1x + b1, b2, b3). +The potential +V (x, y, z) = −λ2 +8 +� +x2 + 4(1 − c2 +1)y2� +− λ2 +4 (c2x + 4c1yz) + c3y + +c4 +(x + c2)2 + F(z − c1y) +(213) +where c1 = b3 +b2 , c2 = b1 +k1 , c3, c4 are arbitrary constants and F is an arbitrary smooth function of its arguments. +We find the independent FIs: +I1 += +eλt +�� +˙x − λ +2 x +�2 +− λc2 +� +˙x − λ +2 x − λ +4 c2 +� ++ +2c4 +(x + c2)2 +� +(214) +I2 += +eλt � +˙y + c1 ˙z − λ(y + c1z) + c3 +λ +� +. +(215) +We note that for c1 = 0 we obtain the separable potential +V (x, y, z) = −λ2 +8 +� +x2 + 4y2� +− λ2 +4 c2x + c3y + +c4 +(x + c2)2 + F(z) +(216) +which is a new maximally superintegrable potential due to the additional time-dependent FIs (214) and (215). +The potential (see Table 7) +V (x, y, z) = −λ2 +8 +� +R2 + 4z2� ++ c4 +x2 + c0 +y2 + c3z. +(217) +is a subcase of (216) for y ↔ z, c2 = 0 and F(z) = − λ2 +8 z2 + c0 +z2 . +7.2 +Parameters a17, a19, a20: The components L(a;b) are constant and non-diagonal +In the following cases, the only non-vanishing parameters are the a17, a19, and a20. +1) Case a17 ̸= 0, a20 = 0 and a19 is free. +The vector La = (0, 2a17x, 2a19x) and the KT L(a;b) = + + +0 +a17 +a19 +a17 +0 +0 +a19 +0 +0 + +. +39 + +Then, equation (184) gives the potential +V (x, y, z) = −λ2 +2 x2 + F (z − cy) +(218) +where c = a19 +a17 and F is an arbitrary smooth function. +The associated QFI is +I = eλt( ˙x − λx)( ˙y + c ˙z). +(219) +From Table 2, the potential (218) admits the additional autonomous FIs: I1 = 1 +2 ˙x2 − λ2 +2 x2 and I2 = ˙y + c ˙z. +Therefore, the QFI (219) contains the independent LFI I3 = eλt( ˙x − λx). +We conclude that (218) is a new minimally superintegrable potential. +2) Case a17 = α +2 ̸= 0, a19 = 0 and a20 = β +2 . +The vector La = (0, αx, βy), where α and β are arbitrary constants and the KT L(a;b) = + + +0 +α +2 +0 +α +2 +0 +β +2 +0 +β +2 +0 + +. +The potential +V (x, y, z) = − +λ2 +2(1 + c2 +1) +� +x2 + c2 +1y2� +− +λ2 +2(1 + c2 +1) (z − 2c1x)2 + c2 (z − 2c1x) +(220) +where c1 = β +α and c2 are arbitrary constants. +The associated QFI is +I = eλt +� +( ˙x − λx) ˙y + c1( ˙y − λy) ˙z − λ2c1 +1 + c2 +1 +(c1x − z)y − c1c2y +� +. +(221) +Moreover, the potential (220) admits the additional autonomous QFI I1 = +1 +2 ˙y2 − +λ2c2 +1 +2(1+c2 +1)y2 because the +y-coordinate is separated from the coordinates x and z. +7.3 +Parameters a2, a5, a8, a11, a12, a15, a16, a18: +The components L(a;b) are linear on +x, y, z +We consider the following cases: +1) a15 is the only non-vanishing parameter. +The vector La = a15(−y2, xy, 0) and the KT L(a;b) = a15 + + +0 +− y +2 +0 +− y +2 +x +0 +0 +0 +0 + +. +The potential +V (x, y, z) = −λ2 +2 R2 + c1x +y2R + c2 +y2 + F(z) +(222) +where c1, c2 are arbitrary constants and F(z) is an arbitrary smooth function. +The associated QFI is +I = eλt +� +M3( ˙y − λy) + 2c2x +y2 ++ c1(y2 + 2x2) +y2R +� +. +(223) +We note that the potential (222) is of the integrable form (see Table 1) V = +F1( y +x) +R2 ++ F2(R) + F3(z) with +F1 +�y +x +� += + + +c1 +� +1 + y2 +x2 ++ c2 + + +� +1 + x2 +y2 +� +, F2(R) = −λ2 +2 R2. +(224) +Therefore, it is a new minimally superintegrable potential due to the additional autonomous QFIs: +I1 = 1 +2 ˙z2 + F3(z), I2 = 1 +2M 2 +3 + (c1R + c2x)x +y2 +. +40 + +Moreover, for F(z) = − λ2 +2 z2 + c3 +z2 , where c3 is an arbitrary constant, the resulting potential +V (x, y, z) = −λ2 +2 r2 + c1x +y2R + c2 +y2 + c3 +z2 +(225) +is a subcase of the minimally superintegrable potential (78) with F1 +� y +x +� +as given in (224). Hence, (225) is a +new maximally superintegrable potential due to the additional autonomous QFI (see Table 6) +I3 = 1 +2M2 + c1xr2 +y2R + c2 +r2 +y2 + c3R2 +z2 . +(226) +2) a2 and a12 are the only non-vanishing parameters. +The vector La = (a2xz, a12yz, −a2x2 − a12y2) and the KT L(a;b) = + + +a2z +0 +− a2 +2 x +0 +a12z +− a12 +2 y +− a2 +2 x +− a12 +2 y +0 + +. +The potential (see Table 3) +V (x, y, z) = −λ2 +2 r2 + k1 +x2 + k2 +y2 +(227) +where k1 and k2 are arbitrary constants. +The associated QFI consists of the independent QFIs: +I1 = eλt +� +M2( ˙x − λx) + 2k1z +x2 +� +, I2 = eλt +� +M1( ˙y − λy) − 2k2z +y2 +� +. +Therefore, the separable potential (227) is maximally superintegrable. +3) Case a2 = a12. +The vector La = a2(xz, yz, −R2) and the KT L(a;b) = a2 + + +z +0 +− x +2 +0 +z +− y +2 +− x +2 +− y +2 +0 + +. +The potential +V (x, y, z) = −λ2 +2 r2 + c1z +rR2 + F +� y +x +� +R2 +(228) +where c1 is an arbitrary constant and F +� y +x +� +is an arbitrary smooth function. +The associated QFI is +I1 = eλt +� +M2 ( ˙x − λx) − M1 ( ˙y − λy) + c1 +r + 2c1z2 +rR2 + 2zF +� y +x +� +R2 +� +. +(229) +We note that the potential (228) belongs to the general family of potentials (74); hence, it admits the +additional autonomous QFI (see Table 4) +I2 = 1 +2M 2 +3 + F +�y +x +� +. +(230) +If c1 = 0, the resulting potential +V (x, y, z) = −λ2 +2 r2 + F +� y +x +� +R2 +(231) +is a new maximally superintegrable potential due to the additional autonomous QFIs (see Table 6): +I3 = 1 +2 ˙z2 − λ2 +2 z2, I4 = 1 +2M2 + r2F +� y +x +� +R2 +. +We note that the potential (231) is of the form (78) for k1 = − λ2 +2 and k2 = 0. +4) a3, a6, a10, a14 are non-vanishing and a2a13 ̸= 0. +41 + +The vector La = + + +a2xz + a3x + a6 +a13y + a14 +−a2x2 + a10 + + and the KT L(a;b) = + + +a2z + a3 +0 +− a2 +2 x +0 +a13 +0 +− a2 +2 x +0 +0 + +. +The potential (see Table 3) +V (x, y, z) = −λ2 +8 (4x2 + 4z2 + y2) − λ2c1z − λ2 +4 c2y + +k +(y + c2)2 +(232) +where k, c1 = a3 +a2 , and c2 = a14 +a13 are arbitrary constants. +The associated QFI consists of the independent FIs: +I1 += +eλt ( ˙x − λx) +(233) +I2 += +eλt +�� +˙y − λ +2 (y + c2) +�2 ++ +2k +(y + c2)2 +� +(234) +I3 += +eλt [ ˙z − λ(z + c1)] +(235) +I4 += +M2 + c1 ˙x. +(236) +We note that {I2, Ip} = 0 where p = 1, 2, 3, 4, {I1, I3} = 0, {I1, I4} = I3 and {I4, I3} = I1. +The potential (232) is integrable because the independent FIs I1, I2, I3 are in involution or, directly, because +it is separable. It is also maximally superintegrable due to the additional independent FIs I4 and H, where H +is the Hamiltonian. +5) Case a2 ̸= 0 and a3, a13 are non-vanishing. +The vector La = + + +a2xz + a3x +a13y +−a2x2 + + and the KT L(a;b) = + + +a2z + a3 +0 +− a2 +2 x +0 +a13 +0 +− a2 +2 x +0 +0 + +. +The potential +V (x, y, z) = −λ2 +2 +� +x2 + z2� +− λ2 +8 y2 + c1 +x2 + c2 +y2 − λ2c3z + +k(z + c3) +x2� +(z + c3)2 + x2 +(237) +where k, c1, c2, and c3 = a3 +a2 are arbitrary constants. +The associated QFI consists of the following independent QFIs: +I1 += +eλt +�� +˙y − λ +2 y +�2 ++ 2c2 +y2 +� +(238) +I2 += +eλt +� +(M2 + c3 ˙x)( ˙x − λx) + 2c1(z + c3) +x2 ++ k +x2 + 2(z + c3)2 +x2� +x2 + (z + c3)2 +� +. +(239) +It is well-known that the dynamical equations (and hence the associated FIs) of a regular Lagrangian system +are preserved if: +a. We add an arbitrary constant c to the potential V of the system. +b. We apply a canonical transformation. +Then, the potential (237) is a subcase of the minimally superintegrable potential (222). Indeed, by adding +the constant c = − λ2 +2 c2 +3 to (237), we obtain the equivalent potential +V (x, y, z) = −λ2 +2 +� +x2 + (z + c3)2� ++ +k(z + c3) +x2� +(z + c3)2 + x2 + c1 +x2 − λ2 +8 y2 + c2 +y2 . +(240) +If we apply the canonical transformation x → y, y → z and z → x − c3, the potential (240) becomes +V (x, y, z) = −λ2 +2 R2 + kx +y2R + c1 +y2 − λ2 +8 z2 + c2 +z2 +(241) +which is a subcase of (222) for F(z) = − λ2 +8 z2 + c2 +z2 . +42 + +The potential (241) is a new maximally superintegrable potential due to the following independent QFIs: +I1 += +eλt +�� +˙z − λ +2 z +�2 ++ 2c2 +z2 +� +(242) +I2 += +eλt +� +M3( ˙y − λy) + 2c1x +y2 ++ k(y2 + 2x2) +y2R +� +. +(243) +I3 += +1 +2 ˙z2 − λ2 +8 z2 + c2 +z2 +(244) +I4 += +1 +2M 2 +3 + (kR + c1x)x +y2 +. +(245) +We recall that the potential (225) is another maximally superintegrable potential which is also a subcase of +(222) but for a different choice of the function F(z). If we rename λ → 2λ, the QFI (242) is admitted also by +(225) because the z-coordinate is separated from x and y. +We collect the results of section 7 in Tables 11 - 13. +43 + +Potential +LFIs and QFIs +V = λ2 +2 +� +c2 +1 − 1 +� +x2 + c2x− +−c1λ2xy + F(y − c1x, z) +I = eλt � +λ ˙x + c1λ ˙y − λ2x − c1λ2y + c2 +� +V = λ2 +2 x2 + kx − λ2(y + z)x+ ++F(x − z, y − z) +I = eλt � +λ( ˙x + ˙y + ˙z) − λ2(x + y + z) + k +� +V = − λ2 +8 r2 − λ2 +4 (c1x + c2y + c3z) + ++ +1 +(z+c3)2 F +� +y+c2 +x+c1 , z+c3 +x+c1 +� +I = eλt �� +˙x − λ +2 x +�2 + +� +˙y − λ +2 y +�2 + +� +˙z − λ +2 z +�2 − +−λ (c1 ˙x + c2 ˙y + c3 ˙z) + ++ λ2 +2 +� +c1x + c2y + c3z + c2 +1 +2 + c2 +2 +2 + c2 +3 +2 +� ++ +2F +(z+c3)2 +� +V = − λ2 +8 r2 + +F( y +x , z +x) +z2 +I = eλt +� +˙x2 + ˙y2 + ˙z2 − λ(x ˙x + y ˙y + z ˙z) + λ2 +4 r2 + +2F( y +x , z +x) +z2 +� +V = − λ2 +8 r2 − λ2 +4 x + +F( +y +x+1 , +z +x+1) +z2 +I = eλt �� +˙x − λ +2 x +�2 + +� +˙y − λ +2 y +�2 + +� +˙z − λ +2 z +�2 − λ ˙x+ ++ λ2 +2 x + λ2 +4 + 2F +z2 +� +V = − λ2 +8 r2 − λ2 +4 (x + y)+ ++ 1 +z2 F +� +y+1 +x+1, +z +x+1 +� +I = eλt �� +˙x − λ +2 x +�2 + +� +˙y − λ +2 y +�2 + +� +˙z − λ +2 z +�2 − +−λ( ˙x + ˙y) + λ2 +2 (x + y + 1) + 2F +z2 +� +V = − λ2 +8 r2 − λ2 +4 (x + y + z) + ++ +1 +(z+1)2 F +� +y+1 +x+1, z+1 +x+1 +� +I = eλt �� +˙x − λ +2 x +�2 + +� +˙y − λ +2 y +�2 + +� +˙z − λ +2 z +�2 − λ ( ˙x + ˙y + ˙z) + ++ λ2 +2 +� +x + y + z + 3 +2 +� ++ +2F +(z+1)2 +� +V = − λ2 +8 r2 − λ2 +4 (c1x + c2y + c3z) + ++ +c0 +(x+c1)2 + +1 +(z+c3)2 F1 +� +y+c2 +z+c3 +� +I1 = eλt �� +˙x − λ +2 x +�2 − λc1 +� +˙x − λ +2 x − λc1 +4 +� ++ +2c0 +(x+c1)2 +� +I2 = eλt �� +˙y − λ +2 y +�2 + +� +˙z − λ +2 z +�2 − λ (c2 ˙y + c3 ˙z) + ++ λ2 +2 +� +c2y + c3z + c2 +2 +2 + c2 +3 +2 +� ++ +2F1 +(z+c3)2 +� +V = − λ2 +8 +� +x2 + 4(1 − c2 +1)y2� +− +− λ2 +4 (c2x + 4c1yz) + ++c3y + +c4 +(x+c2)2 + F(z − c1y) +I1 = eλt �� +˙x − λ +2 x +�2 − λc2 +� +˙x − λ +2 x − λ +4 c2 +� ++ +2c4 +(x+c2)2 +� +I2 = eλt � +˙y + c1 ˙z − λ(y + c1z) + c3 +λ +� +V = − +λ2 +2(1+c2 +1) +� +x2 + c2 +1y2� +− +− +λ2 +2(1+c2 +1) (z − 2c1x)2 + ++c2 (z − 2c1x) +I1 = 1 +2 ˙y2 − +λ2c2 +1 +2(1+c2 +1)y2 +I2 = eλt � +( ˙x − λx) ˙y + c1( ˙y − λy) ˙z − λ2c1 +1+c2 +1 (c1x − z)y − c1c2y +� +V = − λ2 +2 r2 + c1z +rR2 + +F( y +x) +R2 +I1 = eλt +� +M2 ( ˙x − λx) − M1 ( ˙y − λy) + c1 +r + 2c1z2 +rR2 + +2zF( y +x) +R2 +� +I2 = 1 +2M 2 +3 + F +� y +x +� +Table 11: Possibly non-integrable potentials V (x, y, z) in E3 that +admit time-dependent LFIs/QFIs of the form I(3). +44 + +Minimally superintegrable potentials +Potential +Ref [2] +LFIs and QFIs +V = − λ2 +2 x2 + c2x + F1(y) + F2(z) +New +I1 = 1 +2 ˙x2 − λ2 +2 x2 + c2x +I2 = 1 +2 ˙y2 + F1(y) +I3 = 1 +2 ˙z2 + F2(z) +I4 = eλt � +λ ˙x − λ2x + c2 +� +V = − λ2 +2 x2 + F (z − cy) +New +I1 = 1 +2 ˙x2 − λ2 +2 x2 +I2 = ˙y + c ˙z +I3 = eλt( ˙x − λx) +V = − λ2 +2 R2 + c1x +y2R + c2 +y2 + F(z) +New +I1 = 1 +2 ˙z2 + F(z) +I2 = 1 +2M 2 +3 + (c1R+c2x)x +y2 +I3 = eλt � +M3( ˙y − λy) + 2c2x +y2 + c1(y2+2x2) +y2R +� +Table 12: Minimally superintegrable potentials V (x, y, z) in E3 +that admit time-dependent LFIs/QFIs of the form I(3). +45 + +Maximally superintegrable potentials +Potential +Ref [2] +LFIs and QFIs +V = − λ2 +8 r2 + b1x + b2y + b3z +New +Ii = ˙q2 +i +2 − λ2 +8 q2 +i + biqi +Ji = eλt � +λ2 +4 +� +˙qi − λ +2 qi +�2 + λbi ˙qi − λ2 +2 biqi + b2 +i +� +V = − λ2 +8 r2 − λ2 +4 (b1x + b2y + b3z) + ++ +c1 +(x+b1)2 + +c2 +(y+b2)2 + +c3 +(z+b3)2 +New +Ii = ˙q2 +i +2 − λ2 +8 q2 +i − λ2 +4 biqi + +ci +(x+bi)2 +Ji = eλt �� +˙qi − λ +2 qi�2 − λbi +� +˙qi − λ +2 qi − λbi +4 +� ++ +2ci +(qi+bi)2 +� +V = − λ2 +8 +� +x2 + 4y2� +− λ2 +4 c2x + c3y+ ++ +c4 +(x+c2)2 + F(z) +New +I1 = ˙x2 +2 − λ2 +8 x2 − λ2 +4 c2x + +c4 +(x+c2)2 +I2 = ˙y2 +2 − λ2 +2 y2 + c3y +I3 = ˙z2 +2 + F(z) +I4 = eλt �� +˙x − λ +2 x +�2 − λc2 +� +˙x − λ +2 x − λ +4 c2 +� ++ +2c4 +(x+c2)2 +� +I5 = eλt � +˙y − λy + c3 +λ +� +V = − λ2 +2 r2 + c1x +y2R + c2 +y2 + c3 +z2 +New +I1 = 1 +2 ˙z2 − λ2 +2 z2 + c3 +z2 +I2 = 1 +2M 2 +3 + (c1R+c2x)x +y2 +I3 = eλt � +M3( ˙y − λy) + 2c2x +y2 + c1(y2+2x2) +y2R +� +I4 = 1 +2M2 + c1xr2 +y2R + c2 r2 +y2 + c3R2 +z2 +V = − λ2 +2 r2 + k1 +x2 + k2 +y2 +New +I1 = ˙x2 +2 − λ2 +2 x2 + k1 +x2 +I2 = ˙y2 +2 − λ2 +2 y2 + k2 +y2 +I3 = ˙z2 +2 − λ2 +2 z2 +I4 = eλt � +M2( ˙x − λx) + 2k1z +x2 +� +I5 = eλt � +M1( ˙y − λy) − 2k2z +y2 +� +V = − λ2 +2 r2 + +F( y +x) +R2 +New +I1 = eλt +� +M2 ( ˙x − λx) − M1 ( ˙y − λy) + +2zF( y +x) +R2 +� +I2 = 1 +2M 2 +3 + F +� y +x +� +I3 = 1 +2 ˙z2 − λ2 +2 z2 +I4 = 1 +2M2 + +r2F( y +x) +R2 +V = − λ2 +8 (4x2 + 4z2 + y2)− +−λ2c1z − λ2 +4 c2y + +k +(y+c2)2 +New +I1 = eλt ( ˙x − λx) +I2 = eλt �� +˙y − λ +2 (y + c2) +�2 + +2k +(y+c2)2 +� +I3 = eλt [ ˙z − λ(z + c1)] +I4 = M2 + c1 ˙x +V = − λ2 +2 R2 + +kx +y2R + c1 +y2 − +− λ2 +8 z2 + c2 +z2 +New +I1 = eλt �� +˙z − λ +2 z +�2 + 2c2 +z2 +� +I2 = eλt � +M3( ˙y − λy) + 2c1x +y2 + k(y2+2x2) +y2R +� +I3 = 1 +2 ˙z2 − λ2 +8 z2 + c2 +z2 +I4 = 1 +2M 2 +3 + (kR+c1x)x +y2 +Table 13: Maximally superintegrable potentials V (x, y, z) in E3 +that admit time-dependent LFIs/QFIs of the form I(3). +8 +Comparison with existing results +As we have remarked in section 1, the main review works in this topic are the works of Evans in [2] and Kalnins +in [4]. Therefore, it is imperative to discuss how the present review is related to these. +46 + +8.1 +Evans work [2] +Evans in [2], using the separability of the Hamilton-Jacobi equation in E3, determined all minimally and +maximally superintegrable potentials with autonomous QFIs of the form I = Kab(q) ˙qa ˙qb + G(q). The author +did not consider (autonomous or time-dependent) LFIs and time-dependent QFIs. In particular, in Table I of +[2] are given five maximally superintegrable potentials and in Table II of [2] eight minimally superintegrable +potentials. +As it can be seen from Tables 1 - 13, all the results of [2] have been recovered plus new ones. Therefore, the +claim made in [2] that all second order superintegrable potentials in E3 are determined is not valid. +Furthermore, it should be noted that there are misprints in some results of [2]. Indeed, we have: +1) In eq. (3.43) of [2], the leading term of the QFI I4 must be L2P1 − P2L1. +2) In Table II of [2], the leading part of the QFIs I3 associated with the potentials (55) and (56) should be +L2P1 − P2L1. +3) The QFI I2 in eq. (3.57) of [2] should be replaced with the QFI (8). +4) The QFI I3 in eq. (3.57) of [2] should be replaced with the QFI (9). +8.2 +Kalnins et all work [4] +In [4], the authors discussed classical 3d superintegrable nondegenerate (i.e. +four-parameter) potentials on +a conformally flat real or complex space. +They proved that the quadratic algebra always closes at order +six (the ‘5 +=⇒ +6 Theorem’), that is, the space of autonomous QFIs is 6d. +Moreover, using the St¨ackel +transformation (an invertible conformal mapping between superintegrable structures on distinct spaces), they +gave strong evidence (no proof) that all nondegenerate 3d superintegrable systems are St¨ackel transforms of +constant curvature systems (i.e. the complex Euclidean space or the the complex 3-sphere). This means that +in order to obtain all nondegenerate conformally flat superintegrable systems, it is sufficient to classify those in +the complex Euclidean space and on the complex 3-sphere. Finally, they found eight families of superintegrable +systems that are separable in generic coordinates. +Comparing the results of [4] with the results of the present work, we note the following: +1) All seven maximally superintegrable Euclidean potentials given in eqs. (10) - (16) of [4] are recovered (see +Table 7). +2) The potentials given in eqs. (10) and (13) of [4] have been found earlier in Table I of [2]. The potential (13) +is more general from the one found by Evans. +3) It is proved in section 5 that the potentials (11), (12), (14), (16) of [4] are subcases of the more general +potential (28) for specific forms of the arbitrary smooth functions F1(w, z) and F2(w). This justifies the fact +that these potentials admit a QFI of the form I = ˙w2 + G(x, y, z) where w = x + iy. +4) The potential (15) of [4] is a subcase of (31) and hence admits a QFI of the form I = ˙¯w2 + G(x, y, z). +5) The potentials (12), (16) of [4] are of the integrable form (34); therefore, they admit a QFI of the form +I = ˙z ˙w + G(x, y, z). +6) The potentials (11), (12) of [4] are of the integrable form (82); therefore, they admit a QFI of the form +I = (M2 − iM1)2 + G(x, y, z). +7) The potential (15) of [4] is a subcase of the new minimally superintegrable potential (97) for F(z) = k1z2+ k4 +z2 . +For this reason, it admits an additional QFI of the form I = 1 +4 ˙w2 + iM3 ˙¯w + G(x, y, z). +8) The two additional maximally superintegrable potentials given in eq. (17) of [4] are just subcases of the last +maximally superintegrable potential in Table 3 for k1 = k2 = 0 when F(z) = − λ2 +8 z2+c3z and F(z) = − λ2 +32 z2+ c3 +z2 . +Therefore, with the systematic application of Theorem 1, we have found all the results of [4] plus new ones; +especially time-dependent QFIs. +9 +Conclusions +The aim of the present work was twofold: a. To assess the second order integrability of autonomous conser- +vative dynamical systems of the form qa = −V ,a(q) where a = 1, 2, 3 in a systematic, i.e. algorithmic, way; +and b. To enrich, if possible, the existing results of the main sources on this topic which are found in the +review papers [2] and [4]. +Therefore, the present work should be approached as an updated review of the +integrable/superintegrable 3d Newtonian autonomous conservative dynamical systems that admit LFIs/QFIs. +47 + +We have considered two types of integrable and superintegrable 3d Newtonian potentials. Potentials of the +form Φ(x, y)+F(z) which are 2+1 decomposable and hence their QFIs follow from the QFIs of the 2d potentials +Φ(x, y); and non-decomposable potentials V (x, y, z) in E3 which cannot be treated in this way. These latter +potentials we have searched using the algorithm of Theorem 1. +After a detailed study of the three types of QFIs I(1,ℓ), I(2,ℓ), I(3) considered in Theorem 1, we have recovered +all known integrable/superintegrable potentials together with new ones. It has also been shown that many of the +existing results are in fact special cases of more general ones for specific values of the free parameters/functions. +For convenience, the results in each case have been collected in tables which contain the known results with the +appropriate reference and the new ones found in the present work. These results can be used in many ways in +the study of the dynamical systems and, especially, in the case of more complex systems. One such study will +be given elsewhere. +References +[1] V.I. Arnold, ‘Mathematical Methods of Classical Mechanics’, Springer (1989), proof in pp. 272-284. +[2] N.W. Evans, ‘Superintegrability in classical mechanics’, Phys. Rev. A 41(10), 5666 (1990). +[3] W. Miller, ‘Second Order Superintegrable Systems in Three Dimensions’, SIGMA 1, 015 (2005). +[4] E.G. Kalnins, J.M. Kress and W. Miller Jr., ‘Classification of superintegrable systems in three dimensions’, +Quantum Theory and Symmetries IV, ed. V.K. Dobrev, Heron Press, Sofia (2006). +[5] E.G. Kalnins, J.M. Kress and W. Miller Jr., ‘Fine structure for 3D second-order superintegrable systems: +three-parameter potentials’, J. Phys. A: Math. Theor. 40, 5875 (2007). +[6] V.V. Kozlov, ‘Integrability and non-integrability in Hamiltonian mechanics’, Russ. Math. Surv., Turpion, +38(1), pp. 1-76 (1983), see p.17, Theorem 1, Chapter II, Paragraph 2. +[7] T.G. Vozmishcheva, ‘Integrable problems of celestial mechanics in spaces of constant curvature’, J. Math. +Sc. 125(4), 419 (2005), see Theorem 3.4. +[8] M. Tsamparlis and A. Mitsopoulos, ‘Quadratic first integrals of autonomous conservative dynamical sys- +tems’ J. Math. Phys. 61, 072703 (2020). +[9] E.G. Kalnins, J.M. Kress and W. Miller Jr., ‘Nondegenerate three-dimensional complex Euclidean superin- +tegrable systems and algebraic varieties’, J. Math. Phys. 48, 113518 (2007). +[10] A. Mitsopoulos, M. Tsamparlis and A. Paliathanasis, ‘Integrable and Superintegrable Potentials of 2d +Autonomous Conservative Dynamical Systems’, Symmetry 12, 1655 (2020). +[11] M. Tsamparlis and A. Mitsopoulos, ‘First integrals of holonomic systems without Noether symmetries’, J. +Math. Phys. 61, 122701 (2020). +[12] J. Hietarinta, ‘Direct methods for the search of the second invariant’, Phys. Rep. 147(2), 87 (1987). +48 + diff --git a/PdAyT4oBgHgl3EQf7fpb/content/tmp_files/load_file.txt b/PdAyT4oBgHgl3EQf7fpb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6fe35b96d1deadbe883c368dc04bc9610a818f5d --- /dev/null +++ b/PdAyT4oBgHgl3EQf7fpb/content/tmp_files/load_file.txt @@ -0,0 +1,2598 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf,len=2597 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='00839v1 [math-ph] 2 Jan 2023 Integrable and superintegrable 3d Newtonian potentials using quadratic first integrals: A review Antonios Mitsopoulos1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a) and Michael Tsamparlis2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) 1Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Department of Astronomy-Astrophysics-Mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' University of Athens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Panepistemiopolis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Athens 157 83,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Greece 2NITheCS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' National Institute for Theoretical and Computational Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Pietermaritzburg 3201,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' KwaZulu-Natal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' South Africa 3TCCMMP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Theoretical and Computational Condensed Matter and Materials Physics Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' School of Chemistry and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' University of KwaZulu-Natal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Pietermaritzburg 3201,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' KwaZulu-Natal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' South Africa a)Author to whom correspondence should be addressed: antmits@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='uoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='gr b)Email: mtsampa@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='uoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='gr Abstract The determination of the first integrals (FIs) of a dynamical system and the subsequent assessment of their integrability or superintegrability in a systematic way is still an open subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' One method which has been developed along these lines for holonomic autonomous dynamical systems with dynamical equations ¨qa = −Γa bc(q) ˙qb ˙qc − Qa(q), where Γa bc(q) are the coefficients of the Riemannian connection defined by the kinetic metric of the system and −Qa(q) are the generalized forces, is the so-called direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' According to this method, one assumes a general functional form for the FI I and requires the condition dI dt = 0 along the dynamical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This results to a system of partial differential equations (PDEs) to which one adds the necessary integrability conditions of the involved scalar quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' It is found that the final system of PDEs breaks into two sets: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' One set containing geometric elements only and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' A second set with geometric and dynamical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, provided the geometric quantities are known or can be found, one uses the second set to compute the FIs and, accordingly, assess on the integrability of the dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The ‘solution’ of the system of PDEs for quadratic FIs (QFIs) has been given in a recent paper (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Tsamparlis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Mitsopoulos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 61, 122701 (2020) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In the present work, we consider the application of this ‘solution’ to Newtonian autonomous conservative dynamical systems with three degrees of freedom, and compute integrable and superintegrable potentials V (x, y, z) whose integrability is determined via autonomous and/or time-dependent QFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The geometric elements of these systems are the ones of the Euclidean space E3 which are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Setting various values for the parameters determining the geometric elements, we determine in a systematic way all known integrable and superintegrable potentials in E3 together with new ones obtained in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For easy reference, the results are collected in tables so that the present work may act as an updated review of the QFIs of Newtonian autonomous conservative dynamical systems with three degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' It is emphasized that by assuming different values for the parameters, other authors may find more integrable potentials of this type of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Keywords: Integrable potentials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' superintegrable potentials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3d Newtonian potentials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' quadratic first inte- grals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' time-dependent first integrals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' autonomous conservative dynamical systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Killing tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 1 Introduction According to Liouville integrability theorem [1], a three-dimensional (3d) Newtonian autonomous conservative system is (Liouville) integrable if it admits three (functionally) independent first integrals (FIs) in involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 1 Integrable systems that admit five independent FIs are called maximally superintegrable, while if they admit four independent FIs they are called minimally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' A superintegrable potential is always integrable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' however, some authors [2, 3, 4, 5] define superintegrability without the requirement of integrability, that is, they look only for sets of independent FIs whose number exceeds the degrees of freedom of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For 3d Newtonian autonomous conservative systems one quadratic FI (QFI) is the Hamiltonian H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' therefore, one needs two additional independent autonomous1 FIs in involution in order to establish integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' If in addition to these FIs there exist one/two more independent autonomous or time-dependent FIs, then the system is minimally/maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Besides establishing superintegrability, time-dependent FIs can be used also to establish the integrability of a dynamical system provided they are in involution (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' [6, 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The maximum number of independent autonomous FIs of a Hamiltonian dynamical system of n degrees of freedom is 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' However, if time-dependent FIs are considered, this maximum limit can be exceeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For example, the 3d potential V = −kr2, where r = � x2 + y2 + z2 and k is an arbitrary constant, admits the six (five are enough) time-dependent linear FIs (LFIs) I3a±, a = 1, 2, 3 (see Table V in [8]): k > 0 : I3a± = e± √ 2kt � ˙qa ∓ √ 2kqa � k < 0 : I3a± = e±i√−2kt � ˙qa ∓ i √ −2kqa � which are functionally independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Since the three LFIs I3a+ (or I3a−) are also in involution, the considered 3d potential is superinetgarble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Concerning the number of the free parameters that define a 3d superintegrable potential, the following terminology is used (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' [5]): a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The degenerate (or three-parameter) potentials, and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The non-degenerate (or four-parameter) potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In many works [3, 4, 5, 9], the term second order superintegrable potentials is used for potentials that are superintegrable due to QFIs only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Such potentials have the following special properties [3, 4]: 1) Multi-integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' They are integrable in multiple ways and the comparison of ways of integration leads to new facts about the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2) They are multi-separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) The second order symmetries expressed by second order Killing tensors (KTs) generate a closed quadratic algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In the quantum case, the representation of this algebra yields results concerning the spectral resolution of the Schr¨odinger operator and the other symmetry operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' There are two types of integrable potentials in E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The decomposable potentials (or 2+1 separable integrable potentials) generated from integrable potentials in E2 and the non-decomposable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Let V (x, y) be a 2d integrable potential in E2 which admits an additional autonomous FI I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, the 3d Newtonian z-separable potential ¯V (x, y, z) = V (x, y) + F(z), where F is an arbitrary smooth function of z, is a 2 + 1 separable integrable potential in E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The integrability of these potentials is due to the three independent FIs H, I1 and I2 = 1 2 ˙z2 + F(z) which are in involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' If V (x, y) is superintegrable with respect to (wrt) two additional FIs, say J1 and J2, then ¯V (x, y, z) is minimally superintegrable because of the four independent FIs H, J1, J2, and I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' If in addition to J1 and J2 the 2d superintegrable potential V (x, y) admits also a time-dependent FI J3, then ¯V (x, y, z) is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For example, the second potential of Table II in [2] is not minimally superintegrable but maximally superintegrable because it admits in addition the time-dependent FIs I73a and I73b from the last Table of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The non-decomposable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' non-separable) 3d Newtonian integrable potentials V (x, y, z) cannot be written in the form ¯V (x, y, z) = V (x, y) + F(z) where V (x, y) is a 2d Newtonian integrable potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In general, their determination is more difficult and various methods of escalating complexity have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Furthermore, the existing results concern autonomous FIs only and are limited in number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The purpose of the present work is to provide a systematic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' algorithmic) method which enables one to determine integrable and superintegrable potentials in E3 using autonomous and time-dependent QFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The method relies on Theorem 1 [11] (see section 3) which relates the QFIs of the dynamical system with the dynamical elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' the potential) and the geometry defined by the kinetic energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 1These additional FIs must be autonomous because the Poisson bracket (PB) of the Hamiltonian with an arbitrary time- dependent FI J(t, q, ˙q) does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Indeed, we have {H, J} = ∂J ∂t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2 In section 2, we determine the 3d integrable/superintegrable 2+1 decomposable potentials directly from the well-known 2d integrable/superintegrable potentials listed in the reference works [10] and [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The results are presented in tables where the known potentials with the corresponding reference are listed together with the new ones determined in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In section 3, we state Theorem 1 from which follows that there are three types of QFIs to consider, denoted as I(1,ℓ), I(2,ℓ), I(3), which are expressed in terms of the geometric elements of the kinetic metric and the potential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In section 4, we state the geometric quantities of E3 which are required for the application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' It is seen that the number of parameters introduced from the KT components is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This remark and the fact that the associated system of PDEs is overdetermined have the result that one will find special solutions only by assuming particular values of the geometric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In section 5, we consider the QFI I(1,1) (ℓ = 1) and the relevant PDEs for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We consider various values for the parameters and recover all the existing results together with new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For easy reference, the various potentials are grouped in Tables 4 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In section 6, we consider the potentials admitting QFIs of the type I(2,0) (ℓ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' These results are presented in Tables 8 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In section 7, we consider time-dependent LFIs/QFIs of the type I(3) and the results are collected in Tables 11 - 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In section 8, we compare and discuss the results listed in the tables with the existing results of the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Finally, in section 9, we draw our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' List of abbreviations and notations/conventions For the convenience of the reader, we give a list of abbreviations and notations used throughout the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Abbreviations: FI = first integral HV = homothetic vector KT = Killing tensor KV = Killing vector LFI = linear first integral Nd = N-dimensional ODE = ordinary differential equation PB = Poisson bracket PDE = partial differential equation QFI = quadratic first integral Mathematical notations/conventions: En = n-dimensional Euclidean space r = � x2 + y2 + z2, R = � x2 + y2, tan θ = y x, and w = x + iy = Reiθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The angular momentum M ≡ Mi = (M1, M2, M3) = (y ˙z − z ˙y, z ˙x − x ˙z, x ˙y − y ˙x) with square magnitude M2 = M 2 1 + M 2 2 + M 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The kinetic metric γab(q) of the dynamical system is used for lowering and raising the indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' A comma indicates partial derivative and a semicolon Riemannian covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Coordinate systems of E3: Cartesian coordinates: (x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Spherical coordinates: (r, θ, φ) with x = r sin θ cos φ, y = r sin θ sin φ and z = r cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Parabolic cylindrical coordinates: (λ′, µ′, z) with λ′ = R + y and µ′ = R − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Rotational parabolic coordinates: (ζ, η, φ) with ζ = r + z, η = r − z, φ = tan−1 � y x � or, equivalently, x = √ζη cos φ, y = √ζη sin φ, z = 1 2 (ζ − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3 2 Integrable/superintegrable 2+1 separable potentials As it has been remarked, the 2+1 separable integrable/superintegrable potentials in E3 are given in terms of the integrable/superintegrable potentials Φ(x, y) in E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' From the latter potentials, the ones that admit LFIs/QFIs are collected in the review papers [10] and [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Using these results, the 2 + 1 separable potentials in E3 V (x, y, z) = Φ(x, y) + F(z) (1) where F(z) is an arbitrary smooth function, are integrable/superintegrable due to the additional QFI I = 1 2 ˙z2 + F(z) which is in involution with the FIs of Φ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Applying the above procedure to the results of [10, 12], we find the integrable and superintegrable potentials in E3 listed in Tables 1 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The QFI of the Hamiltonian H is not included in the tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In Tables 2 and 3, we compare with the results of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' A similar comparison cannot be done in Table 1 because in [2] only superintegrable potentials are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Concerning the notation, we set r = � x2 + y2 + z2, R = � x2 + y2 and the angular momentum Mi = (y ˙z − z ˙y, z ˙x − x ˙z, x ˙y − y ˙x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 4 Integrable 2 + 1 separable potentials Potential LFIs and QFIs V = F1 � R2 2 + b1y − b2x � + F2(z) I1 = M3 − b1 ˙x − b2 ˙y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I2 = 1 2 ˙z2 + F2(z) V = F1( y x) R2 + F2(R) + F3(z) I1 = M 2 3 + 2F1 � y x � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I2 = 1 2 ˙z2 + F3(z) V = k x2+ℓy2 + F1(R) + F2(z) I1 = M 2 3 + 2k(1−ℓ)y2 x2+ℓy2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F2(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F1(u)−F2(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='u2−v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='u2 = R2 + A + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(R2 + A)2 − 4Ax2�1/2 and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='v2 = R2 + A − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(R2 + A)2 − 4Ax2�1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + A ˙x2 + v2F1(u)−u2F2(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='u2−v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F1(u)−F2(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='u2−v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='u2 = R2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R4 − 4A(x ± iy)2�1/2 and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='v2 = R2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R4 − 4A(x ± iy)2�1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + A( ˙x ± i ˙y)2 + v2F1(u)−u2F2(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='u2−v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F1(R+y)+F2(R−y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = −M3 ˙x + (R+y)F2(R−y)−(R−y)F1(R+y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ¯w−1/2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F1(w + √ ¯w) + F2(w − √ ¯w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w = x + iy and ¯w = x − iy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = −M3( ˙x + i ˙y) + i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8( ˙x − i ˙y)2+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='√ ¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F1(w + √ ¯w)+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='−1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='√ ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F2(w − √ ¯w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F1(w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ F ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2(w) + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 = dF2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='dw and w = x ± iy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = −M3( ˙x ± i ˙y) − iwV + iF2(w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F1(x) + F2(y) + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I2 = 1 2 ˙y2 + F2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I3 = 1 2 ˙z2 + F3 V = F1 � y + b0x + � b2 0 + 1x � + +F2 � y + b0x − � b2 0 + 1x � + F3(z) where b0 ≡ A−B 2C I1 = A ˙x2 + B ˙y2 + 2C ˙x ˙y + (A + B)(F1 + F2)+ +2C � b2 0 + 1(F1 − F2) I2 = 1 2 ˙z2 + F3(z) V (b0 = 0) = F1(y + x) + F2(y − x) + F3(z) I1 = ˙x ˙y + F1 − F2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I2 = 1 2 ˙z2 + F3(z) Table 1: Integrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) = Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y) + F(z) in E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' where Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y) are integrable potentials in E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5 Minimally superintegrable 2 + 1 separable potentials Potential Ref [2] LFIs and QFIs V = cx + F1(y − bx) + F2(z) c ̸= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' d2F1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='dw2 ̸= 0 and w ≡ y − bx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = ˙x + b ˙y + ct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ( ˙x + b ˙y)2 + 2c(x + by) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F2(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F1(y − bx) + F2(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='d2F1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='dw2 ̸= 0 and w ≡ y − bx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = ˙x + b ˙y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = t( ˙x + b ˙y) − (x + by) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F2(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 (x2 + 4y2) + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + k3y + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k3 = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ↔ y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = M3 ˙x + k1yx2 − 2k2y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + 2k1y2 + k3y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R + k3y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Rx2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ↔ y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + 2k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + 2k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Ry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = M3 ˙x − 2k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 − k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R − k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2+2y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Rx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='√R+y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='√R−y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = M3 ˙x − k1y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R − k3(R+y)√R−y−k2(R−y)√R+y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = M3 ˙y + G(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y) I3 = 1 2 ˙z2 + F(z) G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x = −yV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y and G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y = 2xV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y − yV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x V = F1(x) + k (y+c)2 + F2(z) New I1 = 1 2 ˙x2 + F1 I2 = 1 2 ˙y2 + k (y+c)2 I3 = 1 2 ˙z2 + F2 I4 = − t2 2 ˙y2 + t(y + c) ˙y − t2 k (y+c)2 − 1 2y2 − cy V = λ 2 R2 + b1y − b2x + F(z) λ ̸= 0 New I1 = λM3 − b1 ˙x − b2 ˙y I2 = 1 2 ˙x2 + 1 2λx2 − b2x I3 = 1 2 ˙y2 + 1 2λy2 + b1y I4 = ˙x ˙y + λxy + b1x − b2y I5 = 1 2 ˙z2 + F(z) Table 2: Minimally superintegrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) = Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y) + F(z) in E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' where Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y) are superintegrable potentials in E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 6 Maximally superintegrable 2 + 1 separable potentials Potential Ref [2] LFIs and QFIs V = cx + λy + F(z) New I1 = ˙x + ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I2 = ˙y + λt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I3 = 1 2 ˙x2 + cx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I4 = 1 2 ˙y2 + λy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I5 = 1 2 ˙z2 + F(z) V = cx − 1 2λ2y2 + F(z) λ ̸= 0 New I1 = ˙x + ct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I2 = eλt( ˙y − λy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I3 = 1 2 ˙x2 + cx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I4 = 1 2 ˙y2 − 1 2λ2y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I5 = 1 2 ˙z2 + F(z) V = − k2 2 R2 + F(z) k ̸= 0 New I1 = M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I2 = 1 2 ˙x2 − 1 2k2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I3 = 1 2 ˙y2 − 1 2k2y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I4 = ˙x ˙y − k2xy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I5 = 1 2 ˙z2 + F(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I6± = e±kt( ˙x ∓ kx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I7± = e±kt( ˙y ∓ ky),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 4I2I3 = I2 4 − k2M 2 3 V = k 2R2 + b x2 + c y2 + F(z) Table II I1 = M 2 3 + 2b y2 x2 + 2c x2 y2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I2 = 1 2 ˙z2 + F(z) I3 = 1 2 ˙x2 + k 2x2 + b x2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I4 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2y2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='For k = 0: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = − t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + ty ˙y − t2 c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I6 = − t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + tx ˙x − t2 b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='For k = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ̸= 0: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ˙x2 + λx ˙x − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 x2 − 2b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I6 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ˙y2 + λy ˙y − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 y2 − 2c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c1)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+c2)2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+c2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = − t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + t(y + c2) ˙y − t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+c2)2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2y2 − c2y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = − t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + t(x + c1) ˙x − t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c1)2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2x2 − c1x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 R2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 (c1x + c2y) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c1)2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+c2)2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ ̸= 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 x2 − c1λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 x − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 y2 − c2λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+c2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ˙x2 + λ(x + c1) ˙x − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 (x + c1)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ˙y2 + λ(y + c2) ˙y − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 (y + c2)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+c2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table 3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Maximally superintegrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) = Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y) + F(z) in E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' where Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y) are superintegrable potentials in E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Note 1: The results indicated as ‘New’ in Tables 2 and 3 do not appear in [2] where only autonomous QFIs are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Note 2: In Table II of [2], the potential (see Table 3) V = k 2 R2 + b x2 + c y2 + F(z) (2) where k, b, c are arbitrary constants and F(z) is an arbitrary smooth function, is said to be minimally superin- tegrable because of the four independent autonomous QFIs: I1 = M 2 3 + 2by2 x2 + 2cx2 y2 , I2 = 1 2 ˙z2 + F(z), I3 = 1 2 ˙x2 + k 2 x2 + b x2 , I4 = 1 2 ˙y2 + k 2 y2 + c y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' However, using in addition the time-dependent QFIs: For k = 0: I5 = −t2 2 ˙y2 + ty ˙y − t2 c y2 − 1 2y2, I6 = −t2 2 ˙x2 + tx ˙x − t2 b x2 − 1 2x2 7 and For k = −λ2 4 ̸= 0: I5 = eλt � − ˙x2 + λx ˙x − λ2 4 x2 − 2b x2 � , I6 = eλt � − ˙y2 + λy ˙y − λ2 4 y2 − 2c y2 � it is seen that the potential (2) for these values of k is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Moreover, if we assume the canonical transformation x → x+c1 and y → y+c2 where c1 and c2 are arbitrary constants, it is shown that the potential (2) is transformed canonically into the last two potentials of Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Indeed, for k = 0, b = k1 and c = k2, we get the potential V = k1 (x + c1)2 + k2 (y + c2)2 + F(z) while for k = − λ2 4 , b = −k1 and c = −k2, we get the potential V = −λ2 8 R2 − λ2 4 (c1x + c2y) − k1 (x + c1)2 − k2 (y + c2)2 − λ2 8 (c2 1 + c2 2) + F(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The constant term − λ2 8 (c2 1 + c2 2) is overlooked because it does not contribute to the dynamical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Note 3: From Table 2, we observe that the minimally superintegrable potential V = k1 R + k2 √R + y R + k3 √R − y R + F(z) (3) where k1, k2, k3 are arbitrary constants and F(z) is an arbitrary smooth function, admits the two autonomous QFIs: I1 = M3 ˙x − k1y R + k2(R − y)√R + y R − k3(R + y)√R − y R (4) I2 = M3 ˙y + G(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (5) The function G(x, y) must satisfy the system of PDEs: G,x + yV,y = 0 (6) G,y + yV,x − 2xV,y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (7) Using the parabolic cylindrical coordinates (λ′, µ′, z) (see eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='51) in [2]) with λ′ = R + y and µ′ = R − y, the QFI (4) becomes2 I1 = M3 ˙x − 2 λ′ + µ′ �k1 2 (λ′ − µ′) − k2µ′√ λ′ + k3λ′� µ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (8) The QFI I2 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='57) of [2] is not correct and should be replaced by the QFI (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In the parabolic cylindrical coordinates (u, v, z) with u = R + x, v = R − x and3 x, y > 0, the system of PDEs (6) - (7) becomes G,v = uV,v and G,u = −vV,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The solution of this system is G(u, v) = 2 u + v �k1 2 (u − v) − (k2 + k3)v �u 2 + (k2 − k3)u �v 2 � or, equivalently, in Cartesian coordinates G(x, y) = 1 R � k1x − (k2 + k3)(R − x) � R + x 2 + (k2 − k3)(R + x) � R − x 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, the QFI (5) is I2 = M3 ˙y + 2 u + v �k1 2 (u − v) − (k2 + k3)v �u 2 + (k2 − k3)u �v 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (9) 2We recall that the coordinates λ′, µ′ are either positive or zero because λ′ + µ′ = 2R, λ′ − µ′ = 2y, and λ′µ′ = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3For x, y > 0 we have: √R + x + √R − x = √ 2√R + y and √R + x − √R − x = √ 2√R − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 8 There is a misprint in the QFI I3 of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='57) in [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' the correct answer is the QFI (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Note 4: The two superintegrable potentials given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (17) of [4] are subcases of the potential (see Table 2) V = λ 2 R2 + b1y − b2x + F(z) (10) for F(z) = λ 2 z2 + b3z and F(z) = λ 8 z2 + b3 z2 , where b3 is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Note 5: The potential (see Table 2) V1 = cx + F1(y − bx) + F2(z) (11) where c is an arbitrary non-zero constant, w ≡ y − bx and d2F1 dw2 ̸= 0, admits the following LFIs/QFIs (apart from the Hamiltonian H): I1 = ˙x + b ˙y + ct, I2 = t( ˙x + b ˙y) − (x + by) + c 2t2, I3 = ( ˙x + b ˙y)2 + 2c(x + by), I4 = 1 2 ˙z2 + F2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We compute the PBs: {H, I1} = c, {H, I2} = I1, {I1, I2} = 1 + b2, {I1, I3} = −2c(1 + b2), {I2, I3} = −2(1 + b2)I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The three FIs H, I3, I4 are (functionally) independent and in involution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' therefore, the potential (11) is in- tegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The five FIs H, I1, I2, I3, I4 are not independent because I2 1 = I3 + 2cI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' However, the four FIs H, I3, I4, I1, or the H, I3, I4, I2, are independent and, therefore, the potential (11) is minimally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3 The Theorem for QFIs In order to compute in a systematic way the QFIs of non-decomposable potentials, we need to recall a theorem which is proved in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Theorem 1 The independent QFIs of the n-dimensional autonomous holonomic dynamical system ¨qa = −Γa bc(q) ˙qb ˙qc − Qa(q) (12) where qa are the coordinates of the configuration space, ˙qa = dqa dt , t is the time variable, Γa bc(q) are the Rieman- nian connection coefficients of the kinetic metric γab(q) defined by the kinetic energy of the system and −Qa(q) are the generalized forces, are the following: Integral 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I(1,ℓ) = � −t2ℓ 2ℓ L(2ℓ−1)(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' − t4 4 L(3)(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) − t2 2 L(1)(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) + Cab � ˙qa ˙qb + t2ℓ−1L(2ℓ−1)a ˙qa + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' + t3L(3)a ˙qa + +tL(1)a ˙qa + t2ℓ 2ℓ L(2ℓ−1)aQa + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' + t4 4 L(3)aQa + t2 2 L(1)aQa + G(q) where4 Cab(q) and L(M)(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b)(q) for M = 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', 2ℓ−1 are KTs, � L(2ℓ−1)bQb� ,a = −2L(2ℓ−1)(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b)Qb, � L(k−1)bQb� ,a = −2L(k−1)(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b)Qb − k(k + 1)L(k+1)a for k = 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', 2ℓ − 2, and G,a = 2CabQb − L(1)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Integral 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I(2,ℓ) = � − t2ℓ+1 2ℓ + 1L(2ℓ)(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' − t3 3 L(2)(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) − tL(0)(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) � ˙qa ˙qb + t2ℓL(2ℓ)a ˙qa + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' + t2L(2)a ˙qa + +L(0)a ˙qa + t2ℓ+1 2ℓ + 1L(2ℓ)aQa + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' + t3 3 L(2)aQa + tL(0)aQa 4We note that for ℓ = 0 the conditions for the QFI I(1,0) are given by nullifying all the vectors L(M)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 9 where LM(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b)(q) for M = 0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', 2ℓ are KTs, � L(2ℓ)bQb� ,a = −2L(2ℓ)(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b)Qb, and � L(k−1)bQb� ,a = −2L(k−1)(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b)Qb − k(k + 1)L(k+1)a for k = 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', 2ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Integral 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' I(3) = eλt � −L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) ˙qa ˙qb + λLa ˙qa + LaQa� where the vector La(q) is such that L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) is a KT and � LbQb� ,a = −2L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b)Qb − λ2La.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Notation: The Einstein summation convention is used, round (square) brackets indicate symmetrization (antisymmetrization) of the enclosed indices, indices enclosed between vertical lines are overlooked by anti- symmetrization or symmetrization symbols, a comma indicates partial derivative and a semicolon Riemannian covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Before we proceed, we recall the geometric quantities of the Euclidean space E3 required by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 4 The geometric quantities of E3 E3 admits three gradient Killing vectors (KVs) ∂x, ∂y, ∂z whose generating functions are x, y, z, respectively, and three non-gradient KVs y∂x − x∂y, z∂y − y∂z, z∂x − x∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' These vectors are written collectively as La = \uf8eb \uf8ed b1 − b4y + b5z b2 + b4x − b6z b3 − b5x + b6y \uf8f6 \uf8f8 (13) where b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', b6 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='The general second order KT in E3 has independent components: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2 + a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z2 + a4yz + a5y + a2z + a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z2 − a6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xy − a4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xz − a14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 yz − a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x − a15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y + a16z + a17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2 − a4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xy − a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xz − a10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 yz − a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x + a18y − a11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z + a19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 + a7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z2 + a14xz + a15x + a12z + a13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 − a14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xy − a10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xz − a7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 yz − (a16 + a18)x − a12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y − a8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z + a20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 + a7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2 + a10xy + a11x + a8y + a9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='where aK with K = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', 20 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The vector La generating the reducible KT Cab = L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='La = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='−a15y2 − a11z2 + a5xy + a2xz + 2(a16 + a18)yz + a3x + 2a4y + 2a1z + a6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='−a5x2 − a8z2 + a15xy − 2a18xz + a12yz + 2(a17 − a4)x + a13y + 2a7z + a14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='−a2x2 − a12y2 − 2a16xy + a11xz + a8yz + 2(a19 − a1)x + 2(a20 − a7)y + a9z + a10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='and the generated KT is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Cab = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a5y + a2z + a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x − a15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y + a16z + a17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x + a18y − a11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z + a19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x − a15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y + a16z + a17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a15x + a12z + a13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='−(a16 + a18)x − a12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y − a8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z + a20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x + a18y − a11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z + a19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='−(a16 + a18)x − a12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y − a8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z + a20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a11x + a8y + a9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='which is a subcase of the general KT (14) for a1 = a4 = a6 = a7 = a10 = a14 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5 The QFI I(1,1) where ℓ = 1 We set L(1)a = La and the QFI I(1,ℓ) for ℓ = 1 becomes I(1,1) = � −t2 2 L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) + Cab � ˙qa ˙qb + tLa ˙qa + t2 2 LaV ,a + G(x, y, z) (17) 10 where Cab is a second order KT given by (14), the vector La is given by (15), the generated KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) is the (16) and the following conditions must be satisfied: � LbV ,b� ,a = −2L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b)V ,b (18) G,a = 2CabV ,b − La.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (19) Equations (18) and (19) must be supplemented with the three integrability conditions for the function G and the three integrability conditions for the function LaV ,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Finally, we have an overdetermined system of twelve PDEs with unknowns the two functions G(x, y, z) and V (x, y, z), and forty free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Obviously, the general solution is not possible, and we have to look for special solutions which are achieved by introducing simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 Case La = 0 In this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' the QFI (17) is the well-known autonomous QFI I(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1)(La = 0) = Cab ˙qa ˙qb + G(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='where the second order KT Cab has independent components ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2 + a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z2 + a4yz + a5y + a2z + a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z2 − a6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xy − a4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xz − a14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 yz − a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x − a15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y + a16z + a17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2 − a4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xy − a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xz − a10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 yz − a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x + a18y − a11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z + a19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(21) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 + a7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z2 + a14xz + a15x + a12z + a13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 − a14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xy − a10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 xz − a7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 yz − (a16 + a18)x − a12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y − a8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z + a20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='C33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 + a7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2 + a10xy + a11x + a8y + a9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='the parameters a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', a20 are arbitrary constants and the function G(x, y, z) satisfies the condition G,a = 2CabV ,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (22) The integrability condition G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='[ab] = 0 gives: 0 = C12 (V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='yy − V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='xx) + �a6(y2 − x2) 2 + (a1 − a7)z2 2 − (a14x − a4y)z − a15x + a5y + (a2 − a12)z+ +a3 − a13] V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='xy + C13V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='yz − C23V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='xz + 3 2(a6y + a4z + a5)V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x − 3 2(a6x + a14z + a15)V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y + + �3a14 2 y − 3a4 2 x + 2a18 + a16 � V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z (23) 0 = C13 (V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='zz − V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='xx) + �a1(z2 − x2) 2 + (a6 − a7)y2 2 − (a10x − a4z)y − a11x + (a5 − a8)y + a2z+ +a3 − a9] V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='xz + C12V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='yz − C23V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='xy + 3 2(a4y + a1z + a2)V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x + �3a10 2 z − 3a4 2 x + 2a16 + a18 � V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y − −3 2(a1x + a10y + a11)V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z (24) 0 = C23 (V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='zz − V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='yy) + �a7(z2 − y2) 2 + (a6 − a1)x2 2 − (a10y − a14z)x + (a15 − a11)x − a8y + a12z+ +a13 − a9] V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='yz + C12V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='xz − C13V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='xy + �3a10 2 z − 3a14 2 y + a16 − a18 � V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x + 3 2(a14x + a7z + a12)V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y − 11 −3 2(a10x + a7y + a8)V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (25) We note that in the case of 2d Newtonian potentials the integrability condition G,[ab] = 0 leads to just one equation, which is the well-known Bertrand-Darboux PDE (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (28) of [10] and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='5) of [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The system of PDEs (23) - (25) has to be solved in order to find potentials V (x, y, z) that admit autonomous QFIs of the form (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Replacing these potentials in the remaining condition (22), we find the functions G(x, y, x) which determine the associated QFIs (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Since the general solution V (x, y, z) is not possible, we consider again several cases for various values of the twenty free parameters a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', a20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 The components of the KT Cab are constants In this case, the possibly non-zero parameters are the a3, a9, a13, a17, a19, and a20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' A detailed study leads to the following five cases (only non-vanishing parameters are listed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 1) a3 = a, a17 = b 2, and a19 = c 2, where a, b, c are arbitrary constants5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is V (x, y, z) = F1 � cz + by + ( � a2 + b2 + c2 + a)x � + F2 � cz + by − ( � a2 + b2 + c2 − a)x � + F3(bz − cy) (26) where F1, F2, and F3 are arbitrary smooth functions of their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (20) is I(1,1) = (a ˙x + b ˙y + c ˙z) ˙x + a(F1 + F2) + � a2 + b2 + c2(F1 − F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (27) We note that the constants a, b, c are parameters of the potential (26);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' therefore, they cannot generate three distinct QFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2)a3 = a, a13 = −a, and a17 = ia, where a is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is V (x, y, z) = F1(w, z) + F2(w) ¯w (28) where w = x + iy, ¯w = x − iy and F1, F2 are arbitrary smooth functions of their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated autonomous QFI (20) is I(1,1) = ( ˙x + i ˙y)2 + 4 � F2(w)dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (29) If F1(w, z) = F3(w) + F4(z), then V (x, y, z) = F2(w) ¯w + F3(w) + F4(z) (30) is a new integrable potential due to the additional QFI I = 1 2 ˙z2 + F4(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) a3 = a, a13 = −a, and a17 = −ia, where a is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential V (x, y, z) = F1( ¯w, z) + F2( ¯w)w (31) and the associated QFI I(1,1) = ( ˙x − i ˙y)2 + 4 � F2( ¯w)d ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (32) If F1( ¯w, z) = F3( ¯w) + F4(z), then V (x, y, z) = F2( ¯w)w + F3( ¯w) + F4(z) (33) is a new integrable potential due to the additional QFI I = 1 2 ˙z2 + F4(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 4) a19 ̸= 0 and a20 = ia19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is V (x, y, z) = F ′ 2z2 + F3(w)z + F4(w) + F2(w) ¯w (34) 5If instead of the triplet a3, a17, a19 we take as non-vanishing parameters the triplets a13, a17, a20 or a9, a19, a20, the resulting potentials are symmetric up to a cyclic permutation of the coordinates x, y, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 12 where w = x + iy, ¯w = x − iy, F2, F3, F4 are arbitrary smooth functions of their arguments, and F ′ 2 ≡ dF2 dw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated autonomous QFI (20) is I(1,1) = 1 2 ˙z ( ˙x + i ˙y) + F2(w)z + 1 2 � F3(w)dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (35) Because the potential (34) is of the general form (28) for F1(w, z) = F ′ 2z2 + F3(w)z + F4(w), it admits the additional QFI (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, it is integrable because the independent QFIs H, (29) and (35) are in involution6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For F2 = k1w + k2 and F3 = k3 where k1, k2, k3 are arbitrary constants, the potential (34) becomes V (x, y, z) = k1r2 + k2 ¯w + k3z + F4(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (36) This is a new minimally superintegrable potential because it is separable in the coordinate z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5) a19 ̸= 0 and a20 = −ia19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is V (x, y, z) = F ′ 2z2 + F3( ¯w)z + F4( ¯w) + F2( ¯w)w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (37) where w = x + iy and F ′ 2 ≡ dF2 d ¯ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated autonomous QFI (20) is I(1,1) = 1 2 ˙z ( ˙x − i ˙y) + F2( ¯w)z + 1 2 � F3( ¯w)d ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (38) The potential (37) is of the general form (31) for F1( ¯w, z) = F ′ 2z2+F3( ¯w)z+F4( ¯w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' therefore, it is integrable due to the additional QFI (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Moreover, for F2 = k1 ¯w + k2 and F3 = k3, the potential (37) becomes V (x, y, z) = k1r2 + k2w + k3z + F4( ¯w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (39) This is a new minimally superintegrable potential because it is separable in the coordinate z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 The components of the KT Cab are linear functions of x, y, z The possibly non-zero parameters are the a2, a5, a8, a11, a12, a15, a16, and a18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In this case, there are six different combinations which lead to new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 1) a2 = a and a5 = b, where a, b are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is V (x, y, z) = (a2 + b2)x2 + 4(az + by)2 + k1 x2 + k2(az + by) + F(ay − bz) (40) where F is an arbitrary smooth function of its argument and k1, k2 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For a = 0, the potential (40) reduces to the minimally superintegrable potential of the form (see Table 2) V (x, y, z) = k1 2 (x2 + 4y2) + k2 x2 + k3y + F(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (41) The associated QFI (20) is I(1,1) = (aM2 − bM3) ˙x − k2 2 (a2 + b2)x2 − 2(a2 + b2)(az + by)x2 + 2k1(az + by) x2 (42) where Mi = (y ˙z − z ˙y, z ˙x − x ˙z, x ˙y − y ˙x) is the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Since the potential (40) is separable in the coordinate x, it admits the additional QFI I = 1 2 ˙x2 + (a2 + b2)x2 + k1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 6The PB of the QFIs (29) and (35) vanishes because for an integrable function of the form M(w) = � F (w)dw with w = x + iy, it holds that: M′ ≡ dM dw = F , M,x = F and M,y = iF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 13 However, it is not integrable because the PB {I(1,1), I} ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2) a2 = a and a12 = b, where a, b are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We find the fully separable potential V (x, y, z) = k1(x2 + y2 + 4z2) + k2 x2 + k3 y2 + k4z (43) where k1, k2, k3, and k4 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We note that the potential in Table I of [2] is a subcase of the potential (43) for k4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (20) consists of the following two independent QFIs: J1 = M2 ˙x + 2z �k2 x2 − k1x2 � − k4 2 x2 (44) J2 = −M1 ˙y + 2z �k3 y2 − k1y2 � − k4 2 y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (45) Moreover, the potential (43) is of the integrable form V = F1( y x) R2 + F2(R) + F3(z) (see Table 1) for F1 �y x � = k2 � 1 + �y x �2� + k3 � 1 + �x y �2� , F2(R) = k1(x2 + y2), F3(z) = 4k1z2 + k4z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, it admits the additional QFI J3 = 1 2M 2 3 + k2 �y x �2 + k3 �x y �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (46) In order to compare the QFIs (44), (45), (46) with the QFIs I3, I4 of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='43) in [2], we set k4 = 0 and we use the rotational parabolic coordinates (see eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='9) in [2]): ζ = r + z, η = r − z, φ = tan−1 �y x � or equivalently x = � ζη cos φ, y = � ζη sin φ, z = 1 2 (ζ − η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We compute (for k4 = 0): J1 = M2 ˙x + (ζ − η) � k2 ζη cos2 φ − k1ζη cos2 φ � (47) J2 = −M1 ˙y + (ζ − η) � k3 ζη sin2 φ − k1ζη sin2 φ � (48) J3 = 1 2M 2 3 + k2 cos2 φ + k3 sin2 φ − k2 − k3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (49) Then, J3 = I3 − k2 − k3 and J1 + J2 = M2 ˙x − M1 ˙y − (ζ − η) � k1ζη − k2 ζη cos2 φ − k3 ζη sin2 φ � = I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' There is a misprint in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='43) of [2] concerning the leading term of the QFI I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' It must be L2P1 − L1P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We conclude that the potential (43) is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' However, from the seven QFIs (the QFIs H, J1, J2, J3 plus the three QFIs arising from the separability of x, y, z) only five are functionally independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) a2 = a and a8 = b, where a, b are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We find the separable potential V (x, y, z) = k1 4 (x2 + 16y2 + 4z2) + k2 x2 + k3y (50) 14 where k1, k2, and k3 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (20) consists of the following two independent QFIs: J1 = M2 ˙x + z �2k2 x2 − k1 2 x2 � (51) J2 = M1 ˙z − z2 � 2k1y + k3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (52) Therefore, the separable potential (50) is a new maximally superintegrable potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 4) a2 = a and a16 = −a18 = b 2, where a, b are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is V (x, y, z) = k � (ax + by)2 + (a2 + b2)z2 (53) where k is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (20) is I(1,1) = aM2 ˙x − bM1 ˙x + azV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (54) In order to show that the potential (53) is integrable, we need one more independent FI in involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5) a2 = a12 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In this case, we find the following three potentials7: V1(x, y, z) = F1 � y x � R2 + k(R2 + 4z2) (55) V2(x, y, z) = F1 � y x � R2 + k1z rR2 − k2 r (56) V3(x, y, z) = F1 � y x � R2 + kz (57) where k, k1, k2 are arbitrary constants, R = � x2 + y2 and r = � x2 + y2 + z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The first two potentials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' V1 and V2, are included in Table II of [2], whereas the third potential V3 is not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFIs (20) are: For the potential8 (55): I(1,1) = M2 ˙x − M1 ˙y + 2zF1 � y x � x2 + y2 − 2kz(x2 + y2) = M2 ˙x − M1 ˙y + (ζ − η) �F1(tan φ) ζη − kζη � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (58) For the potential (56): I(1,1) = M2 ˙x − M1 ˙y + 2zF1 � y x � x2 + y2 + 2k1z2 r(x2 + y2) + k1 r − k2z r = M2 ˙x − M1 ˙y + (ζ − η)F1(tan φ) ζη + k1(ζ2 + η2) ζη(ζ + η) − k2(ζ − η) ζ + η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (59) For the potential (57): I(1,1) = M2 ˙x − M1 ˙y + 2zF1 � y x � x2 + y2 − k R2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (60) We note that both the potentials (55) and (57) are minimally superintegrable, because they are of the form V = F1( y x) R2 + F2(R) + F3(z) (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 7We note that any linear combination of these potentials is a solution of the system of PDEs (23) - (25) for a2 = a12 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 8In Table II of [2], there is a misprint in the QFIs I3 associated with the potentials (55) and (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The leading part of I3 should be L2P1 − P2L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 15 Moreover, from cases 2) and 3) of the following subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3, the potential (56) becomes minimally superintegrable because it admits the additional QFIs (75) and (81) which are also in involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 6) a2 ̸= 0, a12 = −a2, a16 = ia2 and a18 = −i a2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is V (x, y, z) = k1(R2 + 4z2) + k2z + k3 w2 + k4 ¯w w3 (61) where k1, k2, k3, k4 are arbitrary constants, w = x + iy, and ¯w = x − iy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This result coincides with the potential given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (14) of [4] if we apply the canonical transformation x → y, y → z and z → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated autonomous QFI (20) is I1 = 1 2( ˙x + i ˙y) (M2 − iM1) − k1zw2 − k2 4 w2 − k4 z w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (62) Moreover, the potential (61) admits the additional QFIs: I2 = 1 2 ( ˙x + i ˙y)2 + k1w2 − k4 w2 (63) I3 = 1 2 ˙z2 + 4k1z2 + k2z (64) I4 = 1 2M 2 3 + k3e−2iθ + k4e−4iθ (65) I5 = 1 2 (M2 ˙x − M1 ˙y) + k3 z w2 + k4 z ¯w w3 − k1zR2 − k2 R2 4 (66) because it is of the form (28) for F1 = 4k1z2 + k2z + k3 w2 and F2 = k1w + k4 w3 , and of the form (see subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 and Table 6) V (x, y, z) = F1 � y x � R2 + k1(R2 + 4z2) + k2z (67) for F1 � y x � = k3e−2iθ + k4e−4iθ, where tan θ = y x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the potential (61) is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 The components of the KT Cab depend on xy, xz, yz, x2, y2, z2 In this case, the possibly non-zero parameters are the a1, a4, a6, a7, a10, and a14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' New results are produced for the following six cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 1) a1 = a, a6 = b, and a7 = c, where a, b, c are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is (see Table II in [2]) V (x, y, z) = k1 x2 + k2 y2 + k3 z2 + F(r) (68) where k1, k2, k3 are arbitrary constants, r = � x2 + y2 + z2 and F is an arbitrary smooth function of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (20) consists of the following three independent FIs (one for each parameter a, b, c): I1 = 1 2M 2 1 + k2 z2 y2 + k3 y2 z2 (69) I2 = 1 2M 2 2 + k1 z2 x2 + k3 x2 z2 (70) I3 = 1 2M 2 3 + k1 y2 x2 + k2 x2 y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (71) Using spherical coordinates x = r sin θ cos φ, y = r sin θ sin φ and z = r cos θ, the QFIs (69) - (71) coincide with those found in Table II of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Moreover, by adding the above QFIs, we find the QFI I4 = 1 2M2 + k1 sin2 θ cos2 φ + k2 sin2 θ sin2 φ + k3 cos2 θ (72) where M2 = M 2 1 + M 2 2 + M 2 3 is the square magnitude of the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 16 Even though the potential (68) admits the four independent QFIs H, I1, I2, and I3, it is not integrable because the PBs {Ii, Ij} ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For F(r) = kr2, where k is an arbitrary constant, the potential (68) becomes (see Table I in [2]) V (x, y, z) = k � x2 + y2 + z2� + k1 x2 + k2 y2 + k3 z2 (73) which is maximally superintegrable (see Tables 1 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2) a6 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is V (x, y, z) = F1 � y x � R2 + F2(R, z) (74) where F1 and F2 are arbitrary smooth functions of their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (20) is I(1,1) = 1 2M 2 3 + F1 �y x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (75) If F2(R, z) = F3(R) + F4(z) where F3 and F4 are arbitrary smooth functions, then the potential (74) is integrable (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) a1 = a6 = a7 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is V (x, y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' m) = m � j=1 Fj � y x � R2 Nj � z R � + F(r) (76) where Fj, Nj and F with j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', m are smooth functions of their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (20) is I(1,1) = 1 2M2 + m � j=1 r2Fj � y x � R2 Nj � z R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (77) We note that for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' m = 2, N1 = F2 = 1, N2 = k2 R2 z2 , F(r) = k1r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' m = 2, N1 = F2 = 1, N2 = k1 z R � 1+ z2 R2 , F(r) = − k2 r the potential (76) reduces, respectively, to the following potentials (see Table II in [2]): V1(x, y, z) = F1 � y x � x2 + y2 + k1(x2 + y2 + z2) + k2 z2 (78) V2(x, y, z) = F1 � y x � x2 + y2 + k1z r(x2 + y2) − k2 r (79) where k1 and k2 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Both the above potentials are also of the general form (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFIs (77) are as follows: For the potential (78): I(1,1) = 1 2M2 + r2F1 � y x � x2 + y2 + k2(x2 + y2) z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (80) For the potential (79): I(1,1) = 1 2M2 + r2F1 � y x � x2 + y2 + k1zr x2 + y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (81) We note that the potential (78) is minimally superintegrable because it is separable in the coordinate z and is also of the form (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 4) a1 ̸= 0, a7 = −a1 and a10 = ia1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is V (x, y, z) = F3(w) + z2 w F2(w) + F4 � z w � w2 + F2(w) ¯w (82) 17 where w = x + iy, ¯w = x − iy, and F2, F3, F4 are arbitrary smooth functions of their arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (20) is I1 = 1 2 (M2 − iM1)2 + F4 � z w � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (83) Moreover, the potential (82) admits the additional QFI I2 = ( ˙x + i ˙y)2 + 4 � F2(w)dw (84) because it is of the form (28) with F1 = F3(w) + z2 w F2(w) + F4( z w) w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Since the PB {I1, I2} = 0, the potential (82) is (Liouville) integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Finally, for F2 = k1w and F4 = k2 w2 z2 , the potential (82) becomes V (x, y, z) = k1r2 + k2 z2 + F3(w) (85) which is a new minimally superintegrable potential due to the separability in the z-coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5) a1 ̸= 0, a7 = −a1 and a10 = −ia1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Similarly to the previous case 4), we find the integrable potential V (x, y, z) = F3( ¯w) + z2 ¯w F2( ¯w) + F4 � z ¯ w � ¯w2 + F2( ¯w)w (86) and the associated QFI I1 = 1 2 (M2 + iM1)2 + F4 � z ¯w � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (87) Moreover, the potential (86) admits the additional QFI I2 = ( ˙x − i ˙y)2 + 4 � F2( ¯w)d ¯w (88) because it is of the form (31) for F1 = F3( ¯w) + z2 ¯ w F2( ¯w) + F4( z ¯ w) ¯ w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Finally, for F2 = k1 ¯w and F4 = k2 ¯w2 z2 , the potential (86) becomes V (x, y, z) = k1r2 + k2 z2 + F3( ¯w) (89) which is a new minimally superintegrable potential due to the separability in the z-coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 6) a4 ̸= 0 and a14 = −ia4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential is (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (12) of [4]) V (x, y, z) = k1r2 + k2 w2 + k3 z w3 + k4 R2 − 3z2 w4 (90) where k1, k2, k3, k4 are arbitrary constants and w = x + iy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (20) is I1 = M3 (iM1 − M2) + k3 y w − 2ik2 z w − 3ik3 2 z2 w2 − 4ik4 z(x2 + y2 − z2) w3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (91) Moreover, the potential (90) admits the additional QFIs: I2 = 1 2 (M2 − iM1)2 + k3 z w − 4k4 z2 w2 (92) I3 = 1 2 ˙z ( ˙x + i ˙y) + k1zw − k3 4w2 + k4 z w3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (93) I4 = 1 2 ( ˙x + i ˙y)2 + k1w2 − k4 w2 (94) 18 I5 = 1 2M2 + k2 r2 w2 + k3 zr2 w3 + k4 r2(x2 + y2 − 3z2) w4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (95) Specifically, we have the following: 1) It admits the QFI (92) because it is of the form (82) for F2 = k1w + k4 w3 , F3 = k2 w2 and F4 = k3 z w − 4k4 z2 w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2) It admits the QFI (93) because it is of the form (34) for F2 = k1w + k4 w3 , F3 = k3 w3 and F4 = k2 w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) It admits the QFI (94) because it is of the form (28) for F1 = k1z2 + k2 w2 + k3 z w3 − 3k4 z2 w4 and F2 = k1w + k4 w3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 4) It admits the QFI (95) because it is of the form (76) for m = 4, N1 = 1, F1 = k2e−2iθ, N2 = z R, F2 = k3e−3iθ, N3 = 1, F3 = k4e−4iθ, N4 = z2 R2 , F4 = −3k4e−4iθ and F(r) = k1r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We note that the variable θ = tan−1 � y x � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' hence, w = x + iy = Reiθ and R2 = w ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We conclude that the potential (90) is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Specifically, it is integrable due to the triplet H, I2, I4 and superintegrable because it admits the five independent QFIs H, I1, I2, I3, I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 The components of the KT Cab depend on products of x, y, z of mixed degree In this subsection, we continue by considering mixed combinations of the twenty parameters a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', a20 so that the components of the KT Cab contain products of x, y, z of mixed degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We note that we do not exhaust all possible cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' therefore, other authors could consider other cases and determine new non-decomposable integrable/superintegrable potentials in E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 1) The only non-vanishing parameters are the a3 = iB 4 , a5 = B, a13 = − iB 4 , a17 = − B 4 and a15 = iB, where B is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The KT (21) is Cab = \uf8eb \uf8ed By + iB 4 − B 2 x − iB 2 y − B 4 0 − B 2 x − iB 2 y − B 4 iBx − iB 4 0 0 0 0 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (96) For the KT (96) the system of PDEs (23) - (25) gives the potential V (x, y, z) = 4k1 � R2 − ¯w3 2 � + k2 � 2w − 3 ¯w2� + k3 ¯w + F(z) (97) where k1, k2, k3 are arbitrary constants and F(z) is an arbitrary smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (20) is I1 = 1 4 ( ˙x + i ˙y)2 + iM3 ( ˙x − i ˙y) − 2k1 �3 4 ¯w4 − w2 + R2 ¯w � − −2k2 � ¯w3 + 2R2� + k3 � ¯w2 2 + w � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (98) Moreover, the potential (97) admits the additional QFIs: I2 = 1 8 ( ˙x − i ˙y)2 + k1 ¯w2 + k2 ¯w (99) I3 = 1 2 ˙z2 + F(z) (100) because it is of the form (31) for F1 = −2k1 ¯w3 − 3k2 ¯w2 + k3 ¯w + F(z) and F2 = 4k1 ¯w + 2k2, and it is separable on the z-coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, it is minimally superintegrable due to the four independent QFIs H, I1, I2, I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We note that the potential given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (15) of [4] is a subcase of (97) for F(z) = k1z2 + k4 z2 , where k4 is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' As it will be shown below, in this special case, the resulting potential admits additional QFIs which promote it to a maximally superintegrable potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2) The only non-vanishing parameters are the a1 = C, a7 = −C, a8 = −iD + 2iC, a10 = −iC and a11 = D, where C, D are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The KT (21) is Cab = \uf8eb \uf8ed C 2 z2 − iC 2 z2 − C 2 xz + iC 2 yz − D 2 z − iC 2 z2 − C 2 z2 iC 2 xz + C 2 yz + i � D 2 − C � z − C 2 xz + iC 2 yz − D 2 z iC 2 xz + C 2 yz + i � D 2 − C � z C 2 (x2 − y2) + iCy(2 − x) + D(x − iy) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (101) 19 For the KT (101) the system of PDEs (23) - (25) gives the potential (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (15) of [4]) V (x, y, z) = 4k1 � R2 − ¯w3 2 + z2 4 � + k2 � 2w − 3 ¯w2� + k3 ¯w + k4 z2 (102) where k1, k2, k3, and k4 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (20) consists of the following independent QFIs: I1 = 1 2 (M2 + iM1)2 + 2i ˙zM1 + k1z2 � 3 ¯w2 − 4iy � + 2k2z2 (2 ¯w + 1) − −k3z2 + k4 z2 � ¯w2 + 4iy � (103) I2 = 1 2 ˙z (M2 + iM1) + k1z2 ¯w + k2z2 − k4 ¯w z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (104) Moreover, the potential (102) admits the three additional QFIs (98) - (100) because it is of the form (97) for F(z) = k1z2 + k4 z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, it is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) The only non-vanishing parameters are the: a1 = −2C, a2 = iB − C, a5 = a8 = iA, a7 = 2C, a9 = iB 4 − C 4 , a10 = −2iC, a11 = a15 = A, a12 = −iB + 2C, a13 = C 4 , a16 = −B − 3iC 2 , a18 = B 2 + 3iC 2 , a20 = iA 4 where A, B, and C are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The KT (21) has independent components: C11 = −Cz2 + iAy + (iB − C)z C12 = −iCz2 − iA 2 ¯w − � B + 3iC 2 � z C13 = Cxz − −iCyz − iB − C 2 x + 1 2(B + 3iC)y − A 2 z (105) C22 = Cz2 + Ax − (iB − 2C)z − C 4 C23 = iCxz − Cyz + B 2 x + iB − 2C 2 y − iA 2 z + iA 4 C33 = −Cx2 + Cy2 − 2iCxy + Aw + iB − C 4 where w = x + iy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For the KT (105) the system of PDEs (23) - (25) gives the potential (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (16) of [4]) V (x, y, z) = k1w + k2 � 3w2 + z � + k3 � 4w3 + 3wz + ¯w 4 � + k4 �5 2w4 + r2 2 + 3w2z � (106) where k1, k2, k3, and k4 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='The associated QFI (20) consists of the following independent QFIs: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='M3 ˙w − (M1 + iM2) ˙z − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y ˙z + ik1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w2 − z ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ik4w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2w4 + zw2 − z2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(107) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='M1 (2 ˙x + i ˙y) + iM2 ˙x + M3 ˙z + i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ˙z2 + ik2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z − w2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ik3w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z − w2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ik4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 + 2zw2 − 3w4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(108) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(M1 + iM2)2 + (2iM1 − M2) ˙w − iM3 ˙z + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙y2 − ˙z2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k1zw + ik1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+2k2w (2zw + iy) + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w2 − z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k3w3(6z + 1) + k3zw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2z − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ik3y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3w2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− k3xz + k4w4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4z + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ 2ik4yw3 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+k4z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3w2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 − zR2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (109) We note that the parameter A produces the QFI I1, B the I2, and C the I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Moreover, the potential (106) admits the additional QFIs: I4 = ˙w2 + k3w + k4w2 (110) I5 = ˙z ˙w + � k4w + k3 2 � z + k4w3 + 3k3 2 w2 + k2w (111) because it is of the form (28) for F1 = k1w + k2(3w2 + z) + k3w(4w2 + 3z) + k4 � 5 2w4 + 3w2z + z2 2 � and F2 = k4 2 w+ k3 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' and of the form (34) for F2 = k4 2 w+ k3 4 , F3 = 3k4w2+3k3w+k2 and F4 = 5k4 2 w4+4k3w3+3k2w2+k1w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We compute the PB {I2, I4} = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' therefore, the potential (106) is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='5 Special superintegrable potentials In this subsection, we construct potentials whose form belongs to two or more of the previous general results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We have the following cases: 1) Consider the potential (see Table I in [2]) V (x, y, z) = −c1 r + c2 x2 + c3 y2 (112) where c1, c2, and c3 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This potential is of the general form (68) for F(r) = − c1 r , k1 = c2, k2 = c3 and k3 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' and of the form (56) for k1 = 0, k2 = c1 and F1 � y x � = c2 � 1 + � y x �2� + c3 � 1 + � x y �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, it admits the additional QFIs: I1 = 1 2M 2 1 + c3 z2 y2 (113) I2 = 1 2M 2 2 + c2 z2 x2 (114) I3 = 1 2M 2 3 + c2 y2 x2 + c3 x2 y2 (115) I4 = M2 ˙x − M1 ˙y − 2z � c1 2r − c2 x2 − c3 y2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (116) We conclude that the potential (112) is maximally superintegrable because the QFIs H, I3, I4 are in involution and the five QFIs H, I1, I2, I3, I4 are functionally independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2) Consider the potential (see Table I in [2]) V (x, y, z) = c1y x2R + c2 x2 + c3 z2 (117) where c1, c2, and c3 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This potential is of the form V = k1 x2 + k2 R + k3y Rx2 + F(z) (see Table 2) for k1 = c2, k2 = 0, k3 = c1, and F(z) = c3 z2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' and of the form (78) for k1 = 0, k2 = c3, and F1 � y x � = � c1 � 1+ x2 y2 + c2 � � 1 + y2 x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, it admits the additional QFIs: I1 = 1 2 ˙z2 + c3 z2 (118) 21 I2 = 1 2M 2 3 + c2 y2 x2 + c1 yR x2 (119) I3 = M3 ˙x − 2c2 y x2 − c1 x2 + 2y2 x2R (120) I4 = 1 2M2 + c1 yr2 Rx2 + c2 r2 x2 + c3 R2 z2 (121) and it is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) Another maximally superintegrable potential is the (see Table I in [2]) V (x, y, z) = c1y x2R + c2 x2 + c3z (122) where c1, c2, and c3 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This potential is of the form V = k1 x2 + k2 R + k3y Rx2 + F(z) (see Table 2) for k1 = c2, k2 = 0, k3 = c1, and F(z) = c3z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' and of the form (57) for k = c3, and F1 � y x � = � c1 � 1+ x2 y2 + c2 � � 1 + y2 x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, it admits the additional QFIs: I1 = 1 2 ˙z2 + c3z (123) I2 = 1 2M 2 3 + c2 y2 x2 + c1 yR x2 (124) I3 = M3 ˙x − 2c2 y x2 − c1 x2 + 2y2 x2R (125) I4 = M2 ˙x − M1 ˙y + c1 2yz x2R + c2 2z x2 − c3 R2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (126) 4) Consider the potential (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (11) of [4]) V (x, y, z) = k1r2 + k2 ¯w w3 + k3 w2 + k4 z2 (127) where k1, k2, k3, k4 are arbitrary constants, w = x + iy and ¯w = x − iy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This potential admits the additional QFIs: I1 = 1 2 (M2 − iM1)2 + k4 w2 z2 − k2 z2 w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (128) I2 = 1 2 ( ˙x + i ˙y)2 + k1w2 − k2 w2 (129) I3 = 1 2M 2 3 + k2e−4iθ + k3e−2iθ = 1 2M 2 3 + k2 � ¯w w �2 + k3 ¯w w (130) I4 = 1 2 ˙z2 + k1z2 + k4 z2 (131) I5 = 1 2M2 + k2 r2 ¯w w3 + k3 r2 w2 + k4 r2 z2 (132) because it is of the form (82) for F2 = k1w + k2 w3 , F3 = k3 w2 and F4 = −k2 z2 w2 + k4 w2 z2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' of the form (28) for F1 = k1z2 + k3 w2 + k4 z2 and F2 = k1w + k2 w3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' of the form (74) for F1 = k2e−4iθ + k3e−2iθ and F2 = k1r2 + k4 z2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' separable on the z-coordinate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' and of the form (76) for m = 2, F1 = N2 = 1, N1 = k4 R2 z2 , F2 = k2e−4iθ +k3e−2iθ and F(r) = k1r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The variable θ = tan−1 � y x � and, hence, w = x + iy = Reiθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We recall that w = Reiθ =⇒ einθ = �w R �n =⇒ einθ = \uf8eb \uf8ed 1 + i y x � 1 + � y x �2 \uf8f6 \uf8f8 n 22 where n is an arbitrary real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' If n = 2k ∈ R, then e2ikθ = � w ¯ w �k because R2 = w ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We conclude that the potential (127) is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We collect the results of this section in Tables 4 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='LFIs and QFIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='cz + by + ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a2 + b2 + c2 + a)x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+F2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='cz + by − ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a2 + b2 + c2 − a)x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+F3(bz − cy) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = (a ˙x + b ˙y + c ˙z) ˙x + a(F1 + F2)+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a2 + b2 + c2(F1 − F2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = (a2 + b2)x2 + 4(az + by)2 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+k2(az + by) + F(ay − bz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = aM2 ˙x − bM3 ˙x − k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 (a2 + b2)x2− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='−2(a2 + b2)(az + by)x2 + 2k1(az+by) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + (a2 + b2)x2 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(ax+by)2+(a2+b2)z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = aM2 ˙x − bM1 ˙x + azV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 + F(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 + k2 z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 + k1 z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + k3 x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + k2 x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F1( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2+y2 + F2(x2 + y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) I1 = 1 2M 2 3 + F1 � y x � V = �m j=1 Fj( y x) R2 Nj � z R � + F(r) I = 1 2M2 + �m j=1 r2Fj( y x) R2 Nj � z R � V = F1(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) + F2(w) ¯w I1 = ( ˙x + i ˙y)2 + 4 � F2(w)dw V = F1( ¯w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) + F2( ¯w)w I1 = ( ˙x − i ˙y)2 + 4 � F2( ¯w)d ¯w Table 4: Possibly non-integrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) in E3 that admit one or more QFIs of the type I(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1) which are not in involu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Integrable potentials ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='LFIs and QFIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F2(w) ¯w + F3(w) + F4(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = ( ˙x + i ˙y)2 + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F2(w)dw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F4(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F2( ¯w)w + F3( ¯w) + F4(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = ( ˙x − i ˙y)2 + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F2( ¯w)d ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F4(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2z2 + F3(w)z + F4(w) + F2(w) ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z ( ˙x + i ˙y) + F2(w)z + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F3(w)dw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ( ˙x + i ˙y)2 + 4 � F2(w)dw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2z2 + F3( ¯w)z + F4( ¯w) + F2( ¯w)w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z ( ˙x − i ˙y) + F2( ¯w)z + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F3( ¯w)d ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ( ˙x − i ˙y)2 + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F2( ¯w)d ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F3(w) + z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w F2(w) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F4( z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ F2(w) ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 (M2 − iM1)2 + F4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ( ˙x + i ˙y)2 + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F2(w)dw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = F3( ¯w) + z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w F2( ¯w) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F4( z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ F2( ¯w)w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 (M2 + iM1)2 + F4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ( ˙x − i ˙y)2 + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F2( ¯w)d ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table 5: Integrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) in E3 that admit QFIs of the type I(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Minimally superintegrable potentials ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Ref [2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='LFIs and QFIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F1( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k1(R2 + 4z2) + k2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k2 = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + 4k1z2 + k2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + F1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = M2 ˙x − M1 ˙y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2zF1( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− 2k1zR2 − k2 R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F1( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k1z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='rR2 − k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + F1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r2F1( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k1zr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = M2 ˙x − M1 ˙y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2zF1( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ 2k1z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='rR2 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r − k2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F1( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k1r2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + k1z2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + F1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r2F1( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k2R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = 4k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 − ¯w3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2w − 3 ¯w2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+k3 ¯w + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 ( ˙x − i ˙y)2 + k1 ¯w2 + k2 ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ˙w2 + iM3 ( ˙x − i ˙y) − 2k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ¯w4 − w2 + R2 ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='−2k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='¯w3 + 2R2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='¯w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 + w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1r2 + k2 ¯w + k3z + F4(w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z ( ˙x + i ˙y) + k1wz + k2z + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ( ˙x + i ˙y)2 + 2k1w2 + 4k2w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + k1z2 + k3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1r2 + k2w + k3z + F4( ¯w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z ( ˙x − i ˙y) + k1 ¯wz + k2z + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ( ˙x − i ˙y)2 + 2k1 ¯w2 + 4k2 ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + k1z2 + k3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1r2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 + F3(w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 (M2 − iM1)2 + k2 w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ( ˙x + i ˙y)2 + 2k1w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + k1z2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1r2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 + F3( ¯w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 (M2 + iM1)2 + k2 ¯w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ( ˙x − i ˙y)2 + 2k1 ¯w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + k1z2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table 6: Minimally superintegrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) in E3 that admit QFIs of the type I(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 25 Maximally superintegrable potentials Potential Ref [2] Ref [4] LFIs and QFIs V = k1(R2 + 4z2) + k2 x2 + k3 y2 + k4z Table I k4 = 0 eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z ↔ x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + k1x2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + k1y2 + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + 4k1z2 + k4z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = M2 ˙x + 2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 − k1x2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− k4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = −M1 ˙y + 2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 − k1y2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− k4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I6 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + 16y2 + 4z2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+k3y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 x2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + 4k1y2 + k3y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + k1z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = M2 ˙x + z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� 2k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 − k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = M1 ˙z − z2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2k1y + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = kr2 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + kx2 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + ky2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + kz2 + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 + k2 z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 + k1 z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + k3 x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I6 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + k2 x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='included ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 + c3 z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 + c2 z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + c3 x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = M2 ˙x − M1 ˙y − 2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2r − c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 − c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = c1y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2R + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ↔ y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='included ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='yR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = M3 ˙x − 2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 − c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2+2y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M2 + c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='yr2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Rx2 + c2 r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + c3 R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = c1y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2R + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + c3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ↔ y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='included ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + c3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='yR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = M3 ˙x − 2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 − c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2+2y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = M2 ˙x − M1 ˙y + c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2yz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2R + c2 2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 − c3 R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1r2 + k2 ¯w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w3 + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w2 + k4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w = x + iy = Reiθ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='¯w = x − iy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='included ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (11) I1 = 1 2 (M2 − iM1)2 + k4 w2 z2 − k2 z2 w2 I2 = 1 2 ( ˙x + i ˙y)2 + k1w2 − k2 w2 I3 = 1 2M 2 3 + k2e−4iθ + k3e−2iθ = 1 2M 2 3 + k2 � ¯w w �2 + k3 ¯ w w I4 = 1 2 ˙z2 + k1z2 + k4 z2 I5 = 1 2M2 + k2 r2 ¯w w3 + k3 r2 w2 + k4 r2 z2 V = k1r2 + k2 w2 + k3 z w3 + +k4 R2−3z2 w4 not included eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (12) I1 = 1 2 (M2 − iM1)2 + k3 z w − 4k4 z2 w2 I2 = M3 (iM1 − M2) + k3 y w − 2ik2 z w− − 3ik3 2 z2 w2 − 4ik4 z(x2+y2−z2) w3 I3 = 1 2 ( ˙x + i ˙y)2 + k1w2 − k4 w2 I4 = 1 2 ˙z ( ˙x + i ˙y) + k1zw − k3 4w2 + k4 z w3 I5 = 1 2M2 + k2 r2 w2 + k3 zr2 w3 + k4 r2(x2+y2−3z2) w4 26 V = k1(R2 + 4z2) + k2z + k3 w2 + +k4 ¯w w3 not included eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (14) I1 = 1 2 ( ˙x + i ˙y)2 + k1w2 − k4 w2 I2 = 1 2( ˙x + i ˙y) (M2 − iM1) − k1zw2− − k2 4 w2 − k4 z w2 I3 = 1 2 ˙z2 + 4k1z2 + k2z I4 = 1 2M 2 3 + k3e−2iθ + k4e−4iθ I5 = 1 2 (M2 ˙x − M1 ˙y) + k3 z w2 + k4 z ¯ w w3 − −k1zR2 − k2 R2 4 V = 4k1 � R2 − ¯w3 2 + z2 4 � + +k2 � 2w − 3 ¯w2� + k3 ¯w+ + k4 z2 not included eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (15) I1 = 1 8 ( ˙x − i ˙y)2 + k1 ¯w2 + k2 ¯w I2 = 1 4 ˙w2 + iM3 ˙¯w − 2k1 � 3 4 ¯w4 − w2 + R2 ¯w � − −2k2 � ¯w3 + 2R2� + k3 � ¯ w2 2 + w � I3 = 1 2 ˙z2 + k1z2 + k4 z2 I4 = 1 2 (M2 + iM1)2 + 2i ˙zM1+ +k1z2 � 3 ¯w2 − 4iy � + 2k2z2 (2 ¯w + 1) − −k3z2 + k4 z2 � ¯w2 + 4iy � I5 = 1 2 ˙z (M2 + iM1) + k1z2 ¯w + k2z2 − k4 ¯ w z2 V = k1w + k2 � 3w2 + z � + +k3 � 4w3 + 3wz + ¯ w 4 � + +k4 � 5 2w4 + r2 2 + 3w2z � not included eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = M3 ˙w − (M1 + iM2) ˙z − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y ˙z+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ik1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w2 − z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ik2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2w3 − zw + i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ik3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w2 − z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ik3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3w4 − z2 + iyw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− k4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w2 + z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ik4w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2w4 + zw2 − z2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = M1 (2 ˙x + i ˙y) + iM2 ˙x + M3 ˙z + i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ˙z2+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ik2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z − w2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ik3w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z − w2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ik4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 + 2zw2 − 3w4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = (M1 + iM2)2 + (2iM1 − M2) ˙w− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='−iM3 ˙z + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙y2 − ˙z2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k1zw+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ik1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y + 2k2w (2zw + iy) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='w2 − z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k3w3(6z + 1)+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+k3zw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2z − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ik3y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3w2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='−k3xz + k4w4 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4z + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ 2ik4yw3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+k4z2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3w2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 − zR2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = ˙w2 + k3w + k4w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = ˙z ˙w + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k4w + k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z + k4w3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ 3k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 w2 + k2w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table 7: Maximally superintegrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) in E3 that admit QFIs of the type I(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Remark: The potential (68) admits the four independent QFIs H, I1, I2 and I3 (see Table 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' however, it is not second order integrable because the PBs {Ii, Ij} ̸= 0 for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In Table II of [2], it is claimed that this potential is minimally superintegrable because in that paper superintegrability is defined without the requirement of the integrability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' the vanishing of the PBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Indeed, we have: I4 ≡ {I1, I3} = {I2, I1} = {I3, I2} = M1 � x2 ˙y ˙z + 2k1 yz x2 � +M2 � y2 ˙x ˙z + 2k2 xz y2 � +M3 � z2 ˙x ˙y + 2k3 xy z2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (133) The third order (cubic) FI I4 cannot be used for establishing integrability because the PBs {Ii, I4} ̸= 0, where i, j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 6 The QFI I(2,0) where ℓ = 0 We set L(0)a = La and the QFI I(2,ℓ) for ℓ = 0 becomes I(2,0) = −tL(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) ˙qa ˙qb + La ˙qa + tLaV ,a (134) 27 where the vector La is given by (15), the generated KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) is given by (16) and the following condition is satisfied � LbV ,b� ,a = −2L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b)V ,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (135) Condition (135) is a subcase of the general condition (22) in the case that the function G = −LaV ,a and the general second order KT Cab = L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1, we have computed (not all) pairs of functions (G, V ) which satisfy the condition (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, in order to find potentials V (x, y, z) that admit QFIs of the form (134), it is sufficient to solve the constraint G = −LaV ,a (136) for all pairs (G, V ) for which the KT Cab is given by the reducible form (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' If the constraint (136) is not satisfied for some pairs (G, V ), then the corresponding potentials V of these pairs do not admit QFIs of the type I(2,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Moreover, the QFI (134) is written as I(2,0) = −Jt + La ˙qa where J is the associated autonomous QFI (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The PB {H, I(2,0)} = ∂I(2,0) ∂t = −J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore: The time-dependent QFI I(2,0) generates an autonomous QFI of the type I(1,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This is an interesting connection between (first degree) time-dependent and autonomous QFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We consider the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 1) In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2, we determined the functions: V (x, y, z) = (a2 + b2)x2 + 4(az + by)2 + k1 x2 + k2(az + by) + F(ay − bz) (137) G(x, y, z) = −k2 2 (a2 + b2)x2 − 2ab(ay + bz)x2 − 2(a3z + b3y)x2 + 2k1(by + az) x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (138) Then, the vector La = \uf8eb \uf8ed bxy + axz + 2b2y + 2b1z + b3 −bx2 − 2b2x + 2b4z + b6 −ax2 − 2b1x − 2b4y + b5 \uf8f6 \uf8f8 (139) where b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', b6 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Replacing (137), (138) and (139) in the condition (136), we find: b1 = b2 = b3 = b4 = 0, a = ±ib, b5 = ±b6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the potential (137) becomes9 (see the potential V = F1(y − bx) + F2(z) in Table 2) V (x, y, z) = k1 x2 + 4b(y ± iz)2 + bk2(y ± iz) + F(y ± iz) � �� � =F1(y±iz) = k1 x2 + F1(y ± iz) (140) and the vector La = \uf8eb \uf8ed bx(y ± iz) −bx2 + b6 ±i(−bx2 + b6) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (141) The associated time-dependent QFI (134) is I(2,0) = −bt(y ± iz) ˙x2 + btx ˙x( ˙y ± i ˙z) + b(y ± iz)x ˙x − (bx2 − b6) ˙y ∓ i(bx2 − b6) ˙z − 2k1bt(y ± iz) x2 = b6J1 − bJ2 (142) which contains the independent FIs: J1 = ˙y ± i ˙z, J2 = t � ˙x2 + 2k1 x2 � (y ± iz) − x ˙x(y ± iz) − J1x(t ˙x − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 9The function F (iz ± y) is either the F (y + iz) or the F (y − iz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, we can write F (y ± iz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 28 From section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 we have that the potential (140) admits also the autonomous QFIs: J3 = (±iM2 − M3) ˙x + 2k1(y ± iz) x2 , J4 = 1 2 ˙x2 + k1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We note that J2 = J3t − x ˙x(y ± iz) + J1x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential (140) is maximally superintegrable due to the five linearly independent FIs H, J1, J2, J3, and J4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The autonomous FIs H, J1, J4 are in involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This is a new result which could not be found in [2] because of the additional time-dependent QFI J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The PBs are: {H, J2} = ∂J2 ∂t = J3, {J1, J2} = {J1, J3} = {J1, J4} = 0, {J3, J4} = −J1 � ˙x2 + 2k1 x2 � , {J2, J3} = −2(M3 ∓ iM2)2 − 4k1 x2 (y ± iz)2, {J2, J4} = − (J1t + y ± iz) � ˙x2 + 2k1 x2 � + 2J1x ˙x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2) In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2, we determined the functions: V (x, y, z) = F1 � y x � R2 + k1z rR2 − k2 r (143) G(x, y, z) = a2 2zF1 � y x � R2 + a2 2k1z2 rR2 + a2 k1 r − a2 k2z r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (144) Then, the vector La = \uf8eb \uf8ed axz + 2b2y + 2b1z + b3 ayz − 2b2x + 2b4z + b6 −aR2 − 2b1x − 2b4y + b5 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (145) Replacing (143), (144) and (145) in the condition (136), we get: b1 = b2 = b3 = b4 = b5 = b6 = 0, k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the potential (143) becomes V (x, y, z) = F1 � y x � R2 + k1z rR2 = R−2 � F1 �y x � + k1z r � (146) and the vector Lb = a � xz, yz, −R2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated time-dependent QFI (134) is I(2,0) = −J1t + z(x ˙x + y ˙y) − (x2 + y2) ˙z (147) where J1 is the autonomous QFI J1 = M2 ˙x − M1 ˙y + 2zF1 � y x � x2 + y2 + 2k1z2 r(x2 + y2) + k1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' From Table 6 we have that the potential (146) admits the additional autonomous QFIs: J2 = 1 2M 2 3 + F1 �y x � , J3 = 1 2M2 + r2F1 � y x � x2 + y2 + k1zr x2 + y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, (146) is a new maximally superintegrable potential due to the five independent QFIs H, J1, J2, J3, and (147).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We note that this potential was considered to be minimally superintegrable (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' [2]) because only autonomous QFIs were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The PBs are {H, I(2,0)} = −J1 and {I(2,0), J2} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1, we determined the functions: V (x, y, z) = F1(x) + F2(y) + F3(z) (148) 29 G(x, y, z) = 2a3F1(x) + 2a13F2(y) + 2a9F3(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (149) Then, the vector La = \uf8eb \uf8ed a3x + 2b2y + 2b1z + b3 a13y − 2b2x + 2b4z + b6 a9z − 2b1x − 2b4y + b5 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (150) Replacing (148), (149) and (150) in the condition (136), we obtain the following ordinary differential equation (ODE): 0 = a3 [xF ′ 1 + 2F1(x)] + b3F ′ 1 + a13 [yF ′ 2 + 2F2(y)] + b6F ′ 2 + a9 [zF ′ 3 + 2F3(z)] + b5F ′ 3 + +2b2 (F ′ 1y − F ′ 2x) + 2b1 (F ′ 1z − F ′ 3x) + 2b4 (F ′ 2z − F ′ 3y) (151) where F ′ 1 = dF1 dx , F ′ 2 = dF2 dy and F ′ 3 = dF3 dz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We consider the following subcases: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Subcase b1 = b2 = b4 = 0 and the pairs (a3, b3), (a13, b6), (a9, b5) are not the origin (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, the ODE (151) gives: a3 [xF ′ 1 + 2F1(x)] + b3F ′ 1 = λ1 (152) a13 [yF ′ 2 + 2F2(y)] + b6F ′ 2 = λ2 (153) a9 [zF ′ 3 + 2F3(z)] + b5F ′ 3 = −λ1 − λ2 (154) where λ1 and λ2 are arbitrary constants, and the vector La = \uf8eb \uf8ed a3x + b3 a13y + b6 a9z + b5 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Solving the system of ODEs (152) - (154), we find the functions: F1(x) = λ1 � a3 2 x2 + b3x � + c1 (a3x + b3)2 , F2(y) = λ2 � a13 2 y2 + b6y � + c2 (a13y + b6)2 , F3(z) = −(λ1 + λ2) � a9 2 z2 + b5z � + c3 (a9z + b5)2 where c1, c2, and c3 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, the potential (148) becomes V (x, y, z) = λ1 � a3 2 x2 + b3x � + c1 (a3x + b3)2 + λ2 � a13 2 y2 + b6y � + c2 (a13y + b6)2 − (λ1 + λ2) � a9 2 z2 + b5z � + c3 (a9z + b5)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (155) The associated time-dependent QFI (134) is I(2,0) = −Jt + (a3x ˙x + a13y ˙y + a9z ˙z) + b3 ˙x + b6 ˙y + b5 ˙z (156) where J = 2a3I1 + 2a13I2 + 2a9I3 is the sum of the three separated QFIs: I1 = 1 2 ˙x2 + F1(x), I2 = 1 2 ˙y2 + F2(y), I3 = 1 2 ˙z2 + F3(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (157) Therefore, (155) is a new minimally superintegrable potential due to the four independent QFIs I1, I2, I3, and (156).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We note that (155) depends on the eleven parameters a3, a9, a13, b3, b5, b6, c1, c2, c3, λ1 and λ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' hence, the time-dependent QFI (156) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For λ1 = λ2 = 0 and a3a13a9 ̸= 0 we obtain the potential10 V (x, y, z) = k1 (x + m1)2 + k2 (y + m2)2 + k3 (z + m3)2 (158) where k1, k2, k3, m1, m2, and m3 are new arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, the associated time-dependent QFI (156) consists of the independent QFIs: I4 = −2I1t + (x + m1) ˙x, I5 = −2I2t + (y + m2) ˙y, I6 = −2I3t + (z + m3) ˙z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 10Since a3a13a9 ̸= 0, we can set b3 = m1a3, b6 = m2a13, b5 = m3a9, k1 = c1 a2 3 , k2 = c2 a2 13 and k3 = c3 a2 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 30 Therefore, the potential (158) is maximally superintegrable due to the independent QFIs I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', I6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Because time-dependent FIs are considered, the maximum number of independent FIs is six (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' greater than five).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Subcase b1 = b2 = b4 = 0, a3 ̸= 0 and a9 = a13 = b5 = b6 = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' two pairs of parameters from subcase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 vanish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' From the system of ODEs (152) - (154), we find that λ1 = λ2 = 0 and the potential (148) becomes V (x, y, z) = k1 (x + m1)2 + F2(y) + F3(z) (159) where k1, m1 are arbitrary constants and F2(y), F3(z) are arbitrary smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated time-dependent QFI (134) is I(2,0) = −2I1t + (x + m1) ˙x (160) where the QFI I1 = 1 2 ˙x2 + k1 (x+m1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the potential (159) is minimally superintegrable (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Subcase b1 = b2 = b4 = 0, a9 = b5 = 0 and the pairs (a3, b3), (a13, b6) are not the origin (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' From the system of ODEs (152) - (154), we find that λ2 = −λ1 and the potential (148) becomes V (x, y, z) = λ1 � a3 2 x2 + b3x � + c1 (a3x + b3)2 − λ1 � a13 2 y2 + b6y � + c2 (a13y + b6)2 + F3(z) (161) where F3(z) is an arbitrary smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated time-dependent QFI (134) is I(2,0) = −2(a3I1 + a13I2)t + (a3x + b3) ˙x + (a13y + b6) ˙y (162) where the QFIs I1 and I2 are given by (157).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We note that the potential (161) is a minimally superintegrable potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For λ1 = 0 and a3a13 ̸= 0 we obtain the maximally superintegrable potential (see Table 3) V (x, y, z) = k1 (x + m1)2 + k2 (y + m2)2 + F3(z) (163) which admits the additional time-dependent QFIs: I4 = −2I1t + (x + m1) ˙x, I5 = −2I2t + (y + m2) ˙y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Subcase a3 = a9 = a13 = 0 (autonomous LFIs, L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) = 0 and G = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The ODE (151) becomes 2b2 (F ′ 1y − F ′ 2x) + 2b1 (F ′ 1z − F ′ 3x) + 2b4 (F ′ 2z − F ′ 3y) + b3F ′ 1 + b6F ′ 2 + b5F ′ 3 = 0 (164) and the vector La = \uf8eb \uf8ed 2b2y + 2b1z + b3 −2b2x + 2b4z + b6 −2b1x − 2b4y + b5 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (165) The ODE (164) admits solutions of the form: F1(x) = kx2 + k1x, F2(y) = ky2 + k2y, F3(z) = kz2 + k3z (166) where k, k1, k2, and k3 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, we get the separable potential V (x, y, z) = kr2 + k1x + k2y + k3z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (167) Replacing (166) in (164), we find the following system of equations: k1b3 + k2b6 + k3b5 = 0 (168) kb3 − k2b2 − k3b1 = 0 (169) 31 kb6 + k1b2 − k3b4 = 0 (170) kb5 + k1b1 + k2b4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (171) We consider the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Case k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential (167) becomes V (x, y, z) = k1x + k2y + k3z (172) where k1k2k3 ̸= 0 in order to have a 3d potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Solving the system of equations (168) - (171) for k = 0, we find: b1 = −k2 k1 b4, b2 = k3 k1 b4, b3 = −k2 k1 b6 − k3 k1 b5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI (134) reduces to the LFI I = La ˙qa = −2b4 3 � i=1 kiMi − b5(k3 ˙x − k1 ˙z) − b6(k2 ˙x − k1 ˙y) which consists of the LFIs: J1 = 3 � i=1 kiMi, J2 = k3 ˙x − k1 ˙z, J3 = k2 ˙x − k1 ˙y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the separable potential (172) is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Case k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The system of equations (168) - (171) implies that b3 = k2 k b2+ k3 k b1, b5 = − k1 k b1− k2 k b4, and b6 = k3 k b4− k1 k b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Similarly, we find the LFIs: J1 = 2kM1 + k2 ˙z − k3 ˙y, J2 = 2kM2 + k3 ˙x − k1 ˙z, J3 = 2kM3 + k1 ˙y − k2 ˙x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the separable potential (167) is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We note that the k ̸= 0 introduces the term kr2 which is the oscillator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' therefore, the corresponding change in the FIs is the addition of the components of the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 Case La is a KV We consider that La is a KV in E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) = 0 and the time-dependent QFI (134) becomes the time- dependent LFI I = La ˙qa + st (173) where the arbitrary constant s satisfies the condition LaV ,a = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (174) Replacing the general KV La given by (13) in (174), we find the PDE (b1 − b4y + b5z) ∂V ∂x + (b2 + b4x − b6z) ∂V ∂y + (b3 − b5x + b6y) ∂V ∂z = s (175) where b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', b6 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We consider the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 1) Case b1 ̸= 0 and b4 = b5 = b6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, the PDE (175) gives the potential V = c1x + F(y − c2x, z − c3x) (176) where c1 = s b1 , c2 = b2 b1 , c3 = b3 b1 and F is an arbitrary smooth function of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 32 The associated LFI (173) is I = ˙x + c2 ˙y + c3 ˙z + c1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (177) 2) Case b2 ̸= 0, b1 = 0 and b4 = b5 = b6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We find a subcase of the potential (176) for x ↔ y and c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) Case b3 ̸= 0, b1 = b2 = 0 and b4 = b5 = b6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We find a subcase of the potential (176) for c1 = c, c2 = c3 = 0 and x ↔ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 4) Case b4 ̸= 0 and b5 = b6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, the PDE (175) gives the potential V = c0 tan−1 �x + c2 y + c1 � + F �1 2(x2 + y2) + c2x + c1y, z + c3 tan−1 �x + c2 y + c1 �� (178) where c0 = s b4 , c1 = − b1 b4 , c2 = b2 b4 , c3 = − b3 b4 and F is an arbitrary function of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated LFI (173) is I = M3 − c1 ˙x + c2 ˙y − c3 ˙z + c0t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (179) 5) Case b4 ̸= 0, b6 = 0 and b2 = b3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, the PDE (175) gives the potential V = c0 � 1 + c2 1 tan−1 � y + c1z + c2 � 1 + c2 1x � + F � z − c1y, x2 + (1 − c2 1)y2 + 2c2y + 2c1yz � (180) where c0 = s b4 , c1 = − b5 b4 , c2 = − b1 b4 and F is an arbitrary function of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated LFI (173) is I = M3 − c1M2 − c2 ˙x + c0t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (181) 6) Case b1 = b2 = b3 = s = 0 and b6 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, the PDE (175) gives the potential V = F(r, x − c1y − c2z) (182) where c1 = − b5 b6 , c2 = − b4 b6 and F is an arbitrary function of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated LFI (173) is I = M1 − c1M2 − c2M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (183) We collect the results of section 6 in Tables 8 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Minimally superintegrable potentials ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Ref [2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='LFIs and QFIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ1( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2+b3x)+c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(a3x+b3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ2( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2+b6y)+c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(a13y+b6)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(λ1+λ2)( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z2+b5z)+c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(a9z+b5)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I = −Jt + (a3x ˙x + a13y ˙y + a9z ˙z)+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+b3 ˙x + b6 ˙y + b5 ˙z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='J = 2a3I1 + 2a13I2 + 2a9I3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ1( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2+b3x)+c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(a3x+b3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ2( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2+b6y)+c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(a13y+b6)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(λ1+λ2)( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z2+b5z)+c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(a9z+b5)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+m1)2 + F2(y) + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+m1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + F2(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = −2I1t + (x + m1) ˙x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ1( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2+b3x)+c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(a3x+b3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ1( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2+b6y)+c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(a13y+b6)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ1( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2+b3x)+c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(a3x+b3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ1( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='a13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2+b6y)+c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(a13y+b6)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = −2(a3I1 + a13I2)t + (a3x + b3) ˙x+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+(a13y + b6) ˙y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table 8: Minimally superintegrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) in E3 that admit time-dependent QFIs of the form I(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Maximally superintegrable potentials ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Ref [2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='LFIs and QFIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + F1(y ± iz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='J1 = ˙y ± i ˙z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='J2 = J3t − x ˙x(y ± iz) + J1x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='J3 = (±iM2 − M3) ˙x + 2k1(y±iz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='J4 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = R−2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ k1z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='J1 = M2 ˙x − M1 ˙y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2zF1( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2+y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2k1z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r(x2+y2) + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='J2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + F1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='J3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r2F1( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2+y2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1zr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2+y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='J4 = −J1t + z(x ˙x + y ˙y) − (x2 + y2) ˙z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+m1)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+m2)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(z+m3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+m1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+m2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(z+m3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = −2I1t + (x + m1) ˙x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = −2I2t + (y + m2) ˙y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I6 = −2I3t + (z + m3) ˙z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+m1)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+m2)2 + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+m1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+m2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F3(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = −2I1t + (x + m1) ˙x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = −2I2t + (y + m2) ˙y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = k1x + k2y + k3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + k1x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + k2y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + k3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = �3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='i=1 kiMi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = k3 ˙x − k1 ˙z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I6 = k2 ˙x − k1 ˙y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = kr2 + k1x + k2y + k3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 + kx2 + k1x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + ky2 + k2y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + kz2 + k3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = 2kM1 + k2 ˙z − k3 ˙y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = 2kM2 + k3 ˙x − k1 ˙z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I6 = 2kM3 + k1 ˙y − k2 ˙x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table 9: Maximally superintegrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) in E3 that admit QFIs of the form I(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Potential LFIs and QFIs V = c1x + F(y − c2x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z − c3x) I = ˙x + c2 ˙y + c3 ˙z + c1t V = c0 tan−1 � x+c2 y+c1 � + +F � 1 2(x2 + y2) + c2x + c1y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z + c3 tan−1 � x+c2 y+c1 �� I = M3 − c1 ˙x + c2 ˙y − c3 ˙z + c0t V = c0 √ 1+c2 1 tan−1 � y+c1z+c2 √ 1+c2 1x � + +F � z − c1y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' x2 + (1 − c2 1)y2 + 2c2y + 2c1yz � I = M3 − c1M2 − c2 ˙x + c0t V = F(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' x − c1y − c2z) I = M1 − c1M2 − c2M3 Table 10: Possibly non-integrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) in E3 that admit LFIs of the form I = La ˙qa + st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 35 7 The QFI I(3) In this section, we consider the QFI I(3) = eλt � −L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) ˙qa ˙qb + λLa ˙qa + LaV ,a� where the vector La is given by (15), the generated KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) is given by (16) and the following condition is satisfied � LbV ,b� ,a = −2L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b)V ,b − λ2La.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (184) We consider several cases concerning the parameters a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', a20 which define the vector La given in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 Case containing KVs and the HV: parameters a1, a3, a4, a6, a7, a9, a10, a13, a14 In this case, the vector La given in (15) has the general form La = \uf8eb \uf8ed k1x k2y k3z \uf8f6 \uf8f8 + \uf8eb \uf8ed b1 − b4y + b5z b2 + b4x − b6z b3 − b5x + b6y \uf8f6 \uf8f8 (185) where k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', k3, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=', b6 are arbitrary constants and the generated KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) = diag(k1, k2, k3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We assume k1 = k2 = k3 = k is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, the vector (185) is the linear combination of the homothetic vector (HV) with the gradient and non-gradient KVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) = kδab and the time-dependent QFI I(3) becomes I = eλt (−k ˙qa ˙qa + λLa ˙qa + LaV ,a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (186) The condition (184) is � LbV ,b + 2kV � ,a + λ2La = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (187) From the integrability condition of (187), we get: La,b − Lb,a = 0 =⇒ La,b = kδab =⇒ b4 = b5 = b6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This implies that only the HV and the gradient KVs survive, that is, the vector (185) becomes La = \uf8eb \uf8ed kx + b1 ky + b2 kz + b3 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (188) We consider the following special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 1) Case k = 0, b3 = 0 and b1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The vector La = (b1, b2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, equation (187) gives the potential V (x, y, z) = λ2 2 � c2 1 − 1 � x2 + c2x − c1λ2xy + F(y − c1x, z) (189) where c1 ≡ b2 b1 , c2 are arbitrary constants and F is an arbitrary smooth function of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated time-dependent LFI is I = eλt � λ ˙x + c1λ ˙y − λ2x − c1λ2y + c2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (190) We note that {H, I} = ∂I ∂t = λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For c1 = 0, the potential (189) becomes V (x, y, z) = −λ2 2 x2 + c2x + F(y, z) (191) and the associated LFI (190) is I = eλt � λ ˙x − λ2x + c2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (192) 36 In the case that F(y, z) = F1(y) + F2(z), the potential (191) is separable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' therefore, it is minimally superinte- grable due to the additional independent LFI (192).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2) Case k = 0 and b1 = b2 = b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We have La = (1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential (after the transformation x ↔ z) V (x, y, z) = λ2 2 x2 + kx − λ2(y + z)x + F(x − z, y − z) (193) where k is an arbitrary constant and F is an arbitrary smooth function of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated LFI is I = eλt � λ( ˙x + ˙y + ˙z) − λ2(x + y + z) + k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (194) 3) Case k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We find the potential V (x, y, z) = −λ2 8 r2 − λ2 4 �b1 k x + b2 k y + b3 k z � + 1 � z + b3 k �2 F � y + b2 k x + b1 k , z + b3 k x + b1 k � (195) where F is an arbitrary function of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI is I = eλt �� ˙x − λ 2 x �2 + � ˙y − λ 2 y �2 + � ˙z − λ 2 z �2 − λ �b1 k ˙x + b2 k ˙y + b3 k ˙z � + +λ2 2 �b1 k x + b2 k y + b3 k z + b2 1 2k2 + b2 2 2k2 + b2 3 2k2 � + 2F � z + b3 k �2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (196) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For b1 = b2 = b3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential V (x, y, z) = −λ2 8 r2 + F � y x, z x � z2 (197) and the associated QFI is I = eλt � ˙x2 + ˙y2 + ˙z2 − λ(x ˙x + y ˙y + z ˙z) + λ2 4 r2 + 2F � y x, z x � z2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (198) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For b1 = k and b2 = b3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential V (x, y, z) = −λ2 8 r2 − λ2 4 x + F � y x+1, z x+1 � z2 (199) and the associated QFI is I = eλt �� ˙x − λ 2 x �2 + � ˙y − λ 2 y �2 + � ˙z − λ 2 z �2 − λ ˙x + λ2 2 x + λ2 4 + 2F z2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (200) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For b1 = b2 = k and b3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential V (x, y, z) = −λ2 8 r2 − λ2 4 (x + y) + 1 z2 F �y + 1 x + 1, z x + 1 � (201) and the associated QFI is I = eλt �� ˙x − λ 2 x �2 + � ˙y − λ 2 y �2 + � ˙z − λ 2 z �2 − λ( ˙x + ˙y) + λ2 2 (x + y + 1) + 2F z2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (202) 37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For b1 = b2 = b3 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential V (x, y, z) = −λ2 8 r2 − λ2 4 (x + y + z) + 1 (z + 1)2 F �y + 1 x + 1, z + 1 x + 1 � (203) and the associated QFI is I = eλt �� ˙x − λ 2 x �2 + � ˙y − λ 2 y �2 + � ˙z − λ 2 z �2 − λ ( ˙x + ˙y + ˙z) + +λ2 2 � x + y + z + 3 2 � + 2F (z + 1)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (204) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For F � y+ b2 k x+ b1 k , z+ b3 k x+ b1 k � = F1 � y+ b2 k x+ b1 k x+ b1 k z+ b3 k � + c0 (x+ b1 k ) 2 = F1 � y+ b2 k z+ b3 k � + c0 (x+ b1 k ) 2 , where c0 is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential V (x, y, z) = −λ2 8 r2 − λ2 4 (c1x + c2y + c3z) + c0 (x + c1)2 + 1 (z + c3)2 F1 �y + c2 z + c3 � (205) where ci = bi k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI consists of the independent QFIs: I1 = eλt �� ˙x − λ 2 x �2 − λc1 � ˙x − λ 2 x − λc1 4 � + 2c0 (x + c1)2 � (206) I2 = eλt �� ˙y − λ 2 y �2 + � ˙z − λ 2 z �2 − λ (c2 ˙y + c3 ˙z) + λ2 2 � c2y + c3z + c2 2 2 + c2 3 2 � + 2F1 (z + c3)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (207) 4) Case k1 = k2 = k3 = k and La = V,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We find the potential V (x, y, z) = k 2 r2 + b1x + b2y + b3z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (208) Then, equation (187) gives k = − λ2 4 and the potential (208) becomes V (x, y, z) = −λ2 8 r2 + b1x + b2y + b3z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (209) The associated QFI is I = eλt � λ2 4 3 � i=1 � ˙qi − λ 2 qi �2 + λ(bi ˙qi) − λ2 2 biqi + 3 � i=1 b2 i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (210) This QFI consists of the independent QFIs: I1 = eλt � λ2 4 � ˙x − λ 2 x �2 + λb1 ˙x − λ2 2 b1x + b2 1 � , I2 = eλt � λ2 4 � ˙y − λ 2 y �2 + λb2 ˙y − λ2 2 b2y + b2 2 � , I3 = eλt � λ2 4 � ˙z − λ 2 z �2 + λb3 ˙z − λ2 2 b3z + b2 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the potential (209) is maximally superintegrable (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5) Case k1k2k3 ̸= 0 and b4 = b5 = b6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 38 The potential V (x, y, z) = −λ2 8 r2 − λ2 4 � b1 k1 x + b2 k2 y + b3 k3 z � + c1 � x + b1 k1 �2 + c2 � y + b2 k2 �2 + c3 � z + b3 k3 �2 (211) where c1, c2, and c3 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI gives the following three independent QFIs Ii = eλt \uf8ee \uf8ef\uf8f0 � ˙qi − λ 2 qi �2 − λ bi ki � ˙qi − λ 2 qi − λbi 4ki � + 2ci � qi + bi ki �2 \uf8f9 \uf8fa\uf8fb (212) where i = 1, 2, 3 and qi = (x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the separable potential (211) is maximally superintegrable (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We note that, as expected, for k1 = k2 = k3 = k the resulting potential (211) belongs to the family of potentials (195) if we set F � y + b2 k x + b1 k , z + b3 k x + b1 k � = c1 � z + b3 k x + b1 k �2 + c2 � y + b2 k x + b1 k �−2 � z + b3 k x + b1 k �2 + c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 6) Case k1b2 ̸= 0, k2 = k3 = 0 and b4 = b5 = b6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The vector La = (k1x + b1, b2, b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential V (x, y, z) = −λ2 8 � x2 + 4(1 − c2 1)y2� − λ2 4 (c2x + 4c1yz) + c3y + c4 (x + c2)2 + F(z − c1y) (213) where c1 = b3 b2 , c2 = b1 k1 , c3, c4 are arbitrary constants and F is an arbitrary smooth function of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We find the independent FIs: I1 = eλt �� ˙x − λ 2 x �2 − λc2 � ˙x − λ 2 x − λ 4 c2 � + 2c4 (x + c2)2 � (214) I2 = eλt � ˙y + c1 ˙z − λ(y + c1z) + c3 λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (215) We note that for c1 = 0 we obtain the separable potential V (x, y, z) = −λ2 8 � x2 + 4y2� − λ2 4 c2x + c3y + c4 (x + c2)2 + F(z) (216) which is a new maximally superintegrable potential due to the additional time-dependent FIs (214) and (215).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential (see Table 7) V (x, y, z) = −λ2 8 � R2 + 4z2� + c4 x2 + c0 y2 + c3z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (217) is a subcase of (216) for y ↔ z, c2 = 0 and F(z) = − λ2 8 z2 + c0 z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 Parameters a17, a19, a20: The components L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) are constant and non-diagonal In the following cases, the only non-vanishing parameters are the a17, a19, and a20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 1) Case a17 ̸= 0, a20 = 0 and a19 is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The vector La = (0, 2a17x, 2a19x) and the KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) = \uf8eb \uf8ed 0 a17 a19 a17 0 0 a19 0 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 39 Then, equation (184) gives the potential V (x, y, z) = −λ2 2 x2 + F (z − cy) (218) where c = a19 a17 and F is an arbitrary smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI is I = eλt( ˙x − λx)( ˙y + c ˙z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (219) From Table 2, the potential (218) admits the additional autonomous FIs: I1 = 1 2 ˙x2 − λ2 2 x2 and I2 = ˙y + c ˙z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the QFI (219) contains the independent LFI I3 = eλt( ˙x − λx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We conclude that (218) is a new minimally superintegrable potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2) Case a17 = α 2 ̸= 0, a19 = 0 and a20 = β 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The vector La = (0, αx, βy), where α and β are arbitrary constants and the KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) = \uf8eb \uf8ed 0 α 2 0 α 2 0 β 2 0 β 2 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential V (x, y, z) = − λ2 2(1 + c2 1) � x2 + c2 1y2� − λ2 2(1 + c2 1) (z − 2c1x)2 + c2 (z − 2c1x) (220) where c1 = β α and c2 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI is I = eλt � ( ˙x − λx) ˙y + c1( ˙y − λy) ˙z − λ2c1 1 + c2 1 (c1x − z)y − c1c2y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (221) Moreover, the potential (220) admits the additional autonomous QFI I1 = 1 2 ˙y2 − λ2c2 1 2(1+c2 1)y2 because the y-coordinate is separated from the coordinates x and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 Parameters a2, a5, a8, a11, a12, a15, a16, a18: The components L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) are linear on x, y, z We consider the following cases: 1) a15 is the only non-vanishing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The vector La = a15(−y2, xy, 0) and the KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) = a15 \uf8eb \uf8ed 0 − y 2 0 − y 2 x 0 0 0 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential V (x, y, z) = −λ2 2 R2 + c1x y2R + c2 y2 + F(z) (222) where c1, c2 are arbitrary constants and F(z) is an arbitrary smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI is I = eλt � M3( ˙y − λy) + 2c2x y2 + c1(y2 + 2x2) y2R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (223) We note that the potential (222) is of the integrable form (see Table 1) V = F1( y x) R2 + F2(R) + F3(z) with F1 �y x � = \uf8eb \uf8ed c1 � 1 + y2 x2 + c2 \uf8f6 \uf8f8 � 1 + x2 y2 � , F2(R) = −λ2 2 R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (224) Therefore, it is a new minimally superintegrable potential due to the additional autonomous QFIs: I1 = 1 2 ˙z2 + F3(z), I2 = 1 2M 2 3 + (c1R + c2x)x y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 40 Moreover, for F(z) = − λ2 2 z2 + c3 z2 , where c3 is an arbitrary constant, the resulting potential V (x, y, z) = −λ2 2 r2 + c1x y2R + c2 y2 + c3 z2 (225) is a subcase of the minimally superintegrable potential (78) with F1 � y x � as given in (224).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Hence, (225) is a new maximally superintegrable potential due to the additional autonomous QFI (see Table 6) I3 = 1 2M2 + c1xr2 y2R + c2 r2 y2 + c3R2 z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (226) 2) a2 and a12 are the only non-vanishing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The vector La = (a2xz, a12yz, −a2x2 − a12y2) and the KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) = \uf8eb \uf8ed a2z 0 − a2 2 x 0 a12z − a12 2 y − a2 2 x − a12 2 y 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential (see Table 3) V (x, y, z) = −λ2 2 r2 + k1 x2 + k2 y2 (227) where k1 and k2 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI consists of the independent QFIs: I1 = eλt � M2( ˙x − λx) + 2k1z x2 � , I2 = eλt � M1( ˙y − λy) − 2k2z y2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the separable potential (227) is maximally superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) Case a2 = a12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The vector La = a2(xz, yz, −R2) and the KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) = a2 \uf8eb \uf8ed z 0 − x 2 0 z − y 2 − x 2 − y 2 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential V (x, y, z) = −λ2 2 r2 + c1z rR2 + F � y x � R2 (228) where c1 is an arbitrary constant and F � y x � is an arbitrary smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI is I1 = eλt � M2 ( ˙x − λx) − M1 ( ˙y − λy) + c1 r + 2c1z2 rR2 + 2zF � y x � R2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (229) We note that the potential (228) belongs to the general family of potentials (74);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' hence, it admits the additional autonomous QFI (see Table 4) I2 = 1 2M 2 3 + F �y x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (230) If c1 = 0, the resulting potential V (x, y, z) = −λ2 2 r2 + F � y x � R2 (231) is a new maximally superintegrable potential due to the additional autonomous QFIs (see Table 6): I3 = 1 2 ˙z2 − λ2 2 z2, I4 = 1 2M2 + r2F � y x � R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We note that the potential (231) is of the form (78) for k1 = − λ2 2 and k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 4) a3, a6, a10, a14 are non-vanishing and a2a13 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 41 The vector La = \uf8eb \uf8ed a2xz + a3x + a6 a13y + a14 −a2x2 + a10 \uf8f6 \uf8f8 and the KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) = \uf8eb \uf8ed a2z + a3 0 − a2 2 x 0 a13 0 − a2 2 x 0 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential (see Table 3) V (x, y, z) = −λ2 8 (4x2 + 4z2 + y2) − λ2c1z − λ2 4 c2y + k (y + c2)2 (232) where k, c1 = a3 a2 , and c2 = a14 a13 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI consists of the independent FIs: I1 = eλt ( ˙x − λx) (233) I2 = eλt �� ˙y − λ 2 (y + c2) �2 + 2k (y + c2)2 � (234) I3 = eλt [ ˙z − λ(z + c1)] (235) I4 = M2 + c1 ˙x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (236) We note that {I2, Ip} = 0 where p = 1, 2, 3, 4, {I1, I3} = 0, {I1, I4} = I3 and {I4, I3} = I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential (232) is integrable because the independent FIs I1, I2, I3 are in involution or, directly, because it is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' It is also maximally superintegrable due to the additional independent FIs I4 and H, where H is the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5) Case a2 ̸= 0 and a3, a13 are non-vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The vector La = \uf8eb \uf8ed a2xz + a3x a13y −a2x2 \uf8f6 \uf8f8 and the KT L(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='b) = \uf8eb \uf8ed a2z + a3 0 − a2 2 x 0 a13 0 − a2 2 x 0 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential V (x, y, z) = −λ2 2 � x2 + z2� − λ2 8 y2 + c1 x2 + c2 y2 − λ2c3z + k(z + c3) x2� (z + c3)2 + x2 (237) where k, c1, c2, and c3 = a3 a2 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The associated QFI consists of the following independent QFIs: I1 = eλt �� ˙y − λ 2 y �2 + 2c2 y2 � (238) I2 = eλt � (M2 + c3 ˙x)( ˙x − λx) + 2c1(z + c3) x2 + k x2 + 2(z + c3)2 x2� x2 + (z + c3)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (239) It is well-known that the dynamical equations (and hence the associated FIs) of a regular Lagrangian system are preserved if: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We add an arbitrary constant c to the potential V of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We apply a canonical transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Then, the potential (237) is a subcase of the minimally superintegrable potential (222).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Indeed, by adding the constant c = − λ2 2 c2 3 to (237), we obtain the equivalent potential V (x, y, z) = −λ2 2 � x2 + (z + c3)2� + k(z + c3) x2� (z + c3)2 + x2 + c1 x2 − λ2 8 y2 + c2 y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (240) If we apply the canonical transformation x → y, y → z and z → x − c3, the potential (240) becomes V (x, y, z) = −λ2 2 R2 + kx y2R + c1 y2 − λ2 8 z2 + c2 z2 (241) which is a subcase of (222) for F(z) = − λ2 8 z2 + c2 z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 42 The potential (241) is a new maximally superintegrable potential due to the following independent QFIs: I1 = eλt �� ˙z − λ 2 z �2 + 2c2 z2 � (242) I2 = eλt � M3( ˙y − λy) + 2c1x y2 + k(y2 + 2x2) y2R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (243) I3 = 1 2 ˙z2 − λ2 8 z2 + c2 z2 (244) I4 = 1 2M 2 3 + (kR + c1x)x y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (245) We recall that the potential (225) is another maximally superintegrable potential which is also a subcase of (222) but for a different choice of the function F(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' If we rename λ → 2λ, the QFI (242) is admitted also by (225) because the z-coordinate is separated from x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' We collect the results of section 7 in Tables 11 - 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 43 Potential LFIs and QFIs V = λ2 2 � c2 1 − 1 � x2 + c2x− −c1λ2xy + F(y − c1x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) I = eλt � λ ˙x + c1λ ˙y − λ2x − c1λ2y + c2 � V = λ2 2 x2 + kx − λ2(y + z)x+ +F(x − z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y − z) I = eλt � λ( ˙x + ˙y + ˙z) − λ2(x + y + z) + k � V = − λ2 8 r2 − λ2 4 (c1x + c2y + c3z) + + 1 (z+c3)2 F � y+c2 x+c1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z+c3 x+c1 � I = eλt �� ˙x − λ 2 x �2 + � ˙y − λ 2 y �2 + � ˙z − λ 2 z �2 − −λ (c1 ˙x + c2 ˙y + c3 ˙z) + + λ2 2 � c1x + c2y + c3z + c2 1 2 + c2 2 2 + c2 3 2 � + 2F (z+c3)2 � V = − λ2 8 r2 + F( y x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z x) z2 I = eλt � ˙x2 + ˙y2 + ˙z2 − λ(x ˙x + y ˙y + z ˙z) + λ2 4 r2 + 2F( y x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z x) z2 � V = − λ2 8 r2 − λ2 4 x + F( y x+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z x+1) z2 I = eλt �� ˙x − λ 2 x �2 + � ˙y − λ 2 y �2 + � ˙z − λ 2 z �2 − λ ˙x+ + λ2 2 x + λ2 4 + 2F z2 � V = − λ2 8 r2 − λ2 4 (x + y)+ + 1 z2 F � y+1 x+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z x+1 � I = eλt �� ˙x − λ 2 x �2 + � ˙y − λ 2 y �2 + � ˙z − λ 2 z �2 − −λ( ˙x + ˙y) + λ2 2 (x + y + 1) + 2F z2 � V = − λ2 8 r2 − λ2 4 (x + y + z) + + 1 (z+1)2 F � y+1 x+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I = eλt �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙x − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙y − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙z − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 − λ ( ˙x + ˙y + ˙z) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x + y + z + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(z+1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 r2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 (c1x + c2y + c3z) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='c0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c1)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(z+c3)2 F1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y+c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z+c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = eλt �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙x − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 − λc1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙x − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x − λc1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2c0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = eλt �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙y − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙z − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 − λ (c2 ˙y + c3 ˙z) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='c2y + c3z + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2F1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(z+c3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + 4(1 − c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1)y2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 (c2x + 4c1yz) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+c3y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c2)2 + F(z − c1y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = eλt �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙x − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 − λc2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙x − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙y + c1 ˙z − λ(y + c1z) + c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2(1+c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1y2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2(1+c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1) (z − 2c1x)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+c2 (z − 2c1x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2(1+c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1)y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='( ˙x − λx) ˙y + c1( ˙y − λy) ˙z − λ2c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1+c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 (c1x − z)y − c1c2y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 r2 + c1z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='rR2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = eλt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='M2 ( ˙x − λx) − M1 ( ˙y − λy) + c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r + 2c1z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='rR2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2zF( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table 11: Possibly non-integrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) in E3 that admit time-dependent LFIs/QFIs of the form I(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Minimally superintegrable potentials ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Ref [2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='LFIs and QFIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 + c2x + F1(y) + F2(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 + c2x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙y2 + F1(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F2(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ ˙x − λ2x + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 + F (z − cy) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙x2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ˙y + c ˙z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = eλt( ˙x − λx) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 R2 + c1x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2R + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + (c1R+c2x)x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='M3( ˙y − λy) + 2c2x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 + c1(y2+2x2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table 12: Minimally superintegrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) in E3 that admit time-dependent LFIs/QFIs of the form I(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Maximally superintegrable potentials ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Ref [2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='LFIs and QFIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 r2 + b1x + b2y + b3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Ii = ˙q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='i + biqi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Ji = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙qi − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 qi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 + λbi ˙qi − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 biqi + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 r2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 (b1x + b2y + b3z) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+b1)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+b2)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(z+b3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Ii = ˙q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='i − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 biqi + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='ci ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+bi)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Ji = eλt �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙qi − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 qi�2 − λbi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙qi − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 qi − λbi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2ci ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(qi+bi)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + 4y2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 c2x + c3y+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c2)2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = ˙x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 x2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 c2x + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ˙y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2 + c3y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = ˙z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 + F(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = eλt �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙x − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 − λc2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙x − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(x+c2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙y − λy + c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 r2 + c1x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2R + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 + c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z2 + c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + (c1R+c2x)x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='M3( ˙y − λy) + 2c2x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 + c1(y2+2x2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M2 + c1xr2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2R + c2 r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 + c3R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 r2 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = ˙x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 x2 + k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = ˙y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 y2 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = ˙z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='M2( ˙x − λx) + 2k1z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I5 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='M1( ˙y − λy) − 2k2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 r2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='F( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = eλt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='M2 ( ˙x − λx) − M1 ( ˙y − λy) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2zF( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='r2F( y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 (4x2 + 4z2 + y2)− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='−λ2c1z − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='4 c2y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+c2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = eλt ( ˙x − λx) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = eλt �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙y − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 (y + c2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='(y+c2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = eλt [ ˙z − λ(z + c1)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = M2 + c1 ˙x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='V = − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 R2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='kx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2R + c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='− λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 z2 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='New ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I1 = eλt �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='˙z − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='�2 + 2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I2 = eλt � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='M3( ˙y − λy) + 2c1x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 + k(y2+2x2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I3 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 ˙z2 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='8 z2 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I4 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='3 + (kR+c1x)x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='Table 13: Maximally superintegrable potentials V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' z) in E3 that admit time-dependent LFIs/QFIs of the form I(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 8 Comparison with existing results As we have remarked in section 1, the main review works in this topic are the works of Evans in [2] and Kalnins in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, it is imperative to discuss how the present review is related to these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 46 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='1 Evans work [2] Evans in [2], using the separability of the Hamilton-Jacobi equation in E3, determined all minimally and maximally superintegrable potentials with autonomous QFIs of the form I = Kab(q) ˙qa ˙qb + G(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The author did not consider (autonomous or time-dependent) LFIs and time-dependent QFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' In particular, in Table I of [2] are given five maximally superintegrable potentials and in Table II of [2] eight minimally superintegrable potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' As it can be seen from Tables 1 - 13, all the results of [2] have been recovered plus new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the claim made in [2] that all second order superintegrable potentials in E3 are determined is not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Furthermore, it should be noted that there are misprints in some results of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Indeed, we have: 1) In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='43) of [2], the leading term of the QFI I4 must be L2P1 − P2L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2) In Table II of [2], the leading part of the QFIs I3 associated with the potentials (55) and (56) should be L2P1 − P2L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) The QFI I2 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='57) of [2] should be replaced with the QFI (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 4) The QFI I3 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='57) of [2] should be replaced with the QFI (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='2 Kalnins et all work [4] In [4], the authors discussed classical 3d superintegrable nondegenerate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' four-parameter) potentials on a conformally flat real or complex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' They proved that the quadratic algebra always closes at order six (the ‘5 =⇒ 6 Theorem’), that is, the space of autonomous QFIs is 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Moreover, using the St¨ackel transformation (an invertible conformal mapping between superintegrable structures on distinct spaces), they gave strong evidence (no proof) that all nondegenerate 3d superintegrable systems are St¨ackel transforms of constant curvature systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' the complex Euclidean space or the the complex 3-sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This means that in order to obtain all nondegenerate conformally flat superintegrable systems, it is sufficient to classify those in the complex Euclidean space and on the complex 3-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Finally, they found eight families of superintegrable systems that are separable in generic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Comparing the results of [4] with the results of the present work, we note the following: 1) All seven maximally superintegrable Euclidean potentials given in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (10) - (16) of [4] are recovered (see Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 2) The potentials given in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (10) and (13) of [4] have been found earlier in Table I of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' The potential (13) is more general from the one found by Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 3) It is proved in section 5 that the potentials (11), (12), (14), (16) of [4] are subcases of the more general potential (28) for specific forms of the arbitrary smooth functions F1(w, z) and F2(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' This justifies the fact that these potentials admit a QFI of the form I = ˙w2 + G(x, y, z) where w = x + iy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 4) The potential (15) of [4] is a subcase of (31) and hence admits a QFI of the form I = ˙¯w2 + G(x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 5) The potentials (12), (16) of [4] are of the integrable form (34);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' therefore, they admit a QFI of the form I = ˙z ˙w + G(x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 6) The potentials (11), (12) of [4] are of the integrable form (82);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' therefore, they admit a QFI of the form I = (M2 − iM1)2 + G(x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 7) The potential (15) of [4] is a subcase of the new minimally superintegrable potential (97) for F(z) = k1z2+ k4 z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For this reason, it admits an additional QFI of the form I = 1 4 ˙w2 + iM3 ˙¯w + G(x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 8) The two additional maximally superintegrable potentials given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' (17) of [4] are just subcases of the last maximally superintegrable potential in Table 3 for k1 = k2 = 0 when F(z) = − λ2 8 z2+c3z and F(z) = − λ2 32 z2+ c3 z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, with the systematic application of Theorem 1, we have found all the results of [4] plus new ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' especially time-dependent QFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 9 Conclusions The aim of the present work was twofold: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' To assess the second order integrability of autonomous conser- vative dynamical systems of the form qa = −V ,a(q) where a = 1, 2, 3 in a systematic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' algorithmic, way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' To enrich, if possible, the existing results of the main sources on this topic which are found in the review papers [2] and [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Therefore, the present work should be approached as an updated review of the integrable/superintegrable 3d Newtonian autonomous conservative dynamical systems that admit LFIs/QFIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 47 We have considered two types of integrable and superintegrable 3d Newtonian potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Potentials of the form Φ(x, y)+F(z) which are 2+1 decomposable and hence their QFIs follow from the QFIs of the 2d potentials Φ(x, y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' and non-decomposable potentials V (x, y, z) in E3 which cannot be treated in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' These latter potentials we have searched using the algorithm of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' After a detailed study of the three types of QFIs I(1,ℓ), I(2,ℓ), I(3) considered in Theorem 1, we have recovered all known integrable/superintegrable potentials together with new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' It has also been shown that many of the existing results are in fact special cases of more general ones for specific values of the free parameters/functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' For convenience, the results in each case have been collected in tables which contain the known results with the appropriate reference and the new ones found in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' These results can be used in many ways in the study of the dynamical systems and, especially, in the case of more complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' One such study will be given elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' Arnold, ‘Mathematical Methods of Classical Mechanics’, Springer (1989), proof in pp.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 147(2), 87 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} +page_content=' 48' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAyT4oBgHgl3EQf7fpb/content/2301.00839v1.pdf'} diff --git a/T9E4T4oBgHgl3EQfLwz0/content/tmp_files/2301.04942v1.pdf.txt b/T9E4T4oBgHgl3EQfLwz0/content/tmp_files/2301.04942v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2bb785aa6488d9988abdbc2283b29ad7dd91bf01 --- /dev/null +++ b/T9E4T4oBgHgl3EQfLwz0/content/tmp_files/2301.04942v1.pdf.txt @@ -0,0 +1,3949 @@ +1 + +Hydrogen storage in C14 type Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 high entropy alloy + +Abhishek Kumar1 +, T. P. Yadav1,2*, M.A. Shaz1and N.K. Mukhopadhyay3 +1Hydrogen Energy Centre, Department of Physics, Institute of Science +Banaras Hindu University, Varanasi, Uttar Pradesh, India +2Department of Physics, Faculty of Science, University of Allahabad, Prayagraj-211002, India +3Department of Metallurgical Engineering, Indian Institute of Technology (Banaras Hindu University), +Varanasi-221 005, India + +Abstract +In this present investigation, we discussed the synthesis, microstructure, and hydrogen storage behavior in C14 type +intermetallic Laves phase in a hexanary Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 high entropy alloy (HEA). In this HEA, +three elements are hydride-forming elements (Ti, V, Zr), whereas other three are non-hydride-forming elements (Fe, +Mn, Co). The thermodynamic parameter like enthalpy of mixing was calculated using the Meidma’s model. The +mixing enthalpy (∆Hmix) of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA system was evaluated to be- 23.3472 kJ/mole, and +atomic radius mismatch turned out to be = 7.441%. This alloy was synthesized using 35 kW radio frequency +induction furnace under argon atmosphere. X-ray diffraction technique (XRD) revealed that this system belongs to +the C14 type Laves phase with unit cell parameters a= b =5.0158 Å, c=8.1790 Å, α = β = 90˚, γ = 120˚ under Space +group P63/mmc. Microstructural analysis was carried out with the help of a transmission electron microscope +(TEM). The SEM- EDX data confirms the elemental composition. Hydrogen absorption and desorption of this high +entropy intermetallic was carried out using the PCI apparatus. The total hydrogen storage of this system was +observed around ~0.53 wt%. However, it exhibited better hydrogen and ab/de-sorption kinetics. With the help of the +Van’t Hoff plot, calculated experimental change in enthalpy of Ti0.24-V0.17-Zr0.17-Co0.17-Fe0.08-Mn0.17 HEA for +hydrogen absorption and desorption was found out to be ~ -19.06 ± 1.12 kJ/mol and -34.10 ± 1.32 kJ /mol +respectively. The possibility of developing high entropy Laves phase-based hydrogen storage materials was +advocated. +Corresponding authors: yadavtp@gmail.com + + + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)2 + +Introduction +Recent years have seen a lot of interest in a new class of materials called ‘High Entropy Alloys’ (HEAs) (Marques et +al. 2021, Yadav el al. 2017, Mishra et al. 2019, Mishra et al. 2020). In general, HEAs contain five or more elements, +each with a concentration of five to thirty-five atomic percentages (at.%) or more, in contrast to conventional alloys +based on a single primary element. To improve phase stability, HEAs are understood to exhibit large mixing +entropies of solid solution phases (Murty et al. 2019). The research publication by Yeh et al. (2004a 2004b), Cantor +et al. (2004), and Ranganathan (2003) was published for the first time for launching the field of HEAs. Yeh +independently proposed the single-phase multi-principal element alloy in 1995, making this idea a ground-breaking +success in researching HEAs (Murty et al. 2019). It' is interesting to note that the high mixing entropy in multi- +principal element alloys can dramatically lower the number of phases in high-order alloys, leading to a single phase +solid solution (Tsai et al. 2014). HEA has many functional properties like magnetic,, thermoelectric, catalytic, +hydrogen storage etc. In these functional properties, hydrogen storage is considered to be one of the interesting +areas to explore the HEA as an effective hydrogen storage material. Nowadays, in order to counteract climate +change and the rise in global warming brought on by conventional fossil fuels; people demand innovative, flexible, +clean, and green energy sources. Among many fuels that are readily available worldwide, hydrogen is accepted as +one of the best candidates due to its high energy range per unit mass. Three essential elements that are needed to use +hydrogen as a fuel in the future are (i) hydrogen production, (ii) its storage, and (iii) applications. Hydrogen storage +is one of the most crucial components of using hydrogen as a fuel. One of the safest and most efficient ways to store +hydrogen is in solid-state metal hydrides. Due to the infinite combination of alloy forming possibilities, the HEAs +are novel and promising materials for hydrogen storage (Yadav et al. 2022). In 2010, the first investigation was done +in HEAs to study the hydrogen storage kinetics. This study claimed 0.03-1.80 wt% hydrogen storage in multi- +principal component CoFeMnTixVyZrz (Kao et al. 2010) alloys; after that, in TiZrHfNbV HEA, 2.7wt% hydrogen +storage was reported in 2016 (Sahlberg et al. 2016). There is only one BCC phase in this alloy composition. One +more point common in this system is that this alloy system is designed with all the hydride forming elements, +because of which it has a good hydrogen storage capacity. In recent years hydrogen storage is reported as high as +3.51 wt% in V35Ti30Cr25Fe5Mn5 HEA belonging to a single BCC phase (Liu et al. 2021). On the contrary, the +maximum hydrogen storage in Laves phases is known to be 1.91 wt% (Sarc et al. 2020). It can stated from the +reported data that the Laves phase has less storage properties and better absorption and desorption kinetics + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)3 + +compared to BCC phase. The investigation on low-vanadium TiZrMnCrV-based alloys for high-density hydrogen +storage (Zhou et al. 2021) was reported. Due to its maximal interstitial sites available for absorbing hydrogen in +their voids, C14 Laves phase has been explored as hydrogen storage phase tested in recent study. People have +recently been concentrating on the research of phase stability during hydrogen absorption and desorption of HEAs. +In multi-component HEA for TiZrFeMnCrV (Chen et al. 2022), C14 type Laves phase-based HEA was fabricated +and followed by hydrogen storage testing after mechanical milling. The maximal hydrogen absorption for this alloy +was reported to be 1.80 wt% for the first cycle and 1.76 wt% for the second cycle. According to their findings, the +hydrogen storage capacity varied marginally between each cycle's i.e., 1.76 and 1.73 wt%. In another study, +TiZrCrMnFeNi HEA of C14 Laves phase has exhibited hydrogen absorption as 1.7 weight percent (Edalati et al. +2020). Kumar et al (2022) has shown that TiZrVCrNi Laves phase with 1-52 weight percent hydrogen remains +stable even after 10 cycles of hydrogenation from the perspective of phase stability. The TiZrNbCrFe HEA +consisting of C14 Laves phase as maor and BCC phase as minor was reported by Floriano et al. 2021 to have 1.9 +wt% hydrogen storage capacity.In view of the potential of HEAs for hydrogen storage capability, it was felt worth +pursuing the study of other high entropy based alloys for exploring their structure and hydrogen storage +performance. Accordingly, in the present study, we selected TiZrVMnFeCo nonequiatomic HEAs and investigated +the structure, microstructure, and hydrogen storage kinetics. We chose a HEA system with three hydride forming +elements (TiZrV) and the remaining three non-hydride-forming elements (Mn, Fe, Co).The thermodynamic +calculation for evaluating enthalpy of mixing of this HEA was done using Meidma model. This HEA was +synthesized with the help of a 35 KW Radio Frequency Induction furnace in the argon atmosphere and characterized +by XRD, SEM and TEM techniques Hydrogen storage performance was evaluated using pressure composition +isotherm (PCI) equipment supplied by Advanced Material Corporation (Pittsburgh, USA). + +Material synthesis and characterization methods +The high purity materials powder for the synthesis of the Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA system was procured +from Alfa Aesar with a purity of more than 99.50%. The constituent elements were taken as per their stoichiometry +for making a palette using a cylindrical steel mold equipped with the hydraulic press of acting pressure ~3x105 +N/m2. The palette (~10 g by weight) then used for the as-cast synthesis of multicomponent HEA using the RF +induction melting process under argon atmosphere (purity of more than 99.90%). The ingots are melted four times to + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)4 + +ensure uniformity of chemical composition. The as-cast induction melted ingots of HEA crushed and converted into +powder form to perform further characterization. The first cutting-edge characterization technique used for phase +analysis is the Empyrean x-ray diffraction (XRD) system (Malvern Panalytical) equipped with an area detector +(256x256 pixels) equipped with a graphite monochromator and Cu radiation source (CuKa; = 1.5406, operating at +45 kV and 40 mA) in Bragg-Brentano geometry. The transmission electron microscope (TEM), TECNAI 20 G2, was +used to acquire the microstructures and selected area electron diffraction (SAED) pattern of the samples operating at +200 kV of accelerating voltage.EVO 18 scanning electron microscope at operating voltage of 25 kV (vacuum 10-5 +torr) was used to investigate surface morphology and perform energy dispersive X-ray analysis (EDX) as well as +colour mapping of elements in the as-prepared samples. All de/re-hydrogenation measurements were carried out +with the aid of an automated two-channel volumetric sieverts apparatus (supplied by Advanced Materials +Corporation Pittsburgh, USA). For hydrogen storage testing, we took the 500 mg sample of HEA and placed the +sample in the reactor seized by quartz wool. Before performing hydrogen cycle testing, the powder HEA sample was +activated at 400℃ under a hydrogen pressure of 1/0.1 MPa for hydrogenation/dehydrogenation. After activation, +testing of the hydrogen absorption kinetics at 410 °C under 60 atm H2 pressure was carried out. + +Results and Discussion +The experimental XRD diffraction patterns of the as-cast Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA are shown in figure +2(a). The diffraction profile has been recorded for the gross structural analysis of the as-cast alloy sample by using +the Empyrean x-ray diffraction (XRD; Malvern Panalytical) system. All the diffraction peaks (shown in the figure. +2(a)) are well fitted with the hexagonal C14 Laves phase structure parameters.The XRD pattern was well refined +through Le Bail profile fitting using JANA 2006 software shown in the figure. 2(b). The refinement data validated +the Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA system with unit cell parameters of a=b= 5.0141 Å, c= 8.1756 Å, and the +unit cell volume 178.0 Å3 under the space group of P63/mmc. All the refine parameters are given below in Table 1 +To validate the structure analysis of this XRD, we used another characterization technique by transmission electron +microscopy (TEM) for analyzing the phase and microstructure of this Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA. The +bright field TEM micrograph of as-synthesized HEA shown in figure 3(a) identifies no other phases other than +Laves phase. The corresponding SAD pattern of this as cast HEA shown in figure 3(b) validates that this HEA +system belongs to a C14 type hexagonal structure with a corresponding space group is P63/mmc. + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)5 + +Surface morphology and elemental composition analysis +Scanning electron microscopy (SEM) has been done for surface microstructure and confirming homogeneous +element distribution. Figure4 (a) shows the SEM –BSE, and Energy dispersive X-ray analyses (EDX) mapping +images of as cast Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA with the corresponding region which is located in square +box in figure 4(a). The SEM-BSE image reveals the microstructure of this HEA without any cracks or defects in +this as-cast HEA. Figure 4(b) overlays all the constituent elements present in this HEA. EDAX mapping image +establishes that all the constituent elements are distributed as per atomic percent in this as-cast Ti0.24-V0.17-Zr0.17- +Co0.17-Fe0.08-Mn0.17 HEA. Figure 4(c) shows the SEM-BSE image from another region for the HEA sample, where +no crack is observed, and also no other contrast corresponding another phase. Figure 4(d) shows the EDX elemental +spectra to confirm the stoichiometry of the elements present in this as-cast HEA. All the data indicate that this HEA +has forms a single Laves phase with uniform elemental distribution. + +Hydrogen ab/de-sorption analysis +Hydrogen ab/de-sorption performance in as-cast Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA is studied in this section. The +measurements of hydrogen sorption were carried out with automated two-channel volumetric sieverts instrument. +The results of the absorption kinetic curve of the as-cast Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA are shown in figure +5(a). Before introducing hydrogen into as-cast HEA, we firstly activate the as-cast HEA under 400 ˚C under 10-3 +atm evacuation. We perform hydrogenation at 410˚C under 60 atm hydrogen pressures. The hydrogen desorption +kinetic curve of the as-cast Ti0.24-HEAis shown in figure 5(b). The hydrogen desorption kinetic curve of this as- +cast HEA shows that this as-cast HEA absorbed 0.53 wt% of hydrogen within 15 seconds this curve. In contrast, the +maximum storage capacity is evaluated to be about 0.72 wt% in 150 minutes. This fastest kinetics gives interesting +results to understand the hydrogen storage performance In the case of desorption, we can see that the +dehydrogenated curve shown in figure 5(b) the as cast Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA perform desorption at +410 ˚C under 1 atm hydrogen pressure. According to the hydrogenation desorption curve we can say that this HEA +released 0.28 wt% hydrogen within one minute at 410 ˚C under 1 atm hydrogen pressure. The results suggests that +this HEA shows faster hydrogen ab/desorption kinetics than some other Laves phase based HEAs. + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)6 + +The representative PCI ab/de-sorption of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA has been shown in figure 6(a) the +corresponding represents active Van’t Hoff plots (shown in figure 6(b)). PCI was performed at 395˚C, 410˚Cand +425˚C temperatures under 60 atm hydrogen pressures. With the help of the three different temperatures, we get the +plot corresponding to temperature v/s pressure. Calculations of the entropy and enthalpy changes that occur +throughout the hydrogen ab/de-sorption process typically employ the pressure values of the hydrogen ab/de-sorption +platform at various temperatures. The change in enthalpy (∆H) of hydride formation is given by the well-known +Van’t Hoff equation (Dornheim et al. 2010) + ln 𝑃 = +Δ𝐻 +RT − +Δ𝑆 +R …………(i) +Where P is the previously specified plateau pressure, T is the corresponding temperature, R is the gas constant, and +H and S are the reaction enthalpy and entropy changes, respectively. The alloys' Van't Hoff plots are computed using +the P, as shown in figure 6. (b). The relationship between ln(P) and 1000/T is clearly linear, as can be seen in the +image. The slope of the fitted curves for ln(P) and 1000/T as well as the intercept on the vertical coordinate allow +for the quick calculation of the H and S. The results of the calculations demonstrate that the enthalpy of hydrogen +desorption changes. The change in enthalpy of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA for hydrogen absorption and +desorption has been calculated to be ΔHabs~ -19.06 ± 1.12 kJ/mol and ΔHdes -34.10 ± 1.32 kJ /mol respectively. The +smaller negative enthalpy of mixing in HEA suggests that they are more likely to form stable metal hydrides. The +formation of the metal hydride's absorption and desorption enthalpies are not equal in the current experiment. +Therefore, this system has fewer tendencies to create metal hydride and aids in improving the ab/desorption kinetics. +This suggests that they have a decreased tendency to form a stable metal hydride. +Conclusions +In this study, we have successfully synthesized the hexanary Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA with the help of +an RF induction furnace for the study of hydrogen storage kinetics. The evolution of a single phase of hexagonal +C14 high entropy Laves phase with lattice parameters a = 5.01Å and c =8.17Åwas established following Rietveld +refinement in this multicomponent alloy. On the basis of the kinetics study, Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 shows +good ab/de-desorption kinetics (absorb ~ 0.53 wt.% of H2 within 15 seconds) but poor in hydrogen storage capacity. +The change in enthalpy of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA for hydrogen absorption and desorption has been + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)7 + +calculated to be ~ -19.06 ± 1.12 kJ/mol and -34.10 ± 1.32 kJ /mol respectively. The present investigation suggests +the scope for further study on the hydrogenation kinetics at various temperatures for exploring the potential for +developing Laves phase high entropy alloy for hydrogen storage. + +Acknowledgment +The author (AK) wishes to thank the Council of Scientific and Industrial Research (CSIR) in New Delhi, India, for +financial support for a senior research fellowship (Award No. 09/013(0952)/2020-EMR-I). + +Author contributions +A.K. synthesized the materials and made the characterizations; T.P.Y. conceived, designed the experiments, +organized the data and supervision. M.A.S. advised on the discussion of results; N.K.M. advised on the discussion +of results and editing the manuscript. The manuscript was written through contributions of all authors. All authors +have given approval to the final version of the manuscript. + +Notes +The authors declare no competing financial interests. + + + + + + + + + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)8 + +References +Cantor B, Chang ITH, Knight P, Vincent AJB (2004) Microstructural development in equiatomic multicomponent +alloys. 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Royal Society of Chemistry 14, 5191-5227. https://doi.org/10.1039/D1EE01543E + +Mishra SS, Mukhopadhyay S, Yadav TP, Mukhopadhyay NK, Srivastava ON (2019) Synthesis and characterization +of hexanary Ti–Zr–V–Cr–Ni–Fe high-entropy Laves phase. Journal of Materials Research 34 (5): 807-818. +https://doi.org/10.1557/jmr.2018.502 + +Mishra SS, Yadav TP, Srivastava ON, Mukhopadhyay NK, Biswas K (2020) Formation and stability of C14 type +Laves phase in multi component high-entropy alloys. Journal of Alloys and Compounds 832:153764. +https://doi.org/10.1016/j.jallcom.2020.153764 + +Murty BS, Yeh JW, Ranganathan S, Bhattacharjee PP (2019) High-Entropy Alloys 2nd Edition Elsevier ISBN: +9780128160671. pp 1-388. + +Ranganathan S (2003) Alloyed pleasures: Multimetallic cocktails. Current Science 85: 1404-1406. + +Sahlberg M, Karlsson D, Zlotea C, Jansson U (2016) Superior hydrogen storage in high entropy alloys, Scientific +Reports: 36770. https://doi.org/10.1038/srep36770 +Sarac B, Zadorozhnyy V, Berdonosova E, Lvanov YP, Klyamkin S, Gumrukcu S, Sarac AS, Korol A, Semenov D, +Zadorozhnyy M, Sharma A, Greer AL, Eckert J (2020) Hydrogen storage performance of the multi-principal- +component CoFeMnTiVZr alloy in electrochemical and gas-solid reactions, RSC Advances 10: 24613–24623. +https://doi.org/10.1039/D0RA04089D + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)9 + +Tsai MH, Yeh JW (2014) High-Entropy Alloys: A Critical Review. Materials Research Letters 2 (3): 107–123. +https://doi.org/10.1080/21663831.2014.912690 + +Yadav TP, Kumar A, Verma SK, Mukhopadhyay NK (2022) High-Entropy Alloys for Solid Hydrogen Storage: +Potentials and Prospects. Transactions of the Indian National Academy of Engineering 7: 147-156. +https://doi.org/10.1007/s41403-021-00316-w + +Yadav TP, Mukhopadhyay S, Mishra SS, Mukhopadhyay NK, Srivastava ON (2017) Synthesis of a single phase of +high-entropy Laves intermetallics in the Ti–Zr–V–Cr–Ni equiatomic alloy. Philosophical Magazine Letters 97 (12): +494-503. https://doi.org/10.1080/09500839.2017.1418539 + +Yeh JW, Chen SK, Gan JY, Lin SJ, Chin TS, Shun TT, Tsau CH, Chou SY (2004a) Formation of simple crystal +structures in Cu-Co-Ni-Cr-Al-Fe-Ti-V alloys with multiprincipal metallic elements. Metallurgical and Materials +Transactions A 35:2533-2536. https://doi.org/10.1007/s11661-006-0234-4 + +Yeh JW, Chen SK, Lin SJ, Gan JY, Chin TS, Shun TT, Tsau CH, Chang SY (2004b) Nanostructured high-entropy +alloys with multiple principal elements: novel alloy design concepts and outcomes. Advanced Engineering Materials +6:299303. Zeitschrift für Physikalische Chemie 117: 89-112. https://doi.org/10.1002/adem.200300567 + +Zhou P, Cao Z, Xiao X, Jiang Z, Zhan L, Li Z, Jiang L, Chen L (2022) Study on low-vanadium TiZrMnCrV based +alloys for high-density hydrogen storage. International Journal of Hydrogen Energy 47: 710-1722. +https://doi.org/10.1016/j.ijhydene.2021.10.106 + + +Figure captions + +Figure 1: (a) Schematic diagramof the synthesis protocol for Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA + +Figure 2:(a) XRD pattern of Ti0.24-V0.17-Zr0.17-Co0.17-Fe0.08-Mn0.17 HEA system and (b) Rietveld refinement profile +pattern of all the peaks well fitted with C14 type hexagonal parameters with unit cell parameters a= b =5.0158 Å, +c=8.1790 Å, α = β = 90˚, γ = 120˚ under space group P63/mmc +Figure 3 : (a) TEM bright field micrograph of as-cast HEA synthesized by RF induction melting (b) Corresponding +SAD patterns are shown indexed with hexagonal structure parameter under the space group of P63/mmc +Figure 4: (a) SEM–BSE and energy dispersive X-ray analyses (EDX) mapping images of as +Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA (b) overlays all the constituent elements present in this HEA. (c SEM-BSE +image from another region for the HEA. (d) EDX elemental spectra to validate the atomic percentage of the +elements in this HEA. +Figure 5: (a) Hydrogenation curve of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA at 410 ˚C under 60 atm H2 pressure and +(b) Dehydrogenation curve of hydrogenated Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA at 410 ˚C under 60 atm H2 +pressure +Figure 6: (a) Fig: (a) PCI ab/de-sorption curves of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA and (b) Corresponding +Van’t Hoff plots for PCI ab/de-sorption curves. + + + + + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)10 + +Table 1: +Lattice Parameters and refinement parameters obtained from powder x-ray diffraction data of the +as-cast HEA. +Refined Parameter and phase data +Unit-Cell Parameters a= b =5.0158 Å, c=8.1790 Å, α = β = 90˚, γ = 120˚ + Space Group P63/mmc (Space Group = 194) + R- Factor Rp = 3.23%, wRp = 4.45%, GOF = 1.26%, + Volume V = 178.20Å3 + + + + + + + + + + + + + + + + + + + + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)11 + + + + + +Figure 1: (a) Schematic diagram of the synthesis protocol for Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA + + + + + + + + + + + + + + + + + + + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)12 + + +Figure 2:(a) XRD pattern of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA system and (b) Rietveld refinement profile +pattern of all the peaks well fitted with C14 type hexagonal parameters with unit cell parameters a= b =5.0158 Å, +c=8.1790 Å, α = β = 90˚, γ = 120˚ under Space group P63/mmc. + + + + + + + + + + + + + + + + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)13 + + + +Figure 3: (a) TEM bright field micrograph of as-cast HEA synthesized by RF induction melting (b) Corresponding +SAD pattern are shown indexed with hexagonal structure parameter under the space group of P63/mmc. + + + + + + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)14 + + +Figure 4 : (a) shows the SEM–BSE and energy dispersive X-ray (EDX) analysis mapping images of as cast +Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA (b) overlays all the constituent elements present in this HEA. (c) Shows the +SEM-BSE image from another region for the HEA. (d) Shows the EDX elemental spectra to validate the atomic +presence of the elements in this HEA. + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)15 + + +Figure 5 : (a) Hydrogenation curve of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA at 410 ˚C under 60 atm H2 pressure +and (b) Dehydrogenation curve of hydrogenated Ti0.24-V0.17-Zr0.17-Co0.17-Fe0.08-Mn0.17 HEA at 410 ˚C under 60 atm +H2 pressure. + + + + + + + + + + + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm +Tome.t:20.15 +ZEIS +Le300.8 +Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17 +0.5 +DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at +0.7 +at410cunder60atmH2pressure +Hydrogen absorbed (wt%) +desorbed (wt%) +410Cunder1atmH2pressure +0.6 +0.4 - +0.5 +(b) +(a) +0.3 +0.4 +0.3. +0.2 +0.2 +0.1 +0.0 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +16( +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time (Min.) +Time (Min.).6 +PClabs.at410°C +Van'tHoffplotforPclabsorption +60 +(a) +.5 +(b) +PCI abs. at 425 °C +Van'tHoffplotforPcldesorption +Linear fit +50 +PClabs.at395°C +.4 +PCldes.at410°C +.3 +PCldes.at425°C +40 +(atm) +PCI des. at 395 °C +.2 +30 +Equation +y=a+bx +ressure +.1- +Adj.R-Square +0.99317 +0.997 +Value +Standard Error +PClabs +Intercept +4.15133 +0.19668 +20 +.0 +PClabs +Slope +-2.29308 +0.1342 +PCIdes +Intercept +7.2496 +0.23282 +PCIdes +Slope +-4.10198 +0.159 +6' +P +10 +.8 +0 +.7 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +1.43 +1.44 +1.45 +1.46 +1.47 +1.48 +0.0 +1.49 +1.5 +Hydrogenstoragecapacity (wt%) +1000/T(K)16 + + + +Figure 6:(a) Fig: (a) PCI ab/de-sorption curves of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA and (b) Corresponding +Van’t Hoff plots for PCI ab/de-sorption curves. + + +Co +Mn +Zr +Ti +Melting in R.F.induction Furnace +HEA +(ascastalloy)Hydraulic +Press +3 × 105 N/m² +RF- +Induction +Melting +Melting in R.F. induction Furnace +(Melted under dynamic Argon atmosphere) +35-KW +(as cast alloy) +RF-Induction +Furnace2900 +3000 +(b) +IYobserved +(a) +1500 +C14LavesPhase +Yealculated +2500 +2100 +IBraggPositions +1700 +2000 +(210) +13) +1300 +1500 +5 +2 +- +202) +3 +- +5 +(31 +5 +- +1000 +500 +10 +20 +30 +40 +50 +60 +70 +80 +90 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Angle (20) +Angle 20(a) +(b) +0111 +1101 +100.1/mm +10 1/nm +[1213]a +Mn +Fe +b +ZrLa +Ti Ka +B1 +(d) +ElementWeight% +720 +WYA +ZrL +17.15 +638 +TiK +22.92 +54C +VK +17.46 +MnK +16.93 +MaKa +FeK +8.46 +36 +CoK +17.08 +27 +18 +EMT-20.00AV +XX00SE 6es +De 1 Feo 2922 +WD+ t0.0 mm 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N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Mukhopadhyay3 1Hydrogen Energy Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Institute of Science Banaras Hindu University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Varanasi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Uttar Pradesh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' India 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Faculty of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' University of Allahabad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Prayagraj-211002,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' India 3Department of Metallurgical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Indian Institute of Technology (Banaras Hindu University),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Varanasi-221 005,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' India Abstract In this present investigation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' we discussed the synthesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' microstructure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' and hydrogen storage behavior in C14 type intermetallic Laves phase in a hexanary Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 high entropy alloy (HEA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' In this HEA, three elements are hydride-forming elements (Ti, V, Zr), whereas other three are non-hydride-forming elements (Fe, Mn, Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The thermodynamic parameter like enthalpy of mixing was calculated using the Meidma’s model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The mixing enthalpy (∆Hmix) of Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA system was evaluated to be- 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3472 kJ/mole, and atomic radius mismatch turned out to be = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='441%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' This alloy was synthesized using 35 kW radio frequency induction furnace under argon atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' X-ray diffraction technique (XRD) revealed that this system belongs to the C14 type Laves phase with unit cell parameters a= b =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0158 Å, c=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1790 Å, α = β = 90˚, γ = 120˚ under Space group P63/mmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Microstructural analysis was carried out with the help of a transmission electron microscope (TEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The SEM- EDX data confirms the elemental composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Hydrogen absorption and desorption of this high entropy intermetallic was carried out using the PCI apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The total hydrogen storage of this system was observed around ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='53 wt%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' However, it exhibited better hydrogen and ab/de-sorption kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' With the help of the Van’t Hoff plot, calculated experimental change in enthalpy of Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24-V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17-Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17-Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17-Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08-Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17 HEA for hydrogen absorption and desorption was found out to be ~ -19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='06 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='12 kJ/mol and -34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='10 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='32 kJ /mol respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The possibility of developing high entropy Laves phase-based hydrogen storage materials was advocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Corresponding authors: yadavtp@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='com Co Mn Zr Ti Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='induction Furnace HEA (ascastalloy)Hydraulic Press 3 × 105 N/m² RF- Induction Melting Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' induction Furnace (Melted under dynamic Argon atmosphere) 35-KW (as cast alloy) RF-Induction Furnace2900 3000 (b) IYobserved (a) 1500 C14LavesPhase Yealculated 2500 2100 IBraggPositions 1700 2000 (210) 13) 1300 1500 5 2 202) 3 5 (31 5 1000 500 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Angle (20) Angle 20(a) (b) 0111 1101 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1/mm 10 1/nm [1213]a Mn Fe b ZrLa Ti Ka B1 (d) ElementWeight% 720 WYA ZrL 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 638 TiK 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='92 54C VK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 MnK 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='93 MaKa FeK 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 36 CoK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 27 18 EMT-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='00AV XX00SE 6es De 1 Feo 2922 WD+ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 mm Tome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='t:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 ZEIS Le300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='8 Hydrogenationof Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Coo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Feo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0aMno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 DehydrogenationofhydrogenatedTia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24Va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='Zra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='Coa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Feo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='oMna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17at 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at395°C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='4 PCldes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at410°C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3 PCldes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at425°C 40 (atm) PCI des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' at 395 °C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2 30 Equation y=a+bx ressure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1- Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='R-Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='99317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='997 Value Standard Error PClabs Intercept 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='19668 20 .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 Hydrogenstoragecapacity (wt%) 1000/T(K)2 Introduction Recent years have seen a lot of interest in a new class of materials called ‘High Entropy Alloys’ (HEAs) (Marques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2021, Yadav el al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2017, Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2019, Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' In general, HEAs contain five or more elements, each with a concentration of five to thirty-five atomic percentages (at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='%) or more, in contrast to conventional alloys based on a single primary element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' To improve phase stability, HEAs are understood to exhibit large mixing entropies of solid solution phases (Murty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The research publication by Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' (2004a 2004b), Cantor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' (2004), and Ranganathan (2003) was published for the first time for launching the field of HEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Yeh independently proposed the single-phase multi-principal element alloy in 1995, making this idea a ground-breaking success in researching HEAs (Murty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=" It' is interesting to note that the high mixing entropy in multi- principal element alloys can dramatically lower the number of phases in high-order alloys, leading to a single phase solid solution (Tsai et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' HEA has many functional properties like magnetic,, thermoelectric, catalytic, hydrogen storage etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' In these functional properties, hydrogen storage is considered to be one of the interesting areas to explore the HEA as an effective hydrogen storage material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Nowadays, in order to counteract climate change and the rise in global warming brought on by conventional fossil fuels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' people demand innovative, flexible, clean, and green energy sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Among many fuels that are readily available worldwide, hydrogen is accepted as one of the best candidates due to its high energy range per unit mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Three essential elements that are needed to use hydrogen as a fuel in the future are (i) hydrogen production, (ii) its storage, and (iii) applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Hydrogen storage is one of the most crucial components of using hydrogen as a fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' One of the safest and most efficient ways to store hydrogen is in solid-state metal hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Due to the infinite combination of alloy forming possibilities, the HEAs are novel and promising materials for hydrogen storage (Yadav et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' In 2010, the first investigation was done in HEAs to study the hydrogen storage kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' This study claimed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='03-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='80 wt% hydrogen storage in multi- principal component CoFeMnTixVyZrz (Kao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2010) alloys;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' after that, in TiZrHfNbV HEA, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='7wt% hydrogen storage was reported in 2016 (Sahlberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' There is only one BCC phase in this alloy composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' One more point common in this system is that this alloy system is designed with all the hydride forming elements, because of which it has a good hydrogen storage capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' In recent years hydrogen storage is reported as high as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='51 wt% in V35Ti30Cr25Fe5Mn5 HEA belonging to a single BCC phase (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' On the contrary, the maximum hydrogen storage in Laves phases is known to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='91 wt% (Sarc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' It can stated from the reported data that the Laves phase has less storage properties and better absorption and desorption kinetics Co Mn Zr Ti Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='induction Furnace HEA (ascastalloy)Hydraulic Press 3 × 105 N/m² RF- Induction Melting Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' induction Furnace (Melted under dynamic Argon atmosphere) 35-KW (as cast alloy) RF-Induction Furnace2900 3000 (b) IYobserved (a) 1500 C14LavesPhase Yealculated 2500 2100 IBraggPositions 1700 2000 (210) 13) 1300 1500 5 2 202) 3 5 (31 5 1000 500 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Angle (20) Angle 20(a) (b) 0111 1101 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1/mm 10 1/nm [1213]a Mn Fe b ZrLa Ti Ka B1 (d) ElementWeight% 720 WYA ZrL 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 638 TiK 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='92 54C VK 17.' metadata={'source': 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BCC phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The investigation on low-vanadium TiZrMnCrV-based alloys for high-density hydrogen storage (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2021) was reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Due to its maximal interstitial sites available for absorbing hydrogen in their voids, C14 Laves phase has been explored as hydrogen storage phase tested in recent study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' People have recently been concentrating on the research of phase stability during hydrogen absorption and desorption of HEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' In multi-component HEA for TiZrFeMnCrV (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2022), C14 type Laves phase-based HEA was fabricated and followed by hydrogen storage testing after mechanical milling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The maximal hydrogen absorption for this alloy was reported to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='80 wt% for the first cycle and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='76 wt% for the second cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=" According to their findings, the hydrogen storage capacity varied marginally between each cycle's i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='76 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='73 wt%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' In another study, TiZrCrMnFeNi HEA of C14 Laves phase has exhibited hydrogen absorption as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='7 weight percent (Edalati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Kumar et al (2022) has shown that TiZrVCrNi Laves phase with 1-52 weight percent hydrogen remains stable even after 10 cycles of hydrogenation from the perspective of phase stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The TiZrNbCrFe HEA consisting of C14 Laves phase as maor and BCC phase as minor was reported by Floriano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2021 to have 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='9 wt% hydrogen storage capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='In view of the potential of HEAs for hydrogen storage capability, it was felt worth pursuing the study of other high entropy based alloys for exploring their structure and hydrogen storage performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Accordingly, in the present study, we selected TiZrVMnFeCo nonequiatomic HEAs and investigated the structure, microstructure, and hydrogen storage kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' We chose a HEA system with three hydride forming elements (TiZrV) and the remaining three non-hydride-forming elements (Mn, Fe, Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='The thermodynamic calculation for evaluating enthalpy of mixing of this HEA was done using Meidma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' This HEA was synthesized with the help of a 35 KW Radio Frequency Induction furnace in the argon atmosphere and characterized by XRD, SEM and TEM techniques Hydrogen storage performance was evaluated using pressure composition isotherm (PCI) equipment supplied by Advanced Material Corporation (Pittsburgh, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Material synthesis and characterization methods The high purity materials powder for the synthesis of the Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA system was procured from Alfa Aesar with a purity of more than 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The constituent elements were taken as per their stoichiometry for making a palette using a cylindrical steel mold equipped with the hydraulic press of acting pressure ~3x105 N/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The palette (~10 g by weight) then used for the as-cast synthesis of multicomponent HEA using the RF induction melting process under argon atmosphere (purity of more than 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='90%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The ingots are melted four times to Co Mn Zr Ti Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='induction Furnace HEA (ascastalloy)Hydraulic Press 3 × 105 N/m² RF- Induction Melting Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' induction Furnace (Melted under dynamic Argon atmosphere) 35-KW (as cast alloy) RF-Induction Furnace2900 3000 (b) IYobserved (a) 1500 C14LavesPhase Yealculated 2500 2100 IBraggPositions 1700 2000 (210) 13) 1300 1500 5 2 202) 3 5 (31 5 1000 500 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Angle (20) Angle 20(a) (b) 0111 1101 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1/mm 10 1/nm [1213]a Mn Fe b ZrLa Ti Ka B1 (d) ElementWeight% 720 WYA ZrL 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 638 TiK 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='92 54C VK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 MnK 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='93 MaKa FeK 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 36 CoK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 27 18 EMT-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='00AV XX00SE 6es De 1 Feo 2922 WD+ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 mm Tome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='t:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 ZEIS Le300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='8 Hydrogenationof Tio.' metadata={'source': 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+page_content='1- Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='R-Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='99317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='997 Value Standard Error PClabs Intercept 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='19668 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 PClabs Slope 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 Hydrogenstoragecapacity (wt%) 1000/T(K)4 ensure uniformity of chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The as-cast induction melted ingots of HEA crushed and converted into powder form to perform further characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The first cutting-edge characterization technique used for phase analysis is the Empyrean x-ray diffraction (XRD) system (Malvern Panalytical) equipped with an area detector (256x256 pixels) equipped with a graphite monochromator and Cu radiation source (CuKa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5406, operating at 45 kV and 40 mA) in Bragg-Brentano geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The transmission electron microscope (TEM), TECNAI 20 G2, was used to acquire the microstructures and selected area electron diffraction (SAED) pattern of the samples operating at 200 kV of accelerating voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='EVO 18 scanning electron microscope at operating voltage of 25 kV (vacuum 10-5 torr) was used to investigate surface morphology and perform energy dispersive X-ray analysis (EDX) as well as colour mapping of elements in the as-prepared samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' All de/re-hydrogenation measurements were carried out with the aid of an automated two-channel volumetric sieverts apparatus (supplied by Advanced Materials Corporation Pittsburgh, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' For hydrogen storage testing, we took the 500 mg sample of HEA and placed the sample in the reactor seized by quartz wool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Before performing hydrogen cycle testing, the powder HEA sample was activated at 400℃ under a hydrogen pressure of 1/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1 MPa for hydrogenation/dehydrogenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' After activation, testing of the hydrogen absorption kinetics at 410 °C under 60 atm H2 pressure was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Results and Discussion The experimental XRD diffraction patterns of the as-cast Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA are shown in figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The diffraction profile has been recorded for the gross structural analysis of the as-cast alloy sample by using the Empyrean x-ray diffraction (XRD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Malvern Panalytical) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' All the diffraction peaks (shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2(a)) are well fitted with the hexagonal C14 Laves phase structure parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='The XRD pattern was well refined through Le Bail profile fitting using JANA 2006 software shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The refinement data validated the Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA system with unit cell parameters of a=b= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0141 Å, c= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1756 Å, and the unit cell volume 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 Å3 under the space group of P63/mmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' All the refine parameters are given below in Table 1 To validate the structure analysis of this XRD, we used another characterization technique by transmission electron microscopy (TEM) for analyzing the phase and microstructure of this Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The bright field TEM micrograph of as-synthesized HEA shown in figure 3(a) identifies no other phases other than Laves phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The corresponding SAD pattern of this as cast HEA shown in figure 3(b) validates that this HEA system belongs to a C14 type hexagonal structure with a corresponding space group is P63/mmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Co Mn Zr Ti Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='induction Furnace HEA (ascastalloy)Hydraulic Press 3 × 105 N/m² RF- Induction Melting Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' induction Furnace (Melted under dynamic Argon atmosphere) 35-KW (as cast alloy) RF-Induction Furnace2900 3000 (b) IYobserved (a) 1500 C14LavesPhase Yealculated 2500 2100 IBraggPositions 1700 2000 (210) 13) 1300 1500 5 2 202) 3 5 (31 5 1000 500 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Angle (20) Angle 20(a) (b) 0111 1101 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1/mm 10 1/nm [1213]a Mn Fe b ZrLa Ti Ka B1 (d) ElementWeight% 720 WYA ZrL 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 638 TiK 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='92 54C VK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 MnK 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='93 MaKa FeK 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 36 CoK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 27 18 EMT-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='00AV XX00SE 6es De 1 Feo 2922 WD+ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 mm Tome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='t:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 ZEIS Le300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='8 Hydrogenationof Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Coo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Feo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0aMno.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=" at 425 °C Van'tHoffplotforPcldesorption Linear fit 50 PClabs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at395°C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='4 PCldes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at410°C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3 PCldes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at425°C 40 (atm) PCI des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' at 395 °C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2 30 Equation y=a+bx ressure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1- Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='R-Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='99317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='997 Value Standard Error PClabs Intercept 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='19668 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 PClabs Slope 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='29308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1342 PCIdes Intercept 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='23282 PCIdes Slope 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='10198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content="159 6' P 10 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='8 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 Hydrogenstoragecapacity (wt%) 1000/T(K)5 Surface morphology and elemental composition analysis Scanning electron microscopy (SEM) has been done for surface microstructure and confirming homogeneous element distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Figure4 (a) shows the SEM –BSE, and Energy dispersive X-ray analyses (EDX) mapping images of as cast Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA with the corresponding region which is located in square box in figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The SEM-BSE image reveals the microstructure of this HEA without any cracks or defects in this as-cast HEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Figure 4(b) overlays all the constituent elements present in this HEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' EDAX mapping image establishes that all the constituent elements are distributed as per atomic percent in this as-cast Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24-V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17-Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17- Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17-Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08-Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17 HEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Figure 4(c) shows the SEM-BSE image from another region for the HEA sample, where no crack is observed, and also no other contrast corresponding another phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Figure 4(d) shows the EDX elemental spectra to confirm the stoichiometry of the elements present in this as-cast HEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' All the data indicate that this HEA has forms a single Laves phase with uniform elemental distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Hydrogen ab/de-sorption analysis Hydrogen ab/de-sorption performance in as-cast Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA is studied in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The measurements of hydrogen sorption were carried out with automated two-channel volumetric sieverts instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The results of the absorption kinetic curve of the as-cast Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA are shown in figure 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Before introducing hydrogen into as-cast HEA, we firstly activate the as-cast HEA under 400 ˚C under 10-3 atm evacuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' We perform hydrogenation at 410˚C under 60 atm hydrogen pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The hydrogen desorption kinetic curve of the as-cast Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24-HEAis shown in figure 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The hydrogen desorption kinetic curve of this as- cast HEA shows that this as-cast HEA absorbed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='53 wt% of hydrogen within 15 seconds this curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' In contrast, the maximum storage capacity is evaluated to be about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='72 wt% in 150 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' This fastest kinetics gives interesting results to understand the hydrogen storage performance In the case of desorption, we can see that the dehydrogenated curve shown in figure 5(b) the as cast Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA perform desorption at 410 ˚C under 1 atm hydrogen pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' According to the hydrogenation desorption curve we can say that this HEA released 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='28 wt% hydrogen within one minute at 410 ˚C under 1 atm hydrogen pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The results suggests that this HEA shows faster hydrogen ab/desorption kinetics than some other Laves phase based HEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Co Mn Zr Ti Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='induction Furnace HEA (ascastalloy)Hydraulic Press 3 × 105 N/m² RF- Induction Melting Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' induction Furnace (Melted under dynamic Argon atmosphere) 35-KW (as cast alloy) RF-Induction Furnace2900 3000 (b) IYobserved (a) 1500 C14LavesPhase Yealculated 2500 2100 IBraggPositions 1700 2000 (210) 13) 1300 1500 5 2 202) 3 5 (31 5 1000 500 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Angle (20) Angle 20(a) (b) 0111 1101 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1/mm 10 1/nm [1213]a Mn Fe b ZrLa Ti Ka B1 (d) ElementWeight% 720 WYA ZrL 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 638 TiK 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='92 54C VK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 MnK 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='93 MaKa FeK 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 36 CoK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 27 18 EMT-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='00AV XX00SE 6es De 1 Feo 2922 WD+ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 mm Tome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='t:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 ZEIS Le300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='8 Hydrogenationof Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Coo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Feo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0aMno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 DehydrogenationofhydrogenatedTia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24Va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='Zra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='Coa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Feo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='oMna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='7 at410cunder60atmH2pressure Hydrogen absorbed (wt%) desorbed (wt%) 410Cunder1atmH2pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 (b) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 0 20 40 60 80 100 120 140 16( 0 20 40 60 80 100 120 140 160 Time (Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=') Time (Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='6 PClabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content="at410°C Van'tHoffplotforPclabsorption 60 (a) ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 (b) PCI abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=" at 425 °C Van'tHoffplotforPcldesorption Linear fit 50 PClabs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at395°C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='4 PCldes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at410°C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3 PCldes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at425°C 40 (atm) PCI des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' at 395 °C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2 30 Equation y=a+bx ressure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1- Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='R-Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='99317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='997 Value Standard Error PClabs Intercept 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='19668 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 PClabs Slope 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='29308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1342 PCIdes Intercept 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='23282 PCIdes Slope 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='10198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content="159 6' P 10 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='8 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 Hydrogenstoragecapacity (wt%) 1000/T(K)6 The representative PCI ab/de-sorption of Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA has been shown in figure 6(a) the corresponding represents active Van’t Hoff plots (shown in figure 6(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' PCI was performed at 395˚C, 410˚Cand 425˚C temperatures under 60 atm hydrogen pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' With the help of the three different temperatures, we get the plot corresponding to temperature v/s pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Calculations of the entropy and enthalpy changes that occur throughout the hydrogen ab/de-sorption process typically employ the pressure values of the hydrogen ab/de-sorption platform at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The change in enthalpy (∆H) of hydride formation is given by the well-known Van’t Hoff equation (Dornheim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 2010) ln 𝑃 = Δ𝐻 RT − Δ𝑆 R ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='……(i) Where P is the previously specified plateau pressure, T is the corresponding temperature, R is the gas constant, and H and S are the reaction enthalpy and entropy changes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=" The alloys' Van't Hoff plots are computed using the P, as shown in figure 6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The relationship between ln(P) and 1000/T is clearly linear, as can be seen in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The slope of the fitted curves for ln(P) and 1000/T as well as the intercept on the vertical coordinate allow for the quick calculation of the H and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The results of the calculations demonstrate that the enthalpy of hydrogen desorption changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The change in enthalpy of Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA for hydrogen absorption and desorption has been calculated to be ΔHabs~ -19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='06 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='12 kJ/mol and ΔHdes -34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='10 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='32 kJ /mol respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The smaller negative enthalpy of mixing in HEA suggests that they are more likely to form stable metal hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=" The formation of the metal hydride's absorption and desorption enthalpies are not equal in the current experiment." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Therefore, this system has fewer tendencies to create metal hydride and aids in improving the ab/desorption kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' This suggests that they have a decreased tendency to form a stable metal hydride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Conclusions In this study, we have successfully synthesized the hexanary Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA with the help of an RF induction furnace for the study of hydrogen storage kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The evolution of a single phase of hexagonal C14 high entropy Laves phase with lattice parameters a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='01Å and c =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Åwas established following Rietveld refinement in this multicomponent alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' On the basis of the kinetics study, Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 shows good ab/de-desorption kinetics (absorb ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='53 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='% of H2 within 15 seconds) but poor in hydrogen storage capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The change in enthalpy of Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA for hydrogen absorption and desorption has been Co Mn Zr Ti Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='induction Furnace HEA (ascastalloy)Hydraulic Press 3 × 105 N/m² RF- Induction Melting Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' induction Furnace (Melted under dynamic Argon atmosphere) 35-KW (as cast alloy) RF-Induction Furnace2900 3000 (b) IYobserved (a) 1500 C14LavesPhase Yealculated 2500 2100 IBraggPositions 1700 2000 (210) 13) 1300 1500 5 2 202) 3 5 (31 5 1000 500 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Angle (20) Angle 20(a) (b) 0111 1101 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1/mm 10 1/nm [1213]a Mn Fe b ZrLa Ti Ka B1 (d) ElementWeight% 720 WYA ZrL 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 638 TiK 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='92 54C VK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 MnK 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='93 MaKa FeK 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 36 CoK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 27 18 EMT-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='00AV XX00SE 6es De 1 Feo 2922 WD+ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 mm Tome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='t:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 ZEIS Le300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='8 Hydrogenationof Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Coo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Feo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0aMno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 DehydrogenationofhydrogenatedTia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24Va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='Zra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='Coa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Feo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='oMna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='7 at410cunder60atmH2pressure Hydrogen absorbed (wt%) desorbed (wt%) 410Cunder1atmH2pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 (b) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 0 20 40 60 80 100 120 140 16( 0 20 40 60 80 100 120 140 160 Time (Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=') Time (Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='6 PClabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content="at410°C Van'tHoffplotforPclabsorption 60 (a) ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 (b) PCI abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=" at 425 °C Van'tHoffplotforPcldesorption Linear fit 50 PClabs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at395°C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='4 PCldes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at410°C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3 PCldes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at425°C 40 (atm) PCI des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' at 395 °C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2 30 Equation y=a+bx ressure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1- Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='R-Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='99317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='997 Value Standard Error PClabs Intercept 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='19668 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 PClabs Slope 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 Hydrogenstoragecapacity (wt%) 1000/T(K)7 calculated to be ~ -19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='06 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='12 kJ/mol and -34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='10 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='32 kJ /mol respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The present investigation suggests the scope for further study on the hydrogenation kinetics at various temperatures for exploring the potential for developing Laves phase high entropy alloy for hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Acknowledgment The author (AK) wishes to thank the Council of Scientific and Industrial Research (CSIR) in New Delhi, India, for financial support for a senior research fellowship (Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' 09/013(0952)/2020-EMR-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Author contributions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' synthesized the materials and made the characterizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' conceived, designed the experiments, organized the data and supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' advised on the discussion of results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' advised on the discussion of results and editing the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' The manuscript was written through contributions of all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' All authors have given approval to the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Notes The authors declare no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Co Mn Zr Ti Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='induction Furnace HEA (ascastalloy)Hydraulic Press 3 × 105 N/m² RF- Induction Melting Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' induction Furnace (Melted under dynamic Argon atmosphere) 35-KW (as cast alloy) RF-Induction Furnace2900 3000 (b) IYobserved (a) 1500 C14LavesPhase Yealculated 2500 2100 IBraggPositions 1700 2000 (210) 13) 1300 1500 5 2 202) 3 5 (31 5 1000 500 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Angle (20) Angle 20(a) (b) 0111 1101 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1/mm 10 1/nm [1213]a Mn Fe b ZrLa Ti Ka B1 (d) ElementWeight% 720 WYA ZrL 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 638 TiK 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='92 54C VK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 MnK 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='93 MaKa FeK 8.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Ranganathan S (2003) Alloyed pleasures: Multimetallic cocktails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Current Science 85: 1404-1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Sahlberg M, Karlsson D, Zlotea C, Jansson U (2016) Superior hydrogen storage in high entropy alloys, Scientific Reports: 36770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1038/srep36770 Sarac B, Zadorozhnyy V, Berdonosova E, Lvanov YP, Klyamkin S, Gumrukcu S, Sarac AS, Korol A, Semenov D, Zadorozhnyy M, Sharma A, Greer AL, Eckert J (2020) Hydrogen storage performance of the multi-principal- component CoFeMnTiVZr alloy in electrochemical and gas-solid reactions, RSC Advances 10: 24613–24623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1039/D0RA04089D Co Mn Zr Ti Melting in R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1007/s11661-006-0234-4 Yeh JW, Chen SK, Lin SJ, Gan JY, Chin TS, Shun TT, Tsau CH, Chang SY (2004b) Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Advanced Engineering Materials 6:299303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Zeitschrift für Physikalische Chemie 117: 89-112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1002/adem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='200300567 Zhou P, Cao Z, Xiao X, Jiang Z, Zhan L, Li Z, Jiang L, Chen L (2022) Study on low-vanadium TiZrMnCrV based alloys for high-density hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' International Journal of Hydrogen Energy 47: 710-1722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='ijhydene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='106 Figure captions Figure 1: (a) Schematic diagramof the synthesis protocol for Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA Figure 2:(a) XRD pattern of Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24-V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17-Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17-Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17-Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08-Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17 HEA system and (b) Rietveld refinement profile pattern of all the peaks well fitted with C14 type hexagonal parameters with unit cell parameters a= b =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0158 Å, c=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1790 Å, α = β = 90˚, γ = 120˚ under space group P63/mmc Figure 3 : (a) TEM bright field micrograph of as-cast HEA synthesized by RF induction melting (b) Corresponding SAD patterns are shown indexed with hexagonal structure parameter under the space group of P63/mmc Figure 4: (a) SEM–BSE and energy dispersive X-ray analyses (EDX) mapping images of as Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA (b) overlays all the constituent elements present in this HEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' (c SEM-BSE image from another region for the HEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' (d) EDX elemental spectra to validate the atomic percentage of the elements in this HEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Figure 5: (a) Hydrogenation curve of Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA at 410 ˚C under 60 atm H2 pressure and (b) Dehydrogenation curve of hydrogenated Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA at 410 ˚C under 60 atm H2 pressure Figure 6: (a) Fig: (a) PCI ab/de-sorption curves of Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Co0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 HEA and (b) Corresponding Van’t Hoff plots for PCI ab/de-sorption curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' Co Mn Zr Ti Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='induction Furnace HEA (ascastalloy)Hydraulic Press 3 × 105 N/m² RF- Induction Melting Melting in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' induction Furnace (Melted under dynamic Argon atmosphere) 35-KW (as cast alloy) RF-Induction Furnace2900 3000 (b) IYobserved (a) 1500 C14LavesPhase Yealculated 2500 2100 IBraggPositions 1700 2000 (210) 13) 1300 1500 5 2 202) 3 5 (31 5 1000 500 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Angle (20) Angle 20(a) (b) 0111 1101 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1/mm 10 1/nm [1213]a Mn Fe b ZrLa Ti Ka B1 (d) ElementWeight% 720 WYA ZrL 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 638 TiK 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='92 54C VK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 MnK 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='93 MaKa FeK 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='46 36 CoK 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='08 27 18 EMT-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='00AV XX00SE 6es De 1 Feo 2922 WD+ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 mm Tome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='t:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 ZEIS Le300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='8 Hydrogenationof Tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24Vo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Zro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Coo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Feo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0aMno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 DehydrogenationofhydrogenatedTia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='24Va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='Zra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='Coa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17Feo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='oMna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='17at 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' induction Furnace (Melted under dynamic Argon atmosphere) 35-KW (as cast alloy) RF-Induction Furnace2900 3000 (b) IYobserved (a) 1500 C14LavesPhase Yealculated 2500 2100 IBraggPositions 1700 2000 (210) 13) 1300 1500 5 2 202) 3 5 (31 5 1000 500 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Angle (20) Angle 20(a) (b) 0111 1101 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1/mm 10 1/nm [1213]a Mn Fe b ZrLa Ti Ka B1 (d) ElementWeight% 720 WYA ZrL 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='15 638 TiK 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='92 54C VK 17.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='0 0 20 40 60 80 100 120 140 16( 0 20 40 60 80 100 120 140 160 Time (Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=') Time (Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='6 PClabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content="at410°C Van'tHoffplotforPclabsorption 60 (a) ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='5 (b) PCI abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=" at 425 °C Van'tHoffplotforPcldesorption Linear fit 50 PClabs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at395°C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='4 PCldes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at410°C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='3 PCldes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='at425°C 40 (atm) PCI des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content=' at 395 °C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='2 30 Equation y=a+bx ressure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='1- Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='R-Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} +page_content='99317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfLwz0/content/2301.04942v1.pdf'} 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https://git-lfs.github.com/spec/v1 +oid sha256:e8d3c8b8fae3618a73a265a1813afd001054cc2c8e69e5fe7cc3f35536b141e5 +size 247812 diff --git a/YNE1T4oBgHgl3EQfwAWP/content/tmp_files/2301.03406v1.pdf.txt b/YNE1T4oBgHgl3EQfwAWP/content/tmp_files/2301.03406v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..244c3b39c7e045a56c1880f5c6559b1777c7f3fa --- /dev/null +++ b/YNE1T4oBgHgl3EQfwAWP/content/tmp_files/2301.03406v1.pdf.txt @@ -0,0 +1,790 @@ +Astronomy & Astrophysics manuscript no. output +©ESO 2023 +January 10, 2023 +Cosmic rate of type IIn supernovae +and its evolution with redshift +C. Cold1 and J. Hjorth1 +DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen N, Denmark +Received month day, year; accepted month day, year +ABSTRACT +Context. Type IIn supernovae potentially constitute a large fraction of the gravitationally lensed supernovae predicted to be found +with upcoming facilities. However, the local rate is used for these estimates, which is assumed to be independent of properties such +as the host galaxy mass. Some studies hint that a host galaxy mass bias may exist for IIn supernovae. +Aims. This paper aims to provide an updated local IIn supernova-to-core-collapse ratio based on data from the Palomar Transient +Factory (PTF) and the Zwicky Transient Facility (ZTF) Bright Transient Survey (BTS). Furthermore, the goal is to investigate the +dependency of the IIn supernova peak magnitude on the host galaxy mass and the consequences of a possible host galaxy mass +preference on the volumetric rate of type IIn supernovae. +Methods. We constructed approximately volume-limited subsamples to determine the local IIn supernova-to-core-collapse ratio. We +investigated the absolute peak magnitude of a subsample of type IIn and superluminous II or IIn supernovae exploring how this relates +to the i-band magnitude of the host galaxies (as a proxy for stellar mass). We presented a method to quantify the effect of a potential +preference for low-mass host galaxies utilizing the UniverseMachine algorithm. +Results. The IIn supernova-to-core-collapse ratios for PTF and BTS are 0.046 ± 0.013 and 0.048 ± 0.011, respectively, which results +in a ratio of 0.047±0.009, which is consistent with the ratio of 0.05 currently used to estimate the number of gravitationally lensed IIn +supernovae. We report fainter host galaxy median absolute magnitudes for type IIn brighter than −20.5 mag with a 3 σ significance. +If the IIn supernova-to-core-collapse ratio were described by the power law model IIn/CC = 0.15 · log(M/M⊙)−0.05, we would expect +a slightly elevated volumetric rate for redshifts beyond 3.2. +Conclusions. +Key words. supernovae: general. +1. Introduction +Type IIn supernovae (SNe IIn) exhibit narrow hydrogen emis- +sion lines in their spectra (Schlegel 1990). The distinct features +of this SN class arise from the slow-moving and dense circum- +stellar material (CSM) ejected by the star prior to explosion. This +means that the SN IIn subtype is very diverse, as these SNe can +emerge whenever CSM indications are present in their spectra, +whether it is early or late in the lifespan of the SN, or whatever +lies beneath the veil of the CSM (Smith 2017). Narrow hydrogen +features may also arise from flash ionization of local CSM fol- +lowing shock breakout. However, such emission lines disappear +shortly after peak magnitude to reveal the underlying SN type +(Yaron et al. 2017; Bruch et al. 2021; Jacobson-Galán et al. 2022; +Terreran et al. 2022). Nevertheless, flash ionization in combina- +tion with the complexity of the CSM structure can complicate +the classification of SNe IIn (Ransome et al. 2021). +Luminous blue variables, extreme red super giants, and yel- +low hyper giants have all been proposed as progenitors due to +their recurring violent mass-loss episodes. The intervals of mass +loss can range from a period of months to thousands of years, +which is necessary to produce the amount of CSM required to +make SNe IIn (Smith 2017). +Brighter and more long-lived than other SN types, super- +luminous supernovae (SLSNe) are recognized as their own class +of SNe (Moriya et al. 2018; Gal-Yam 2012). These very lu- +minous objects differ from other SNe by their optical absolute +magnitudes of around −21 or less, although SLSNe have been +classified at around −19 mag at peak for the faintest objects +(Moriya et al. 2018; Angus et al. 2019). SLSNe, as the classic +SN classes, are also further categorized into subtypes. The super- +luminous counterpart to the SNe IIn, SLSNe-IIn, also feature +narrow emission lines of the hydrogen Balmer series similar to +the regular SNe IIn, and these constitute a significant percentage +of all hydrogen-rich SLSNe (Gal-Yam 2019). This is indicative +of CSM interaction partly powering very bright transients. It is +not yet clear whether SLSNe-IIn and SNe IIn are two distinctive +populations or if they form a continuum in luminosity. However, +in this work, we considered the SLSNe-IIn as the brightest SNe +IIn. +Models indicate that SNe IIn along with SNe Ia will dom- +inate the observed rates of lensed supernovae. Estimates from +the upcoming Legacy Survey of Time and Space (LSST) with +the Vera C. Rubin Observatory (Wojtak et al. 2019; Goldstein +et al. 2019) predict on the order of 100 SNe IIn per year to be +gravitationally lensed. For the Roman Space Telescope, the grav- +itationally lensed SNe predictions are comparable (Pierel et al. +2021). +The lensed SNe IIn predictions are based on the observed lo- +cal rate of SNe IIn. Data from the Lick Observatory Supernova +Search (LOSS) yielded an SNe-IIn-to-CC ratio of 8.8%+3.3% +−2.9% +SNe IIn out of all CC SNe (Li et al. 2011; Smith et al. 2011). +Article number, page 1 of 8 +arXiv:2301.03406v1 [astro-ph.HE] 9 Jan 2023 + +A&A proofs: manuscript no. output +Fig. 1. Overview of redshift and host galaxy mass distributions. Left panel: Redshift distribution of the SNe IIn in Nyholm et al. (2020) and +SLSN-IIn from PTF. Both are subsamples of the full PTF SNe IIn and SLSNe-IIn samples. The redshifts can be found in Schulze et al. (2021). +Right panel: Distribution of the host galaxy masses corresponding to the same SNe IIn from Nyholm et al. (2020) and a subsample of SLSNe-IIn +from the PTF. These subsample distributions are consistent with the full sample distributions of SNe IIn and SLSNe-IIn from Schulze et al. (2021). +Fig. 2. Overview of redshift and host galaxy magnitude distributions. Left panel: Redshift distribution of the SNe IIn and SLSN-II from BTS. Right +panel: Distribution of the host galaxy i-band absolute magnitudes. For two SNe IIn and 1 SLSN-II, the i-band magnitudes were not available. The +i-band magnitude is used as a proxy for the host stellar mass. +Several of these SNe IIn were subsequently identified as SN +Impostors such that the IIn rate from LOSS is now considered +to be around 5% (Graur et al. 2017). Other examples of local +rate measurements include studies by Smartt et al. (2009), who +present a volume-limited (28 Mpc) sample compiled from all lo- +cal SNe with a named host galaxy discovered over a ten-year +period. This sample includes 3.8% SNe IIn out of all CC SNe +in the sample. Eldridge et al. (2013) updated the study done by +Smartt et al. (2009), searching for SNe discovered over a 14-year +period, yielding 2.4 ± 1.4%. +The existing rate estimates of SNe IIn assume that the local +fraction of IIn to all other CC SNe does not evolve with redshift +and so is not affected by large-scale changes in compositions +or characteristics of galaxies and stellar populations over time. +Several studies show a bias toward less massive host galaxies for +SLSNe in general (e.g., Leloudas et al. 2015; Angus et al. 2016; +Schulze et al. 2018; Taggart & Perley 2021), and this dearth of +Article number, page 2 of 8 + +8 +42lln(Nyholmetal.2020) +14 +9 SLSN-lln from PTE +7 +12 +5 +8 +Numberof +4 +Number +6 +3 +4 +2 +1 +2 +0 +0 +0.000.05 +0.100.150.200.25 +0.300.35 +5 +9 +101112 +13 +z +Host Galaxy Stellar Mass [log(M)]92lnfromBTS +90linfromBTS +12 +18 SLSN-11from BTS +17 SLSN-Lfrom BTS +20 +10 +Numberof Galaxies +15 +8 +6 +10 +4 +5 +2 +0 +0.00 +00.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +-14 +-16 +-18 +-20 +-22 +Host Galaxy Magnitude [M;]C. Cold and J. Hjorth: Cosmic rate of type IIn supernovae and its evolution with redshift +massive hosts suggests some dependence on the characteristics +of the environment. In a study by Graur et al. (2017), based on +data from LOSS, SNe IIn seem to be more common in less mas- +sive galaxies. However, other studies do not come to the same +conclusion (e.g., Kelly & Kirshner 2012). The Palomar Tran- +sient Factory (PTF) CC SN sample presented in Schulze et al. +(2021) also does not reveal a bias for SNe IIn toward low-mass +host galaxies, and is inconclusive regarding the host galaxy mass +preference of SLSNe-IIn. +The Zwicky Transient Facility (ZTF) Bright Transient Sur- +vey (BTS) (Fremling et al. 2020; Perley et al. 2020) will be part +of the analysis in this paper in addition to the PTF CC sample +(Schulze et al. 2021). The paper is structured as follows. In Sec- +tion 2, we briefly introduced the data used for the analysis. In +Section 3, we created an approximately volume-limited sample +from the PTF and BTS data, followed by a presentation of an +updated SNe-IIn-to-CC ratio (SNe IIn fraction) for both sam- +ples. In Section 4, we compare the absolute magnitude and the +i-band magnitude of the host galaxies of a subsample of SNe IIn, +SLSNe-IIn, and SLSN-II from PTF and BTS. In Section 5, we +presented a generic method for inferring the rate and its evolu- +tion with redshift and studying the consequences of a possible +mass-biased SNe IIn rate, before the discussion in Section 6 and +conclusions in Section 7. +2. Data +The PTF CC SN sample from Schulze et al. (2021) and the ZTF +Bright Transient Survey (Fremling et al. 2020; Perley et al. 2020) +constitute the basis of the analysis presented in this paper. The +PTF was a deep, wide-field survey followed by the intermediate +PTF (iPTF) survey. The PTF CC sample contains 888 objects, +of which 111 are SNe IIn and 16 SLSNe-IIn. Redshifts and host +galaxy stellar masses are available for all objects in the sample. +For later analysis, we will use the host galaxy i-band (either Pan- +STARRS1 (PS1) or Sloan Digital Sky Survey (SDSS) i-band) +absolute magnitude. However, two SNe IIn and one SLSN-IIn +have no reported host i-band magnitude. The sample contains +only host photometry, and so we used a subsample of IIn and +SLSNe-IIn where the peak absolute magnitude is available. The +subsample of 42 SNe IIn with available peak magnitudes is de- +scribed in Nyholm et al. (2020). These SNe were chosen based +on the amount of available light-curve data to allow an analy- +sis of both the rise times and decline rates of the SNe IIn, all +with at least one available low-resolution spectrum. The red- +shifts in Nyholm et al. (2020) differ slightly from the redshifts +in Schulze et al. (2021), which are the galaxy redshifts estimated +by one of four possible methods: taken from SDSS, taken from +the NASA Extragalactic Database, estimated from galaxy lines +in the spectra or estimated from SN-template matching (Schulze +et al. 2021). In Nyholm et al. (2020), the redshifts are estimated +from the Hα emission lines in the SNe spectra. In this work, +we used the data published in Schulze et al. (2021) as well as +SLSNe-IIn peak magnitudes (Leloudas, priv. comm.). The red- +shift and host galaxy stellar mass distributions of the SNe IIn +from Nyholm et al. (2020) along with the SLSN-IIn sample from +the PTF are shown in Fig. 1. +The BTS is currently the largest spectroscopic survey of +SNe. The survey is magnitude-limited in the g and r (<19 mag- +nitude) bands. The sample is 97% spectroscopically complete at +<18 mag, 93% at <18.5 mag, and 75% at <19 mag (Perley et al. +2020). The survey is updated daily as new observations come +in. For this paper, we chose to use all available CC SNe and IIn +from BTS regardless of magnitude as of May 16, 2022. The to- +tal number of CC SNe adds up to 949, of which 92 are IIn. We +also included the SLSN-II in our analysis, of which there are 18. +In BTS, the SLSNe-II are not further divided into subclasses. +However, as most SLSNe-II exhibit IIn-like features, we chose +to include all of the SLSN-II in our analysis. The parameters +we used in our analysis in this paper are redshift, peak magni- +tude and host i-band absolute magnitude. Redshifts are available +for all but six CC SNe, none of which are SNe IIn or SLSN- +II. Since the host galaxy stellar masses are not available in this +sample, we instead utilized the absolute i-band magnitude of the +hosts where available as these magnitudes are a good proxy for +the stellar masses, as is seen in Fig. 3. Distributions of redshift +and host galaxy i-band magnitude are shown in Fig. 2. +Fig. 3. PTF CC SNe host galaxy absolute i-band magnitudes are plotted +against host galaxy stellar masses. Due to the standard deviation of the +residuals being 0.4, which is comparable to the uncertainty on the stellar +mass, the i-band magnitude can be used as a proxy for the stellar mass +in our analysis. +3. Inferred IIn fractions +In Frohmaier et al. (2021), the CC SN rate for the PTF is de- +termined while taking all the survey limitations into account +through extensive modeling. This method yields a total of 86 CC +SNe and three SNe IIn. Unfortunately, inferring relative rates +of SNe IIn with only three sources will be dominated by low- +number statistics. For the purpose of estimating the SNe IIn +fraction, we will alternatively create an approximately volume- +limited sample for the recent PTF data released in Schulze et al. +(2021) and also from BTS (Fremling et al. 2020; Perley et al. +2020) under the assumption of fair spectroscopic classification. +A simple way to estimate the distance, or redshift, at which +the PTF CC sample is approximately complete is to compare +with a known complete sample. The LOSS sample is com- +plete out to 60 Mpc for CC SNe (Li et al. 2011; Graur et al. +2017). With the limiting magnitude of LOSS having a median +of 18.8 ± 0.5 mag (Leaman et al. 2011), 60 Mpc is also the dis- +tance to the furthest SNe LOSS could theoretically observe as- +suming the faintest SNe to have an absolute magnitude of around +−15.1. The PTF has a limiting magnitude of 20.5 mag in the R +band, which implies a distance of up to 131.8 Mpc for creating a +volume-limited sample, assuming the same −15.1 mag for faint +SNe. This corresponds to a redshift cut-off of 0.031 when adopt- +ing a Hubble constant of H0 = 73 kms−1Mpc−1 as in LOSS. This +Article number, page 3 of 8 + +24 +[M;] +22 +Magnitude +20 +/Absolute +-18 +-16 +Host Galaxy +-14 +-12 +PTF CC SNe Host Galaxies +-10 +4 +5 +6 +1 +8 +9 +10 +11 +12 +HostGalaxyStellarMass[log(M*/M)]A&A proofs: manuscript no. output +Fig. 4. Redshift evolution of SNe-IIn-to-CC ratio. The gray data points represent the cumulative SNe-IIn-to-CC ratio from PTF, indicated on the +left y-axis, as a function of redshift limits out to maximum redshift of the sample. SLSNe-IIn are not included in the IIn-to-CC ratio. The vertical +gray dashed line represents the chosen redshift cut. The number of CC SNe, SNe IIn, and SLSNe-IIn as a function of redshift are represented by +the colored curves indicated on the right y-axis. The inset shows a zoomed-in image of a plot of the cumulative SNe-IIn-to-CC ratio up to redshift +0.06 as well as the chosen redshift cut. +Fig. 5. Redshift evolution of SNe-IIn-to-CC ratio. The gray data points represent the cumulative SNe-IIn-to-CC ratio from BTS, indicated on the +left y-axis, as a function of redshift limits. The vertical gray dashed line represents the chosen redshift cut. The number of CC SNe and SNe IIn +are also depicted as the colored curves, which are indicated on the right y-axis. The inset shows a zoomed-in image of a plot of the cumulative +SNe-IIn-to-CC ratio up to redshift 0.06 as well as the chosen redshift cut. +estimate can be tested visually, as is done in Fig. 4, where we +plot the SNe-IIn-to-CC ratio as a function of redshift limit. The +estimated cut of 0.031 is indicated in the plot by a dashed line. +The SNe IIn fraction for lower redshift cut-offs is dominated by +noise due to the small number of supernovae found at these red- +shifts, whereas the ratio increases above 0.031 as more SNe IIn +are observed at this range. This indicates that the estimated red- +shift cut is located where the noise from few observations has +started to diminish, but we do not yet see the effect of a larger +volume wherein to observe SNe IIn and CC SNe in general. As +such, imposing a redshift cut-off of 0.031 on the PTF CC sample +Article number, page 4 of 8 + +0.14 +Z=0.031 +PTF +800 +0.12 +0.10 +Supemova +0.075 +IIn/CC +0.08 +IIn/CC +0.050 +4006 +0.06 +Z=0.031 +Number +0.025 +PTF +0.04 +0.000 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +200 +z limit +0.02 +SLSN IIn +IIn +0.00 +CC +0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +z limitZ=0.033 +BTS +0.10 +800 +0.08 +Number of Supemovae +0.100 +600 +0.075 +IIn/CC +0.050 +400 +0.04 +0.025 +Z=0.033 +PTF +0.000 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +200 +0.02 +z limit +IIn +0.00 +CC +0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +z lirmitC. Cold and J. Hjorth: Cosmic rate of type IIn supernovae and its evolution with redshift +is a reasonable approach for creating an approximately volume- +limited sample to use for estimating the SNe IIn fraction. +Fig. 6. Distributions of host galaxy stellar mass of the complete PTF CC +sample using z < 0.031. No SLSNe-IIn in the PTF sample are found +within this redshift. Results from a KS test show no significant SNe +IIn host galaxy mass preference for the volume-limited sample with a p +value of 0.002. +We employ the same method for the BTS sample. Using +−15.1 mag for the faintest CC SNe is consistent with the mean of +the 20 faintest CC SNe in the BTS sample. According to Bellm +et al. (2019), the limiting magnitudes for ZTF are 20.8 mag in +the g band, 20.6 mag in the r band, and 19.9 in the i band. We +will use the r-band value to be consistent with LOSS. From the +distance modulus, this yields a distance of 138 Mpc, correspond- +ing to a redshift cut-off of about 0.033, as illustrated in Fig. 5. +The volume effect is less obvious in the BTS data, and as such +the resulting IIn fraction from BTS is more robust toward any +uncertainties in the redshift cut compared to the result from PTF. +However, as the redshift cut of 0.033 occurs before a slow rise in +the SNe-IIn-to-CC value and after the inital large uncertainties, +we deem 0.033 to be a good estimate for creating an approxi- +mately volume-limited sample. +For the PTF, a redshift cut of 0.031 leaves us with an ap- +proximately complete CC sample of 263 CC SNe in total. This +includes 12 SNe IIn, but none of the SLSNe-IIn are observed +within this redshift. Therefore, we find that the ratio of SNe IIn to +CC SNe for our subsample of PTF data to be 0.046 ± 0.013. The +uncertainty on the resulting SNe-IIn ratio is propagated from the +Poisson error on the individual number of CC and SNe IIn. The +resulting histogram of the host galaxy stellar masses of this vol- +ume limited sample of PTF data, as illustrated in Fig. 6, reveals +no obvious SNe IIn preference for less massive host galaxies, +which is in agreement with the analysis done by Schulze et al. +(2021) on the full PTF sample. +We can compare this value to the BTS data. A redshift cut- +off of 0.033 yields a sample of 440 CC SNe, of which 21 are +SNe IIn. This results in an SNe-IIn-to-CC ratio of 0.048±0.011. +The uncertainty on this number is similarly determined using er- +ror propagation. As host galaxy masses are not available in BTS, +and a significant amount of CC hosts do not have i-band magni- +tudes either, we will not compare the host galaxy distributions of +the SNe IIn and CC SNe from BTS. As these resulting fractions +are independent, we combine them and get a SNe IIn relative +fraction of 0.047 ± 0.009. +4. Brightness of SNe IIn +In this section, we investigate whether the peak brightness of the +IIn is influenced by the host galaxy stellar mass when consider- +ing the SLSN-IIn as the brighest members of the IIn class. We +know from several studies that SLSNe prefer lower mass host +galaxies. According to Schulze et al. (2021), this phenomenon is +not significant when only studying the SLSNe-IIn, as the objects +are still too few. We note that for redshifts on the order of 0.03 +the influence of peculiar velocities on calculating peak absolute +magnitudes of the SNe is decreasing compared to SN samples, +which are mostly comprised of local sources. +The distributions of redshift and host galaxy mass of the sub- +sample of IIn and SLSN-IIn from PTF are displayed in Fig. 1. +In Fig. 2, we show the distributions of redshift and host galaxy +i-band magnitude from BTS. For the comparison between these +two data sets, we use the i-band magnitude of the host galaxies +as a proxy for the stellar mass. Only three sources from either +data set do not have available host i-band magnitudes. +In Fig. 7, we compare the absolute magnitude at peak and the +i-band magnitude of the host galaxies of the SNe IIn and SLSNe- +IIn or SLSN-II from the PTF as well as BTS. As these objects are +chosen based on data availability, the subsamples seen in Fig. 7 +are not complete. However, Nyholm et al. (2020) state that the +host galaxy mass distribution of their subsample is in agreement +with the distribution of the full PTF SNe IIn sample, such that +the distribution of the i-band magnitudes should follow a similar +distribution. The data in Fig. 7 show no clear trend regarding the +effect of the host galaxy magnitude on the peak absolute mags +of the SNe IIn. To further investigate, we divide the combined +PTF and BTS subsamples shown in Fig. 7 in two, namely a faint +sample and a bright sample, and subsequently calculate the me- +dian and uncertainty on the median as 1.48·MAD/ +√ +N − 1 of the +i-band magnitudes for the host galaxies, where MAD is the me- +dian absolute deviation. This division of the subsample is done +for several different SN IIn peak magnitudes. We employ Mpeak +values from −17.5 to −21 as the dividing lines between the faint +and the bright sub-samples and compare the medians, as can be +seen in Fig. 8. We find that the median i-band magnitude (and +thus the stellar mass) of the host galaxies becomes fainter with +a 3σ significance when choosing a sample of SNe IIn brighter +than −20.5 mag. +Fig. 7. Absolute peak magnitude versus host galaxy i-band magnitude +for SNe IIn and SLSNe-IIn/SLSNe-II from PTF and BTS. +Article number, page 5 of 8 + +102 +263 CC +12 IIn +Supernovae +Numberof +101 +100 +5 +6 +7 +8 +9 +10 +011 +12 +13 +HostGalaxyStellarMass[log(M)]-23 +22 +-21 +20 +Peak +19 +Absolute +-18 +-17 +BTS IIn +BTS SLSN-II +-16 +PTF IIn +PTF SLSN-IIn +-15 +10 +-12 +-14 +-16 +-18 +-20 +-22 +Host Galaxy Absolute Mag [Mi]A&A proofs: manuscript no. output +Fig. 8. Median host Mi as a function of different cuts on the supernova +peak magnitude for combined BTS and PTF samples. The median i- +band magnitude of the host galaxies becomes fainter for the brighter +SNe in the combined sample. +5. Consequences of a host-mass-dependent IIn +fraction +While the evidence for a host-mass-dependent IIn fraction is not +strong, we next explore the consequences of a hypothetical pref- +erence for low-mass host galaxies. We parametrize the IIn frac- +tion as a power-law function of the stellar mass of the host galaxy +and investigate the impact on the volumetric rate as a function of +redshift. This will be affected since more low-mass galaxies are +present in the earlier Universe. +In general, the volumetric SNe IIn rate can be expressed as +IInrate = IIn +CC · kCC · S FR. +(1) +The CC constant, kCC, is set to 0.0091M−1 +⊙ +following Strol- +ger et al. (2015). This model is consistent with observational +constraints from Dahlen et al. (2012) and Madau & Dickinson +(2014) as demonstrated in Strolger et al. (2015). Here, a model +for the star-formation rate (SFR) is taken from the UniverseMa- +chine algorithm by Behroozi et al. (2019). The results of this +code are best-fitting models of stellar-mass functions (SMFs), +cosmic star formation rates (CSFRs), specific star formation +rates (sSFRs), and UV luminosity functions (UVLFs) to obser- +vations. One can determine the SFR from the output of the Uni- +verseMachine algorithm: +S FR = +� Mmax +Mmin +S MF · M · sS FR dM. +(2) +When computing the SFR this way, it is possible to split it into +different mass bins in order to infer the contribution to the total +SFR from galaxies of different masses and how this changes with +redshift, as is shown in the top panel of Fig. 9. The UniverseMa- +chine resulting models have mass ranges of 107M⊙ to 1013M⊙, +and we choose to split these into five different bins, as indicated +in Fig. 9. The bin containing the 1011M⊙ to 1013M⊙ galaxies is +chosen to be wider than the other bins, as the contribution from +the 1012M⊙ to 1013M⊙ galaxies is negligible. We parametrize the +IIn-to-CC ratio as a power law: +IIn +CC (log(M/M⊙)) = 0.15 · log(M/M⊙)−0.05. +(3) +This power-law model is chosen to have a higher SNe-IIn-to- +CC ratio than 0.047 for host galaxies below 1010M⊙ and a lower +ratio for more massive galaxies. For this specific example, the +ratio will be 0.057 for galaxies with stellar masses of 107M⊙, and +0.043 for 1012M⊙. The motivation for this kind of model comes +from the LOSS data in Graur et al. (2017), showing a preference +for low-mass hosts for SNe IIn, which can be modeled with a +power law with different sets of parameters (Hede 2021). +To calculate the volumetric rate, we use the central value of +the IIn/CC model for every mass bin in log as the IIn/CC factor +in Eq. (1), and thus compute a separate SN IIn rate for each mass +bin as well as the combined rate. Since the power-law model and +the constant model do not predict the same number of SNe given +a different area under the curve, we normalize the resulting rate +from the power law model to 4.77 · 10−6 yr−1Mpc−3, which is +the SNe IIn rate at redshift zero for a constant SNe IIn fraction of +0.047 calculated from Eq. (1), where the UniverseMachine is the +source of the SFR. We do this to be consistent with the constant +model. The resulting SNe IIn rate is plotted in the bottom panel +of Fig. 9 for both the example power-law model and the constant +model of IIn/CC = 0.047. The overall IIn rate is slightly lower +for a redshift below four, and slightly higher for one above four. +This plot also shows how the contribution from the different host +mass bins differs for the constant ratio to the power-law model. It +is evident that the overall rate from low-mass galaxies is higher, +and since these contribute a larger fraction of the total rate at +higher redshift, we also see an increase in the rate in this high- +redshift domain as expected. +6. Discussion +In this section, we discuss and reflect on some of the shortcom- +ings and consequences of the methods and results of this paper +as well as some widely used assumptions. First, we note that the +SNe-IIn fractions from the PTF and BTS are based on approxi- +mate volume-limited samples. The identification of a sweet spot +in the IIn fraction in the PTF sample (Fig. 4) supports this ap- +proach, although we note that, ultimately, IIn fractions will have +to be based on carefully defined volume-limited samples of a +large number of core-collapse supernovae. +From Fig. 8, we see a statistically significant connection be- +tween the brightness of the SNe IIn and the host galaxy stellar +mass when choosing a sample of SNe IIn brighter than −20.5 +mag. For the faint SNe IIn sample at the −18 mag cut-off, the +median i-band magnitude of the host galaxies drops below the +bright IIn sample median host i-band magnitude. However, the +subsamples employed in this part of our analysis could be sub- +ject to different selection effects. Observing faint SNe in bright +galaxies is challenging as the light from the galaxy can hide the +SNe and so we could expect that some were missed in this area. +This effect will influence the median i-band host magnitude for +the faintest SNe IIn and could explain the slight shift in median +host i-band magnitude for the faintest SNe in Fig. 8. However, +the drop in median i-band host-galaxy magnitude is not as sig- +nificant as the drop for the brightest supernovae. +Using the UniverseMachine algorithm DR1 as the input for +the SFR gives insight into the contributions from host galaxies +of different stellar masses to the resulting SNe IIn rate whether a +preference for low-mass hosts exists or not. One limitation, how- +ever, is the lower mass boundary on these galaxies. As is evident +from both Figs. 6 and 1, several SNe IIn have host galaxies less +massive than 107M⊙, but using UniverseMachine it is not pos- +sible to see the contribution from these galaxies. Another limi- +tation is the mass resolution of UniverseMachine. In this case, +Article number, page 6 of 8 + +21.0 +Bright SNe +20.5 +Faint SNe +-20.0 +M +-19.5 +Median Host I +19.0 +-18.5 +18.0 +17.5 +17.0. +17.0-17.5-18.0-18.5-19.0-19.5-20.0-20.5-21.0-21.5 +Mpeak CutC. Cold and J. Hjorth: Cosmic rate of type IIn supernovae and its evolution with redshift +Fig. 9. Overview of the cosmic SFR, volumetric SNe IIn rate and rel- +ative SNe IIn rate. Top panel: SFR calculated from UniverseMachine +DR1 (Behroozi et al. 2019). Middle panel: Resulting SNe-IIn rate de- +termined using Eq. (1). The solid lines denote rates from the power-law +IIn-to-CC ratio, and the dashed lines represent the rate for a constant +ratio of 0.047. The contribution from each mass bin is plotted for com- +parison. Bottom panel: Total relative volumetric SNe IIn rate. The gray +line represents a line of agreement between the two models. +we have four or five data points per mass bin, which prevents us +from further dividing the host galaxies into smaller bins to obtain +a more detailed overview. +In Fig. 9, we show the resulting SNe-IIn volumetric rates. For +illustration, the volumetric rate is computed for a uniform fixed +SNe-IIn fraction of 0.047 next to a model in which the SNe IIn +prefer lower mass host galaxies, here represented by a power law +model. The two models agree at z = 0 and the normalization of +the curves is uncertain by 20 %. The middle and bottom panels +show that the relative volumetric rate of SNe IIn increases over +the default constant model for redshifts beyond 3.2. +Introducing the example power-law model to describe the +SNe-IIn-to-CC ratio produces a minimal effect on the volumet- +ric rate compared to the constant ratio as seen in Fig. 9: A lower +rate for redshifts below 3.2 and higher for redshifts beyond. The +LSST or Roman Space Telescope are not expected to be able to +observe lensed SNe beyond redshifts of three and four, respec- +tively (Wojtak et al. 2019; Goldstein et al. 2019; Pierel et al. +2021), and so the possibility of testing such a model is currently +limited. On the other hand, we show that a limited mass depen- +dence of the IIn rate should not affect predicted volumetric rates +of SNe IIn significantly. +7. Conclusions +We studied the PTF and BTS SNe IIn and SLSNe-IIn/SLSNe- +II populations throughout this work and now present our main +conclusions. +Creating a complete sample of CC SNe from PTF and +BTS, we find the SNe IIn to CC ratios of 0.046 ± 0.013 and +0.048 ± 0.011 for the PTF and BTS, respectively. The combined +resulting SNe-IIn fraction is 0.047 ± 0.009. We see a marginally +significant (3 σ) bias towards low-mass host galaxies for SNe IIn +brighter than −20.5 mag. We present a general method to evalu- +ate the consequences of a SNe IIn to CC ratio that is nonconstant +and is instead described by a power-law model on the resulting +volumetric rate. The example model chosen here can be freely +replaced by another model as required. We find that the example +power law model of IIn/CC = 0.15 · M−0.05 results in a slightly +lower volumetric rate below a redshift of four and a higher rate +beyond a redshift of 3.2 when comparing to a constant ratio of +0.047. Neither the LSST nor the Roman Space Telescope are pre- +dicted to find lensed SNe beyond redshifts of three or four. We +emphasize that our method is generic and can be applied to other +CC subtypes if needed. +Acknowledgements. We gratefully acknowledge Giorgios Leloudas for sharing +peak absolute magnitudes for a subsample of PTF SLSNe-IIn, without which a +large part of the analysis in this paper would not have been possible, as well as +invaluable comments on the paper draft. 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A., Gal-Yam, A., et al. 2017, Nature Physics, 13, 510 +Article number, page 8 of 8 + diff --git a/YNE1T4oBgHgl3EQfwAWP/content/tmp_files/load_file.txt b/YNE1T4oBgHgl3EQfwAWP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..99efc0a85e008d13368932d1de67c45f27e1df61 --- /dev/null +++ b/YNE1T4oBgHgl3EQfwAWP/content/tmp_files/load_file.txt @@ -0,0 +1,723 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf,len=722 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' output ©ESO 2023 January 10, 2023 Cosmic rate of type IIn supernovae and its evolution with redshift C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Cold1 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Hjorth1 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen N, Denmark Received month day, year;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' accepted month day, year ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Type IIn supernovae potentially constitute a large fraction of the gravitationally lensed supernovae predicted to be found with upcoming facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' However, the local rate is used for these estimates, which is assumed to be independent of properties such as the host galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Some studies hint that a host galaxy mass bias may exist for IIn supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This paper aims to provide an updated local IIn supernova-to-core-collapse ratio based on data from the Palomar Transient Factory (PTF) and the Zwicky Transient Facility (ZTF) Bright Transient Survey (BTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Furthermore, the goal is to investigate the dependency of the IIn supernova peak magnitude on the host galaxy mass and the consequences of a possible host galaxy mass preference on the volumetric rate of type IIn supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We constructed approximately volume-limited subsamples to determine the local IIn supernova-to-core-collapse ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We investigated the absolute peak magnitude of a subsample of type IIn and superluminous II or IIn supernovae exploring how this relates to the i-band magnitude of the host galaxies (as a proxy for stellar mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We presented a method to quantify the effect of a potential preference for low-mass host galaxies utilizing the UniverseMachine algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The IIn supernova-to-core-collapse ratios for PTF and BTS are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='046 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='013 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='048 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='011, respectively, which results in a ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='047±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='009, which is consistent with the ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='05 currently used to estimate the number of gravitationally lensed IIn supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We report fainter host galaxy median absolute magnitudes for type IIn brighter than −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 mag with a 3 σ significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' If the IIn supernova-to-core-collapse ratio were described by the power law model IIn/CC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='15 · log(M/M⊙)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='05, we would expect a slightly elevated volumetric rate for redshifts beyond 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' supernovae: general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Introduction Type IIn supernovae (SNe IIn) exhibit narrow hydrogen emis- sion lines in their spectra (Schlegel 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The distinct features of this SN class arise from the slow-moving and dense circum- stellar material (CSM) ejected by the star prior to explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This means that the SN IIn subtype is very diverse, as these SNe can emerge whenever CSM indications are present in their spectra, whether it is early or late in the lifespan of the SN, or whatever lies beneath the veil of the CSM (Smith 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Narrow hydrogen features may also arise from flash ionization of local CSM fol- lowing shock breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' However, such emission lines disappear shortly after peak magnitude to reveal the underlying SN type (Yaron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Bruch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Jacobson-Galán et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Terreran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Nevertheless, flash ionization in combina- tion with the complexity of the CSM structure can complicate the classification of SNe IIn (Ransome et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Luminous blue variables, extreme red super giants, and yel- low hyper giants have all been proposed as progenitors due to their recurring violent mass-loss episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The intervals of mass loss can range from a period of months to thousands of years, which is necessary to produce the amount of CSM required to make SNe IIn (Smith 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Brighter and more long-lived than other SN types, super- luminous supernovae (SLSNe) are recognized as their own class of SNe (Moriya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Gal-Yam 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' These very lu- minous objects differ from other SNe by their optical absolute magnitudes of around −21 or less, although SLSNe have been classified at around −19 mag at peak for the faintest objects (Moriya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Angus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' SLSNe, as the classic SN classes, are also further categorized into subtypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The super- luminous counterpart to the SNe IIn, SLSNe-IIn, also feature narrow emission lines of the hydrogen Balmer series similar to the regular SNe IIn, and these constitute a significant percentage of all hydrogen-rich SLSNe (Gal-Yam 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This is indicative of CSM interaction partly powering very bright transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' It is not yet clear whether SLSNe-IIn and SNe IIn are two distinctive populations or if they form a continuum in luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' However, in this work, we considered the SLSNe-IIn as the brightest SNe IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Models indicate that SNe IIn along with SNe Ia will dom- inate the observed rates of lensed supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Estimates from the upcoming Legacy Survey of Time and Space (LSST) with the Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Rubin Observatory (Wojtak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Goldstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2019) predict on the order of 100 SNe IIn per year to be gravitationally lensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' For the Roman Space Telescope, the grav- itationally lensed SNe predictions are comparable (Pierel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The lensed SNe IIn predictions are based on the observed lo- cal rate of SNe IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Data from the Lick Observatory Supernova Search (LOSS) yielded an SNe-IIn-to-CC ratio of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='8%+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='3% −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='9% SNe IIn out of all CC SNe (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Article number, page 1 of 8 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='03406v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='HE] 9 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' output Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Overview of redshift and host galaxy mass distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Left panel: Redshift distribution of the SNe IIn in Nyholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2020) and SLSN-IIn from PTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Both are subsamples of the full PTF SNe IIn and SLSNe-IIn samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The redshifts can be found in Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Right panel: Distribution of the host galaxy masses corresponding to the same SNe IIn from Nyholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2020) and a subsample of SLSNe-IIn from the PTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' These subsample distributions are consistent with the full sample distributions of SNe IIn and SLSNe-IIn from Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Overview of redshift and host galaxy magnitude distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Left panel: Redshift distribution of the SNe IIn and SLSN-II from BTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Right panel: Distribution of the host galaxy i-band absolute magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' For two SNe IIn and 1 SLSN-II, the i-band magnitudes were not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The i-band magnitude is used as a proxy for the host stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Several of these SNe IIn were subsequently identified as SN Impostors such that the IIn rate from LOSS is now considered to be around 5% (Graur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Other examples of local rate measurements include studies by Smartt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2009), who present a volume-limited (28 Mpc) sample compiled from all lo- cal SNe with a named host galaxy discovered over a ten-year period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This sample includes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='8% SNe IIn out of all CC SNe in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Eldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2013) updated the study done by Smartt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2009), searching for SNe discovered over a 14-year period, yielding 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The existing rate estimates of SNe IIn assume that the local fraction of IIn to all other CC SNe does not evolve with redshift and so is not affected by large-scale changes in compositions or characteristics of galaxies and stellar populations over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Several studies show a bias toward less massive host galaxies for SLSNe in general (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=', Leloudas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Angus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Taggart & Perley 2021), and this dearth of Article number, page 2 of 8 8 42lln(Nyholmetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='2020) 14 9 SLSN-lln from PTE 7 12 5 8 Numberof 4 Number 6 3 4 2 1 2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='35 5 9 101112 13 z Host Galaxy Stellar Mass [log(M)]92lnfromBTS 90linfromBTS 12 18 SLSN-11from BTS 17 SLSN-Lfrom BTS 20 10 Numberof Galaxies 15 8 6 10 4 5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='00 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='35 14 16 18 20 22 Host Galaxy Magnitude [M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=']C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Cold and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Hjorth: Cosmic rate of type IIn supernovae and its evolution with redshift massive hosts suggests some dependence on the characteristics of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In a study by Graur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2017), based on data from LOSS, SNe IIn seem to be more common in less mas- sive galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' However, other studies do not come to the same conclusion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=', Kelly & Kirshner 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The Palomar Tran- sient Factory (PTF) CC SN sample presented in Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2021) also does not reveal a bias for SNe IIn toward low-mass host galaxies, and is inconclusive regarding the host galaxy mass preference of SLSNe-IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The Zwicky Transient Facility (ZTF) Bright Transient Sur- vey (BTS) (Fremling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2020) will be part of the analysis in this paper in addition to the PTF CC sample (Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In Sec- tion 2, we briefly introduced the data used for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In Section 3, we created an approximately volume-limited sample from the PTF and BTS data, followed by a presentation of an updated SNe-IIn-to-CC ratio (SNe IIn fraction) for both sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In Section 4, we compare the absolute magnitude and the i-band magnitude of the host galaxies of a subsample of SNe IIn, SLSNe-IIn, and SLSN-II from PTF and BTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In Section 5, we presented a generic method for inferring the rate and its evolu- tion with redshift and studying the consequences of a possible mass-biased SNe IIn rate, before the discussion in Section 6 and conclusions in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Data The PTF CC SN sample from Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2021) and the ZTF Bright Transient Survey (Fremling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2020) constitute the basis of the analysis presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The PTF was a deep, wide-field survey followed by the intermediate PTF (iPTF) survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The PTF CC sample contains 888 objects, of which 111 are SNe IIn and 16 SLSNe-IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Redshifts and host galaxy stellar masses are available for all objects in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' For later analysis, we will use the host galaxy i-band (either Pan- STARRS1 (PS1) or Sloan Digital Sky Survey (SDSS) i-band) absolute magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' However, two SNe IIn and one SLSN-IIn have no reported host i-band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The sample contains only host photometry, and so we used a subsample of IIn and SLSNe-IIn where the peak absolute magnitude is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The subsample of 42 SNe IIn with available peak magnitudes is de- scribed in Nyholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' These SNe were chosen based on the amount of available light-curve data to allow an analy- sis of both the rise times and decline rates of the SNe IIn, all with at least one available low-resolution spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The red- shifts in Nyholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2020) differ slightly from the redshifts in Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2021), which are the galaxy redshifts estimated by one of four possible methods: taken from SDSS, taken from the NASA Extragalactic Database, estimated from galaxy lines in the spectra or estimated from SN-template matching (Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In Nyholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2020), the redshifts are estimated from the Hα emission lines in the SNe spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In this work, we used the data published in Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2021) as well as SLSNe-IIn peak magnitudes (Leloudas, priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The red- shift and host galaxy stellar mass distributions of the SNe IIn from Nyholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2020) along with the SLSN-IIn sample from the PTF are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The BTS is currently the largest spectroscopic survey of SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The survey is magnitude-limited in the g and r (<19 mag- nitude) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The sample is 97% spectroscopically complete at <18 mag, 93% at <18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 mag, and 75% at <19 mag (Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The survey is updated daily as new observations come in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' For this paper, we chose to use all available CC SNe and IIn from BTS regardless of magnitude as of May 16, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The to- tal number of CC SNe adds up to 949, of which 92 are IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We also included the SLSN-II in our analysis, of which there are 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In BTS, the SLSNe-II are not further divided into subclasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' However, as most SLSNe-II exhibit IIn-like features, we chose to include all of the SLSN-II in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The parameters we used in our analysis in this paper are redshift, peak magni- tude and host i-band absolute magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Redshifts are available for all but six CC SNe, none of which are SNe IIn or SLSN- II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Since the host galaxy stellar masses are not available in this sample, we instead utilized the absolute i-band magnitude of the hosts where available as these magnitudes are a good proxy for the stellar masses, as is seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Distributions of redshift and host galaxy i-band magnitude are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' PTF CC SNe host galaxy absolute i-band magnitudes are plotted against host galaxy stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Due to the standard deviation of the residuals being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='4, which is comparable to the uncertainty on the stellar mass, the i-band magnitude can be used as a proxy for the stellar mass in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Inferred IIn fractions In Frohmaier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2021), the CC SN rate for the PTF is de- termined while taking all the survey limitations into account through extensive modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This method yields a total of 86 CC SNe and three SNe IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Unfortunately, inferring relative rates of SNe IIn with only three sources will be dominated by low- number statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' For the purpose of estimating the SNe IIn fraction, we will alternatively create an approximately volume- limited sample for the recent PTF data released in Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2021) and also from BTS (Fremling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Perley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2020) under the assumption of fair spectroscopic classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' A simple way to estimate the distance, or redshift, at which the PTF CC sample is approximately complete is to compare with a known complete sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The LOSS sample is com- plete out to 60 Mpc for CC SNe (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Graur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' With the limiting magnitude of LOSS having a median of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 mag (Leaman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2011), 60 Mpc is also the dis- tance to the furthest SNe LOSS could theoretically observe as- suming the faintest SNe to have an absolute magnitude of around −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The PTF has a limiting magnitude of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 mag in the R band, which implies a distance of up to 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='8 Mpc for creating a volume-limited sample, assuming the same −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='1 mag for faint SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This corresponds to a redshift cut-off of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='031 when adopt- ing a Hubble constant of H0 = 73 kms−1Mpc−1 as in LOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This Article number, page 3 of 8 24 [M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='] 22 Magnitude 20 /Absolute 18 16 Host Galaxy 14 12 PTF CC SNe Host Galaxies 10 4 5 6 1 8 9 10 11 12 HostGalaxyStellarMass[log(M*/M)]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' output Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Redshift evolution of SNe-IIn-to-CC ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The gray data points represent the cumulative SNe-IIn-to-CC ratio from PTF, indicated on the left y-axis, as a function of redshift limits out to maximum redshift of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' SLSNe-IIn are not included in the IIn-to-CC ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The vertical gray dashed line represents the chosen redshift cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The number of CC SNe, SNe IIn, and SLSNe-IIn as a function of redshift are represented by the colored curves indicated on the right y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The inset shows a zoomed-in image of a plot of the cumulative SNe-IIn-to-CC ratio up to redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='06 as well as the chosen redshift cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Redshift evolution of SNe-IIn-to-CC ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The gray data points represent the cumulative SNe-IIn-to-CC ratio from BTS, indicated on the left y-axis, as a function of redshift limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The vertical gray dashed line represents the chosen redshift cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The number of CC SNe and SNe IIn are also depicted as the colored curves, which are indicated on the right y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The inset shows a zoomed-in image of a plot of the cumulative SNe-IIn-to-CC ratio up to redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='06 as well as the chosen redshift cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' estimate can be tested visually, as is done in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 4, where we plot the SNe-IIn-to-CC ratio as a function of redshift limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The estimated cut of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='031 is indicated in the plot by a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The SNe IIn fraction for lower redshift cut-offs is dominated by noise due to the small number of supernovae found at these red- shifts, whereas the ratio increases above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='031 as more SNe IIn are observed at this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This indicates that the estimated red- shift cut is located where the noise from few observations has started to diminish, but we do not yet see the effect of a larger volume wherein to observe SNe IIn and CC SNe in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' As such, imposing a redshift cut-off of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='031 on the PTF CC sample Article number, page 4 of 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='14 Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='031 PTF 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='10 Supemova 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='075 IIn/CC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='08 IIn/CC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='050 4006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='06 Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='031 Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='025 PTF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='06 200 z limit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='02 SLSN IIn IIn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='00 CC 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='30 z limitZ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='033 BTS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='10 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='08 Number of Supemovae 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='100 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='075 IIn/CC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='050 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='025 Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='033 PTF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='06 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='02 z limit IIn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='00 CC 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='30 z lirmitC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Cold and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Hjorth: Cosmic rate of type IIn supernovae and its evolution with redshift is a reasonable approach for creating an approximately volume- limited sample to use for estimating the SNe IIn fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Distributions of host galaxy stellar mass of the complete PTF CC sample using z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' No SLSNe-IIn in the PTF sample are found within this redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Results from a KS test show no significant SNe IIn host galaxy mass preference for the volume-limited sample with a p value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We employ the same method for the BTS sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Using −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='1 mag for the faintest CC SNe is consistent with the mean of the 20 faintest CC SNe in the BTS sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' According to Bellm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2019), the limiting magnitudes for ZTF are 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='8 mag in the g band, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='6 mag in the r band, and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='9 in the i band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We will use the r-band value to be consistent with LOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' From the distance modulus, this yields a distance of 138 Mpc, correspond- ing to a redshift cut-off of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='033, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The volume effect is less obvious in the BTS data, and as such the resulting IIn fraction from BTS is more robust toward any uncertainties in the redshift cut compared to the result from PTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' However, as the redshift cut of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='033 occurs before a slow rise in the SNe-IIn-to-CC value and after the inital large uncertainties, we deem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='033 to be a good estimate for creating an approxi- mately volume-limited sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' For the PTF, a redshift cut of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='031 leaves us with an ap- proximately complete CC sample of 263 CC SNe in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This includes 12 SNe IIn, but none of the SLSNe-IIn are observed within this redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Therefore, we find that the ratio of SNe IIn to CC SNe for our subsample of PTF data to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='046 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The uncertainty on the resulting SNe-IIn ratio is propagated from the Poisson error on the individual number of CC and SNe IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The resulting histogram of the host galaxy stellar masses of this vol- ume limited sample of PTF data, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 6, reveals no obvious SNe IIn preference for less massive host galaxies, which is in agreement with the analysis done by Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2021) on the full PTF sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We can compare this value to the BTS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' A redshift cut- off of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='033 yields a sample of 440 CC SNe, of which 21 are SNe IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This results in an SNe-IIn-to-CC ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='048±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The uncertainty on this number is similarly determined using er- ror propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' As host galaxy masses are not available in BTS, and a significant amount of CC hosts do not have i-band magni- tudes either, we will not compare the host galaxy distributions of the SNe IIn and CC SNe from BTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' As these resulting fractions are independent, we combine them and get a SNe IIn relative fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='047 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Brightness of SNe IIn In this section, we investigate whether the peak brightness of the IIn is influenced by the host galaxy stellar mass when consider- ing the SLSN-IIn as the brighest members of the IIn class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We know from several studies that SLSNe prefer lower mass host galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' According to Schulze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2021), this phenomenon is not significant when only studying the SLSNe-IIn, as the objects are still too few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We note that for redshifts on the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='03 the influence of peculiar velocities on calculating peak absolute magnitudes of the SNe is decreasing compared to SN samples, which are mostly comprised of local sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The distributions of redshift and host galaxy mass of the sub- sample of IIn and SLSN-IIn from PTF are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2, we show the distributions of redshift and host galaxy i-band magnitude from BTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' For the comparison between these two data sets, we use the i-band magnitude of the host galaxies as a proxy for the stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Only three sources from either data set do not have available host i-band magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 7, we compare the absolute magnitude at peak and the i-band magnitude of the host galaxies of the SNe IIn and SLSNe- IIn or SLSN-II from the PTF as well as BTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' As these objects are chosen based on data availability, the subsamples seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 7 are not complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' However, Nyholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2020) state that the host galaxy mass distribution of their subsample is in agreement with the distribution of the full PTF SNe IIn sample, such that the distribution of the i-band magnitudes should follow a similar distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 7 show no clear trend regarding the effect of the host galaxy magnitude on the peak absolute mags of the SNe IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' To further investigate, we divide the combined PTF and BTS subsamples shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 7 in two, namely a faint sample and a bright sample, and subsequently calculate the me- dian and uncertainty on the median as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='48·MAD/ √ N − 1 of the i-band magnitudes for the host galaxies, where MAD is the me- dian absolute deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This division of the subsample is done for several different SN IIn peak magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We employ Mpeak values from −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 to −21 as the dividing lines between the faint and the bright sub-samples and compare the medians, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We find that the median i-band magnitude (and thus the stellar mass) of the host galaxies becomes fainter with a 3σ significance when choosing a sample of SNe IIn brighter than −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Absolute peak magnitude versus host galaxy i-band magnitude for SNe IIn and SLSNe-IIn/SLSNe-II from PTF and BTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Article number, page 5 of 8 102 263 CC 12 IIn Supernovae Numberof 101 100 5 6 7 8 9 10 011 12 13 HostGalaxyStellarMass[log(M)]-23 22 21 20 Peak 19 Absolute 18 17 BTS IIn BTS SLSN-II 16 PTF IIn PTF SLSN-IIn 15 10 12 14 16 18 20 22 Host Galaxy Absolute Mag [Mi]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' output Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Median host Mi as a function of different cuts on the supernova peak magnitude for combined BTS and PTF samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The median i- band magnitude of the host galaxies becomes fainter for the brighter SNe in the combined sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Consequences of a host-mass-dependent IIn fraction While the evidence for a host-mass-dependent IIn fraction is not strong, we next explore the consequences of a hypothetical pref- erence for low-mass host galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We parametrize the IIn frac- tion as a power-law function of the stellar mass of the host galaxy and investigate the impact on the volumetric rate as a function of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This will be affected since more low-mass galaxies are present in the earlier Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In general, the volumetric SNe IIn rate can be expressed as IInrate = IIn CC · kCC · S FR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (1) The CC constant, kCC, is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0091M−1 ⊙ following Strol- ger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This model is consistent with observational constraints from Dahlen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2012) and Madau & Dickinson (2014) as demonstrated in Strolger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Here, a model for the star-formation rate (SFR) is taken from the UniverseMa- chine algorithm by Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The results of this code are best-fitting models of stellar-mass functions (SMFs), cosmic star formation rates (CSFRs), specific star formation rates (sSFRs), and UV luminosity functions (UVLFs) to obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' One can determine the SFR from the output of the Uni- verseMachine algorithm: S FR = � Mmax Mmin S MF · M · sS FR dM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2) When computing the SFR this way, it is possible to split it into different mass bins in order to infer the contribution to the total SFR from galaxies of different masses and how this changes with redshift, as is shown in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The UniverseMa- chine resulting models have mass ranges of 107M⊙ to 1013M⊙, and we choose to split these into five different bins, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The bin containing the 1011M⊙ to 1013M⊙ galaxies is chosen to be wider than the other bins, as the contribution from the 1012M⊙ to 1013M⊙ galaxies is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We parametrize the IIn-to-CC ratio as a power law: IIn CC (log(M/M⊙)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='15 · log(M/M⊙)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (3) This power-law model is chosen to have a higher SNe-IIn-to- CC ratio than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='047 for host galaxies below 1010M⊙ and a lower ratio for more massive galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' For this specific example, the ratio will be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='057 for galaxies with stellar masses of 107M⊙, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='043 for 1012M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The motivation for this kind of model comes from the LOSS data in Graur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (2017), showing a preference for low-mass hosts for SNe IIn, which can be modeled with a power law with different sets of parameters (Hede 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' To calculate the volumetric rate, we use the central value of the IIn/CC model for every mass bin in log as the IIn/CC factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (1), and thus compute a separate SN IIn rate for each mass bin as well as the combined rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Since the power-law model and the constant model do not predict the same number of SNe given a different area under the curve, we normalize the resulting rate from the power law model to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='77 · 10−6 yr−1Mpc−3, which is the SNe IIn rate at redshift zero for a constant SNe IIn fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='047 calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (1), where the UniverseMachine is the source of the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We do this to be consistent with the constant model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The resulting SNe IIn rate is plotted in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 9 for both the example power-law model and the constant model of IIn/CC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The overall IIn rate is slightly lower for a redshift below four, and slightly higher for one above four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This plot also shows how the contribution from the different host mass bins differs for the constant ratio to the power-law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' It is evident that the overall rate from low-mass galaxies is higher, and since these contribute a larger fraction of the total rate at higher redshift, we also see an increase in the rate in this high- redshift domain as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Discussion In this section, we discuss and reflect on some of the shortcom- ings and consequences of the methods and results of this paper as well as some widely used assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' First, we note that the SNe-IIn fractions from the PTF and BTS are based on approxi- mate volume-limited samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The identification of a sweet spot in the IIn fraction in the PTF sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 4) supports this ap- proach, although we note that, ultimately, IIn fractions will have to be based on carefully defined volume-limited samples of a large number of core-collapse supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 8, we see a statistically significant connection be- tween the brightness of the SNe IIn and the host galaxy stellar mass when choosing a sample of SNe IIn brighter than −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' For the faint SNe IIn sample at the −18 mag cut-off, the median i-band magnitude of the host galaxies drops below the bright IIn sample median host i-band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' However, the subsamples employed in this part of our analysis could be sub- ject to different selection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Observing faint SNe in bright galaxies is challenging as the light from the galaxy can hide the SNe and so we could expect that some were missed in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This effect will influence the median i-band host magnitude for the faintest SNe IIn and could explain the slight shift in median host i-band magnitude for the faintest SNe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' However, the drop in median i-band host-galaxy magnitude is not as sig- nificant as the drop for the brightest supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Using the UniverseMachine algorithm DR1 as the input for the SFR gives insight into the contributions from host galaxies of different stellar masses to the resulting SNe IIn rate whether a preference for low-mass hosts exists or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' One limitation, how- ever, is the lower mass boundary on these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' As is evident from both Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 6 and 1, several SNe IIn have host galaxies less massive than 107M⊙, but using UniverseMachine it is not pos- sible to see the contribution from these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Another limi- tation is the mass resolution of UniverseMachine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In this case, Article number, page 6 of 8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0 Bright SNe 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 Faint SNe 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0 M 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 Median Host I 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 Mpeak CutC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Cold and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Hjorth: Cosmic rate of type IIn supernovae and its evolution with redshift Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Overview of the cosmic SFR, volumetric SNe IIn rate and rel- ative SNe IIn rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Top panel: SFR calculated from UniverseMachine DR1 (Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Middle panel: Resulting SNe-IIn rate de- termined using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The solid lines denote rates from the power-law IIn-to-CC ratio, and the dashed lines represent the rate for a constant ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The contribution from each mass bin is plotted for com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Bottom panel: Total relative volumetric SNe IIn rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The gray line represents a line of agreement between the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' we have four or five data points per mass bin, which prevents us from further dividing the host galaxies into smaller bins to obtain a more detailed overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 9, we show the resulting SNe-IIn volumetric rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' For illustration, the volumetric rate is computed for a uniform fixed SNe-IIn fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='047 next to a model in which the SNe IIn prefer lower mass host galaxies, here represented by a power law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The two models agree at z = 0 and the normalization of the curves is uncertain by 20 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The middle and bottom panels show that the relative volumetric rate of SNe IIn increases over the default constant model for redshifts beyond 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Introducing the example power-law model to describe the SNe-IIn-to-CC ratio produces a minimal effect on the volumet- ric rate compared to the constant ratio as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 9: A lower rate for redshifts below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='2 and higher for redshifts beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The LSST or Roman Space Telescope are not expected to be able to observe lensed SNe beyond redshifts of three and four, respec- tively (Wojtak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Goldstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Pierel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2021), and so the possibility of testing such a model is currently limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' On the other hand, we show that a limited mass depen- dence of the IIn rate should not affect predicted volumetric rates of SNe IIn significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Conclusions We studied the PTF and BTS SNe IIn and SLSNe-IIn/SLSNe- II populations throughout this work and now present our main conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Creating a complete sample of CC SNe from PTF and BTS, we find the SNe IIn to CC ratios of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='046 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='013 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='048 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='011 for the PTF and BTS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The combined resulting SNe-IIn fraction is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='047 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We see a marginally significant (3 σ) bias towards low-mass host galaxies for SNe IIn brighter than −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We present a general method to evalu- ate the consequences of a SNe IIn to CC ratio that is nonconstant and is instead described by a power-law model on the resulting volumetric rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' The example model chosen here can be freely replaced by another model as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We find that the example power law model of IIn/CC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='15 · M−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='05 results in a slightly lower volumetric rate below a redshift of four and a higher rate beyond a redshift of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='2 when comparing to a constant ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Neither the LSST nor the Roman Space Telescope are pre- dicted to find lensed SNe beyond redshifts of three or four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We emphasize that our method is generic and can be applied to other CC subtypes if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We gratefully acknowledge Giorgios Leloudas for sharing peak absolute magnitudes for a subsample of PTF SLSNe-IIn, without which a large part of the analysis in this paper would not have been possible, as well as invaluable comments on the paper draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We also gratefully acknowledge Steve Schulze and Radek Wojtak for helpful conversations about type IIn and statis- tical conundrums, respectively, and Wynn Jacobson-Galán and Doogesh Kodi Ramanah for reading and commenting on the paper before submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' We also thank the referee for their useful and thorough comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' This work was supported by a VILLUM FONDEN Investigator grant to JH (project number 16599).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' References Angus, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=', Levan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' J.' 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Leaman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=', Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=', Chornock, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=', & Filippenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' 2011, MNRAS, 412, 1419 Leloudas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=', Schulze, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='047 7 ≤log(M)<8 5 6>(W)60|≤8 0T>(W)601≤6 4 10 ≤log(M)<11 11 ≤log(M)<13 m Total 1 Relative Rate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content='0 0 1 2 E 4 5 6 ZA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' output Ransome, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf'} +page_content=', Habergham-Mawson, S.' metadata={'source': 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b/YNE5T4oBgHgl3EQfdQ-k/content/tmp_files/2301.05610v1.pdf.txt @@ -0,0 +1,1327 @@ +Accelerating greedy algorithm for model reduction of complex +systems by multi-fidelity error estimation +Lihong Feng ∗, Luigi Lombardi†, Giulio Antonini‡, and Peter Benner§ +January 16, 2023 +Abstract +Model order reduction usually consists of two stages: the offline stage and the online stage. The +offline stage is the expensive part that sometimes takes hours till the final reduced-order model is +derived, especially when the original model is very large or complex. Once the reduced-order model +is obtained, the online stage of querying the reduced-order model for simulation is very fast and +often real-time capable. This work concerns a strategy to significantly speed up the offline stage of +model order reduction for large and complex systems. In particular, it is successful in accelerating +the greedy algorithm that is often used in the offline stage for reduced-order model construction. +We propose multi-fidelity error estimators and replace the high-fidelity error estimator in the greedy +algorithm. Consequently, the computational complexity at each iteration of the greedy algorithm is +reduced and the algorithm converges more than 3 times faster without incurring noticeable accuracy +loss. +1 +Introduction +Model order reduction (MOR) has achieved much success in many areas of computational science with +its capability of realizing real-time simulation and providing accurate results. Different MOR methods, +their applications and the promising results they produce can be found in the survey papers [2, 4, 12] +and books [27, 3, 8, 9, 10, 11]. +MOR needs an offline stage for constructing the ROM. For many intrusive MOR methods that are +based on projection, the offline stage is usually realized via a greedy algorithm. The greedy algorithm +is used to properly select important parameter samples that contribute most to the solution space. The +offline computational time is basically the runtime of the greedy algorithm. For large-scale systems, +the offline computation is expensive and the runtime is often longer than several hours even when +run on a high-performance server. Sometimes, the system is not very large, for example, the number +of degrees of freedom is only O(105), but the system structure is complicated, so that the greedy +algorithm still takes long time to converge. +It is known that an efficient error estimator makes the greedy algorithm successful in producing +an accurate ROM without running many iterations. +Therefore, many efforts have been made in +this direction to develop computable error estimators for different problems [15, 16, 18, 19, 20, 21, +22, 23, 24, 25, 28, 34, 33, 35]. +However, more attention has been paid to improve the effectivity +∗Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, 39106 Magdeburg, Ger- +many feng@mpi-magdeburg.mpg.de +†Luigi Lombardi is with Micron Semiconductor, 67051 Avezzano, Italy. luigilombardi89@gmail.com +‡Giulio Antonini is with the UAq EMC Laboratory, Department of Industrial and Information Engineering and +Economics, University of L’Aquila, I-67100 L’Aquila, Italy. giulio.antonini@univaq.it +§Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany and Fakult¨at f¨ur Mathe- +matik, Otto-von-Guericke-Universit¨at Magdeburg, Germany. benner@mpi-magdeburg.mpg.de +1 +arXiv:2301.05610v1 [math.NA] 13 Jan 2023 + +or accuracy of the error estimator than to develop more efficient strategies to accelerate the greedy +process [36, 13, 34, 25, 33]. Recently, some techniques are proposed to improve the adaptivity of the +greedy algorithm [7, 13, 6, 26]. +In [13, 14], we proposed a surrogate model for error estimation, and proposed an adaptive greedy +algorithm by alternatively using this surrogate error estimator and the original error estimator during +the greedy algorithm. The focus in [13, 14] was to make the greedy process adaptive by starting from a +coarse training set of a small number of parameter samples, and adaptively update the coarse training +set with the aid of a surrogate error estimator. The original error estimator is computed only over +the coarse training set, while the surrogate error estimator helps to pick out candidates of important +parameter samples from a fine training set, which are then collected in the coarse training set. +In this work, we emphasize the role of the surrogate error estimator and propose the concept of +bi-fidelity error estimation and multi-fidelity error estimation. In fact, a bi-fidelity error estimation has +been used in the adaptive greedy algorithm proposed in [13, 14] without being formally defined, i.e., +the original (high-fidelity) error estimator, and the surrogate (low-fidelity) error estimator. To further +improve the convergence speed of the greedy algorithm, we propose multi-fidelity error estimation +built upon the bi-fidelity error estimation. Here, we use a more efficient high-fidelity error estimator +than the two different high-fidelity error estimators used in [13, 14]. Although the proposed multi- +fidelity error estimation is dependent on the original high-fidelity error estimator, the idea of using +multi-fidelity error estimation is general and can be extended to develop multi-fidelity error estimation +associated with other high-fidelity error estimators. +Unlike the problems considered in [13, 14], whose ROMs can be constructed by standard greedy +algorithms within seconds to minutes, we consider in this work much more complicated problems. +On the same computer, the standard greedy algorithm takes more than half a day to converge for +such problems. By using the proposed multi-fidelity error estimator, the greedy algorithm achieves +4x speed-up and produces ROMs with little loss of accuracy. The speed-up is also higher than those +reported in [13, 14] by using the bi-fidelity error estimation, which is usually 2x. +In the next section, we present the greedy algorithm in the standard form. +Then we analyze +some ingredients of the algorithm, which contribute most to the computational cost. Starting from +those computationally expensive parts, we develop possible strategies to reduce the computational +complexity in Section 3. As a consequence, it becomes clear that the resulting strategy develops a +greedy algorithm with multi-fidelity error estimation. The proposed algorithm is then applied to large +time-delay systems with many delays. The numerical tests on three large time-delay systems with +more than 100 delays are demonstrated in Section 4. Conclusions are given in the end. +2 +Standard greedy algorithm +The standard greedy algorithm was first proposed for steady systems without time evolution. Then +it was extended to POD-greedy for dynamical systems, which is used to construct the ROM using +snapshots in the time domain. Later the greedy algorithm found its capability in adaptively choosing +interpolation points for frequency-domain MOR methods [15, 16]. The greedy algorithm for steady +systems and frequency-domain MOR has the same formulation, whereas POD-greedy for time-domain +MOR of time-dependent systems needs an SVD step at each greedy iteration. In this work, we focus +on the greedy algorithm, though the proposed scheme can be easily extended to POD-greedy. We +consider constructing a ROM for the following full-order model (FOM) using the greedy algorithm, +F(x(µ), µ) = B(µ). +(1) +Here, F(x(µ), µ) ∈ Cn×nI, x(µ) ∈ Cn×nI, and B(µ) ∈ Cn×nI. µ ∈ P is a parameter in the parameter +domain P. The variable n is the order of the FOM, which can be the number of degrees of freedom +2 + +Algorithm 1 Standard greedy algorithm +Input: the FOM, a training set Ξ composed of parameter samples taken from the parameter domain +P, error tolerance tol< 1, ∆(µ) to estimate the error. +Output: Projection matrix V . +1: Choose initial parameter µ∗ ∈ Ξ. +2: V ← ∅, ε = 1. +3: while ε >tol do +4: +Compute the snapshot(s) x(µ∗) by solving the FOM at µ = µ∗. +5: +Update V by V = orth{V, x(µ∗)}, (e.g., using the modified Gram-Schmidt process with defla- +tion.) +6: +Compute µ∗ such that µ∗ = arg max +µ∈Ξ ∆(µ). +7: +ε = ∆(µ∗). +8: end while +after numerical discretization of PDEs describing a physical phenomenon. The proposed algorithms +are also applicable to problems with nI > 1. +The ROM can be obtained via Galerkin projection using a projection matrix V ∈ Rn×r, r ≪ n, +as below, +ˆF(V z(µ), µ) = ˆb(µ), +(2) +where ˆF(V z(µ), µ) = V T f(V z(µ), µ) ∈ Cr×nI, z(µ) ∈ Cr×nI, and ˆB(µ) = V T b(µ) ∈ Cr×nI. +The standard greedy process used to compute the projection matrix V is described in Algorithm 1. +Step 4 in Algorithm 1 solves the FOM at µ∗, and Step 6 computes an error estimator ∆(µ) at all µ in Ξ. +These two steps constitute the most computational expensive part of the greedy algorithm. However, +Step 4 is unavoidable, since x(µ∗) is needed for the reduced basis construction.The computational +complexity of Step 6 could be reduced, if the cardinality of Ξ, i.e., |Ξ| is kept small, so that ∆(µ) +needs not be evaluated at many parameter samples. This is the motivation of the surrogate error +estimator proposed in [13, 14]. +We call the surrogate error estimator ∆l(µ) the low-fidelity error +estimator as compared to the original error estimator ∆(µ), since ∆l(µ) is only an approximation to +∆(µ), but is much cheaper to compute. +In the next section, we present a greedy algorithm using bi-fidelity error estimation, where the low- +fidelity error estimator is computed following the method in [13, 14]. Based on this, a greedy algorithm +using multi-fidelity error estimation associated with a particular high-fidelity error estimator for MOR +of linear parametric systems, is proposed. +3 +Greedy algorithm with bi-fidelity and multi-fidelity error estima- +tion +This section presents greedy algorithms with bi-fidelity and multi-fidelity error estimation, respectively. +3.1 +Greedy algorithm with bi-fidelity error estimation +Algorithm 2 is the greedy algorithm with bi-fidelity error estimation. +Its original version using a +different high-fidelity error estimator was firstly proposed in [13]. The key step of Algorithm 2 is Step +8, where the low-fidelity error estimator ∆l(µ) is computed using values of ∆(µ) at the samples of +µ in the small parameter set Ξc. Basically, ∆l(µ) is represented by a weighted sum of radial basis +3 + +Algorithm 2 Greedy algorithm with bi-fidelity error estimation +Input: the FOM, a training set Ξc composed of a small number of parameter samples taken from the +parameter domain P, a set Ξf composed a large number of parameter samples of µ from P, error +tolerance tol< 1, ∆(µ) to estimate the error. +Output: Projection matrix V . +1: Choose initial parameter µ∗ ∈ Ξc. +2: V ← ∅, ε = 1. +3: while ε >tol do +4: +Compute the snapshot(s) x(µ∗) by solving the FOM at µ = µ∗. +5: +Update V by V = orth{V, x(µ∗)}, (e.g., using the modified Gram-Schmidt process with defla- +tion.) +6: +Compute µ∗ such that µ∗ = arg max +µ∈Ξc ∆(µ). +7: +Compute µo such that µo = arg min +µ∈Ξc ∆(µ). +8: +Compute the low-fidelity error estimator ∆l(µ) using values of ∆(µ) corresponding to the sam- +ples of µ in Ξc via (3) and (4). +9: +Evaluate ∆l(µ) over Ξf and pick out a parameter µc from the large parameter set Ξf corre- +sponding to the largest value of ∆l(µ), i.e., µc = arg max +µ∈Ξf from Ξf. +10: +Update the small parameter set Ξc: if ∆l(µc) >tol, enrich Ξc with µc, i.e., Ξc = {Ξc, µc}, if +∆(µo) |Ξc|. +Finally, at each iteration of the bi-fidelity algorithm, the total computational cost of +Steps 6-8: computing the high-fidelity error estimator ∆(µ) over Ξc, solving the linear system (4) and +evaluating the low-fidelity erorr estimator ∆l(µ) over Ξf is still much cheaper than computing the +4 + +high-fidelity error estimator ∆(µ) over a training set Ξ, whose cardinality is, e.g., twice that of |Ξc|, +as shown in the numerical tests. +Besides computing µ∗ corresponding to the maximal value of the error estimator ∆(µ) over Ξc, +the minimal value of ∆(µ) is also computed in Step 7. The corresponding parameter µo could be +deleted from Ξc if ∆(µo) is already below the tolerance tol, see Step 10. In this way, the cardinality +of the training set Ξc remains almost constant, and can further save computations as compared with +enriching Ξc only. We will show in the numerical results that adding and removing samples to and +from Ξc gets ROMs with similar accuracy (even smaller) as only adding samples to Ξc, but leads to +even faster convergence of the greedy algorithm. +Remark 3.1 In Step 9, it is also possible to choose more than one parameter from Ξf by modifying +Step 9 as: choose nadd samples from Ξf corresponding to nadd largest values of ∆l(µ). Similarly, +In Step 7, one can also choose ndel > 1 parameter samples corresponding to ndel smallest values of +∆(µ) from Ξc. However, this will more or less increase the computational time at each iteration, +since more computations are needed to choose those samples. Furthermore, to make sure that only +samples at which ∆l(µ) is larger than the tolerance tol are added to Ξc, and only samples at which +∆(µ) is smaller than tol are removed, additional calculations are necessary to check if all the selected +samples meet the above criteria and should be finally selected or removed (see Step 10). Therefore, +adding/removing at most one parameter sample each time should be more efficient. In the numerical +tests, we also show results when nadd = ndel = 2 and nadd = ndel = 5 at each iteration of Algorithm 2. +The bi-fidelity error estimation is general and can be applied to any high-fidelity error estimators. +For example, the high-fidelity error estimator in [13] estimates the error of the ROM for nonlinear +time-dependent parametric systems in the time domain, while the high-fidelity error estimator in [14] +estimates the error of the ROM in the frequency-domain for linear parametric systems. +3.2 +Greedy algorithm with multi-fidelity error estimation +The multi-fidelity error estimation we are going to introduce depends on the formulation of the high- +fidelity error estimator ∆(µ). To illustrate the basic concept, we use an error estimator proposed +in [16] as the high-fidelity error estimator and discuss how to further reduce the computational load +by using multi-fidelity error estimation. +3.2.1 +An error estimator for linear parametric systems +The error estimator is applicable to estimating the output error of the ROM for FOMs in the following +linear parametric form, +M(µ)x(µ) += +B(µ), +y(µ) += +C(µ)x(µ). +(5) +Here, M(µ) ∈ Rn×n, B(µ) ∈ Rn×nI, C(µ) ∈ RnO×n , x(µ) ∈ Rn, y(µ) ∈ RnO×nI. We consider the +general case that both B(µ) and C(µ) are matrices, i.e. systems with multiple inputs and multiple +outputs. The ROM of the above linear parametric system can be derived via Galerkin projection +using a projection matrix V composed of the reduced basis. That is, +ˆ +M(µ)z(µ) += +ˆB(µ), +ˆy(µ) += +ˆC(µ)z(µ), +(6) +where ˆ +M(µ) = V T M(µ)V , ˆB(µ) = V T B(µ), ˆC(µ) = C(µ)V . +5 + +For the general situation when both B(µ) and C(µ) are matrices, the error of the i, j-th entry of +the output matrix ˆy(µ) is +|yij(µ) − ˆyij(µ)| += |Ci(µ)(M−1(µ)B(µ) − V ˆ +M−1(µ) ˆBj(µ))| += |Ci(µ)M−1(µ)(Bj(µ) − M(µ) V ˆ +M−1(µ) ˆBj(µ)) +� +�� +� +ˆxj(µ):=V zj(µ)| += |Ci(µ)M−1(µ)rj(µ)|, +(7) +where Ci(µ) is the i-th row of C(µ) and Bj(µ) is the j-th column of B(µ). Here, we have defined: +zj(µ) = ˆ +M−1(µ) ˆBj(µ), i.e., ˆ +M(µ)zj(µ) = ˆBj(µ), ˆxj(µ) := V zj(µ) and rj(µ) := Bj(µ) − M(µ)ˆxj(µ). +It is clear that +ˆ +M(µ)zj(µ) = ˆBj(µ) +is a reduced-order model of +M(µ)xj(µ) = Bj(µ), +(8) +and ˆxj(µ) ≈ xj(µ), the j-th column of x(µ). Finally, rj(µ) is the residual induced by ˆxj(µ). +From the last equation in (7), it is clear that to compute the absolute error of ˆyij, we need to solve +a residual system: +M(µ)xrj(µ) = rj(µ). +(9) +Instead, we construct a ROM of it: +V T +r M(µ)Vrzrj(µ) = V T +r rj(µ), +(10) +so that xrj(µ) ≈ ˆxrj(µ) = Vrzrj(µ) . Finally, +|yij(µ) − ˆyij(µ)| ≈ |Ci(µ)ˆxrj(µ)|. +Note that ˆxrj(µ) depends on Bj(µ), since rj(µ) depends on Bj(µ). Each column Bj(µ) is associated +with a ˆxrj(µ). The overall error of ˆy(µ) as a matrix can be estimated as: +∥y(µ) − ˆy(µ)∥max := max +i,j |yij(µ) − ˆyij(µ)| ≈ max +i,j |Ci(µ)ˆxrj(µ)| =: ˜∆(µ). +(11) +˜∆(µ) defined in (11) is one of the error estimators proposed in [16], where the proposed error +estimators were shown to outperform other existing error estimators in the literature [34, 15] in terms +of both accuracy and computational efficiency. Furthermore, it has been discussed in [16] that ˜∆(µ) is +even more accurate but has less computational complexity than other proposed estimators, including +the one used in [14]. Even with this error estimator, the greedy algorithm could take several hours +to converge for some complex systems, for example, the time-delay systems we consider in this work. +For such systems, although the standard greedy algorithm can already be accelerated by the bi- +fidelity greedy algorithm, we suggest a possibility to further improve the bi-fidelity greedy algorithm +by introducing multi-fidelity error estimation. +We notice that in order to compute ˜∆(µ), an extra projection matrix Vr has to be constructed +for ˆxrj(µ). Although ˆxrj(µ) is dependent on the individual column of B(µ), the matrix Vr can be +uniformly constructed based on the whole matrix B(µ). Then Vr is valid for any column of B(µ). It +is proved in [16] that taking Vr = V leads to ˜∆(µ) identically zero for all µ. Therefore, Vr should be +additionally computed. +6 + +3.2.2 +Standard greedy algorithm using ˜∆(µ) +For easy understanding of the multi-fidelity error estimation, we first present Algorithm 3, the standard +greedy algorithm using ˜∆(µ) in (11) as the error estimator. There, some additional steps are added +to compute Vr, see Step 5, Steps 7-8. In Step 7 of Algorithm 3, Vr is not only updated by x(µr), but +also by V . This is due to the fact that the solution xrj(µ) to the residual system in (9) can be written +as +xrj(µ) += +M(µ)−1rj(µ) += +M(µ)−1(Bj(µ) − M(µ)ˆxj(µ)) += +M(µ)−1Bj(µ) − V zj(µ) +≈ +˜Vrzrj − V zj(µ). +(12) +It is clear that xrj(µ) is a linear combination of (M(µ))−1Bj(µ) and the columns of V . Therefore, +V contributes to the subspace approximating the solution space of xrj(µ) and cannot be neglected. +It is also noticed that (M(µ))−1Bj(µ) is in fact the solution xj(µ) in (8), while V zj(µ) is ˆxj(µ) +that approximates xj(µ). This means xrj(µ) is the difference between xj(µ) and ˆxj(µ), which is a +nonzero vector. +Therefore, we should compute another matrix ˜Vr, so that xj(µ) ≈ ˜Vrzrj(µ), but +˜Vrzrj(µ) ̸= ˆxj(µ) = V zj(µ). Finally, xrj is approximated by the difference between ˜Vrzrj(µ) and +V zj(µ). In other words, it is approximately represented as the linear combination of the columns of +both Vr and V . This approximation also explains Step 5 and Step 7 of Algorithm 3: Step 5 computes +the reduced basis vectors contributing to ˜Vr, Step 7 computes the complete reduced basis vectors +contributing to Vr. New reduced basis vectors for both V and Vr are computed at each iteration of +the greedy algorithm. Step 8 and Step 9 compute the new important parameter samples for Vr and V , +respectively. In general, µr should be different from µ∗, since ˜∆(µ) ̸= +max +j=1,...,nI ∥rj(µ) − M(µ)ˆxrj(µ)∥. +Here, rj(µ) − M(µ)ˆxrj(µ) is nothing but the residual induced by the approximate solution (ˆxrj(µ)) to +the residual system (9). +Algorithm 3 Standard greedy algorithm using ˜∆(µ) for linear parametric systems. +Input: the FOM, a training set Ξ composed of parameter samples taken from the parameter domain +µ ∈ P, error tolerance tol< 1. +Output: Projection matrix V . +1: Choose initial parameter µ∗ ∈ Ξ for V , and initial parameter µr ̸= µ∗ ∈ Ξ for Vr. +2: V ← ∅, Vr ← ∅, ε = 1. +3: while ε >tol do +4: +Compute the snapshot(s) x(µ∗) by solving the FOM, i.e. x(µ∗) = (M(µ∗))−1B(µ∗). +5: +Compute the snapshot(s) x(µr) by solving the FOM, i.e. x(µr) = (M(µr))−1B(µr). +6: +Update V by V = orth{V, x(µ∗)}, (e.g., using the modified Gram-Schmidt process with defla- +tion.) +7: +Update Vr by Vr = orth{V, Vr, x(µr)}. +8: +Compute µr such that µr = arg max +µ∈Ξ +max +j=1,...,nI ∥rj(µ) − M(µ)ˆxrj(µ)∥, (nI is the total number of +columns of B(µ)). +9: +Compute µ∗ such that µ∗ = arg max +µ∈Ξ +˜∆(µ). +10: +ε = ˜∆(µ∗). +11: end while +7 + +3.2.3 +Greedy algorithm with multi-fidelity error estimation +The computational complexity of Algorithm 3 using the error estimator ˜∆(µ) comes from Steps 4-9. +Efficiency of Step 9 can be improved by using the bi-fidelity error estimation as shown in Algorithm 2. +The computations in Step 4, 6 are unavoidable, since V is used to compute the ROM of the original +FOM and should be updated till acceptable error tolerance is satisfied. In contrast, Vr in Step 7 needs +not be updated at every iteration. This implies that the ROM of the residual system does not have to +be very accurate, since it is not the ROM that we seek, but an auxiliary ROM aiding the computation +of ˜∆(µ). +An immediate consequence of Theorem 4.2 in [16] for single-input and single-output systems is the +following Lemma for systems with multiple inputs and multiple outputs: +Lemma 3.1 The error of the output ˆy(µ) of the ROM (6) can be bounded as +˜∆(µ) − δ(µ) ≤ ∥y(µ) − ˆy(µ)∥max ≤ ˜∆(µ) + δ(µ), +(13) +where δ(µ) := max +i,j |Ci(µ)(xrj(µ) − ˆxrj(µ))| ≥ 0. +Proof From (7), we know +|yij(µ) − ˆyij(µ)| = |Ci(µ)xrj(µ)| ≈ |Ci(µ)ˆxrj(µ)|. +Then +|yij(µ) − ˆyij(µ)| += +|Ci(µ)xrj(µ)| + |Ci(µ)ˆxrj(µ)| − |Ci(µ)ˆxrj(µ)| +≤ +|Ci(µ)ˆxrj(µ)| + |Ci(µ)xrj(µ) − Ci(µ)ˆxrj(µ)| +� +�� +� +δij(µ) +. +(14) +On the other hand, +|Ci(µ)ˆxrj(µ)| += +|Ci(µ)ˆxrj(µ)| + |Ci(µ)xrj(µ)| − |Ci(µ)xrj(µ)| +≤ +|Ci(µ)xrj(µ)| + δij(µ). +(15) +From (11), (15) and the definition of δ(µ), we have +˜∆(µ) = max +i,j |Ci(µ)ˆxrj(µ)| ≤ ∥y(µ) − ˆy(µ)∥max + δ(µ). +Similarly, from (14), we get +∥y(µ) − ˆy(µ)∥max ≤ ˜∆(µ) + δ(µ). +This completes the proof. +From the definition of δ(µ), it is seen that the more accurate the ROM of the residual system, the +smaller δ(µ) is. As a result, ˜∆(µ) should measure the true error more accurately so that the important +parameters it selects are closer to those selected by the true error, given the same training set Ξ. +On the contrary, if the ROM of the residual system is less accurate, ˜∆(µ) will be less accurate, too. +However, at a certain stage, when ˜∆(µ) is already small, the right-hand side of the residual system +rj(µ) will also be small, so that it can be expected that both xrj(µ) and ˆxrj(˜µ) are close to zero. This +leads to a small δ(µ). Variation of a small δ(µ) will not cause big variation of the difference between +˜∆(µ) and the true error ∥y(µ) − ˆy(µ)∥max. The trend, though not the exact route, of error decay +could still be anticipated so that important parameters corresponding to the error peaks can also be +detected. The above analyses are also justified by the numerical results in the next section, see, e.g., +Figure 3 and Figure 5. +8 + +This motivates the multi-fidelity error estimation. We set a second tolerance ϵ >tol, and when +˜∆(µ) < ϵ < 1, we stop updating the ROM of the residual system, i.e., stop implementing Step 5, Step +7 and Step 8 of Algorithm 3. The error estimator ˜∆(µ) after this stage may not be as accurate as it +would be when keep updating the ROM of the residual system. However, the difference should be small +as ˜∆(µ) is already below a small value ϵ. Without implementing Step 5, we have saved computations +of simulating the FOM. For large and complex systems, solving the FOM even once is not fast. The +computation in Step 7 is relatively cheap if the system is not very large. The computational cost in +Step 8 is not low for certain complex problems, though some µ-independent parts of rj(µ) and M(µ) +can be pre-computed. For example, this is the case for the time-delay systems in the next section. +Stop updating the ROM of the residual system gives rise to a low-fidelity error estimator at later +iteration steps of the greedy algorithm. +When this low-fidelity error estimator is combined with +∆l(µ) in Algorithm 2, we obtain the multi-fidelity error estimation. This is detailed in Algorithm 4. +Compared with the standard greedy algorithm, the overall saving in computational costs is noticeable, +which can be seen from the numerical results in the next section. +The concept of multi-fidelity error estimation could also be applied to other high-fidelity error +estimators. For example, Step 15 could be modified as “Stop updating partial information of ∆(µ)”, +if some parts of the high-fidelity error estimator ∆(µ) are not “essential” for computing ∆(µ). +Algorithm 4 Greedy algorithm with multi-fidelity error estimation +Input: the FOM, a training set Ξc composed of a small number of parameter samples taken from the +parameter domain µ ∈ P, a set Ξf composed of a large number of parameter samples of µ from +P, error tolerance tol< 1. +Output: Projection matrix V . +1: Choose initial parameter µ∗ ∈ Ξc for V , and initial parameter µr ̸= µ∗ ∈ Ξc for Vr. +2: V ← ∅, Vr ← ∅, ε = 1. +3: while ε >tol do +4: +Compute the snapshot(s) x(µ∗) by solving the FOM, i.e. x(µ∗) = (M(µ∗))−1B(µ∗). +5: +Compute the snapshot(s) x(µr) by solving the FOM, i.e. x(µr) = (M(µr))−1B(µr). +6: +Update V by V = orth{V, x(µ∗)} (e.g., using the modified Gram-Schmidt process with defla- +tion). +7: +Update Vr by Vr = orth{V, Vr, x(µr)}. +8: +Compute µ∗ such that µ∗ = arg max +µ∈Ξc +˜∆(µ). +9: +Compute µo such that µo = arg min +µ∈Ξc +˜∆(µ). +10: +Compute µr such that µr = arg max +µ∈Ξc +max +j=1,...,nI ∥rj(µ) − M(µ)ˆxrj(µ)∥, +% nI is the total number +of columns of B(µ). +11: +Compute the low-fidelity error estimator ˜∆l(µ) using values of ˜∆(µ) corresponding to the sam- +ples of µ in Ξc via (3) and (4). +12: +Evaluate ˜∆l(µ) over Ξf and pick out a parameter µc from the large parameter set Ξf corre- +sponding to the largest value of ˜∆l(µ), i.e., µc = arg max +µ∈Ξf from Ξf. +13: +Update the small parameter set Ξc: if ∆l(µc) >tol, enrich Ξc with µc, i.e., Ξc = {Ξc, µc}, if +∆(µo) 1 samples are added or removed from the small training set Ξc at each +12 + +Table 1: Three-port divider: n = 10, 626, d = 93 delays, tol=0.001, adding/removing a single sample +at each iteration. +Method +Iter. +Runtime (h) +r +Valid.err +Alg. 3 (standard, |Ξ| = 40) +14 +5.6 +84 +9.2 × 10−4 +Alg. 2 (bi-fidelity, add only, |Ξc| = 20) +14 +3.6 +84 +6 × 10−4 +Alg. 2 (bi-fidelity, add-remove, |Ξc| = 20) +14 +2.7 +84 +0.0022 +Alg. 4 (multi-fidelity, add only, |Ξc| = 15) +15 +2.4 +90 +6.2 × 10−4 +Alg. 4 (multi-fidelity, add-remove, |Ξc| = 15) +15 +1.8 +90 +6.2 × 10−4 +Table 2: +Three-port divider: +n = 10, 626, d = 93 delays, tol=0.001, smaller |Ξ| and |Ξc|, +adding/removing a single sample at each iteration. +Method +Iter. +Runtime (h) +r +Valid.err +Alg. 3 (standard, |Ξ| = 30) +14 +4.2 +84 +0.0017 +Alg. 2 (bi-fidelity, add only,|Ξc| = 15) +13 +2.5 +78 +0.0026 +Alg. 2 (bi-fidelity, add-remove, |Ξc| = 15) +13 +1.9 +78 +0.0088 +Table 3: Three-port divider: n = 10, 626, d = 93 delays, tol=0.001, adding/removing nadd = ndel > 1 +samples at each iteration. +Method +Iter. +Runtime (h) +r +Valid.err +Alg. 2 (bi-fidelity, add-remove, |Ξc| = 15, nadd = 2) +14 +2.0 +84 +0.0022 +Alg. 2 (bi-fidelity, add-remove, |Ξc| = 20, nadd = 2) +14 +2.7 +84 +0.0022 +Alg. 2 (bi-fidelity, add-remove, |Ξc| = 20, nadd = 5) +14 +2.7 +84 +0.0022 +Alg. 4 (multi-fidelity, add-remove, |Ξc| = 15, nadd = 2) +14 +1.7 +84 +0.0039 +Alg. 4 (multi-fidelity, add-remove, |Ξc| = 15, nadd = 5) +14 +1.7 +84 +0.0039 +iteration of the algorithm. In general, they produce similar results as those in Table 1 and Table 2 +given the same Ξc. For |Ξc| = 15, the bi-fidelity greedy algorithm with nadd = ndel = 2 converges in 14 +iterations, running one more iteration than with nadd = ndel = 1 as shown in Table 2, and generates +a ROM with slightly higher accuracy. On the contrary, given |Ξc| = 15, the multi-fidelity greedy +algorithm with either nadd = ndel = 2 or nadd = ndel = 5 runs one iteration less than in the case +of adding/removing a single sample as shown in Table 1, and constructs ROMs with lower accuracy. +Furthermore, it is seen that increasing nadd = ndel from 2 to 5 did not change the results for both +algorithms. In general, adding/removing a single sample keeps the algorithms simple but efficient. +To illustrate the behavior of the error estimators further, we plot the decay of error estimators and +their corresponding true errors during the greedy iterations. Since different µ∗ are chosen according +to different error estimators, the projection matrix V is updated with different snapshots, leading to +ROMs with different accuracy. Consequently, the true errors of the ROMs are expected to be different. +Figures 2-3 are the results of the algorithms in Table 1. The left part of Figure 2 shows the error +of the high-fidelity error estimator at each iteration of Algorithm 3 and the decay of the corresponding +true error. The error estimator almost exactly matches the true error at all the iterations. The right +part of Figure 2 plots the decay of the bi-fidelity error estimator with respect to the true error. The +bi-fidelity error estimator in both of the two cases: only adding (add-only) samples to Ξc, adding +and removing (add-remove) samples to and from Ξc, can accurately catch the true error. Both cases +converge in 14 iterations, but the case “add-only” is more accurate as can be seen from Table 1. +Figure 3 plots the decay of the multi-fidelity error estimator and the corresponding true error +decay. For clarity, the two cases “add-only” and “add-remove” are plotted in two separate figures. +13 + +The multi-fidelity error estimator is not as accurate as the bi-fidelity error estimator. This is indicated +by the error decay from the 10-th iteration to the end in both figures. From the 10-th iteration, the +error estimator is below ϵ = 0.1, the multi-fidelity error estimation at Step 15 of Algorithm 4 begins +to be implemented. For this example, the multi-fidelity error estimator overestimates the true error +more often than the bi-fidelity error estimator, it did not choose the interpolation points that lead to +error decay as fast as those chosen by the bi-fidelity error estimator. Finally, it uses more iteration +steps to converge. Whereas, they still produce ROMs with best accuracy. +Figure 2: Error decay. Left: true error vs high-fidelity error estimator. Right: true error vs bi-fidelity +error estimators. +Figure 3: Error decay. Left: true error vs multi-fidelity error estimator by only adding samples to Ξc. +Right: true error vs multi-fidelity error estimator by adding and deleting samples to and from Ξc. +4.2 +Test 2: results for a model of coplanar microstrips +The second example is a model of a three coplanar microstrips structure shown in Fig. 4. The width +of the metal strips is mw = 0.178 mm, the thickness of metal strips and ground plane is mt = 0.035 +mm while the left and right wing of the microstrips are wd = 3 mm. Finally, the length of each +strip is ℓ = 5 cm, the thickness of the dielectric is dt = 0.8 mm, and the spacing between 2 strips is +s = 0.3 mm. The relative dielectric constant is set to be εr = 4 and the conductivity of the metal is +assumed to be σ = 5.87 S/m. The six ports, located between the ends of each strip and the ground +14 + +10 +.... High.-fidelity estimator +@.... True error +10 +10 +2 +4 +6 +8 +10 +12 +Number of iterations102 +10 +10 + -G - True error +... Bi-fidelity estimator (add only) +-- - True error ++... Bi-fidelity estimator (add-remove) +2 +4 +8 +10 +12 +14 +6 +i-th iteration102 + -G - True error +.... Multi-fidelity estimator (add only) +10-4 +2 +4 +6 +8 +10 +12 +14 +i-th iteration10° +G- True error +...... Multi-fidelity estimator (add-remove) +2 +4 +6 +8 +10 +12 +14 +i-th iterationwd +mw +s +mw +s +mw +wd +mt +dt +mt +ℓ +P1 +Figure 4: Three coplanar microstrips +plane below, are terminated on load resistors Rload = 50 Ω. The order of the FOM is n = 16, 644, and +there are d = 168 delays. The frequency band of interest is [0, 10]GHz. +For this model, we take fl = 1×106, fh = 1×1010. We set 30 samples for Ξ in the standard greedy +Algorithm 3, i.e., |Ξ| = 30. For Algorithm 2 and Algorithm 4, |Ξc| = 10 or |Ξc| = 15, and |Ξf| = 100. +The 1000 samples used for validating the ROM accuracy are generated using the MATLAB function +linspace, with fl = 100 and the given fh. +The results of the three algorithms are listed in Table 4. The standard greedy Algorithm 3 takes +19 hours, resulting in a ROM of order r = 132 with validated error below the tolerance tol. During the +greedy iteration, if the small parameter set Ξc is enriched only (add only), the greedy algorithm with +bi-fidelity error estimation and that with multi-fidelity error estimation converge within the same +number of iterations, producing ROMs with the same sizes and validated errors. +But the greedy +algorithm with multi-fidelity error estimation is almost one hour faster. Similar phenomenon happens +to the case “add-remove”. The greedy algorithm with bi-fidelity error estimation and that with multi- +fidelity error estimation also converge within the same number of iterations and construct ROMs with +the same sizes and accuracy. The runtimes of both algorithms are much less as compared to their +“add only” versions. +Finally, the greedy algorithm with multi-fidelity error estimation by adding +and deleting samples to and from Ξc (“add-remove”) is most efficient in terms of both runtime and +accuracy. It is more than 3 times faster than the standard greedy algorithm resulting in a speed-up +of 4.2x, and produces a ROM with even the smallest validated error. +We note that using |Ξc| = 10, the ROMs constructed by the bi-fidelity greedy algorithm and the +multi-fidelity greedy algorithm with adding the samples only have validated errors larger than the +tolerance. If we increase |Ξc| from 10 to 15, both algorithms generate ROMs with improved accuracy. +The results are presented in Tabel 5. However, the computational time also increases a lot. Again, +the multi-fidelity greedy algorithm outperforms the bi-fidelity one w.r.t. both accuracy and runtime. +In contrast to the results in Tables 1-2 for the divider model, the results for the coplanar microstrips +model in both Tables 4-5 show that the bi-fidelity greedy algorithm (“add-remove”) is more accurate +than its “add-only” version. +Table 6 shows the results of the bi-fidelity greedy algorithm and the multi-fidelity greedy algorithm +based on adding/removing multiple samples at each iteration. For both cases, i.e., nadd = ndel = 2 +and nadd = ndel = 5, the algorithms using |Ξc| = 10, converge in 10 iterations, one less iteration +than they did with nadd = ndel = 1 in Table 4, resulting in ROMs with smaller order r but with +larger validated errors. If we increase |Ξc| to 15, then the multi-fidelity greedy algorithm generates a +ROM with reduced error, but takes longer time to converge. The bi-fidelity greedy algorithm behaves +similarly and its results for |Ξc| = 15 is not presented to avoid repetition. This example again shows +that adding/removing a single parameter at each iteration outperforms the cases with nadd = ndel > 1, +and produces ROMs with desired accuracy. +15 + +Table 4: Three coplanar microstrips: n = 16, 644, d = 168 delays, tol=0.001, adding/removing a single +sample at each iteration. +Method +Iter. +Runtime (h) +r +Valid.err +Alg. 3 (standard, |Ξ| = 30) +11 +15 +132 +8.5 × 10−4 +Alg. 2 (bi-fidelity, add only, |Ξc| = 10) +11 +6.2 +132 +0.0033 +Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10) +11 +5.3 +132 +8.2 × 10−4 +Alg. 4 (multi-fidelity, add only, |Ξc| = 10) +11 +5.3 +132 +0.0033 +Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10) +11 +4.5 +132 +8.2 × 10−4 +Table 5: +Three coplanar microstrips: +n = 16, 644, d = 168 delays, tol=0.001, larger |Ξc|, +adding/removing a single sample at each iteration. +Method +Iter. +Runtime (h) +r +Valid.err +Alg. 2 (bi-fidelity, add only, |Ξc| = 15) +11 +10 +132 +0.0011 +Alg. 4 (multi-fidelity, add only, |Ξc| = 15) +12 +9.3 +144 +4.4 × 10−4 +Table 6: Three coplanar microstrips: n = 16, 644, d = 168 delays, tol=0.001, adding/removing +nadd = ndel > 1 samples at each iteration. +Method +Iter. +Runtime (h) +r +Valid.err +Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10, nadd = 2) +10 +4.7 +120 +0.019 +Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10, nadd = 5) +10 +4.7 +120 +0.019 +Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10, nadd = 2) +10 +4.2 +120 +0.019 +Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10, nadd = 5) +10 +4.3 +120 +0.019 +Alg. 4 (multi-fidelity, add-remove, |Ξc| = 15, nadd = 2) +13 +7.6 +156 +0.0027 +In Figure 5, we show the important frequency samples of f selected by the greedy algorithms in +Table 4. For the case “add-remove”, we find that the greedy algorithm with bi-fidelity error estimation +and the one with multi-fidelity error estimation select the same important frequency samples. Therefore +we only plot one group of samples for both algorithms, see the plot “bi-(multi-) add-remove” in the +figure. For the case “add-only”, both algorithms also select the same important frequency samples, +see the plot “bi-(multi-) add-only” in the figure. +This is in agreement with the results given in +Table 4 where both algorithms for either case produce the same results. The important frequency +samples selected by the high-fidelity error estimator are mostly different from those selected by the +other algorithms. It is seen that the important frequency samples selected by the (bi-)multi-fidelity +estimator could be different from those selected by the high-fidelity estimator. However, both can +derive ROMs with good accuracy. +The left part of Figure 6 gives the error-peak frequencies detected by the multi-fidelity error +estimator and the true error, respectively, at each iteration of the greedy algorithm. Those frequencies +correspond to the largest values of the error estimator/true error. The error-peak frequency detected +by the error estimator at the i-th iteration is then selected as the important frequency sample at the +next iteration to update the reduced basis space. From iteration 5, the error-peak frequencies detected +by the error estimator are exactly the same as those selected by the true error. This can be explained +by the error decay in the right part of the figure. From the 5-th iteration, the error estimator tightly +catches the true error. Although it is less tight at the first 4 iterations, it still follows the overall trend +of the error decay and therefore, can still detect reasonable error-peak frequencies. This example, +once again, supports our theoretical analysis and demonstrates the efficacy of the proposed greedy +algorithms with bi-(multi-) fidelity error estimation. +16 + +Figure 5: Important parameters selected by the greedy algorithms. +Figure 6: Left: Frequencies causing error/estimator peaks. Right: true error vs multi-fidelity error +estimator. +4.3 +Test 3: results for a model of microstrip filter +The third example is a model of a microstrip filter. The 3D structure of a microstrip filter is depicted in +Fig. 7. The physical dimensions for the geometry of the 3D structure are: wzl = 0.5 mm, wz0 = 1.125 +mm, wzC = 4 mm, ℓzl = 18.3 mm, ℓz0 = 1 mm, ℓzC = 14.1 mm, w = 2.4 cm, ℓ = 2ℓzl + 2ℓz0 + ℓzC, +tm = 100 µm, ts = 100 µm, td = 508 µm. The two ends of the microstrip are terminated on 50 Ω +resistors. +The order of the FOM is n = 12, 132, and there are d = 190 delays. +The interesting +frequency band is [0, 5]GHz. +We take fl = 1 × 105, fh = 5 × 109 to generate frequency samples in Ξc and Ξ. We use |Ξ| = 30 +for the standard greedy Algorithm 3. For Algorithm 2 and Algorithm 4, |Ξc| = 10, and |Ξf| = 100. +The 1000 samples used for computing the validated error are generated using the MATLAB function +logspace, with fl = 10 and the given fh. +The results of the high-fidelity greedy algorithm, and the bi-(multi-)fidelity greedy algorithms by +adding/removing a single sample at each iteration, are listed in Table 7. All the bi-(multi-)fidelity +greedy algorithms produce similar results. The runtime of each is around 1 hour, 3 hours faster than +the high-fidelity greedy algorithm. All the ROMs have similar accuracy, with validated errors below +the tolerance. +Table 8 further shows the performance of the bi-(multi-)fidelity greedy algorithms by adding and +removing multiple samples at each iteration. For this model, all these algorithms behave similarly as +17 + +X109 +10 +8 +Frequency (Hz) +6 +4 +2 +--- hi-fidelity +..bi-(.multi)..add-remove +.. bi-(.multi), add-only. +0 +2 +4 +6 +8 +10 +12 +0 +Number of iterations10 ++ +9.5 +Frequency (GHz) +9 +8.5 +Estimator-peak frequency +True-error-peak frequency +8 +2 +4 +6 +8 +10 +0 +i-th iteration40 +35 +30 +25 +.....Multi-fidelity estimator ("add-remove" +--O- - . True errror +20 +15 +10 +5 +0 +米 +0 +2 +4 +6 +8 +10 +12 +i-th iterationwz0 +wzl +wzC +ℓzC +ℓzl +ℓz0 +ℓ +w +tm td ts +Figure 7: Microstrip filter. +they did by adding/removing a single sample at each iteration. The multi-fidelity greedy algorithm +produces ROMs with slightly larger sizes. The ROMs also have larger validated errors, but still fulfill +the accuracy requirement. All algorithms converge within 8 iterations, much faster than for the first +two examples. This may be due to the much smaller frequency band of interest [0, 5]GHz making the +problem much easier to solve and leading to the most efficient performance of all algorithms. +In summary, for all the tested examples, the multi-fidelity algorithm by adding/removing a single +sample at each iteration behaves the best w.r.t. both runtime and accuracy. +Table 7: Microstrip filter: n = 12, 132, d = 190 delays, tol=0.001, adding/removing a single sample +at each iteration. +Method +Iter. +Runtime (h) +r +Valid.err +Alg. 3 (standard, |Ξ| = 30) +8 +2.5 +32 +5.6 × 10−4 +Alg. 2 (bi-fidelity, add only, |Ξc| = 10) +7 +1.1 +28 +4.6 × 10−4 +Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10) +7 +1.1 +28 +4.6 × 10−4 +Alg. 4 (multi-fidelity, add only, |Ξc| = 10) +7 +1.0 +28 +4.6 × 10−4 +Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10) +8 +1.1 +32 +5.7 × 10−4 +Table 8: Microstrip filter: n = 12, 132, d = 190 delays, tol=0.001, adding/removing nadd = ndel > 1 +samples at each iteration. +Method +Iter. +Runtime (h) +r +Valid.err +Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10, nadd = 2) +7 +1.1 +28 +4.6 × 10−4 +Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10, nadd = 5) +7 +1.1 +28 +4.6 × 10−4 +Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10, nadd = 2) +8 +1.1 +32 +9.1 × 10−4 +Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10, nadd = 5) +8 +1.1 +32 +9.1 × 10−4 +18 + +5 +Conclusions +Concepts of bi-fidelity error estimation and multi-fidelity error estimation are proposed in this work. +The concept of bi-fidelity error estimation is general and can be applied to any high-fidelity estima- +tor. Although the multi-fidelity error estimation is dependent on the high-fidelity error estimation in +consideration, the framework is general to a certain extend and could also be combined with other +high-fidelity error estimators. The robustness of the proposed greedy algorithms with bi-fidelity and +multi-fidelity error estimation is tested on three large time-delay systems with many delays. Although +the standard greedy algorithm converges in a few iterations, the computational complexity in each +iteration is high. As a consequence, the runtime is long for such systems. The proposed (bi-)multi- +fidelity greedy processes have significantly accelerated the standard greedy algorithm with no loss of +accuracy in most cases. +References +[1] D. Alfke, L. Feng, L. Lombardi, G. Antonini, and P. Benner. Model order reduction for delay +systems by iterative interpolation. Internat. J. Numer. Methods Engrg., 122(3):684–706, 2021. +[2] A. C. Antoulas. 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Comput., 37(6):B910–B936, 2015. +21 + diff --git a/YNE5T4oBgHgl3EQfdQ-k/content/tmp_files/load_file.txt b/YNE5T4oBgHgl3EQfdQ-k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b6241cc53d7462195532c23401861df18303c78 --- /dev/null +++ b/YNE5T4oBgHgl3EQfdQ-k/content/tmp_files/load_file.txt @@ -0,0 +1,1000 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf,len=999 +page_content='Accelerating greedy algorithm for model reduction of complex systems by multi-fidelity error estimation Lihong Feng ∗, Luigi Lombardi†, Giulio Antonini‡, and Peter Benner§ January 16, 2023 Abstract Model order reduction usually consists of two stages: the offline stage and the online stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The offline stage is the expensive part that sometimes takes hours till the final reduced-order model is derived, especially when the original model is very large or complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Once the reduced-order model is obtained, the online stage of querying the reduced-order model for simulation is very fast and often real-time capable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' This work concerns a strategy to significantly speed up the offline stage of model order reduction for large and complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' In particular, it is successful in accelerating the greedy algorithm that is often used in the offline stage for reduced-order model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' We propose multi-fidelity error estimators and replace the high-fidelity error estimator in the greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Consequently, the computational complexity at each iteration of the greedy algorithm is reduced and the algorithm converges more than 3 times faster without incurring noticeable accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 1 Introduction Model order reduction (MOR) has achieved much success in many areas of computational science with its capability of realizing real-time simulation and providing accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Different MOR methods, their applications and the promising results they produce can be found in the survey papers [2, 4, 12] and books [27, 3, 8, 9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' MOR needs an offline stage for constructing the ROM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' For many intrusive MOR methods that are based on projection, the offline stage is usually realized via a greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The greedy algorithm is used to properly select important parameter samples that contribute most to the solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The offline computational time is basically the runtime of the greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' For large-scale systems, the offline computation is expensive and the runtime is often longer than several hours even when run on a high-performance server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Sometimes, the system is not very large, for example, the number of degrees of freedom is only O(105), but the system structure is complicated, so that the greedy algorithm still takes long time to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' It is known that an efficient error estimator makes the greedy algorithm successful in producing an accurate ROM without running many iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Therefore, many efforts have been made in this direction to develop computable error estimators for different problems [15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 28, 34, 33, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' However, more attention has been paid to improve the effectivity ∗Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, 39106 Magdeburg, Ger- many feng@mpi-magdeburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='de †Luigi Lombardi is with Micron Semiconductor, 67051 Avezzano, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' luigilombardi89@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='com ‡Giulio Antonini is with the UAq EMC Laboratory, Department of Industrial and Information Engineering and Economics, University of L’Aquila, I-67100 L’Aquila, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' giulio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='antonini@univaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='it §Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany and Fakult¨at f¨ur Mathe- matik, Otto-von-Guericke-Universit¨at Magdeburg, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' benner@mpi-magdeburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='de 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='05610v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='NA] 13 Jan 2023 or accuracy of the error estimator than to develop more efficient strategies to accelerate the greedy process [36, 13, 34, 25, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Recently, some techniques are proposed to improve the adaptivity of the greedy algorithm [7, 13, 6, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' In [13, 14], we proposed a surrogate model for error estimation, and proposed an adaptive greedy algorithm by alternatively using this surrogate error estimator and the original error estimator during the greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The focus in [13, 14] was to make the greedy process adaptive by starting from a coarse training set of a small number of parameter samples, and adaptively update the coarse training set with the aid of a surrogate error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The original error estimator is computed only over the coarse training set, while the surrogate error estimator helps to pick out candidates of important parameter samples from a fine training set, which are then collected in the coarse training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' In this work, we emphasize the role of the surrogate error estimator and propose the concept of bi-fidelity error estimation and multi-fidelity error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' In fact, a bi-fidelity error estimation has been used in the adaptive greedy algorithm proposed in [13, 14] without being formally defined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=', the original (high-fidelity) error estimator, and the surrogate (low-fidelity) error estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' To further improve the convergence speed of the greedy algorithm, we propose multi-fidelity error estimation built upon the bi-fidelity error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Here, we use a more efficient high-fidelity error estimator than the two different high-fidelity error estimators used in [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Although the proposed multi- fidelity error estimation is dependent on the original high-fidelity error estimator, the idea of using multi-fidelity error estimation is general and can be extended to develop multi-fidelity error estimation associated with other high-fidelity error estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Unlike the problems considered in [13, 14], whose ROMs can be constructed by standard greedy algorithms within seconds to minutes, we consider in this work much more complicated problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' On the same computer, the standard greedy algorithm takes more than half a day to converge for such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' By using the proposed multi-fidelity error estimator, the greedy algorithm achieves 4x speed-up and produces ROMs with little loss of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The speed-up is also higher than those reported in [13, 14] by using the bi-fidelity error estimation, which is usually 2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' In the next section, we present the greedy algorithm in the standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Then we analyze some ingredients of the algorithm, which contribute most to the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Starting from those computationally expensive parts, we develop possible strategies to reduce the computational complexity in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' As a consequence, it becomes clear that the resulting strategy develops a greedy algorithm with multi-fidelity error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The proposed algorithm is then applied to large time-delay systems with many delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The numerical tests on three large time-delay systems with more than 100 delays are demonstrated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Conclusions are given in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 2 Standard greedy algorithm The standard greedy algorithm was first proposed for steady systems without time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Then it was extended to POD-greedy for dynamical systems, which is used to construct the ROM using snapshots in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Later the greedy algorithm found its capability in adaptively choosing interpolation points for frequency-domain MOR methods [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The greedy algorithm for steady systems and frequency-domain MOR has the same formulation, whereas POD-greedy for time-domain MOR of time-dependent systems needs an SVD step at each greedy iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' In this work, we focus on the greedy algorithm, though the proposed scheme can be easily extended to POD-greedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' We consider constructing a ROM for the following full-order model (FOM) using the greedy algorithm, F(x(µ), µ) = B(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' (1) Here, F(x(µ), µ) ∈ Cn×nI, x(µ) ∈ Cn×nI, and B(µ) ∈ Cn×nI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' µ ∈ P is a parameter in the parameter domain P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The variable n is the order of the FOM, which can be the number of degrees of freedom 2 Algorithm 1 Standard greedy algorithm Input: the FOM, a training set Ξ composed of parameter samples taken from the parameter domain P, error tolerance tol< 1, ∆(µ) to estimate the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Output: Projection matrix V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 1: Choose initial parameter µ∗ ∈ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 2: V ← ∅, ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 3: while ε >tol do 4: Compute the snapshot(s) x(µ∗) by solving the FOM at µ = µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 5: Update V by V = orth{V, x(µ∗)}, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=', using the modified Gram-Schmidt process with defla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=') 6: Compute µ∗ such that µ∗ = arg max µ∈Ξ ∆(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 7: ε = ∆(µ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 8: end while after numerical discretization of PDEs describing a physical phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The proposed algorithms are also applicable to problems with nI > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The ROM can be obtained via Galerkin projection using a projection matrix V ∈ Rn×r, r ≪ n, as below, ˆF(V z(µ), µ) = ˆb(µ), (2) where ˆF(V z(µ), µ) = V T f(V z(µ), µ) ∈ Cr×nI, z(µ) ∈ Cr×nI, and ˆB(µ) = V T b(µ) ∈ Cr×nI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The standard greedy process used to compute the projection matrix V is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Step 4 in Algorithm 1 solves the FOM at µ∗, and Step 6 computes an error estimator ∆(µ) at all µ in Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' These two steps constitute the most computational expensive part of the greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' However, Step 4 is unavoidable, since x(µ∗) is needed for the reduced basis construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='The computational complexity of Step 6 could be reduced, if the cardinality of Ξ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=', |Ξ| is kept small, so that ∆(µ) needs not be evaluated at many parameter samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' This is the motivation of the surrogate error estimator proposed in [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' We call the surrogate error estimator ∆l(µ) the low-fidelity error estimator as compared to the original error estimator ∆(µ), since ∆l(µ) is only an approximation to ∆(µ), but is much cheaper to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' In the next section, we present a greedy algorithm using bi-fidelity error estimation, where the low- fidelity error estimator is computed following the method in [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Based on this, a greedy algorithm using multi-fidelity error estimation associated with a particular high-fidelity error estimator for MOR of linear parametric systems, is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 3 Greedy algorithm with bi-fidelity and multi-fidelity error estima- tion This section presents greedy algorithms with bi-fidelity and multi-fidelity error estimation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='1 Greedy algorithm with bi-fidelity error estimation Algorithm 2 is the greedy algorithm with bi-fidelity error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Its original version using a different high-fidelity error estimator was firstly proposed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' The key step of Algorithm 2 is Step 8, where the low-fidelity error estimator ∆l(µ) is computed using values of ∆(µ) at the samples of µ in the small parameter set Ξc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Basically, ∆l(µ) is represented by a weighted sum of radial basis 3 Algorithm 2 Greedy algorithm with bi-fidelity error estimation Input: the FOM, a training set Ξc composed of a small number of parameter samples taken from the parameter domain P, a set Ξf composed a large number of parameter samples of µ from P, error tolerance tol< 1, ∆(µ) to estimate the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' Output: Projection matrix V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 1: Choose initial parameter µ∗ ∈ Ξc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 2: V ← ∅, ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 3: while ε >tol do 4: Compute the snapshot(s) x(µ∗) by solving the FOM at µ = µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 5: Update V by V = orth{V, x(µ∗)}, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=', using the modified Gram-Schmidt process with defla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=') 6: Compute µ∗ such that µ∗ = arg max µ∈Ξc ∆(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 7: Compute µo such that µo = arg min µ∈Ξc ∆(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 8: Compute the low-fidelity error estimator ∆l(µ) using values of ∆(µ) corresponding to the sam- ples of µ in Ξc via (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 9: Evaluate ∆l(µ) over Ξf and pick out a parameter µc from the large parameter set Ξf corre- sponding to the largest value of ∆l(µ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=', µc = arg max µ∈Ξf from Ξf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=' 10: Update the small parameter set Ξc: if ∆l(µc) >tol, enrich Ξc with µc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfdQ-k/content/2301.05610v1.pdf'} +page_content=', Ξc = {Ξc, µc}, if ∆(µo) 0 +pk +k tr( ¯ψψ)k + N 2tr( ¯ψψ)] +� +dψd ¯ψ exp(N 2tr ¯ψψ) +6 + += +� +R +(−1 +N )|R| DR(N)DR(−N) +dR +SR, +(25) +where ψ and ¯ψ are independent complex Grassmann-valued N × N matrices, and DR(N) = +SR{pk = N}, dR = SR{pk = δk,1} are respectively the dimension of representation R for the +linear group GL(N). +Let us extend (25) to the fermionic two-matrix model, +Z2F += +� +dψd ¯ψdχd¯χ exp[−µtr ¯ψψ − µtr¯χχ + +∞ +� +k=0 +tktr( ¯ψψ)k + +∞ +� +k=0 +gktr(¯χχ)k ++ +∞ +� +l=1 +∞ +� +k1,···k2l=1 +tk1,···k2ltr( ¯ψψ)k1(¯χχ)k2( ¯ψψ)k3 · · · ( ¯ψψ)k2l−1(¯χχ)k2l], +(26) +where χ and ¯χ are independent complex Grassmann-valued N × N matrices. +There are the Virasoro constraints +ˇLnZ2F = 0, +(27) +where +ˇLn += +∞ +� +n=0 +n−1 +� +s=1 +∂2 +∂ts∂tn−s +− δn,0N 2 − µ +∞ +� +n=0 +∂ +∂tn+1 ++ +∞ +� +n=0 +∞ +� +k=0 +ktk +∂ +∂tn+k ++ +∞ +� +n=0 +∞ +� +l=1 +∞ +� +k1,··· ,k2l=1 +l +� +a=1 +k2a−1tk1,···k2l +∂ +∂tk1,··· ,n+k2a−1,k2a,··· ,k2l +. +(28) +Similar to the complex two-matrix case, by considering the following infinitesimal transfor- +mations in the integral (26), respectively, +(i) ψ −→ ψ + ǫ +∞ +� +n=0 +(n + 1)tn+1ψ( ¯ψψ)n, +(ii) ψ −→ ψ + ǫ +∞ +� +n,m=0 +[(n + 1) + (m + 1)]tn+1,m+1ψ(¯χχ)m+1( ¯ψψ)n, +(iii) ψ −→ ψ + ǫ +∞ +� +r=1 +∞ +� +n1,··· ,n2r+1=0 +ˇ +N ˇTψ(¯χχ)n2+1( ¯ψψ)n3+1 · · · (¯χχ)n2r+1( ¯ψψ)n2r+1, +(iv) χ −→ χ + ǫ +∞ +� +m=0 +(m + 1)gm+1χ(¯χχ)m, +where ˇ +N ˇT = (n2 + 1) + (n3 + 1) + · · · + (n2r+1 + 1)tn2r+1,n2+1,··· ,n2r+1, we finally obtain +µ ˇDZ2F = ˇWZ2F , +(29) +where ˇD = �4 +i=1 ˇDi, ˇW = �4 +i=1 ˇWi, the operators ˇDi and ˇWi are +ˇ +D1 = +∞ +� +n=0 +(n + 1)tn+1 +∂ +∂tn+1 +, +ˇ +D2 = +∞ +� +n,m=0 +ˇT1 +∂ +∂tn+1,m+1 +, +ˇ +D3 = +∞ +� +n1,··· ,n2r+1=0 +ˇT2 +∂ +∂tn1+1,··· ,n2r+1+1 +, +ˇ +D4 = +∞ +� +m=0 +(m + 1)gm+1 +∂ +∂gm+1 +, +(30) +7 + +ˇ +W1 += +∞ +� +n=0 +(n + 1)tn+1[ +∞ +� +k=0 +ktk +∂ +∂tn+k ++ +n−1 +� +a=1 +∂ +∂ta +∂ +∂tn−a ++ +∞ +� +l=1 +l +� +a=1 +∞ +� +k1,··· ,k2l=1 +k2a−1 ˇT0 +· +∂ +∂tk1,··· ,k2a−1+n,k2a,··· ,k2l +] − N 2t1, +ˇ +W2 += +∞ +� +n,m=0 +n−1 +� +a=1 +ˇT1{ +∞ +� +l=1 +∞ +� +k1,··· ,k2l=1 +ˇT0[ +l +� +a=2 +k2a−1 +� +s=0 +∂ +∂tk1,··· ,k2a−2,ˇξ0,k2a,··· ,k2l ++ +k1 +� +s=0 +∂ +∂tˇξ1,k2,··· ,k2l +] ++ +∂ +∂ta,m+1 +∂ +∂tn−a +− Nδn,0 +∂ +∂gm+1 ++ +∞ +� +k=0 +ktk +∂ +∂tn+k,m+1 +}, +ˇ +W3 += +∞ +� +n1,··· ,n2r+1=0 +ˇT2{[ +n2r+1−1 +� +s=1 +∂ +∂ts,n2+1,··· ,n2r+1 +∂ +∂tn2r+1−s ++ +r−1 +� +b=1 +n2b+1 +� +s=1 +∂ +∂ts,n2+1,··· ,n2b+1 +· +∂ +∂tˇξ2,n2b+2+1,··· ,n2r+1 +] − Nδn2r+1,0 +∂ +∂tn3+1,··· ,n2r−1+1,ˇξ3 ++ +∞ +� +k=0 +ktk +∂ +∂tn2r+1+k,n2+1,··· ,n2r+1 ++ +∞ +� +l=1 +∞ +� +k1,··· ,k2l=1 +ˇT0[ +k1 +� +s=0 +∂ +∂ts+1,n2+1,··· ,ˇξ4,k2,··· ,k2l ++ +l +� +a=2 +k2a−1 +� +s=0 +∂ +∂tk1,··· ,k2a−2,s+1,ˇξ5,k2a,··· ,k2l +]}, +ˇ +W4 += +∞ +� +m=0 +(m + 1)gm+1[ +m−1 +� +a=1 +∂ +∂ga +∂ +∂gm−a ++ +∞ +� +k=0 +kgk +∂ +∂gm+k ++ +∞ +� +l=1 +l +� +a=1 +∞ +� +k1,··· ,k2l=1 +k2a ˇT0 +· +∂ +∂tk1,··· ,k2a+m,k2a+1,··· ,k2l +] − N 2g1, +(31) +where ˇT0 = tk1,··· ,k2l, ˇT1 = (n + m + 2)tn+1,m+1, ˇT2 = ¯ +Ntn2r+1+1,n2+1,··· ,n2r+1 and ˇξ0 = (a + +1, m + 1, n + k2a−1 − k − 1), +ˇξ1 = (s + 1, m + 1, n + k1 − s − 1), +ˇξ2 = n2r+1 + n2b+1 + 1 − s, +ˇξ3 = n2r + n2 + 2, ˇξ4 = n2r+1 + k1 − 1 − s, ˇξ5 = (n2 + 1, · · · , n2r + 1, n2r+1 + k2a−1 − s − 1). +We find that the fermionic two-matrix model (26) can be realized by the W-representation +Z2F = e +1 +µ ˇ +W eN(t0+g0). +(32) +The compact expression of correlators is +⟨ +l1 +� +i=1 +tr( ¯ψψ)ki +l2 +� +j=1 +tr(¯χχ)rj +l3 +� +n=1 +tr( ¯ψψ)Sn,1(¯χχ)Sn,2 · · · ( ¯ψψ)Sn,2pn−1 (¯χχ)Sn,2pn ⟩ += +l1!l2!l3! +m+1 +� +ρ1+ρ2+ρ3=1 +� +σ +ˇP +(σ(S1,1),··· ,σ(S1,2p1);··· ;σ(Sl3,1),··· ,σ(Sl3,2pl3 )) +(σ(k1),··· ,σ(kl1));(σ(r1),··· ,σ(rl2)) +µm+1(m + 1)!λ(k1,··· ,kl1)λ(r1,··· ,rl2)λ(S1,1,··· ,S1,2p1;··· ;Sl3,1,··· ,Sl3,2pl3 ) +, +(33) +where ρ1 = +l1 +� +i=1 +ki, ρ2 = +l2 +� +i=1 +ri, ρ3 = +l3 +� +i=1 +(Si,1 + · · · + Si,2pi), and ˇP +(σ(S1,1),··· ;··· ;··· ,σ(Sl3,2pl3 )) +(σ(k1),··· ,σ(kl1));(σ(r1),··· ,σ(rl2)) is +the coefficient of tk1 · · · tkl1gr1 · · · grl2tS1,1,··· ,S1,2p1 · · · tSl3,1,··· ,Sl3,2p3 in ˇW m+1. +8 + +Here we list some correlators +⟨tr ¯ψψ⟩ = ⟨tr¯χχ⟩ = − 1 +µN 2, +⟨tr ¯ψψtr¯χχ⟩ = +2 +µ2 N 4, +⟨tr ¯ψψtr ¯ψψ⟩ = ⟨tr¯χχtr¯χχ⟩ = +1 +µ2 (N 2 − 1)N 2, +⟨tr ¯ψψ ¯χχ⟩ = − 2 +µ2 N 3, +⟨tr( ¯ψψ)3⟩ = ⟨tr(¯χχ)3⟩ = +6 +µ3 (−N 2 + N 4), +⟨tr ¯ψψtr ¯ψψtr ¯ψψ⟩ = ⟨tr¯χχtr¯χχtr¯χχ⟩ = +1 +µ3 (N 2 + 2)(N 2 − 1)N 2, +⟨tr ¯ψψtr ¯ψψtr¯χχ⟩ = ⟨tr¯χχtr ¯ψψtr¯χχ⟩ = +3 +µ3 (N 4 − N 6), +⟨tr ¯ψψtr ¯ψψ ¯χχ⟩ = ⟨tr¯χχtr ¯ψψ ¯χχ⟩ = − 1 +µ3(6N 3 + 2N 5). +(34) +5 +Conclusion +We have constructed the Hermitian, complex and fermionic two-matrix models with infinite +set of variables and presented their Virasoro constraints. +W-representation is important for +understanding matrix model, since it provides a dual formula for partition function through dif- +ferentiation. By considering the particular infinitesimal transformations of integration variables +in the partition functions, we finally derived the desired operators preserving and increasing the +grading. Thus it can be shown that the two-matrix models constructed in this paper can be +realized by the W-representations. Moreover, by means of the W-representations, we derived +the compact expressions of correlators for these two-matrix models. It should be noted that +there are the infinite set of variables in these two-matrix models. It leads to that we can not +give their character expansions. For further research, it would be interesting to study the case +of β-deformed two-matrix models. +Appendix A +The operators ˆWi in (5) +ˆ +W1 = +∞ +� +l=1 +∞ +� +k1,··· ,k2l=1 +{t1T1[δk1,1 +∂ +∂tk3,··· ,k2l−1,k2l+k2 ++ +l +� +a=2 +δk2a−1,1 +∂ +∂tk1,··· ,k2a−2+k2a,··· ,k2l +] ++ +∞ +� +n=0 +l +� +a=1 +k2a−1(n + 1)tn+1 +∂ +∂tk1,··· ,n+k2a−1−1,··· ,k2l +} + +∞ +� +k2=1 +t1t1,k2 +∂ +∂gk2 ++ t2 +1N + 2t2N 2 ++ +∞ +� +n=0 +(n + 1)tn+1[ +n−2 +� +b=1 +∂2 +∂tb∂tn−1−b ++ +∞ +� +k=0 +ktk +∂ +∂tn+k−1 +] + +∞ +� +n=2 +2N(n + 1)tn+1 +∂ +∂tn−1 +, +ˆ +W2 = +∞ +� +l,n=1 +∞ +� +k1,··· ,k2l=1 +T2{(1 − δk1,1)( +∂ +∂tk1−1,k2,··· ,k2l+n ++ +∂ +∂tk1−1,k2+n,··· ,k2l +) ++δk1,1 +∂ +∂tk3,··· ,k2l+n+k2 ++ +l +� +a=2 +(1 − δk2a−1,1) +∂ +∂tk1,··· ,k2a−2+n,k2a−1−1,k2a,··· ,k2l ++ +l−1 +� +a=2 +δk2a−1,1 +∂ +∂tk1,··· ,k2a−2+k2a+n,··· ,k2l +} + +∞ +� +l=2, +n=1 +∞ +� +k1,··· ,k2l=1 +T2δk2l−1,1 +∂ +∂tk1,··· ,k2l−2+n+k2l ++ +∞ +� +l,n=1 +l +� +a=1 +∞ +� +k2a−1=3 +k2a−1−2 +� +b=1 +∞ +� +k1,··· ,k2a−2, +k2a,··· ,k2l=1 +T2[(1 − δa,1) +∂ +∂tk1,··· ,k2a−2,b,n,k2a−1−1−b,··· ,k2l ++δa,1 +∂ +∂tb,n,k1−1−b,··· ,k2l +] + +∞ +� +n=1 +(n + 1)t1,n[ +∞ +� +k=2 +ktk +∂ +∂tk−1,n ++ +∞ +� +k2=1 +t1,k2 +∂ +∂gk2+n ++ t1 +∂ +∂gn +], +9 + +ˆ +W3 = +∞ +� +r=1 +∞ +� +n1,··· ,n2r=1 +T3{Nδn1,1 +∂ +∂tn3,··· ,n2r+n2 ++ (1 − δn1,1)( +n1−2 +� +a=1 +∂ +∂ta +∂ +∂tn1−1−a,··· ,n2r ++N +∂ +∂tn1−1,n2,··· ,n2r ++ +∂ +∂tn1−1 +∂ +∂tn3,··· ,n2r +) + +r +� +s=2 +n2s−1−2 +� +a=0 +(1 − δn2s−1,1) +∂ +∂tn1+a,n2,··· ,n2s−2 +· +∂ +∂tn2s−1−1−a,··· ,n2r ++ +r−1 +� +s=2 +[δn2s−1,1 +∂ +∂tn1,··· ,n2s−2 ++ (1 − δn1,1) +∂ +∂tn1+n2s−1−1,n2,··· ,n2s−2 +· +∂ +∂tn2s+1,··· ,n2r+n2s +] + (1 − δn2r−1,1) +∂ +∂tn1+n2r−1−1,··· ,n2r−2 +∂ +∂gn2r ++ +∞ +� +k=0 +ktk +∂ +∂tn1+k−1,··· ,n2r ++δn2r−1,1 +∂2 +∂tn1,··· ,n2r−2∂gn2r ++ +∞ +� +k1,··· ,k2l=1 +T1[ +l +� +i=1 +k2i−1−2 +� +s=0 +∂ +∂tk1,··· ,k2i−2,s+n1,··· ,n2r,ξ1,··· ,k2l ++ +∂ +∂tn1+k1−1,n2··· ,n2r+k2,··· ,k2l ++ +∂ +∂tk1,··· ,n1+k2i−1−1,··· ,n2r+k2i,··· ,k2l +} ++ +∞ +� +n1=2 +∞ +� +n2=1 +(n1 + 1 + n2)tn1+1,n2 +∂2 +∂tn1−1∂gn2 +, +ˆ +W4 = +∞ +� +r=1 +∞ +� +n2,··· ,n2r=1 +T4{ +n3−2 +� +b=1 +∂2 +∂tb,n2∂tn3−1−b,··· ,n2r ++ (1 − δn3,1)( +∂2 +∂gn2∂tn3−1,··· ,n2r ++ +∂2 +∂tn3−1,n2,∂tn5,··· ,n2r+n4 +) + δn3,1 +∂2 +∂gn2∂tn5,··· ,n2r+n4 ++ +r−1 +� +a=3 +[δn2a−1,1 +∂ +∂tn3,··· ,n2a−2+n2 +· +∂ +∂tn2a+1,··· ,n2r ++ (1 − δn2a−1,1)( +∂ +∂tn3,··· ,n2a−1−1,n2a +∂ +∂tn2a+1,··· ,n2r+n2a ++ +∂ +∂tn3,··· ,n2a−2+n2 +· +∂ +∂tn2a−1−1,n2a··· ,n2r +)] + [δn2r−1,1 +∂ +∂tn3,··· ,n2r−2+n2 ++ (1 − δn2r−1,1) +∂ +∂tn3,··· ,n2r−1−1,n2 +] +∂ +∂gn2r ++ +r +� +a=3 +n2a−1−2 +� +b=1 +∂ +∂tn3,··· ,n2a−2,b,n2 +∂ +∂tn2a−1−1−b,··· ,n2r ++ t1 +∂ +∂tn3,··· ,n2r+n2 ++ +∞ +� +k=2 +ktk +∂ +∂tk−1,n2,··· ,n2r ++ +∞ +� +l=1 +∞ +� +k1,··· ,k2l=1 +T1[δk1,1 +∂ +∂tn3,··· ,n2r+k2,··· ,k2l+n2 ++ (1 − δk1,1)( +∂ +∂tk1−1,n2,··· ,n2r+k2,··· ,k2l ++ +∂ +∂tn3,··· ,n2r,k1−1,··· ,k2l+n2 +) + +l +� +b=2 +(1 − δk2b−1,1)( +∂ +∂tk1,··· ,k2b−3,ξ2,··· ,k2l ++ +∂ +∂tk1,··· ,ξ3,··· ,n2r+k2l +) ++ +l +� +b=2 +δk2b−1,1 +∂ +∂tk1,··· ,k2b−1+n2,ξ4,··· ,k2l +] + +∞ +� +l=1 +l +� +a=1 +∞ +� +k2a−1=3 +∞ +� +k1,··· ,k2a−2, +k2a,··· ,k2l=1 +k2a−1−2 +� +s=1 +T1 +∂ +∂tk1,··· ,k2a−2,s,ξ5,··· ,k2l +}, +ˆ +W5 = g2 +1N + +∞ +� +n,k=0 +(n + 1)gn+1kgk +∂ +∂gn+k−1 ++ +∞ +� +kl,k2,k3=1 +g1tk1,k2,k3,1 +∂ +∂tk1+k3,k2 ++ +∞ +� +k1=1 +tk1,1g1 +∂ +∂tk1 ++ +∞ +� +l=1 +∞ +� +n=0 +∞ +� +k1,··· ,k2l=1 +T5[δk2l,1 +l−1 +� +a=1 +∂ +∂tk1+k2l−1,··· ,k2l−2 +k2a +∂ +∂tk1,··· ,ξ6,··· ,k2l ++ +∂ +∂tk1+k2l−1,··· ,k2l−2 +· +· +∂ +∂tk1,··· ,k2l−1,ξ7 ++ δn,0(1 − δk2a,1) +∂ +∂tk1,k2−1,··· ,k2l ++ δn,0 +l−1 +� +a=1 +δk2a,1 +∂ +∂tk1,··· ,k2a−1+k2a+1,··· ,k2l +] +10 + ++ +∞ +� +n=1 +n−2 +� +s=1 +(n + 1)gn+1 +∂ +∂gs +∂ +∂gn−1−s ++ +∞ +� +n=2 +2N(n + 1)gn+1 +∂ +∂gn−1 ++ 2g2N 2, +(A.1) +where T1 = tk1,··· ,k2l, +T2 = (n + 1)t1,ntk1,··· ,k2l, +T3 = N1tn1+1,n2,··· ,n2r, +T4 = N2t1,n2,··· ,n2r, +T5 = (n + 1)gn+1tk1,···k2l and ξ1 = k2i−1 − 1 − s, ξ2 = (k2b−2 + n2, · · · , n2r, k2b−1 − 1), ξ3 = +(k2b−1 − 1, n2), ξ4 = (n3, · · · , n2r + k2b), ξ5 = (n2, · · · , n2r, k2a−1 − 1 − s), ξ6 = k2a + n − 1, +ξ7 = k2l + n − 1. +Acknowledgment +This work is supported by the National Natural Science Foundation of China (No. 11875194). +References +[1] F. 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Wang, W-representations for +multi-character partition functions and their β-deformations, arXiv:2301.12763. +12 + diff --git a/YtFRT4oBgHgl3EQf_ziQ/content/tmp_files/load_file.txt b/YtFRT4oBgHgl3EQf_ziQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a56d340f55f6ce12b710c67be26d6689e1623fa --- /dev/null +++ b/YtFRT4oBgHgl3EQf_ziQ/content/tmp_files/load_file.txt @@ -0,0 +1,837 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf,len=836 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='13696v1 [hep-th] 31 Jan 2023 W-representations of two-matrix models with infinite set of variables Lu-Yao Wanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='∗ Yu-Sen Zhua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='† Ying Chenb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='‡ Bei Kangc§ a School of Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Capital Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Beijing 100048,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' China bSchool of Mathematics and Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Jiangsu Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Xuzhou 221116,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Jiangsu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' China c School of Mathematics and Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' North China University of Water Resources and Electric Power,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Zhengzhou 450046,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Henan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' China Abstract The Hermitian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' complex and fermionic two-matrix models with infinite set of variables are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' We show that these two-matrix models can be realized by the W-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' In terms of the W-representations, we derive the compact expressions of correlators for these two-matrix models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Keywords: Two-matrix models, Conformal and W Symmetry 1 Introduction Matrix models have been developed to solve non-perturbative two-dimensional gravity and pro- vide a rich set of approaches to physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' For two-matrix model, there is the interaction between the two matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Hence it possesses a richer mathematical structure than single ma- trix models, and thus produces more applications in physics and mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' The two-matrix models have been studied as an important solvable example of statistical mechanical systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=', Ising spins [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' For fermionic two-matrix model, the complete sets of loop equations can be derived [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' The Ward identities in Kontsevich-like one-matrix models are used to relate the degree of potential in Kontsevich-like two-matrix model to the W-constraints [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' The spec- tral curves, loop equations and topological expansion for Hermitian two-matrix models were presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' For W-representation of matrix model, it realizes partition function by acting on elemen- tary functions with exponents of the given W-operator [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Since W-representation plays an important role in understanding the structures of matrix models, much interest has been at- tributed to this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' A variety of matrix models have been realized by W-representations and their correlators can be exactly calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Recently the (super) partition function hier- archies with W-representations were constructed [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Some well known superintegrable matrix models were contained in these superintegrable hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' In addition, the progress of W-representation has been made on tensor models [12–15] and super-eigenvalue models [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Recently, the two-matrix models with multi-set of variables were proposed [18–21], which are the superintegrable matrix models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Their W-representations and character expansions were well investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' In this paper, we’ll construct the new two-matrix models with infinite set of variables and derive their W-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ∗wangly100@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='com †zhuyusen@cnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='cn ‡chenying math@jsnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='cn §Corresponding author:kangbei@ncwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='cn 1 2 W-representation of new Hermitian two-matrix model Let us construct the Hermitian two-matrix model Z2H = � dAdB exp(−1 2trA2 − 1 2trB2 + ∞ � k=0 tktrAk + ∞ � k=0 gktrBk + ∞ � l=1 ∞ � k1,···k2l=1 tk1,··· ,2ltrAk1Bk2Ak3Bk4 · · · Ak2l−1Bk2l), (1) where A and B are N × N matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' When B = 0 in (1), it reduces to the well known Gaussian Hermitian matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' By requiring the invariance of the integral (1) under the infinitesimal transformation A → A + ǫAn (n ≥ 0) or B → B + ǫBn (n ≥ 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' we obtain the Virasoro constraints Ln−1Z2H = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (2) where the constraint operators are given by Ln−1 = ∞ � n=0 2N ∂ ∂tn−1 + ∞ � n=0 n−1 � s=1 ∂2 ∂ts∂tn−1−s + δn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1N 2 + ∞ � n=0 ∞ � k=0 ktk ∂ ∂tn+k−1 + ∞ � k2=1 t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2 ∂ ∂gk2 + ∞ � l=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='2l[ l � a=1 ∞ � n=0 k2a−1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n+k2a−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + δk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tk3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l+k2 + l � a=2 δk2a−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−2+k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (3) which obey the Virasoro algebra [Ln−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Lm−1] = (n − m)Ln+m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (4) Let us now consider the following five infinitesimal transformations, respectively, (i) A −→ A + ǫ ∞ � n=0 (n + 1)tn+1An, (ii) A −→ A + ǫ ∞ � n=1 (n + 1)t1,nBn, (iii) A −→ A + ǫ ∞ � r=1 ∞ � n1,···n2r=1 N1tn1+1,n2,··· ,n2rAn1Bn2 · · · An2r−1Bn2r, (iv) A −→ A + ǫ ∞ � r=1 ∞ � n1,···n2r=1 N2t1,n2,··· ,n2rBn2An3 · · · An2r−1Bn2r, (v) B −→ B + ǫ ∞ � n=0 (n + 1)gn+1Bn, where N1 = n1 + 1 + n2 + · · · + n2r and N2 = 1 + n2 + · · · + n2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' From the invariance of the integral (1), it gives ˆDiZ2H = ˆWiZ2H, i = 1, 2, · · · 5, (5) 2 where the operators ˆWi are listed in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1) and ˆDi are ˆD1 = ∞ � i=1 iti ∂ ∂ti , ˆD2 = ∞ � n=1 (n + 1)t1,n ∂ ∂t1,n , ˆD3 = ∞ � r=1 ∞ � n1,···n2r=1 N1tn1+1,n2,··· ,n2r ∂ ∂tn1+1,n2,··· ,n2r , ˆD4 = ∞ � r=1 ∞ � n1,···n2r=1 N2t1,n2,··· ,n2r ∂ ∂t1,n2,··· ,n2r , ˆD5 = ∞ � i=1 igi ∂ ∂gi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (6) In the following, we’ll focus on the sum of (5) ˆDZ2H = ˆWZ2H, (7) where ˆD = �5 i=1 Di and ˆW = �5 i=1 Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Let us write the partition function (1) as the grading form Z2H = �∞ d=0 Z(d) 2H and Z(d) 2H = eN(t0+g0) ∞ � l=0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' � l1+l2+l3=l ρ1+ρ2+ρ3=d ⟨ l1 � i=1 trAki l2 � j=1 trBrj l3 � n=1 trASn,1BSn,2 · · · ASn,2pn−1BSn,2pn⟩ l1 � i=1 tki l2 � j=1 grj l3 � n=1 tSn,1,··· ,Sn,2pn · � dAdB exp(−1 2trA2 − 1 2trB2), (8) where ρ1 = �l1 i=1 ki, ρ2 = �l2 i=1 ri, ρ3 = �l3 i=1(Si,1 + · · · + Si,2pi) correlators ⟨· · · ⟩ are defined as ⟨· · · ⟩ = � dAdB · · · exp(− 1 2trA2 − 1 2trB2) � dAdB exp(− 1 2trA2 − 1 2trB2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (9) We denote the degrees of operators as deg(tk) = deg(gk) = k, deg( ∂ ∂tk ) = deg( ∂ ∂gk ) = −k, deg( ∂ ∂tk1,k2,··· ,k2l−1,k2l ) = −(k1+· · ·+k2l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Then it is easy to see that deg( ˆD) = 0 and deg( ˆ W ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Due to the operators ˆD and ˆD − ˆW being invertible and ˆDeN(t0+g0) = 0, from (7), we have ∞ � s=1 Z(s) 2H = ( ˆD − ˆW)−1 ˆWeN(t0+g0) = ∞ � k=1 ( ˆD−1 ˆW)keN(t0+g0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (10) Note that ˆW is an homogeneous operator with degree 2, and ˆDf = deg(f)·f for any homogeneous function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' We may give the W-representation of the Hermitian two-matrix model (1) Z2H = e 1 2 ˆ W eN(t0+g0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (11) Let us formally write the (m + 1)-th power of the operator ˆW as ˆW (m+1) = 2(m+1) � l1+l2+l3=1 � ρ1+ρ2+ρ3=2(m+1) P (S1,1,··· ,S1,2p1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='Sl3,1,··· ,Sl3,2pl3 ) (k1,··· ,kl1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='(r1,··· ,rl2) tk1 · · · tkl1gr1 · · · grl2 tS1,1,··· ,S1,2p1 · · · tSl3,1,··· ,Sl3,2pl3 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (12) 3 By means of the W-representation of (1), we derive the compact expression of correlators ⟨ l1 � i=1 trAki l2 � j=1 trBrj l3 � n=1 trASn,1BSn,2 · · · ASn,2pn−1BSn,2pn⟩ = l1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='l2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='l3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' 2(m+1) � ρ1+ρ2+ρ3=1 � σ P (σ(S1,1),··· ,σ(S1,2p1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='σ(Sl3,1),··· ,σ(Sl3,2pl3 )) (σ(k1),··· ,σ(kl1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='(σ(r1),··· ,σ(rl2)) 2m+1(m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='λ(k1,··· ,kl1)λ(r1,··· ,rl2)λ(S1,1,··· ,S1,2p1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='Sl3,1,··· ,Sl3,2pl3 ) , (13) where (σ(k1), · · · , σ(kl1)) denotes all distinct permutations of (kl1, · · · , kl1), and λ(k1,··· ,kl1) is the number of distinct permutations (k1, · · · , kl1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' For example, let us consider the cases ˆW = t2 1N + 2t2N 2 + g2 1N + 2g2N 2 + · · · , ˆW 2 = 8t2 1t2N + 24t1t3N 2 + 12t4N 3 + 8t2 2N 2 + 8t1t1,2N 2 + 8t1g1t1,1N + 8g1t2,1N 2 +4t2 1,1N 2 + 8t2,2N 3 + 8g2 1g2N + 24g1g3N 2 + 12g4N 3 + 8g2 2N 2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (14) We may give some correlators in (13) as follows ⟨trAtrA⟩ = ⟨trBtrB⟩ = N, ⟨trA2⟩ = ⟨trB2⟩ = N 2, ⟨trAtrBtrA2⟩ = 2N + N 3, ⟨trAtrA3⟩ = ⟨trBtrB3⟩ = 3N 2, ⟨trA4⟩ = ⟨trB4⟩ = 3N 2, ⟨trA2B2⟩ = N 2, ⟨trABtrAB⟩ = N 2, ⟨trAtrBtrAB⟩ = N, ⟨trAtrAB2⟩ = ⟨trAtrA2B⟩ = N 2, ⟨trA2trA2⟩ = ⟨trB2trB2⟩ = 2N 2 + N 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (15) 3 W-representation of complex two-matrix model Let us construct the complex two-matrix model Z2C = � d2M1d2M2 exp[−µtrM1M† 1 − µtrM2M† 2 + ∞ � k=0 tktr(M1M† 1)k + ∞ � k=0 gktr(M2M† 2)k + ∞ � l=1 ∞ � k1,···k2l=1 tk1,···k2ltr(M1M† 1)k1(M2M† 2)k2 · · · (M1M† 1)k2l−1(M2M† 2)k2l], (16) where M1 and M2 are N × N complex matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' By requiring the invariance of the integral (16) under the infinitesimal transformation M1 −→ M1 + ǫ(M1M† 1)nM1 (n ≥ 0) or M2 −→ M2 + ǫ(M2M† 2)nM2 (n ≥ 0), it gives the Virasoro constraints ¯LnZ2C = 0, (17) where ¯Ln = ∞ � n=0 2N ∂ ∂tn + ∞ � n=0 n−1 � s=1 ∂2 ∂ts∂tn−s + δn,0N 2 − µ ∞ � n=0 ∂ ∂tn+1 + ∞ � n=0 ∞ � k=0 ktk ∂ ∂tn+k + ∞ � n=0 ∞ � l=1 ∞ � k1,··· ,k2l=1 l � a=1 k2a−1tk1,··· ,k2l ∂ ∂tk1,k2,··· ,n+k2a−1,k2a,··· ,k2l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (18) 4 Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' the four constraints of (16) can be derived from the invariance of the integral under the following four infinitesimal transformations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (i) M1 −→ M1 + ǫ ∞ � n=0 (n + 1)tn+1(M1M† 1)nM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (ii) M1 −→ M1 + ǫ ∞ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m=0 [(n + 1) + (m + 1)]tn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1(M2M† 2)m+1(M1M† 1)nM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (iii) M1 −→ M1 + ǫ ∞ � r=1 ∞ � n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1=0 ¯ Ntn2r+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1(M2M† 2)n2+1(M1M† 1)n3+1 · · · · · (M2M† 2)n2r+1(M1M† 1)n2r+1M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (iv) M2 −→ M2 + ǫ ∞ � m=0 (m + 1)gm+1(M2M† 2)mM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' where ¯ N = (n2 + 1) + (n3 + 1) + · · · + (n2r+1 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' The sum of these constraints are µ ¯DZ2C = ¯WZ2C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (19) where ¯D = �4 i=1 ¯Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ¯W = �4 i=1 ¯Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' the operators ¯Di are ¯D1 = ∞ � n=0 (n + 1)tn+1 ∂ ∂tn+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ¯D2 = ∞ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m=0 ¯T1 ∂ ∂tn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ¯D2 = ∞ � n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1=0 ¯T2 ∂ ∂tn1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ¯D4 = ∞ � m=0 (m + 1)gm+1 ∂ ∂gm+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (20) and the operators ¯Wi are ¯W1 = ∞ � n=0 (n + 1)tn+1[2N ∂ ∂tn (1 − δn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='0) + n−1 � a=1 ∂ ∂ta ∂ ∂tn−a + ∞ � k=0 ktk ∂ ∂tn+k ] + N 2t1 + ∞ � n=0 ∞ � l=1 l � a=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 (n + 1)tn+1k2a−1tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−1+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ¯W2 = ∞ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m=0 ¯T1{(N ∂ ∂tn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1 + ∂2 ∂gm+1∂tn )(1 − δn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='0) + n−1 � a=1 ∂2 ∂ta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1∂tn−a + Nδn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='0 ∂ ∂gm+1 + ∞ � k=0 ktk ∂ ∂tn+k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1 + ∞ � l=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l[k1 ∂ ∂tn+k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l+m+1 +k2l−1 l � a=2 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−1+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + l � a=1 k2a−1−1 � s=1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n+k2a−1−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ]} ¯W3 = ∞ � n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1=0 ¯T2{[N ∂ ∂gn2+1 ∂ ∂tn3+n2r+1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n4+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1 + N ∂ ∂tn3+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+n2+2 ∂ ∂tn2r+1 +N r−1 � b=2 ∂ ∂tn3+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2b+n2+2 ∂ ∂tn2b+1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1 + n2r+1−1 � s=1 ∂ ∂ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1 ∂ ∂tn2r+1−s + r−1 � b=1 n2b+1 � s=1 ∂ ∂ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2b+1 ∂ ∂t¯ξ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2b+2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1 + N ∂ ∂tn2r+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1 ] · (1 − δn2r+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='0) + δn2r+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='0 ∂ ∂tn3+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r−1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+n2+2 + ∞ � k=0 ktk ∂ ∂tn2r+1+k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1 5 + ∞ � l=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l[k1 ∂ ∂tn3+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='¯ξ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='¯ξ3 + l � a=2 (k2a−1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='¯ξ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + k2a−1−1 � s=1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='¯ξ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l )]},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ¯W4 = N 2g1 + ∞ � m=0 (m + 1)gm+1[2N ∂ ∂gm (1 − δm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='0) + m−1 � a=1 ∂ ∂ga ∂ ∂gm−a + ∞ � k=0 kgk ∂ ∂gm+k ] + ∞ � m=0 ∞ � l=1 l � a=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 (m + 1)gm+1k2atk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a+m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (21) where ¯T1 = (n + m + 2)tn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ¯T2 = ¯ Ntn2r+1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1 and ¯ξ1 = n2r+1 + n2b+1 + 1 − s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ¯ξ2 = n2r+1 + k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ¯ξ3 = k2l + n2 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ¯ξ4 = (k2a−2 + n2 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' n3 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' n2r + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' n2r+1 + k2a−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ¯ξ5 = n2r+1 + k2a−1 − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Similar to the case of the Hermitian two-matrix model (11), the complex two-matrix model (16) can be realized by the W-representation Z2C = e 1 µ ¯ WeN(t0+g0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (22) There is also the compact expression of correlators ⟨ l1 � i=1 tr(M1M† 1)ki l2 � j=1 tr(M2M† 2)rj l3 � n=1 tr(M1M† 1)Sn,1(M2M† 2)Sn,2 · · · (M2M† 2)Sn,2pn ⟩ = l1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='l2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='l3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' m+1 � ρ1+ρ2+ρ3=1 � σ ¯P (σ(S1,1),··· ,σ(S1,2p1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='σ(Sl3,1),··· ,σ(Sl3,2pl3 )) (σ(k1),··· ,σ(kl1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='(σ(r1),··· ,σ(rl2)) µm+1(m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='λ(k1,··· ,kl1)λ(r1,··· ,rl2)λ(S1,1,··· ,S1,2p1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='Sl3,1,··· ,Sl3,2pl3 ) , (23) where ρ1 = l1 � i=1 ki, ρ2 = l2 � i=1 ri, ρ3 = l3 � i=1 (Si,1 + · · · + Si,2pi), and ¯P (σ(S1,1),··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,σ(Sl3,2pl3 )) (σ(k1),··· ,σ(kl1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='(σ(r1),··· ,σ(rl2)) is the coefficient of tk1 · · · tkl1gr1 · · · grl2tS1,1,··· ,S1,2p1 · · · tSl3,1,··· ,Sl3,2p3 in ¯W m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' we list some correlators ⟨trM1M† 1⟩ = ⟨trM2M† 2⟩ = 1 µN 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ⟨trM1M† 1trM2M† 2⟩ = 2 µ2 N 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ⟨trM1M† 1trM1M† 1⟩ = ⟨trM2M† 2trM2M† 2⟩ = 1 µ2 (N 2 + 1)N 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ⟨trM1M† 1M2M† 2⟩ = 2 µ2 N 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ⟨tr(M1M† 1)3⟩ = ⟨tr(M2M† 2)3⟩ = 6 µ3 (N 2 + N 4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ⟨trM1M† 1trM1M† 1trM1M† 1⟩ = ⟨trM2M† 2trM2M† 2trM2M† 2⟩ = 1 µ3 (N 2 + 2)(N 2 + 1)N 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ⟨trM1M† 1trM1M† 1trM2M† 2⟩ = ⟨trM1M† 1trM2M† 2trM2M† 2⟩ = 3 µ3 (N 4 + N 6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ⟨trM1M† 1trM1M† 1M2M† 2⟩ = ⟨trM2M† 2trM1M† 1M2M† 2⟩ = 6 µ3 (N 3 + N 5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ⟨trM1M† 1tr(M2M† 2)2⟩ = ⟨tr(M1M† 1)2trM2M† 2⟩ = 8 µ3 N 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ⟨trM1M† 1tr(M1M† 1)2⟩ = ⟨trM2M† 2tr(M2M† 2)2⟩ = 8 µ3 (N 3 + N 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (24) 4 W-representation of fermionic two-matrix model The fermionic matrix model ZF with the super integrability is given by [14] ZF = � dψd ¯ψ exp[N � k>0 pk k tr( ¯ψψ)k + N 2tr( ¯ψψ)] � dψd ¯ψ exp(N 2tr ¯ψψ) 6 = � R (−1 N )|R| DR(N)DR(−N) dR SR, (25) where ψ and ¯ψ are independent complex Grassmann-valued N × N matrices, and DR(N) = SR{pk = N}, dR = SR{pk = δk,1} are respectively the dimension of representation R for the linear group GL(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Let us extend (25) to the fermionic two-matrix model, Z2F = � dψd ¯ψdχd¯χ exp[−µtr ¯ψψ − µtr¯χχ + ∞ � k=0 tktr( ¯ψψ)k + ∞ � k=0 gktr(¯χχ)k + ∞ � l=1 ∞ � k1,···k2l=1 tk1,···k2ltr( ¯ψψ)k1(¯χχ)k2( ¯ψψ)k3 · · · ( ¯ψψ)k2l−1(¯χχ)k2l], (26) where χ and ¯χ are independent complex Grassmann-valued N × N matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' There are the Virasoro constraints ˇLnZ2F = 0, (27) where ˇLn = ∞ � n=0 n−1 � s=1 ∂2 ∂ts∂tn−s − δn,0N 2 − µ ∞ � n=0 ∂ ∂tn+1 + ∞ � n=0 ∞ � k=0 ktk ∂ ∂tn+k + ∞ � n=0 ∞ � l=1 ∞ � k1,··· ,k2l=1 l � a=1 k2a−1tk1,···k2l ∂ ∂tk1,··· ,n+k2a−1,k2a,··· ,k2l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (28) Similar to the complex two-matrix case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' by considering the following infinitesimal transfor- mations in the integral (26),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (i) ψ −→ ψ + ǫ ∞ � n=0 (n + 1)tn+1ψ( ¯ψψ)n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (ii) ψ −→ ψ + ǫ ∞ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m=0 [(n + 1) + (m + 1)]tn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1ψ(¯χχ)m+1( ¯ψψ)n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (iii) ψ −→ ψ + ǫ ∞ � r=1 ∞ � n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1=0 ˇ N ˇTψ(¯χχ)n2+1( ¯ψψ)n3+1 · · · (¯χχ)n2r+1( ¯ψψ)n2r+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (iv) χ −→ χ + ǫ ∞ � m=0 (m + 1)gm+1χ(¯χχ)m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' where ˇ N ˇT = (n2 + 1) + (n3 + 1) + · · · + (n2r+1 + 1)tn2r+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' we finally obtain µ ˇDZ2F = ˇWZ2F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (29) where ˇD = �4 i=1 ˇDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇW = �4 i=1 ˇWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' the operators ˇDi and ˇWi are ˇ D1 = ∞ � n=0 (n + 1)tn+1 ∂ ∂tn+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇ D2 = ∞ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m=0 ˇT1 ∂ ∂tn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇ D3 = ∞ � n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1=0 ˇT2 ∂ ∂tn1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇ D4 = ∞ � m=0 (m + 1)gm+1 ∂ ∂gm+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (30) 7 ˇ W1 = ∞ � n=0 (n + 1)tn+1[ ∞ � k=0 ktk ∂ ∂tn+k + n−1 � a=1 ∂ ∂ta ∂ ∂tn−a + ∞ � l=1 l � a=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 k2a−1 ˇT0 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−1+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ] − N 2t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇ W2 = ∞ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m=0 n−1 � a=1 ˇT1{ ∞ � l=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 ˇT0[ l � a=2 k2a−1 � s=0 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='ˇξ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + k1 � s=0 ∂ ∂tˇξ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ] + ∂ ∂ta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1 ∂ ∂tn−a − Nδn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='0 ∂ ∂gm+1 + ∞ � k=0 ktk ∂ ∂tn+k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1 },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇ W3 = ∞ � n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1=0 ˇT2{[ n2r+1−1 � s=1 ∂ ∂ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1 ∂ ∂tn2r+1−s + r−1 � b=1 n2b+1 � s=1 ∂ ∂ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2b+1 ∂ ∂tˇξ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2b+2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1 ] − Nδn2r+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='0 ∂ ∂tn3+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r−1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='ˇξ3 + ∞ � k=0 ktk ∂ ∂tn2r+1+k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1 + ∞ � l=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 ˇT0[ k1 � s=0 ∂ ∂ts+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='ˇξ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + l � a=2 k2a−1 � s=0 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='s+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='ˇξ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ]},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇ W4 = ∞ � m=0 (m + 1)gm+1[ m−1 � a=1 ∂ ∂ga ∂ ∂gm−a + ∞ � k=0 kgk ∂ ∂gm+k + ∞ � l=1 l � a=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 k2a ˇT0 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a+m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ] − N 2g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (31) where ˇT0 = tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇT1 = (n + m + 2)tn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇT2 = ¯ Ntn2r+1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+1 and ˇξ0 = (a + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' m + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' n + k2a−1 − k − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇξ1 = (s + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' m + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' n + k1 − s − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇξ2 = n2r+1 + n2b+1 + 1 − s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇξ3 = n2r + n2 + 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇξ4 = n2r+1 + k1 − 1 − s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˇξ5 = (n2 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' n2r + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' n2r+1 + k2a−1 − s − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' We find that the fermionic two-matrix model (26) can be realized by the W-representation Z2F = e 1 µ ˇ W eN(t0+g0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (32) The compact expression of correlators is ⟨ l1 � i=1 tr( ¯ψψ)ki l2 � j=1 tr(¯χχ)rj l3 � n=1 tr( ¯ψψ)Sn,1(¯χχ)Sn,2 · · · ( ¯ψψ)Sn,2pn−1 (¯χχ)Sn,2pn ⟩ = l1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='l2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='l3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' m+1 � ρ1+ρ2+ρ3=1 � σ ˇP (σ(S1,1),··· ,σ(S1,2p1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='σ(Sl3,1),··· ,σ(Sl3,2pl3 )) (σ(k1),··· ,σ(kl1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='(σ(r1),··· ,σ(rl2)) µm+1(m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='λ(k1,··· ,kl1)λ(r1,··· ,rl2)λ(S1,1,··· ,S1,2p1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='Sl3,1,··· ,Sl3,2pl3 ) , (33) where ρ1 = l1 � i=1 ki, ρ2 = l2 � i=1 ri, ρ3 = l3 � i=1 (Si,1 + · · · + Si,2pi), and ˇP (σ(S1,1),··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,σ(Sl3,2pl3 )) (σ(k1),··· ,σ(kl1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='(σ(r1),··· ,σ(rl2)) is the coefficient of tk1 · · · tkl1gr1 · · · grl2tS1,1,··· ,S1,2p1 · · · tSl3,1,··· ,Sl3,2p3 in ˇW m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' 8 Here we list some correlators ⟨tr ¯ψψ⟩ = ⟨tr¯χχ⟩ = − 1 µN 2, ⟨tr ¯ψψtr¯χχ⟩ = 2 µ2 N 4, ⟨tr ¯ψψtr ¯ψψ⟩ = ⟨tr¯χχtr¯χχ⟩ = 1 µ2 (N 2 − 1)N 2, ⟨tr ¯ψψ ¯χχ⟩ = − 2 µ2 N 3, ⟨tr( ¯ψψ)3⟩ = ⟨tr(¯χχ)3⟩ = 6 µ3 (−N 2 + N 4), ⟨tr ¯ψψtr ¯ψψtr ¯ψψ⟩ = ⟨tr¯χχtr¯χχtr¯χχ⟩ = 1 µ3 (N 2 + 2)(N 2 − 1)N 2, ⟨tr ¯ψψtr ¯ψψtr¯χχ⟩ = ⟨tr¯χχtr ¯ψψtr¯χχ⟩ = 3 µ3 (N 4 − N 6), ⟨tr ¯ψψtr ¯ψψ ¯χχ⟩ = ⟨tr¯χχtr ¯ψψ ¯χχ⟩ = − 1 µ3(6N 3 + 2N 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (34) 5 Conclusion We have constructed the Hermitian, complex and fermionic two-matrix models with infinite set of variables and presented their Virasoro constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' W-representation is important for understanding matrix model, since it provides a dual formula for partition function through dif- ferentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' By considering the particular infinitesimal transformations of integration variables in the partition functions, we finally derived the desired operators preserving and increasing the grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Thus it can be shown that the two-matrix models constructed in this paper can be realized by the W-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Moreover, by means of the W-representations, we derived the compact expressions of correlators for these two-matrix models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' It should be noted that there are the infinite set of variables in these two-matrix models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' It leads to that we can not give their character expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' For further research, it would be interesting to study the case of β-deformed two-matrix models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Appendix A The operators ˆWi in (5) ˆ W1 = ∞ � l=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 {t1T1[δk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tk3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l+k2 + l � a=2 δk2a−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−2+k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ] + ∞ � n=0 l � a=1 k2a−1(n + 1)tn+1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n+k2a−1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l } + ∞ � k2=1 t1t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2 ∂ ∂gk2 + t2 1N + 2t2N 2 + ∞ � n=0 (n + 1)tn+1[ n−2 � b=1 ∂2 ∂tb∂tn−1−b + ∞ � k=0 ktk ∂ ∂tn+k−1 ] + ∞ � n=2 2N(n + 1)tn+1 ∂ ∂tn−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˆ W2 = ∞ � l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 T2{(1 − δk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1)( ∂ ∂tk1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l+n + ∂ ∂tk1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ) +δk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tk3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l+n+k2 + l � a=2 (1 − δk2a−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1) ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−2+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + l−1 � a=2 δk2a−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−2+k2a+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l } + ∞ � l=2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' n=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 T2δk2l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l−2+n+k2l + ∞ � l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n=1 l � a=1 ∞ � k2a−1=3 k2a−1−2 � b=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 T2[(1 − δa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1) ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−1−1−b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l +δa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k1−1−b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ] + ∞ � n=1 (n + 1)t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n[ ∞ � k=2 ktk ∂ ∂tk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n + ∞ � k2=1 t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2 ∂ ∂gk2+n + t1 ∂ ∂gn ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' 9 ˆ W3 = ∞ � r=1 ∞ � n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r=1 T3{Nδn1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+n2 + (1 − δn1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1)( n1−2 � a=1 ∂ ∂ta ∂ ∂tn1−1−a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r +N ∂ ∂tn1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r + ∂ ∂tn1−1 ∂ ∂tn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r ) + r � s=2 n2s−1−2 � a=0 (1 − δn2s−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1) ∂ ∂tn1+a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2s−2 ∂ ∂tn2s−1−1−a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r + r−1 � s=2 [δn2s−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tn1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2s−2 + (1 − δn1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1) ∂ ∂tn1+n2s−1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2s−2 ∂ ∂tn2s+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+n2s ] + (1 − δn2r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1) ∂ ∂tn1+n2r−1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r−2 ∂ ∂gn2r + ∞ � k=0 ktk ∂ ∂tn1+k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r +δn2r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂2 ∂tn1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r−2∂gn2r + ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 T1[ l � i=1 k2i−1−2 � s=0 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2i−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='s+n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='ξ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + ∂ ∂tn1+k1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n1+k2i−1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+k2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l } + ∞ � n1=2 ∞ � n2=1 (n1 + 1 + n2)tn1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2 ∂2 ∂tn1−1∂gn2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˆ W4 = ∞ � r=1 ∞ � n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r=1 T4{ n3−2 � b=1 ∂2 ∂tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2∂tn3−1−b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r + (1 − δn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1)( ∂2 ∂gn2∂tn3−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r + ∂2 ∂tn3−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='∂tn5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+n4 ) + δn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂2 ∂gn2∂tn5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+n4 + r−1 � a=3 [δn2a−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2a−2+n2 ∂ ∂tn2a+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r + (1 − δn2a−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1)( ∂ ∂tn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2a−1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2a ∂ ∂tn2a+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+n2a + ∂ ∂tn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2a−2+n2 ∂ ∂tn2a−1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2a··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r )] + [δn2r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r−2+n2 + (1 − δn2r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1) ∂ ∂tn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r−1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2 ] ∂ ∂gn2r + r � a=3 n2a−1−2 � b=1 ∂ ∂tn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2a−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2 ∂ ∂tn2a−1−1−b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r + t1 ∂ ∂tn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+n2 + ∞ � k=2 ktk ∂ ∂tk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r + ∞ � l=1 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 T1[δk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l+n2 + (1 − δk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1)( ∂ ∂tk1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + ∂ ∂tn3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k1−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l+n2 ) + l � b=2 (1 − δk2b−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1)( ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2b−3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='ξ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='ξ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='n2r+k2l ) + l � b=2 δk2b−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2b−1+n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='ξ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ] + ∞ � l=1 l � a=1 ∞ � k2a−1=3 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' k2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 k2a−1−2 � s=1 T1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='ξ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' ˆ W5 = g2 1N + ∞ � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k=0 (n + 1)gn+1kgk ∂ ∂gn+k−1 + ∞ � kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k3=1 g1tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tk1+k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2 + ∞ � k1=1 tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1g1 ∂ ∂tk1 + ∞ � l=1 ∞ � n=0 ∞ � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l=1 T5[δk2l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 l−1 � a=1 ∂ ∂tk1+k2l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l−2 k2a ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='ξ6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + ∂ ∂tk1+k2l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l−2 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='ξ7 + δn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='0(1 − δk2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1) ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l + δn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='0 l−1 � a=1 δk2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1 ∂ ∂tk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2a−1+k2a+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='k2l ] 10 + ∞ � n=1 n−2 � s=1 (n + 1)gn+1 ∂ ∂gs ∂ ∂gn−1−s + ∞ � n=2 2N(n + 1)gn+1 ∂ ∂gn−1 + 2g2N 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='1) where T1 = tk1,··· ,k2l, T2 = (n + 1)t1,ntk1,··· ,k2l, T3 = N1tn1+1,n2,··· ,n2r, T4 = N2t1,n2,··· ,n2r, T5 = (n + 1)gn+1tk1,···k2l and ξ1 = k2i−1 − 1 − s, ξ2 = (k2b−2 + n2, · · · , n2r, k2b−1 − 1), ξ3 = (k2b−1 − 1, n2), ξ4 = (n3, · · · , n2r + k2b), ξ5 = (n2, · · · , n2r, k2a−1 − 1 − s), ξ6 = k2a + n − 1, ξ7 = k2l + n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' Acknowledgment This work is supported by the National 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for multi-character partition functions and their β-deformations, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content='12763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf'} diff --git a/ZNA0T4oBgHgl3EQfFv_a/content/tmp_files/2301.02038v1.pdf.txt b/ZNA0T4oBgHgl3EQfFv_a/content/tmp_files/2301.02038v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..51f2fdf691e581bbc062f3c8f870e48c77cb3fea --- /dev/null +++ b/ZNA0T4oBgHgl3EQfFv_a/content/tmp_files/2301.02038v1.pdf.txt @@ -0,0 +1,1651 @@ +arXiv:2301.02038v1 [math.DG] 5 Jan 2023 +Contemporary Mathematics +Vinogradov’s Cohomological Geometry +of Partial Differential Equations +Fabrizio Pugliese, Giovanni Sparano, and Luca Vitagliano +to the memory of our teacher and colleague Alexandre Mikhailovich Vinogradov +Abstract. Secondary Calculus is a formal replacement for differential cal- +culus on the space of solutions of a system of possibly non-linear partial dif- +ferential equations and it is essentially due to Alexandre M. Vinogradov and +his collaborators. Many coordinate free properties of PDEs find their natu- +ral place in Secondary Calculus including: symmetries and conservation laws, +variational principles and the coordinate free aspects of the calculus of vari- +ations, recursion operators and Hamiltonian structures, etc. +The building +blocks of this language are horizontal cohomologies of diffieties, i.e. infinite +prolongations of PDEs, and their versions with local coefficients. The main +paradigm of Secondary Calculus is the principle, due to A. M. Vinogradov, +roughly stating that: differential calculus on the space of solutions of a PDE is +calculus up to homotopy on the horizontal De Rham algebra of the associated +diffiety. We will review the fundamentals of Secondary Calculus including its +main motivations. In the last part of the paper, we will try to explain the role +of homotopy in the theory. +Introduction +Systems of algebraic equations can be encoded geometrically in an algebraic +variety. This apparently harmless observation is the starting point for the develop- +ment of such a successful branch of modern Mathematics as Algebraic Geometry. +One of the leading principles behind most of the mathematical work of Alexandre +M. Vinogradov is that Partial Differential Equations (PDEs) should be treated in +a similar way. Namely, let x = (x1, . . . , xn) be some independent variables, let +u = (u1, . . . , um) be some dependent variables, and let +(0.1) +E0 : Fa(x, . . . , uI, . . .) = 0, +a = 1, . . . , r, be a system of possibly non-linear PDEs, where a multi-index I +denotes multiple partial derivatives with respect to the x’s. Adding to (0.1) all its +2020 Mathematics Subject Classification. Primary 58A15, 58A20; Secondary 17B56, 55T25, +58A12, 70S05. +Key words and phrases. Partial differential equations, jet spaces, horizontal cohomology, +C -spectral sequence, secondary calculus. +1 + +2 +FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO +total derivatives results in a new but equivalent system of infinitely many PDEs +(0.2) +DJFa(x, . . . , uI, . . .) = 0. +Here DJ denotes multiple total derivatives of arbitrarily high order with respect +to the x’s. Now, interpret (x, . . . , uI, . . .) as independent coordinates on an ∞- +dimensional manifold J∞. Then (0.2) can be interpreted as defining a (generically +∞-dimensional) submanifold E ⊆ J∞, the ∞-prolongation of E0. This submani- +fold is canonically equipped with an involutive distribution C , called the Cartan +distribution, spanned by the total derivatives Di|E , i = 1, . . . , n. The pair (E , C ) +is, in the terminology of Vinogradov and his school, a diffiety1 (from the words +diff erential equation and variety) and contains most of the relevant information +about the original system E0 (see, e.g., the extensive monographs [2, 11, 27] on +the subject). According to Vinogradov, the diffiety (E , C ) should be considered as +the ultimate geometric portrait of E0. In other words, diffieties are to PDEs what +algebraic varieties are to algebraic equations. However the geometry of a diffiety +is much more intricate than that of an algebraic variety. Namely, while solutions +of a system of algebraic equations are points of an algebraic variety and they do +not possess any internal structure, solutions of a system of PDEs E0 are integral +submanifolds of the Cartan distribution on the diffiety (E , C ) and they do possess +internal structure. As a consequence the space of solutions of a PDE share some +features of the leaf space of a foliation (and it is the leaf space of a foliation when +dim E < ∞) and should be treated as a stack rather than as an ordinary space +[17, 1, 4]. This is exactly where homological and homotopical algebra come into +play when trying to develop a general theory of PDEs. A fundamental achievement +of Vinogradov is that there is a cohomological replacement for differential calculus +on the space of solutions of a PDEs. Vinogradov used to call such formal calculus +Secondary Calculus, because of an analogy with secondary quantization in physical +field theory [26, 2, 27]. The main building blocks of Secondary Calculus are the +leaf-wise cohomologies of the Cartan distribution, often called horizontal De Rham +cohomologies. Most of the coordinate free properties of a PDE can be expressed in +terms of horizontal cohomologies: symmetries, conservation laws, variational prin- +ciples, etc., to name a few. Additionally, horizontal cohomologies can be interpreted +as smooth functions, vector fields, differential forms on the space of solutions of the +PDE. One of the main features of horizontal cohomologies supporting this interpre- +tation is that they possess the correct algebraic structures (those that we expect +from functions, vector fields, differential forms). In summary, Secondary Calculus +is the collection of all these algebraic structures. +In the ultimate idea of Vinogradov this fascinating constructions should be +studied in their homotopy aspects. More precisely, Vinogradov conjectured that +what is important is not really horizontal cohomology but rather the homotopy +type of the horizontal De Rham algebra, and the homotopy category (or even the +derived category) of differential graded modules over it [27, Chapter 5]. Notice that +this conjecture (together with the interpretation of the space of solutions of a PDE +as a stack), is somehow complementary to the interpretation of a PDE as a derived +zero locus and, in the particular case of an Euler-Lagrange PDE, as a derived +critical locus (see, e.g., [23], and references therein), and suggests a relationship +to Homotopical and Derived Geometry [20, 21, 19] that, unfortunately, to the +1To the best of our knowledge, the term diffiety appeared for the first time in [24]. + +VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES +3 +best of our knowledge, has not been yet investigated. At this stage, we can only +speculate that the space of solutions of a PDE should be ultimately treated as a +higher derived stack [18]. +In this exposition we will focus on the theory rather than on applications or +the explicit computation of the involved cohomologies. We will omit the proofs. +They can be found in the cited references. The paper is divided into three sections: +In Section 1 (Geometry of PDEs) we explain how to encode a system of PDEs +in a geometric object: a diffiety. We also discuss the main geometric structure +possessed by a diffiety, namely its Cartan distribution. Our main sources for this +section are [2, 11, 10]. +In Section 2 (Homology of PDEs) we show that there +is a natural cohomology attached to a diffiety (hence to a PDE), the horizontal +cohomology. This is the core of the paper, where Vinogradov’s Secondary Calculus +is discussed. Horizontal cohomology is a generalization of the leaf-wise cohomology +of a foliated manifold and contains important coordinate free information on a +PDE. We provide an interpretation of the main horizontal cohomologies. Horizontal +cohomologies can also be seen as the building blocks of a differential calculus on +the space of solutions of a PDE, what we call Secondary Calculus. One of the main +supporting facts for the latter interpretation is that horizontal cohomologies are +equipped with the correct algebraic structures for smooth functions, vector fields, +differential forms, etc. Our main sources here are [2, 11, 27], see also [25, 28]. In +Section 3 (Homotopy of PDEs) we present the latest developments. This section +is an extremely compact review of the papers [29, 30] by the third author. We +show that the main algebraic structures on horizontal cohomologies do all come +from appropriate structures up to homotopy on horizontal cochains. This supports +Vinogradov’s idea that the appropriate category to develop Secondary Calculus +is (a subcategory of) the derived category of DG modules over the horizontal De +Rham algebra. +Contents +Introduction +1 +1. +Geometry of PDEs +4 +1.1. +jet spaces +4 +1.2. +the Cartan distribution +6 +1.3. +diffieties +8 +2. +Cohomology of PDEs +10 +2.1. +horizontal cohomology +10 +2.2. +interpreting horizontal cohomologies +12 +2.3. +algebraic structures on horizontal cohomologies +15 +3. +Homotopy of PDEs +18 +3.1. +homotopy algebras +18 +3.2. +the LR∞-algebra of secondary vector fields +20 +3.3. +the A∞-algebra of secondary differential operators +22 +References +25 + +4 +FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO +1. Geometry of PDEs +In this section we show that a system of, generically non-linear, PDEs can be +encoded into a geometric object: a diffiety. We will work in the rather general +setting of a PDEs imposed on submanifolds N of a fixed dimension n in a manifold +P (see [10]), in contrast with most of the literature where only PDEs imposed on +sections of a fibration (or a fiber bundle) are considered. +1.1. jet spaces. Let n, m be non-negative integers, and let P be a manifold +of dimension n + m. We begin providing an intrinsic definition of derivatives of +an n-dimensional submanifold N ⊆ P. In the following, n should be interpreted +as the number of independent variables, m as the number of dependent variables, +and P as the space parameterizing both independent and dependent variables. In +other words, in this theory, dependent and independent variables can be mixed +and swapped. +Around every point p ∈ N there are coordinates (xi, uα) on P, +i = 1, . . . , n, α = 1, . . . , m, adapted to N in the sense that, in these coordinates, N +looks like the graph of a map: +(1.1) +N : uα = f α(x). +We will need to take multiple partial derivatives of the f’s with respect to the x’s. +We adopt the following notation. Let h be a non-negative integer and let I = i1 · · · ih +be a multiindex, i.e. a (possibly empty) word containing h letters iℓ ∈ {1, . . . , n}, +ℓ = 1, . . . , h. We identify two words if they only differ by a permutation of their +letters. We also compose two words by concatenation. Denote |I| := h and call it +the lenght of the multiindex I. Finally denote +∂|I|f α +∂xI +:= +∂hf α +∂xi1 . . . ∂xih . +If there is another n-dimensional submanifolds ˜N through p, i.e. p ∈ N ∩ ˜N, +then around p there are coordinates (xi, uα) on P adapted to both N, ˜N: +N : uα = f α(x), +and +˜N : uα = ˜f α(x). +In this case we say that N and ˜N are tangent up to order k = 0, 1, . . . , ∞ at p if +∂|I|f α +∂xI (p) = ∂|I| ˜fα +∂xI (p), +for all multiindexes I with |I| ≤ k, +and we write N ∼k +p ˜N. Tangency up to order k at p is a well-defined equivalence +relation on submanifolds through p. In particular, it is independent of the choice of +adapted coordinates. The equivalence class of N is denoted N k +p and called the k-jet +of N at p. In other words, k-jets at p encode partial derivatives of n-dimensional +submanifolds at p up to order k. The space of tangency classes is denoted Jk +p (P, n). +Finally put +Jk(P, n) = +� +p∈P +Jk +p (P, n). +We call Jk(P, n) the k-jet space of n-dimensional submanifolds of P. It is clear +that J0(P, n) identifies naturally with P and J1(P, n) identify with the bundle of +n-Grassmannians in the fibers of the tangent bundle T P. More precisely, the 0-jet +of N at p identifies with p itself, while the 1-jet of N at p identifies with the tangent +space TpN. For a general k, the space Jk(P, n) can be coordinatized as follows. +Choose coordinates (xi, uα) on P, and let U ⊆ Jk(P, n) be the subset consisting + +VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES +5 +of k-jets of submanifolds for which (xi, uα) are adapted coordinates. We define +coordinates (xi, uα +I ) on U by putting +xi� +N k +p +� +:= xi(p), +and +uα +I +� +N k +p +� += ∂|I|f α +∂xI (p), +for all N locally given by (1.1), i = 1, . . . , n, α = 1, . . . , m, and |I| ≤ k. The (xi, uα) +are called jet coordinates, they cover Jk(P, n) and, for k < ∞, they give to Jk(P, n) +the structure of a smooth manifold. For all h ≤ k ≤ ∞ the are projections +Jk(P, n) → Jh(P, n), +N k +p �→ N h +p , +and by Borel Lemma J∞(P, n) identifies with the inverse limit of the tower of +surjective submersions +J0(P, n) ←− · · · ←− Jk−1(P, n) ←− Jk(P, n) ←− · · · . +This gives to the ∞-jet space the structure of a pro-finite dimensional manifold +(see, e.g., [2, 27], see also [3]). +Remark 1.1. For the reader more familiar with jets of sections, we mention +that, when P is fibered over some n-dimensional manifold M, then, for all k ≤ ∞, +the space Jk(P, M) of k-jets of sections of P over M can be recovered as the open +and dense submanifold of Jk(P, n) consisting of jets of n-dimensional submanifolds +transverse to the projection P → M. The open embedding Jk(P, M) ֒→ Jk(P, n) +maps the k-jet at x ∈ M of a (possibly local) section s of P → M to the k-jet of +its image at s(x). +Differential calculus on J∞(P, n) can be defined algebraically. For instance, +we define smooth functions C∞(J∞(P, n)) on J∞(P, n) as the direct limit of the +algebra filtration +(1.2) +C∞� +J0(P, n) +� +֒→ · · · ֒→ C∞� +Jk−1(P, n) +� +֒→ C∞� +Jk(P, n) +� +֒→ · · · . +In other words a smooth function on J∞(P, n) is a smooth function of the inde- +pendent variables xi, the dependent variables uα, and their derivatives up to some +finite order k, but k can be arbitrarily high (sometimes they are defined as func- +tions which are only locally of this type). Vector fields, differential forms, etc., on +J∞(P, n) are defined in a way compatible with the filtration (1.2). We will not +insist on these technicalities and we refer to [2] for details. +Remark 1.2. We stress that, in this ∞-dimensional setting, some important +results in finite dimensional differential geometry might fail. To mention a few: the +inverse function theorem, existence and uniqueness of the flow of a vector field, the +Frobenius theorem, etc. +Any n-dimensional submanifold N ⊆ P defines a new n-dimensional submani- +fold +N k := +� +N k +p : p ∈ N +� +in Jk(P, n), called the k-jet prolongation of N. +If N locally looks like (1.1) in +coordinates, then N k locally looks like +N k : uα +I = ∂|I|f α +∂xI (x), +|I| ≤ k, +in jet coordinates. + +6 +FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO +Remark 1.3. Let N ⊆ P be an n-dimensional submanifold. For all k ≤ ∞, +the map +jk : N → N k, +p �→ N k +p +is a diffeomorphism. In particular N embeds into Jk(P, n) for all k. When P is +fibered over some n-dimensional manifold M, and s : M → P is a (possibly local) +section of P over N, then the usual k-jet prolongation jks of s can be recovered as +the composition +jks : M +s +−→ im s +jk +−→ (im s)k ֒→ Jk(P, M). +Remark 1.4. Let r be a non-negative integer and let Q ⊆ P be an (n + +r)-dimensional submanifold. +Then, for all k ≥ 0, the k-jet space Jk(Q, n) is a +submanifold in the k-jet space Jk(P, n) in the obvious natural way. +1.2. the Cartan distribution. The main geometric structure on jet spaces +is the Cartan distribution. +There are several equivalent ways to define it. +The +one we present here will be also useful in defining the prolongation of a PDE in +the next subsection. We begin noticing that a section of the fibration J1(P, n) → +J0(P, n) ∼= P can be interpreted as an n-dimensional distribution on P. Now, for +every 1 ≤ k ≤ ∞, the k-jet space is canonically equipped with a rank n distribution +along the projection Jk(P, n) → Jk−1(P, n), i.e. a smooth map +C : Jk(P, n) −→ J1� +Jk−1(P, n), n +� +such that the following diagram commutes +J1� +Jk−1(P, n), n +� +� +Jk(P, n) +C +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +� Jk−1(P, n) +. +The Cartan distribution C is defined as follows. For z = N k +p ∈ Jk(P, n), denote by +¯z = N k−1 +p +∈ Jk−1(P, n) its projection. Now put +Cz := +� +N k−1�1 +¯z = T¯z +� +N k−1� +∈ J1� +Jk−1(P, n), n +� +. +In other words Cz is the 1-jet at ¯z of the (k − 1)-jet prolongation of N. In local +coordinates +(1.3) +C ∗(uα +I,j) = uα +Ij, +|I| ≤ k − 1, +j = 1, . . . , n, +where, in the left hand side, we denoted by (xi, uα +I,j) the jet coordinates on the 1-jet +space J1� +Jk−1(P, n), n +� +(corresponding to the coordinates (xi, uα +I ) on Jk−1(P, n)), +and, in the right hand side, we are using concatenation of multiindexes. +Remark 1.5. The Cartan distribution C : Jk(P, n) −→ J1� +Jk−1(P, n), n +� +is an +embedding that we often use to interpret the k-jet space Jk(P, n) as a submanifold +in the jet space J1� +Jk−1(P, n), n +� +. More generally for any two non-negative integers +h, k, there is an embedding +Jh+k(P, n) ֒→ Jh� +Jk(P, n), n +� +given by +N h+k +p +�→ (N k)h +p, +p = N k +p . + +VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES +7 +In the next subsection we will use this embedding to interpret Jh+k(P, n) as a +submanifold in Jh� +Jk(P, n), n +� +. +For k = ∞, the Cartan distribution is a honest rank n distribution on J∞(P, n), +and (1.3) shows that it is locally spanned by the total derivatives, i.e. the following +vector fields +Di := +∂ +∂xi + +� +|I|<∞ +uα +Ii +∂ +∂uα +I +, +i = 1, . . . , n. +Dually, the annihilator C 0 ⊆ T ∗J∞(P, n) is spanned by the Cartan forms +duα +I − uα +Iidxi, +α = 1, . . . , m, +|I| ≤ ∞. +Here, and in the rest of the paper, we adopt the Einstein summation conven- +tion over pairs of lowercase-uppercase indexes. +We will not adopt the Einstein +convention for multiindexes. +Proposition 1.6. The Cartan distribution on J∞(P, n) is involutive, i.e. the +commutator of two sections of C lays in C as well. Additionally it detects ∞-jet +prolongations in the following sense: a submanifold S ⊆ J∞(P, n) is locally of the +form S = N ∞ for some n-dimensional submanifold N ⊆ P if and only if it is an +n-dimensional integral submanifold of C . +So, morally, the ∞-jet prolongation construction N �→ N ∞ identifies n-dimensional +submanifolds of P with n-dimensional integral submanifolds of C . In the following, +Cartan distribution will always mean the Cartan distribution on J∞(P, n) (unless +otherwise stated). +For their importance in the next section, we now discuss infinitesimal sym- +metries of the Cartan distribution C , i.e. vector fields Y on J∞(P, n) such that +[Y, Γ(C )] ⊆ Γ(C ). Denote by XC the Lie algebra of infinitesimal symmetries of C +and notice that, by involutivity, Γ(C ) ⊆ XC is an ideal. We want to show that +the quotient Lie algebra XC /Γ(C ) identifies with sections of an appropriate vector +bundle V → J∞(P, n). We first define V . So let z = N ∞ +p +∈ J∞(P, n). The fiber of +V over z is then the m-dimensional vector space +Vz := TpP/TpN. +In the following, we denote by π : J∞(P, n) → P the projection. Now, let Y ∈ XC. +We define a section Y of V as follows. For z = N ∞ +p +∈ J∞(P, n) put +Y z := dπ(Yz) mod TpN ∈ TpP/TpN = Vz. +Proposition 1.7. The assignment Y +�→ Y defines an R-linear surjection +XC → Γ(V ) with kernel given by Γ(C ). Hence there is a canonical short exact +sequence of vector spaces +0 −→ Γ(C ) −→ XC −→ Γ(V ) −→ 0. +In particular, Γ(V ) inherits from XC a Lie bracket. +The Lie bracket on Γ(V ) is sometimes called the higher Jacobi bracket and +denoted by {−, −}. We conclude this section describing it in local coordinates. +First notice that Γ(V ) is locally spanned by the sections vα defined as follows. For +z = N ∞ +p +∈ J∞(P, n) put +vα|z := +∂ +∂uα |p mod TpN ∈ TpP/TpN = Vz. + +8 +FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO +The isomorphism Γ(V ) ∼= XC/Γ(C ) is now locally given by +χ �→ Зχ mod Γ(C ) +where, for χ = χαvα ∈ Γ(V ), we denoted by Зχ the (local) infinitesimal symmetry +of C locally given by +(1.4) +Зχ = +� +|I|<∞ +DIχα ∂ +∂uα +I +, +and, for any multiindex i1 · · · ih, +Di1···ih := Di1 ◦ · · · ◦ Dih. +We mention for the reader unfamiliar with the cyrillic alphabet, that the script +З appearing in (1.4) is pronounced [e], like in set. +A vector field of the form Зχ is sometimes called an evolutionary vector field. +Notice that evolutionary vector fields are only locally defined on ∞-jets of subman- +ifolds (while one can give a coordinate free meaning to (1.4) on ∞-jets of sections +of a fibration, see e.g. [2, Chapter 4, Section 2.4] or [11, Section 3.9] ). It easily +follows from (1.4) that, if χ and ψ are sections of V locally given by χ = χαvα and +ψ = ψβvβ, then their higher Jacobi bracket {χ, ψ} is the section locally given by +{χ, ψ} = +� +Зχψα − Зψχα� +vα = +� +|I|<∞ +� +DIχβ ∂ψα +∂uβ +I +− DIψβ ∂χα +∂uβ +I +� +vα. +1.3. diffieties. We now use jets to define PDEs and present their geometric +portraits: diffieties. Let P be a manifold of dimension n + m. +Definition 1.8. A system of k-th order partial differential equations imposed +on n-dimensional submanifolds of P, or simply a PDE, is a closed submanifold +E0 ⊆ Jk(P, n) of the k-jet space. A solution of a PDE E0 ⊆ Jk(P, n) is an n- +dimensional submanifold N ⊆ P such that N k ⊆ E0. +Remark 1.9. As a minimal regularity condition on a PDE E0 ⊆ Jk(P, n) we +will always assume dim E0 ≥ n so that the existence of solutions is not excluded a +priori by trivial dimensional reasons. +For a PDE E0 ⊆ Jk(P, n), locally, in jet coordinates, we have +(1.5) +E0 : Fa(x, . . . , uI, . . .) = 0, +|I| ≤ k, +for some local smooth functions Fa ∈ C∞(Jk(P, n)). If N ⊆ P is an n-dimensional +submanifold locally given by +N : uα = f α(x), +then N is a solution of E0 iff +Fa +� +x, . . . , ∂|I|f +∂xI , . . . +� += 0. +This motivates Definition 1.8. +In Algebraic Geometry, given a system of algebraic equations, it is natural to +take all their algebraic consequences. In other words, given a subset of polynomials, +it is natural to consider the ideal that they span. Similarly, given a PDE (1.5), it +is natural (besides their smooth consequences) to take all their total derivatives, +this can be done in a coordinate free way via a construction known as prolongation +that we now describe. Let E0 ⊆ Jk(P, n) be a PDE, and let q = 0, 1, . . . , ∞. + +VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES +9 +Definition 1.10. The q-prolongation of E0 is the subset Eq ⊆ Jk+q(P, n) of +the (k + q)-jet space defined by +(1.6) +Eq := Jq(E0, n) ∩ Jk+q(P, n). +The ∞-prolongation will be often denoted simply E . +Definition 1.10 requires some little explanations. As E0 is a submanifold in +Jk(P, n), its jet space Jq(E0, n) is a submanifold in the jet space Jq(Jk(P, n), n) +as in Remark 1.4. +The jet space Jk+q(P, n) can be seen as a submanifold in +Jq(Jk(P, n), n) as well via the embedding in Remark 1.5. This explains the in- +tersection in (1.6). We stress that we will always understand the prolongation Eq +as a subset in Jk+q(P, n). If E0 is locally given by (1.5), then its q-prolongation is +locally given by +Eq : DJFa(x, . . . , uI, . . .) = 0, +|J| ≤ q. +This shows that Eq encodes in a coordinate free way the total derivatives of E0 up +to order q, as desired. +Remark 1.11. For a generic E0 ⊆ Jk(P, n) its q-th prolongation Eq is not a +submanifold in Jk+h(P, n). When it is a submanifold, then it is clearly a PDE with +the same solutions as E0. In the following, we will always assume that the ∞-th +prolongation E = E∞ is a submanifold in J∞(P, n). This happens, e.g., when E0 is a +formally integrable PDE. Our assumptions are not a great loss of generality accord- +ing to Cartan-Kuranishi Prolongation Theorem which roughly states that, under +mild regularity conditions, every PDE becomes formally integrable after finitely +many 1-prolongations. +Now on we concentrate on the ∞-prolongation E ⊆ J∞(P, n). +Proposition 1.12. Let E0 ⊆ Jk(P, n) be a PDE. Assume that its ∞-prolongation +E ⊆ J∞(P, n) is a submanifold. Then the Cartan distribution restricts to E in the +sense that +Cz ⊆ TzE +for all z ∈ E . +Hence the restriction C : E → J1(E , n) is a well-defined rank n involutive dis- +tribution on E . Additionally it detects solutions of E0 in the following sense: a +submanifold S ⊆ E is locally of the form S = N ∞ for some solution N of E0 if and +only if it is an n-dimensional integral submanifold of C . +The restriction of the Cartan distribution to the ∞-prolongation of a PDE will +be denoted by C as well and called the Cartan distribution. +Definition 1.13. A diffiety (from the words differential equation and variety) +is a pair (E , C ) where E is the ∞-prolongation of some PDE and C is the Cartan +distribution on E . +Proposition 1.12 now shows that a diffiety (E , C ) contains most of the relevant +information about the PDE E0 defining it. +We conclude this section briefly discussing infinitesimal symmetries of a PDE. +Let (E , C ) be the diffiety corresponding to a PDE E0. +Definition 1.14. An infinitesimal symmetry of E0 is a vector field Y ∈ X(E ) +such that [Y, Γ(C )] ⊆ Γ(C ). A non-trivial infinitesimal symmetry is an infinitesimal +symmetry modulo trivial ones, i.e. sections of C . + +10 +FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO +Remark 1.15. The terminology “trivial, non-trivial symmetries” in Definition +1.14 is motivated by the following facts. Let X be a section of the Cartan dis- +tribution on J∞(P, n) and let (E , C ) be any diffiety, then X is tangent to E and +X|E is an infinitesimal symmetry of E0. In other words, sections of the Cartan +distribution are infinitesimal symmetries of all PDEs (regardless what is the PDE +we are considering). +There is an efficient description of (non-trivial) infinitesimal symmetries in +local coordinates. Consider the vector bundle V → J∞(P, n) from the previous +subsection. Its sections are sometimes called generating sections of symmetries for +the following reasons. Take the restriction VE → E of V to E . Sections of VE are +locally spanned by the restrictions vα|E , α = 1, . . . , m. Take a section χ ∈ Γ(VE ) +and let it be locally given by χ = χαvα|E for some local functions χα ∈ C∞(E ). +Moreover, let ˜χ ∈ Γ(V ) be an extension of χ, i.e. χ = ˜χ|E . Finally, let Y ∈ XC +be any infinitesimal symmetry of the Cartan distribution on J∞(P, n) mapping to +˜χ under the projection XC → Γ(V ) (if we are working locally, we can take, e.g., +the evolutionary vector field З˜χ). If E0 is locally given by (1.5), one can show that +Y is tangent to E if and only if locally χ is a solution of the following universal +linearization of E0: +ℓE (χ) = 0 +where +ℓE : Γ(VE) → C∞(J∞(P, n))r, +r = codim E0 = number of PDEs +is the locally defined operator given by +ℓE (χ) = +� +ℓE (χ)1, . . . , ℓE (χ)r +� +, +ℓE (χ)a := (З˜χFa) |E = ∂Fa +∂uα +I +|E DI|E χα. +and we used that all the total derivatives are tangent to E (a more intrinsic and +global interpretation of the universal linearization can be provided for PDEs im- +posed on sections of a fibration, particularly when E0 is specified as the zero locus of +a nonlinear differential operator, see e.g. [2, Chapter 4, Section 2.7] or [11, Section +3.9]) . In this case, the restriction Y |E is a infinitesimal symmetry of E0 +Proposition 1.16. The assignment χ �→ Y |E mod Γ(C ) establishes a bijection +between solutions of the universal linearization +ℓE (χ) = 0 +of E0 and infinitesimal symmetries of E0. +Remark 1.17. The universal linearization is a linear PDE imposed on sections +of the vector bundle VE → E . As it only involves total derivatives, for any solution +N of E0, it can be pulled-back along the embedding j∞ : N → E of Remark 1.3. If +we do so we get a new linear PDE for sections of the pull-back bundle j∞∗V . The +latter PDE is just the linearization of E0 around the solution N. This explain the +terminology “universal linearization”. +2. Cohomology of PDEs +2.1. horizontal cohomology. In this section we attach a cochain complex to +every diffiety (E , C ). The associated cohomology is called the horizontal cohomology +of E and it contains important coordinate free information on E0. + +VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES +11 +Let (E , C ) be a diffiety. By involutivity, the vector subbundle C → E of the +tangent bundle T E is actually a Lie algebroid. Accordingly, there is an associated +Cartan calculus. There are also associated De Rham cohomologies (with coeffi- +cients). Specifically, denote by +Ω•(E ) := Γ(∧•C ∗) +the exterior algebra of the dual of C . In other words, Ω•(E ) consists of differential +forms on E acting on vector fields in C . Elements in Ω•(E ) are called horizontal +differential forms. +There is a canonical differential d : Ω•(E ) → Ω•+1(E ), the +horizontal De Rham differential, defined by the usual Chevalley-Eilenberg formula: +for all ω ∈ Ωq(E ), +dω(X1, . . . , Xq+1) += +q+1 +� +i=1 +(−)i+1Xi +� +ω(X1, . . . � +Xi, . . . , Xq+1) +� ++ +� +i 3. The +graded Lie-Rinehart algebra structure induced in cohomology (Remark 3.4) agrees +with that of Theorem 2.6. +Remark 3.9. The degree 1 shift in the statement of Theorem 3.8 is due to +our convention on LR∞-algebra structure and can be removed choosing a different +convention (see [29] for more details). +The graded Ω•(E )-module Ω•(E , V )[1] is projective and finitely generated. Ac- +cordingly, the LR∞-algebra structure of Theorem 3.8 can be also encoded into +the associated Chevalley-Eilenberg algebra. +The Chevalley-Eilenberg algebra of +� +Ω•(E ), Ω•(E , V )[1] +� +is described in the next theorem. +Theorem 3.10. The Chevalley-Eilenberg algebra of the LR∞-algebra of Theo- +rem 3.8 is +Ω•(E , ∧•V ∗) +with structure derivations given by +d1 = d +d2 = d − d + ιR +d3 = −ιR + +22 +FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO +while dk = 0 for k > 3. In particular the total derivation D = d1+d2+d3 is just the +De Rham differential d. The DG algebra structure induced in cohomology (Remark +3.6) agrees with that of Theorem 2.7. +Remark 3.11. The LR∞-algebra of Theorem 3.8 is independent of the choice +of the splitting 2.5 up to LR∞-isomorphisms. We will not provide here a notion +of LR∞-morphism. We just mention that the independence of the LR∞-algebra +� +Ω•(E ), Ω•(E , V )[1] +� +of the splitting is rigorously expressed by the fact that the +total derivation D = d1 + d2 + · · · in the Chevalley-Eilenberg algebra of Theorem +3.10 is always the same, i.e. the De Rham differential, regardless of the splitting +we have chosen. See [29] for more details. +Theorems 3.8 and 3.10 provide indications towards Vinogradov’s conjecture +that the correct category where differential calculus on the space of solutions of a +PDE E0 should be developed is the homotopy category of DG-modules over the +horizontal De Rham algebra (Ω•(E ), d). They also suggest to add one further step +to the recipe presented at the end of Section 2. Namely +Secondarization Recipe (Addendum). +(5) Show that the cochains Ω•(E )⊗ V Φ support a homotopy algebra structure +responsible for the algebraic structure on Φ. +3.3. the A∞-algebra of secondary differential operators. In this final +subsection we discuss secondary scalar differential operators. +In order to define +them we follow the recipe presented at the end of Section 2. +Let (E , C ) be a +diffiety. The construction Φ for which we want to find a replacement on the space +of solutions of E0 is “scalar differential operators”, i.e. linear differential operators +from functions to functions (just DOs, for simplicity, in what follows). +For Step (1) in our recipe, we have to define transverse DOs. Intuitively, they +should be DOs taking derivatives just in the direction transverse to C . Begin with +the (non-commutative) R-algebra DO(E ) of DOs +∆ : C∞(E ) → C∞(E ) +on E . +As vector fields are DOs (of order 1), sections of C span a right ideal +Γ(C ) · DO(E ) in DO(E ). Denote +V DO := +DO(E ) +Γ(C ) · DO(E ) +the quotient left DO(E )-module. By definition, V DO is the space of transverse +DOs. +For Step (2) in the recipe, we have to notice that V DO is equipped with an +action of the Lie algebroid C . This is indeed the case: left composition with a +section of C is indeed such action. Accordingly, the Ω•(E )-module +Ω•(E , V DO) := Ω•(E ) ⊗ V DO +is a DG (Ω•(E ), d)-module, whose differential we denote again by d. +In Step (3) we define secondary differential operators as the cohomology of +(Ω•(E , V DO), d): +DO := H•� +Ω(E , V DO), d +� +. +For Step (4) we have to show that DO is equipped with the appropriate alge- +braic structure for DOs. Actually, we can show that DO is a graded associative + +VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES +23 +algebra (notice that, on the contrary, Ω•(E , V DO) is not a DG algebra in gen- +eral, as it does not possess a natural associative product). To do this, it is a good +idea to perform simultaneously Step (5) of the recipe (see the end of the previous +subsection). The relevant homotopy algebras here are A∞-algebras. +Definition 3.12. An A∞-algebra [13, 12] is a graded vector space U equipped +with a sequence a = {ak}k∈N of degree 1 multilinear maps: +ak : +� �k U +�• → U •+1, +(u1, . . . , uk) �→ ak(u1, . . . , uk) +satisfying the following higher associativity conditions: +� +r+s=k +r+s +� +j=1 +(−)|u1|+···+|uj|as+1 +� +u1, . . . , uj, as(uj+1, . . . , uj+r), uj+r+1, . . . , ur+s +� += 0 +for all k ∈ N, and all u1, . . . , uk ∈ U. +It follows from the above definition that a1 is a differential, and it is a derivation +with respect to a2. Moreover a2 is associative up to a homotopy encoded by a3. In +particular a2 induces a honest associative product in cohomology H•(U, a1) (up to +décalage). +Theorem 3.13. The graded vector space Ω•(E , V DO)[1] can be equipped with +an A∞-algebra structure a such that a1 = d. Moreover, the induced associative +product in the cohomology DO is canonical. +The above theorem supports the interpretation of DO as secondary DOs, i.e. +(an appropriate replacement for) DOs on the space of solutions of E0. +We conclude the paper by sketching the proof of Theorem 3.13. As we will see, +the proof will also suggest an alternative Secondarization Recipe. From now on we +assume some familiarity with graded geometry (see, e.g., [15]). We begin recalling +a standard technique in homotopical algebra, namely the Homotopy Transfer (see +[14, 22]). The idea is that homotopy algebras can be transferred along contraction +data. A set of contraction data is a diagram +(3.4) +(A, δA) +h +� +p +� (B, δB) +j +� +where +(1) (A, δA) and (B, δB) are cochain complexes, +(2) p, j are cochain maps, +(3) h : A• → A•−1 is a homotopy, +such that +pj = idB +and +jp = [δA, h] +and, moreover, the following side conditions are satisfied: +ph = hj = h2 = 0. +In particular p, j are mutually homotopy inverse homotopy equivalences and +H•(A, δA) ∼= H•(B, δB). +Theorem 3.14 (Homotopy Transfer). Let (A, δA) be an associative DG alge- +bra and let (3.4) be a set of contraction data over a cochain complex (B, δB). Then +B[1] can be equipped with an A∞-algebra structure a, uniquely determined by the + +24 +FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO +associative product in A and the contraction data, such that a1 = δB. Moreover +(A, δA) and (B, a) induce the same graded associative algebra structure in cohomol- +ogy H•(A, δA) ∼= H•(B, δB). +There is a version of the Homotopy Transfer Theorem for L∞-algebras [6, 22] +and one, slightly more involved, for LR∞-algebras (see [30] for details). Actually +the LR∞-algebra of Theorem 2.6 can be obtained via Homotopy Transfer. Namely, +let (E , C ) be a diffiety. Consider the DG manifold C [1] obtained by the Lie algebroid +C → E by shifting by one the fiber degree. Clearly C∞(C [1]) = Ω•(E ) and the +cohomological vector field on C [1] is the horizontal De Rham differential d. The +pair +� +C∞(C [1]), X(C [1]) +� +is a DG Lie-Rinehart algebra. The differential in X(C [1]) +is the graded commutator of graded vector fields with the horizontal De Rham +differential. Moreover, a splitting (2.5) uniquely determines contraction data +(3.5) +X(C [1]) +h +� +p +� Ω•(E , V ) +j +� +that we can use to construct an LR∞-algebra structure on Ω•(E , V )[1] from the DG +Lie-Rinehart algebra structure on (C∞(C [1]), X(C [1])) (see [30] for details). This +means that the Lie-Rinehart algebra structure on secondary vector fields (C∞, X) +is, equivalently, the one induced in cohomology by the DG Lie-Rinehart algebra +(C∞(C [1]), X(C [1])). A similar result holds for secondary DOs. Namely, consider +the space DO(C [1]) of graded DOs over the DG manifold C [1]. It is an associa- +tive DG algebra whose differential is the graded commutator of graded DOs with +the horizontal De Rham differential. Now, a splitting (2.5), together with certain +additional data that we will not describe, uniquely determine contraction data +(3.6) +DO +� +C [1] +� +h +� +p +� Ω•(E , V DO) +j +� +that we can use to construct an A∞-algebra structure on Ω•(E , V DO)[1] from the +associative DG algebra structure on DO(C [1]). +In particular there is a graded +associative algebra structure on secondary differential operators +DO = H•� +Ω(E , V DO), d +� ∼= H•� +DO(C [1]) +� +induced by either the associative DG algebra structure on DO(C [1]) or the A∞- +algebra structure on Ω•(E , V DO). +This discussion suggests the following alternative recipe to find the secondary +analogue of a given construction Φ in differential geometry. +Secondarization Recipe. +(1) Apply the construction Φ to the DG manifold C [1] (when Φ is “vector +fields”, resp. “DO” we get X(C [1]), resp. DO(C [1])). +(2) Notice that Φ(C [1]) is canonically equipped with a differential induced by +the cohomological vector field d on C [1] (when Φ is “vector fields”, resp. +“DOs”, such differential is the graded commutator of vector fields, resp. +DOs, with d). +(3) Define Φ as the cohomology of Φ(C [1]) (when Φ is “vector fields”, resp. +“DOs”, then Φ is X, resp. DO). +(4) Notice that, being compatible with the differential, the algebraic structure +on Φ(C [1]) induces the same algebraic structure on Φ (when Φ is “vector + +VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES +25 +fields”, resp. +“DOs”, then Φ = X, resp. +Φ = DO, is a graded Lie- +Rinehart algebra, resp. a graded associative algebra). +Remark 3.15. Notice that, in the alternative Secondarization Recipe just +above there is no analogue of the Addendum (5) at p. 22. +The reason is that +the cochain complex Φ(C [1]) does possess the same (honest) algebraic structure as +Φ by definition, and there is no need to work with algebraic structures up to homo- +topy at the level of cochains in this case. Nonetheless, by the Homotopy Transfer +Theorem, cohomologies possess algebraic structures of the same type but only up to +homotopy. Moreover, cohomologies with their algebraic structures up to homotopy +are quasi-isomorphic to cochains with their algebraic structures. In other words, +one can either choose to work with a larger space (cochains) and a simpler algebraic +structure, or with a smaller space (cohomologies) and a more complicated algebraic +structure (algebraic structure up to homotopy). +References +[1] K. Behrend, and P. Xu, Differentiable stacks and gerbes, J. Sympl. 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Phil. +Soc. 158 (2015), 155–191; e-print: arXiv:1304.4353. +DipMat, Università degli Studi di Salerno, via Giovanni Paolo II n◦123, 84084 +Fisciano (SA), Italy. +Email address: fpugliese@unisa.it +DipMat, Università degli Studi di Salerno, via Giovanni Paolo II n◦123, 84084 +Fisciano (SA), Italy. +Email address: sparano@unisa.it +DipMat, Università degli Studi di Salerno, via Giovanni Paolo II n◦123, 84084 +Fisciano (SA), Italy. +Email address: lvitagliano@unisa.it + diff --git a/ZNA0T4oBgHgl3EQfFv_a/content/tmp_files/load_file.txt b/ZNA0T4oBgHgl3EQfFv_a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e25d0da7b616a62c458ac2885564ffd1a15763a3 --- /dev/null +++ b/ZNA0T4oBgHgl3EQfFv_a/content/tmp_files/load_file.txt @@ -0,0 +1,1285 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf,len=1284 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='02038v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='DG] 5 Jan 2023 Contemporary Mathematics Vinogradov’s Cohomological Geometry of Partial Differential Equations Fabrizio Pugliese, Giovanni Sparano, and Luca Vitagliano to the memory of our teacher and colleague Alexandre Mikhailovich Vinogradov Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Secondary Calculus is a formal replacement for differential cal- culus on the space of solutions of a system of possibly non-linear partial dif- ferential equations and it is essentially due to Alexandre M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Vinogradov and his collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Many coordinate free properties of PDEs find their natu- ral place in Secondary Calculus including: symmetries and conservation laws, variational principles and the coordinate free aspects of the calculus of vari- ations, recursion operators and Hamiltonian structures, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The building blocks of this language are horizontal cohomologies of diffieties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' infinite prolongations of PDEs, and their versions with local coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The main paradigm of Secondary Calculus is the principle, due to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Vinogradov, roughly stating that: differential calculus on the space of solutions of a PDE is calculus up to homotopy on the horizontal De Rham algebra of the associated diffiety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We will review the fundamentals of Secondary Calculus including its main motivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In the last part of the paper, we will try to explain the role of homotopy in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Introduction Systems of algebraic equations can be encoded geometrically in an algebraic variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This apparently harmless observation is the starting point for the develop- ment of such a successful branch of modern Mathematics as Algebraic Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' One of the leading principles behind most of the mathematical work of Alexandre M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Vinogradov is that Partial Differential Equations (PDEs) should be treated in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Namely, let x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , xn) be some independent variables, let u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , um) be some dependent variables, and let (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='1) E0 : Fa(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , uI, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=') = 0, a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , r, be a system of possibly non-linear PDEs, where a multi-index I denotes multiple partial derivatives with respect to the x’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Adding to (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='1) all its 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Primary 58A15, 58A20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Secondary 17B56, 55T25, 58A12, 70S05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Partial differential equations, jet spaces, horizontal cohomology, C -spectral sequence, secondary calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' 1 2 FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO total derivatives results in a new but equivalent system of infinitely many PDEs (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='2) DJFa(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , uI, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=') = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Here DJ denotes multiple total derivatives of arbitrarily high order with respect to the x’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Now, interpret (x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , uI, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=') as independent coordinates on an ∞- dimensional manifold J∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Then (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='2) can be interpreted as defining a (generically ∞-dimensional) submanifold E ⊆ J∞, the ∞-prolongation of E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This submani- fold is canonically equipped with an involutive distribution C , called the Cartan distribution, spanned by the total derivatives Di|E , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The pair (E , C ) is, in the terminology of Vinogradov and his school, a diffiety1 (from the words diff erential equation and variety) and contains most of the relevant information about the original system E0 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=', the extensive monographs [2, 11, 27] on the subject).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' According to Vinogradov, the diffiety (E , C ) should be considered as the ultimate geometric portrait of E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In other words, diffieties are to PDEs what algebraic varieties are to algebraic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' However the geometry of a diffiety is much more intricate than that of an algebraic variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Namely, while solutions of a system of algebraic equations are points of an algebraic variety and they do not possess any internal structure, solutions of a system of PDEs E0 are integral submanifolds of the Cartan distribution on the diffiety (E , C ) and they do possess internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' As a consequence the space of solutions of a PDE share some features of the leaf space of a foliation (and it is the leaf space of a foliation when dim E < ∞) and should be treated as a stack rather than as an ordinary space [17, 1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This is exactly where homological and homotopical algebra come into play when trying to develop a general theory of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' A fundamental achievement of Vinogradov is that there is a cohomological replacement for differential calculus on the space of solutions of a PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Vinogradov used to call such formal calculus Secondary Calculus, because of an analogy with secondary quantization in physical field theory [26, 2, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The main building blocks of Secondary Calculus are the leaf-wise cohomologies of the Cartan distribution, often called horizontal De Rham cohomologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Most of the coordinate free properties of a PDE can be expressed in terms of horizontal cohomologies: symmetries, conservation laws, variational prin- ciples, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=', to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Additionally, horizontal cohomologies can be interpreted as smooth functions, vector fields, differential forms on the space of solutions of the PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' One of the main features of horizontal cohomologies supporting this interpre- tation is that they possess the correct algebraic structures (those that we expect from functions, vector fields, differential forms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In summary, Secondary Calculus is the collection of all these algebraic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In the ultimate idea of Vinogradov this fascinating constructions should be studied in their homotopy aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' More precisely, Vinogradov conjectured that what is important is not really horizontal cohomology but rather the homotopy type of the horizontal De Rham algebra, and the homotopy category (or even the derived category) of differential graded modules over it [27, Chapter 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Notice that this conjecture (together with the interpretation of the space of solutions of a PDE as a stack), is somehow complementary to the interpretation of a PDE as a derived zero locus and, in the particular case of an Euler-Lagrange PDE, as a derived critical locus (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=', [23], and references therein), and suggests a relationship to Homotopical and Derived Geometry [20, 21, 19] that, unfortunately, to the 1To the best of our knowledge, the term diffiety appeared for the first time in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES 3 best of our knowledge, has not been yet investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' At this stage, we can only speculate that the space of solutions of a PDE should be ultimately treated as a higher derived stack [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In this exposition we will focus on the theory rather than on applications or the explicit computation of the involved cohomologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We will omit the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' They can be found in the cited references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The paper is divided into three sections: In Section 1 (Geometry of PDEs) we explain how to encode a system of PDEs in a geometric object: a diffiety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We also discuss the main geometric structure possessed by a diffiety, namely its Cartan distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Our main sources for this section are [2, 11, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In Section 2 (Homology of PDEs) we show that there is a natural cohomology attached to a diffiety (hence to a PDE), the horizontal cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This is the core of the paper, where Vinogradov’s Secondary Calculus is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Horizontal cohomology is a generalization of the leaf-wise cohomology of a foliated manifold and contains important coordinate free information on a PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We provide an interpretation of the main horizontal cohomologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Horizontal cohomologies can also be seen as the building blocks of a differential calculus on the space of solutions of a PDE, what we call Secondary Calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' One of the main supporting facts for the latter interpretation is that horizontal cohomologies are equipped with the correct algebraic structures for smooth functions, vector fields, differential forms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Our main sources here are [2, 11, 27], see also [25, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In Section 3 (Homotopy of PDEs) we present the latest developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This section is an extremely compact review of the papers [29, 30] by the third author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We show that the main algebraic structures on horizontal cohomologies do all come from appropriate structures up to homotopy on horizontal cochains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This supports Vinogradov’s idea that the appropriate category to develop Secondary Calculus is (a subcategory of) the derived category of DG modules over the horizontal De Rham algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Contents Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Geometry of PDEs 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' jet spaces 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' the Cartan distribution 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' diffieties 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Cohomology of PDEs 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' horizontal cohomology 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' interpreting horizontal cohomologies 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' algebraic structures on horizontal cohomologies 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Homotopy of PDEs 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' homotopy algebras 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' the LR∞-algebra of secondary vector fields 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' the A∞-algebra of secondary differential operators 22 References 25 4 FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Geometry of PDEs In this section we show that a system of, generically non-linear, PDEs can be encoded into a geometric object: a diffiety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We will work in the rather general setting of a PDEs imposed on submanifolds N of a fixed dimension n in a manifold P (see [10]), in contrast with most of the literature where only PDEs imposed on sections of a fibration (or a fiber bundle) are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' jet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Let n, m be non-negative integers, and let P be a manifold of dimension n + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We begin providing an intrinsic definition of derivatives of an n-dimensional submanifold N ⊆ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In the following, n should be interpreted as the number of independent variables, m as the number of dependent variables, and P as the space parameterizing both independent and dependent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In other words, in this theory, dependent and independent variables can be mixed and swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Around every point p ∈ N there are coordinates (xi, uα) on P, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , n, α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , m, adapted to N in the sense that, in these coordinates, N looks like the graph of a map: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='1) N : uα = f α(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We will need to take multiple partial derivatives of the f’s with respect to the x’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We adopt the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Let h be a non-negative integer and let I = i1 · · · ih be a multiindex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' a (possibly empty) word containing h letters iℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , n}, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We identify two words if they only differ by a permutation of their letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We also compose two words by concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Denote |I| := h and call it the lenght of the multiindex I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Finally denote ∂|I|f α ∂xI := ∂hf α ∂xi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' ∂xih .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' If there is another n-dimensional submanifolds ˜N through p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' p ∈ N ∩ ˜N, then around p there are coordinates (xi, uα) on P adapted to both N, ˜N: N : uα = f α(x), and ˜N : uα = ˜f α(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In this case we say that N and ˜N are tangent up to order k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , ∞ at p if ∂|I|f α ∂xI (p) = ∂|I| ˜fα ∂xI (p), for all multiindexes I with |I| ≤ k, and we write N ∼k p ˜N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Tangency up to order k at p is a well-defined equivalence relation on submanifolds through p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In particular, it is independent of the choice of adapted coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The equivalence class of N is denoted N k p and called the k-jet of N at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In other words, k-jets at p encode partial derivatives of n-dimensional submanifolds at p up to order k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The space of tangency classes is denoted Jk p (P, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Finally put Jk(P, n) = � p∈P Jk p (P, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We call Jk(P, n) the k-jet space of n-dimensional submanifolds of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' It is clear that J0(P, n) identifies naturally with P and J1(P, n) identify with the bundle of n-Grassmannians in the fibers of the tangent bundle T P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' More precisely, the 0-jet of N at p identifies with p itself, while the 1-jet of N at p identifies with the tangent space TpN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For a general k, the space Jk(P, n) can be coordinatized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Choose coordinates (xi, uα) on P, and let U ⊆ Jk(P, n) be the subset consisting VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES 5 of k-jets of submanifolds for which (xi, uα) are adapted coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We define coordinates (xi, uα I ) on U by putting xi� N k p � := xi(p), and uα I � N k p � = ∂|I|f α ∂xI (p), for all N locally given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='1), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , n, α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , m, and |I| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The (xi, uα) are called jet coordinates, they cover Jk(P, n) and, for k < ∞, they give to Jk(P, n) the structure of a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For all h ≤ k ≤ ∞ the are projections Jk(P, n) → Jh(P, n), N k p �→ N h p , and by Borel Lemma J∞(P, n) identifies with the inverse limit of the tower of surjective submersions J0(P, n) ←− · · · ←− Jk−1(P, n) ←− Jk(P, n) ←− · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This gives to the ∞-jet space the structure of a pro-finite dimensional manifold (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=', [2, 27], see also [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For the reader more familiar with jets of sections, we mention that, when P is fibered over some n-dimensional manifold M, then, for all k ≤ ∞, the space Jk(P, M) of k-jets of sections of P over M can be recovered as the open and dense submanifold of Jk(P, n) consisting of jets of n-dimensional submanifolds transverse to the projection P → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The open embedding Jk(P, M) ֒→ Jk(P, n) maps the k-jet at x ∈ M of a (possibly local) section s of P → M to the k-jet of its image at s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Differential calculus on J∞(P, n) can be defined algebraically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For instance, we define smooth functions C∞(J∞(P, n)) on J∞(P, n) as the direct limit of the algebra filtration (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='2) C∞� J0(P, n) � ֒→ · · · ֒→ C∞� Jk−1(P, n) � ֒→ C∞� Jk(P, n) � ֒→ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In other words a smooth function on J∞(P, n) is a smooth function of the inde- pendent variables xi, the dependent variables uα, and their derivatives up to some finite order k, but k can be arbitrarily high (sometimes they are defined as func- tions which are only locally of this type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Vector fields, differential forms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=', on J∞(P, n) are defined in a way compatible with the filtration (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We will not insist on these technicalities and we refer to [2] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We stress that, in this ∞-dimensional setting, some important results in finite dimensional differential geometry might fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' To mention a few: the inverse function theorem, existence and uniqueness of the flow of a vector field, the Frobenius theorem, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Any n-dimensional submanifold N ⊆ P defines a new n-dimensional submani- fold N k := � N k p : p ∈ N � in Jk(P, n), called the k-jet prolongation of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' If N locally looks like (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='1) in coordinates, then N k locally looks like N k : uα I = ∂|I|f α ∂xI (x), |I| ≤ k, in jet coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' 6 FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Let N ⊆ P be an n-dimensional submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For all k ≤ ∞, the map jk : N → N k, p �→ N k p is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In particular N embeds into Jk(P, n) for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' When P is fibered over some n-dimensional manifold M, and s : M → P is a (possibly local) section of P over N, then the usual k-jet prolongation jks of s can be recovered as the composition jks : M s −→ im s jk −→ (im s)k ֒→ Jk(P, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Let r be a non-negative integer and let Q ⊆ P be an (n + r)-dimensional submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Then, for all k ≥ 0, the k-jet space Jk(Q, n) is a submanifold in the k-jet space Jk(P, n) in the obvious natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' the Cartan distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The main geometric structure on jet spaces is the Cartan distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' There are several equivalent ways to define it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The one we present here will be also useful in defining the prolongation of a PDE in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We begin noticing that a section of the fibration J1(P, n) → J0(P, n) ∼= P can be interpreted as an n-dimensional distribution on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Now, for every 1 ≤ k ≤ ∞, the k-jet space is canonically equipped with a rank n distribution along the projection Jk(P, n) → Jk−1(P, n), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' a smooth map C : Jk(P, n) −→ J1� Jk−1(P, n), n � such that the following diagram commutes J1� Jk−1(P, n), n � � Jk(P, n) C �♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ � Jk−1(P, n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The Cartan distribution C is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For z = N k p ∈ Jk(P, n), denote by ¯z = N k−1 p ∈ Jk−1(P, n) its projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Now put Cz := � N k−1�1 ¯z = T¯z � N k−1� ∈ J1� Jk−1(P, n), n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In other words Cz is the 1-jet at ¯z of the (k − 1)-jet prolongation of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In local coordinates (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='3) C ∗(uα I,j) = uα Ij, |I| ≤ k − 1, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , n, where, in the left hand side, we denoted by (xi, uα I,j) the jet coordinates on the 1-jet space J1� Jk−1(P, n), n � (corresponding to the coordinates (xi, uα I ) on Jk−1(P, n)), and, in the right hand side, we are using concatenation of multiindexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The Cartan distribution C : Jk(P, n) −→ J1� Jk−1(P, n), n � is an embedding that we often use to interpret the k-jet space Jk(P, n) as a submanifold in the jet space J1� Jk−1(P, n), n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' More generally for any two non-negative integers h, k, there is an embedding Jh+k(P, n) ֒→ Jh� Jk(P, n), n � given by N h+k p �→ (N k)h p, p = N k p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES 7 In the next subsection we will use this embedding to interpret Jh+k(P, n) as a submanifold in Jh� Jk(P, n), n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For k = ∞, the Cartan distribution is a honest rank n distribution on J∞(P, n), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='3) shows that it is locally spanned by the total derivatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' the following vector fields Di := ∂ ∂xi + � |I|<∞ uα Ii ∂ ∂uα I , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Dually, the annihilator C 0 ⊆ T ∗J∞(P, n) is spanned by the Cartan forms duα I − uα Iidxi, α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , m, |I| ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Here, and in the rest of the paper, we adopt the Einstein summation conven- tion over pairs of lowercase-uppercase indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We will not adopt the Einstein convention for multiindexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The Cartan distribution on J∞(P, n) is involutive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' the commutator of two sections of C lays in C as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Additionally it detects ∞-jet prolongations in the following sense: a submanifold S ⊆ J∞(P, n) is locally of the form S = N ∞ for some n-dimensional submanifold N ⊆ P if and only if it is an n-dimensional integral submanifold of C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' So, morally, the ∞-jet prolongation construction N �→ N ∞ identifies n-dimensional submanifolds of P with n-dimensional integral submanifolds of C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In the following, Cartan distribution will always mean the Cartan distribution on J∞(P, n) (unless otherwise stated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For their importance in the next section, we now discuss infinitesimal sym- metries of the Cartan distribution C , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' vector fields Y on J∞(P, n) such that [Y, Γ(C )] ⊆ Γ(C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Denote by XC the Lie algebra of infinitesimal symmetries of C and notice that, by involutivity, Γ(C ) ⊆ XC is an ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We want to show that the quotient Lie algebra XC /Γ(C ) identifies with sections of an appropriate vector bundle V → J∞(P, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We first define V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' So let z = N ∞ p ∈ J∞(P, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The fiber of V over z is then the m-dimensional vector space Vz := TpP/TpN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In the following, we denote by π : J∞(P, n) → P the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Now, let Y ∈ XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We define a section Y of V as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For z = N ∞ p ∈ J∞(P, n) put Y z := dπ(Yz) mod TpN ∈ TpP/TpN = Vz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The assignment Y �→ Y defines an R-linear surjection XC → Γ(V ) with kernel given by Γ(C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Hence there is a canonical short exact sequence of vector spaces 0 −→ Γ(C ) −→ XC −→ Γ(V ) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In particular, Γ(V ) inherits from XC a Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The Lie bracket on Γ(V ) is sometimes called the higher Jacobi bracket and denoted by {−, −}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We conclude this section describing it in local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' First notice that Γ(V ) is locally spanned by the sections vα defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For z = N ∞ p ∈ J∞(P, n) put vα|z := ∂ ∂uα |p mod TpN ∈ TpP/TpN = Vz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' 8 FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO The isomorphism Γ(V ) ∼= XC/Γ(C ) is now locally given by χ �→ Зχ mod Γ(C ) where, for χ = χαvα ∈ Γ(V ), we denoted by Зχ the (local) infinitesimal symmetry of C locally given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='4) Зχ = � |I|<∞ DIχα ∂ ∂uα I , and, for any multiindex i1 · · · ih, Di1···ih := Di1 ◦ · · · ◦ Dih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We mention for the reader unfamiliar with the cyrillic alphabet, that the script З appearing in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='4) is pronounced [e], like in set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' A vector field of the form Зχ is sometimes called an evolutionary vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Notice that evolutionary vector fields are only locally defined on ∞-jets of subman- ifolds (while one can give a coordinate free meaning to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='4) on ∞-jets of sections of a fibration, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' [2, Chapter 4, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='4] or [11, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='9] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' It easily follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='4) that, if χ and ψ are sections of V locally given by χ = χαvα and ψ = ψβvβ, then their higher Jacobi bracket {χ, ψ} is the section locally given by {χ, ψ} = � Зχψα − Зψχα� vα = � |I|<∞ � DIχβ ∂ψα ∂uβ I − DIψβ ∂χα ∂uβ I � vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' diffieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We now use jets to define PDEs and present their geometric portraits: diffieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Let P be a manifold of dimension n + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' A system of k-th order partial differential equations imposed on n-dimensional submanifolds of P, or simply a PDE, is a closed submanifold E0 ⊆ Jk(P, n) of the k-jet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' A solution of a PDE E0 ⊆ Jk(P, n) is an n- dimensional submanifold N ⊆ P such that N k ⊆ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' As a minimal regularity condition on a PDE E0 ⊆ Jk(P, n) we will always assume dim E0 ≥ n so that the existence of solutions is not excluded a priori by trivial dimensional reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For a PDE E0 ⊆ Jk(P, n), locally, in jet coordinates, we have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='5) E0 : Fa(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , uI, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=') = 0, |I| ≤ k, for some local smooth functions Fa ∈ C∞(Jk(P, n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' If N ⊆ P is an n-dimensional submanifold locally given by N : uα = f α(x), then N is a solution of E0 iff Fa � x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , ∂|I|f ∂xI , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This motivates Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In Algebraic Geometry, given a system of algebraic equations, it is natural to take all their algebraic consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In other words, given a subset of polynomials, it is natural to consider the ideal that they span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Similarly, given a PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='5), it is natural (besides their smooth consequences) to take all their total derivatives, this can be done in a coordinate free way via a construction known as prolongation that we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Let E0 ⊆ Jk(P, n) be a PDE, and let q = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES 9 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The q-prolongation of E0 is the subset Eq ⊆ Jk+q(P, n) of the (k + q)-jet space defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='6) Eq := Jq(E0, n) ∩ Jk+q(P, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The ∞-prolongation will be often denoted simply E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='10 requires some little explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' As E0 is a submanifold in Jk(P, n), its jet space Jq(E0, n) is a submanifold in the jet space Jq(Jk(P, n), n) as in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The jet space Jk+q(P, n) can be seen as a submanifold in Jq(Jk(P, n), n) as well via the embedding in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This explains the in- tersection in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We stress that we will always understand the prolongation Eq as a subset in Jk+q(P, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' If E0 is locally given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='5), then its q-prolongation is locally given by Eq : DJFa(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , uI, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=') = 0, |J| ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This shows that Eq encodes in a coordinate free way the total derivatives of E0 up to order q, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' For a generic E0 ⊆ Jk(P, n) its q-th prolongation Eq is not a submanifold in Jk+h(P, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' When it is a submanifold, then it is clearly a PDE with the same solutions as E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In the following, we will always assume that the ∞-th prolongation E = E∞ is a submanifold in J∞(P, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This happens, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=', when E0 is a formally integrable PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Our assumptions are not a great loss of generality accord- ing to Cartan-Kuranishi Prolongation Theorem which roughly states that, under mild regularity conditions, every PDE becomes formally integrable after finitely many 1-prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Now on we concentrate on the ∞-prolongation E ⊆ J∞(P, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Let E0 ⊆ Jk(P, n) be a PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Assume that its ∞-prolongation E ⊆ J∞(P, n) is a submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Then the Cartan distribution restricts to E in the sense that Cz ⊆ TzE for all z ∈ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Hence the restriction C : E → J1(E , n) is a well-defined rank n involutive dis- tribution on E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Additionally it detects solutions of E0 in the following sense: a submanifold S ⊆ E is locally of the form S = N ∞ for some solution N of E0 if and only if it is an n-dimensional integral submanifold of C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The restriction of the Cartan distribution to the ∞-prolongation of a PDE will be denoted by C as well and called the Cartan distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' A diffiety (from the words differential equation and variety) is a pair (E , C ) where E is the ∞-prolongation of some PDE and C is the Cartan distribution on E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='12 now shows that a diffiety (E , C ) contains most of the relevant information about the PDE E0 defining it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' We conclude this section briefly discussing infinitesimal symmetries of a PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Let (E , C ) be the diffiety corresponding to a PDE E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' An infinitesimal symmetry of E0 is a vector field Y ∈ X(E ) such that [Y, Γ(C )] ⊆ Γ(C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' A non-trivial infinitesimal symmetry is an infinitesimal symmetry modulo trivial ones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' sections of C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' 10 FABRIZIO PUGLIESE, GIOVANNI SPARANO, AND LUCA VITAGLIANO Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The terminology “trivial, non-trivial symmetries” in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='14 is motivated by the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Let X be a section of the Cartan dis- tribution on J∞(P, n) and let (E , C ) be any diffiety, then X is tangent to E and X|E is an infinitesimal symmetry of E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In other words, sections of the Cartan distribution are infinitesimal symmetries of all PDEs (regardless what is the PDE we are considering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' There is an efficient description of (non-trivial) infinitesimal symmetries in local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Consider the vector bundle V → J∞(P, n) from the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Its sections are sometimes called generating sections of symmetries for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Take the restriction VE → E of V to E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Sections of VE are locally spanned by the restrictions vα|E , α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Take a section χ ∈ Γ(VE ) and let it be locally given by χ = χαvα|E for some local functions χα ∈ C∞(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Moreover, let ˜χ ∈ Γ(V ) be an extension of χ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' χ = ˜χ|E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Finally, let Y ∈ XC be any infinitesimal symmetry of the Cartan distribution on J∞(P, n) mapping to ˜χ under the projection XC → Γ(V ) (if we are working locally, we can take, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=', the evolutionary vector field З˜χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' If E0 is locally given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='5), one can show that Y is tangent to E if and only if locally χ is a solution of the following universal linearization of E0: ℓE (χ) = 0 where ℓE : Γ(VE) → C∞(J∞(P, n))r, r = codim E0 = number of PDEs is the locally defined operator given by ℓE (χ) = � ℓE (χ)1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , ℓE (χ)r � , ℓE (χ)a := (З˜χFa) |E = ∂Fa ∂uα I |E DI|E χα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' and we used that all the total derivatives are tangent to E (a more intrinsic and global interpretation of the universal linearization can be provided for PDEs im- posed on sections of a fibration, particularly when E0 is specified as the zero locus of a nonlinear differential operator, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' [2, Chapter 4, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='7] or [11, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='9]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In this case, the restriction Y |E is a infinitesimal symmetry of E0 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The assignment χ �→ Y |E mod Γ(C ) establishes a bijection between solutions of the universal linearization ℓE (χ) = 0 of E0 and infinitesimal symmetries of E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The universal linearization is a linear PDE imposed on sections of the vector bundle VE → E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' As it only involves total derivatives, for any solution N of E0, it can be pulled-back along the embedding j∞ : N → E of Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' If we do so we get a new linear PDE for sections of the pull-back bundle j∞∗V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The latter PDE is just the linearization of E0 around the solution N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' This explain the terminology “universal linearization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Cohomology of PDEs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' horizontal cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In this section we attach a cochain complex to every diffiety (E , C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' The associated cohomology is called the horizontal cohomology of E and it contains important coordinate free information on E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' VINOGRADOV’S COHOMOLOGICAL GEOMETRY OF PDES 11 Let (E , C ) be a diffiety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' By involutivity, the vector subbundle C → E of the tangent bundle T E is actually a Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Accordingly, there is an associated Cartan calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' There are also associated De Rham cohomologies (with coeffi- cients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Specifically, denote by Ω•(E ) := Γ(∧•C ∗) the exterior algebra of the dual of C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' In other words, Ω•(E ) consists of differential forms on E acting on vector fields in C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' Elements in Ω•(E ) are called horizontal differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' There is a canonical differential d : Ω•(E ) → Ω•+1(E ), the horizontal De Rham differential, defined by the usual Chevalley-Eilenberg formula: for all ω ∈ Ωq(E ), dω(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , Xq+1) = q+1 � i=1 (−)i+1Xi � ω(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' � Xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNA0T4oBgHgl3EQfFv_a/content/2301.02038v1.pdf'} +page_content=' , Xq+1) � + � i∼ 2 × 10−3);21 this +sudden detachment creates a vortex ring which strongly mod- +ifies the flow pressure: fluid is transported back into the neck +which in turn reopens. It is remarkable that the same happens +in the case of superfluid 4He in spite of the lack of viscos- +ity. At t = 330 ps, another pair of vortex rings is nucleated +at the droplet-filament indentation preventing pinching again. +Finally, vortex-antivortex rings annihilate and disappear from +the system producing as a result a burst of density waves. +The movie in the supplementary information shows the ap- +pearance of surface protrusions at t = 452 ps which act as +vortex nucleation sites, and their collapse yields a high den- +sity spot. Eventually, the contracted filament is permeated by +a large number of vortex rings at t = 488 ps. This is at variance +with the classical, inviscid fluid description. +The evolution of this filament can be compared to that cor- +responding to L0 = 8.0 shown in Fig. 5 of Ref. 19. Besides the +vortex rings phenomenology, which is absent in the simula- +tions of that reference, in our case end-pinching strictly never +happens. The closer the 4He filament gets to it is at t = 4.1τc, +whereas the time for the filament breakup by end-pinching +read from Fig. 5 of Ref. 19 is t ∼ 4.6τc. +5. +Filament with Γ = 10.5 +Similarly to the previous cases, end drops develop as shown +in Fig. 10. A more violent approach is expected because the +filament is longer and end drops have more time to acceler- +ate under the traction exerted by surface tension. The filament +connecting the end drops contracts and necks appear at the +drop-filament contact region, as shown at t = 250 ps, which +start pinching. The neck shrinks to a minimum at t = 256 ps, +escaping from pinch-off again because vortex rings are nucle- +ated at t ∼ 260 ps. +Vortex rings detach from the neck and move towards the +bulk of the end drops (frame at t = 300 ps). The remaining +filament develops bulges, which evolve to a more complex + +12 +structure (frame at t = 400 ps). +The snapshot at t = 440 ps shows an almost complete frag- +mentation. However, due to the opposite velocities acquired +during the early stages of the contraction, the three fragments +merge again. Other vortex rings are created in the process, +nucleated at the necks during the re-merging, as shown in the +frame at t = 480 ps. The streamlines of the superflow are +drawn in the bottom panel of Fig. 9 for the configuration at +t = 462 ps. +Vortex ring annihilation at later times (see movie in the sup- +plementary material) produces density waves arising from the +collapse of their cores. This is a phenomenon that we have not +observed in the merging of He droplets,41,42 nor the shrink- +ing of a vortex ring up to it collapses. It is interesting to see +that these small vortex rings travel towards the tips of the fil- +ament, evaporating from them. Eventually, vortex rings dis- +appear and the contracted filament enters a complex dynamic +regime, hosting plenty of density waves until the end of the +simulation. +The evolution of this filament should be similar to that cor- +responding to L0 = 10.0 shown in Fig. 5 of Ref. 19. Be- +sides the vortex rings phenomenology and wave dynamics, in +our case end-pinching strictly never occurs. The filament gets +close to it at t = 4.1τc (254 ps) and especially at t = 7.0τc (432 +ps), whereas the breakup time by end pinching read from Fig. +5 of Ref. 19 is t ∼ 4.8τc. +6. +Filament with Γ = 15 +This is the largest filament we have investigated. In clas- +sical simulations of sufficiently long filaments (like the one +shown in Fig. 11) and small Oh numbers, as the filament +contracts it will succumb to end pinching18,20,43 even in cases +where the Rayleigh-Plateau instability is expected to develop, +subsequently resulting in the filament to break up into several +drops. However, this instability does not occur, suggesting +that the timescale for the Rayleigh-Plateau instability to grow +is much larger than the timescale for the filament to fully con- +tract even for long filaments. +In the case of superfluid 4He, the sequence is similar to the +Γ = 8 and Γ = 10.5 cases, except that the number of necks +has increased. Well developed end drops appear at t = 100 +ps, with a well developed necks at t = 160 ps. Figure 11 +shows that end drops nearly pinch-off at t = 248 ps, but at +t = 264 ps one may see vortex rings appearing at the necks, +hindering pinch-off. The vortex rings detach from the neck +and move towards the bulk of the end drops and bulges ap- +pear in the filament close to the end drops (panel at t = 290 +ps). Bulges evolve to bulbs and, similarly to the Γ = 8 and +Γ = 10.5 cases, intermediate drops develop during the time +evolution whose number increases with the length of the fil- +ament, as also observed in the simulations of classical low +viscosity (0.003 ≤ Oh ≤ 0.02) filaments.44 +The evolution of this filament should be compared to that +corresponding to L0 = 15.0 shown in Fig. 5 of Ref. 19. Be- +sides the phenomenology of vortex rings proliferation, also in +this case end-pinching never occurs. End drops are close to +detach at t = 4.02τc (247 ps) but escape pinch off because +of vortex ring nucleation, whereas the filament breakup time +read from Fig. 5 of that reference is t ∼ 4.8τc. +Finally, we have computed the contraction velocity for all +the investigated filaments. We have defined the position of the +tip of the filament as the location of its sharp surface (that at +which the density equals ρ0/2) on the x-axis. +Figure 12 shows the displacement of the tip position as a +function of time for the studied filaments. It appears that all +curves collapse onto the same curve up to t ∼ 170 ps (2.76 +τc). Consequently, within this range of time the retracting +velocity is independent from the aspect ratio Γ. For times +in the 50ps ≤ t ≤ 170ps range, all filaments accurately fol- +low the line with the slope equal to the Taylor-Culick velocity +v = R0/τc = 0.348 Å/ps, which is the relevant velocity scale +expected for the retraction process, originally proposed45,46 +as the steady-state velocity of a capillary-driven retracting in- +viscid planar liquid where inertia effects balance the capil- +lary forces acting on the system. For longer times the be- +havior changes because there are either filament oscillations, +changes in the tip shape, or both. The shorter the filament, +the earlier these deviations start to show up. The retracting +velocity of liquid filaments has been studied for Ohnesorge +numbers Oh ≥ 0.1,47 finding that the tip dynamics is charac- +terized by an oscillating velocity whose mean value is close +to the Taylor-Culick prediction. These oscillations have also +been found for Oh = 0.05 in the Γ = 20 case.47 In superfluid +helium, though, we do not observe any oscillation with time +of the tip retraction velocity. +IV. +SUMMARY +We have studied the instability and breakup of nanoscopic +superfluid 4He jets and filaments within He-DFT at zero tem- +perature. We find that the fragmentation of long cylindrical +jets closely follows the predictions of linear theory for inviscid +fluids, resulting in the formation of larger droplets intercalated +with smaller satellite droplets. +While some of our results for the contraction of free- +standing filaments are consistent with those obtained in the +inviscid regime which corresponds to Ohnesorge numbers +smaller than 2 × 10−3,19 the novelty with respect to previous +calculations for classical inviscid filaments is the appearance +of quantized vortex rings in filaments with aspect ratio Γ > 5. +Non-quantized vortex ring nucleation in the region connect- +ing the end drops with the rest of the filament plays a central +role in escaping filament breakup in the low-to-intermediate +viscosity regime characterized by Ohnesorge numbers in the +0.002 < Oh < 0.1 range.21 Our simulations show that a similar +mechanism, associated with quantized vortex rings, is active +in the superfluid regime at zero temperature, mostly prevent- +ing the droplet formation through end-pinching. Vortices are +also nucleated at surface protrusions appearing in the course +of filament oscillations, similar to those found in the merging +of He droplets. As a result, filaments are permeated by vortex- +antivortex ring pairs whose annihilation yields phonon/roton +bursts which may leave the filament in a turbulent state.41,42 + +13 +A key question is why vortex rings, which have appeared +in the solution of the Navier-Stokes equation in the 0.002 < +Oh < 0.1 regime, cease to appear in the inviscid regime19,21 +whereas we have found them in the superfluid regime within +the He-DFT approach. It is known that the Gross-Pitaevskii +and He-TDDFT equations, appropriated for superfluids, do +not reduce to the zero-viscosity limit of the Navier-Stokes +equation (Euler equation) for a barotropic fluid in irrotational +flow.27 In the superfluid case, an extra term appears involving +the gradient of the expression +Q = ¯h2 +2m +∇2ρ1/2 +ρ1/2 +(13) +the so-called quantum pressure term. +This term, which is +missing in any classical approach, plays an important role +when the density is highly inhomogeneous, as it happens near +the core of a quantized vortex. At variance, it is an ingredient +naturally included in the Schrödinger He-TDDFT Eq. (3). +We have thus seen that the He-DFT approach, which is a +suitable method to describe pure and doped superfluid He nan- +odroplets, can also address superfluid 4He jet breaking and the +contraction of superfluid 4He filaments. Yet, we have found +that upon filament breaking, the resulting fragments have a +tendency to merge again. Two effects combine to favor this +behavior. On the one hand, fragments, which are nanoscopic, +have a non-zero surface width that helps recombination due to +the overlap of the densities tails. On the other hand, the con- +traction velocity acquired by the filament in the early stages +of the contraction tends to push together the two highly de- +formed drops even if they are temporarily apart. One should +also consider the role of long-range van der Waals attractive +interaction between separated fragments, which may also con- +tribute to their merging. For the much larger sizes in the ex- +periments, however, the vdW forces are expected to be neg- +ligible. In fact, the force between two spherical particles of +diameter D made of q atoms per unit volume interacting via +the two-body vdW interaction λ/r6 is48 F ∝ − ˜F(x)/D, where +x = d/D, d being the distance of closest approach between the +spheres surfaces and ˜F(x) ∼ −1/(24x2) (x ≪ 1). Therefore, +for the sizes encountered in experiments the vdW attraction +between fragments will be much reduced if not negligible, +meaning that once a filament breaks into two fragments, re- +combination into a single droplet due to the vdW attraction is +unlike. +SUPPLEMENTARY MATERIAL +See supplementary material for the video files showing the +real time evolution of the processes discussed in the present +work. +ACKNOWLEDGMENTS +We thank Rico Tanyag for useful exchanges. This work +has been performed under Grant No. PID2020-114626GB- +I00 from the MICIN/AEI/10.13039/501100011033. +AUTHOR DECLARATIONS +Conflict of Interest +The authors have no conflicts to disclose. +Author Contributions +All authors contributed equally to this work. +DATA AVAILABILITY +The data that support the findings of this study are available +from the corresponding author upon reasonable request +1 K. K. Lehmann and G. Scoles, Science 279, 2065 (1998). +2 M. Y. Choi, G. E. Douberly, T. M. Falconer, W. K. 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Hamaker, Physica 4, 1058 (1937). + diff --git a/ZdFST4oBgHgl3EQfAjil/content/tmp_files/load_file.txt b/ZdFST4oBgHgl3EQfAjil/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5fcaa78a1b8744ecf0d7785b59199dffc3324006 --- /dev/null +++ b/ZdFST4oBgHgl3EQfAjil/content/tmp_files/load_file.txt @@ -0,0 +1,1005 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf,len=1004 +page_content='Nanoscopic jets and filaments of superfluid 4He at zero temperature: a DFT study Francesco Ancilotto,1, 2 Manuel Barranco,3, 4 and Martí Pi3, 4 1Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM, Università di Padova, via Marzolo 8, 35122 Padova, Italy 2CNR-IOM Democritos, via Bonomea, 265 - 34136 Trieste, Italy 3Departament FQA, Facultat de Física, Universitat de Barcelona, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Diagonal 645, 08028 Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 4Institute of Nanoscience and Nanotechnology (IN2UB), Universitat de Barcelona, Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (Dated: February 1, 2023) Helium droplets produced by the instability of a cryogenic helium jet exiting a source chamber leads to the formation of He drops which are considered as ideal matrices for spectroscopic studies of embedded atoms and molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Here, we present a He-DFT description of droplet formation resulting from jet breaking and contraction of superfluid 4He filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Whereas the fragmentation of long jets closely follows the predictions of linear theory for inviscid fluids, leading to droplet trains interspersed with smaller satellite droplets, the contraction of filaments with an aspect ratio larger than a threshold value leads to the nucleation of vortex rings which hinder their breakup into droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' INTRODUCTION Liquid helium droplets at low temperature offer a unique environment for molecular spectroscopy1–3 and the study of superfluidity on the atomic scale,4–6 including the study of quantum vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='7–10 Usually, helium droplets are produced by expansion of cooled helium gas or by instability of a cryogenic helium jet exiting a source chamber into vacuum throughout a nozzle, whose temperature and pressure deter- mine the appearance of the liquid jet and the droplet size and velocity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='11,12 Eventually, helium drops undergo evaporative cooling and become superfluid at a temperature of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='4 K11 on a µs time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='13 Understanding the dynamical properties of liquid 4He jets and the instabilities leading to their fragmentation is a rele- vant issue in the production and characterization of droplets made of 4He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This unique fluid allows for a large variation of non-dimensional parameters related to the fluid viscosity and the velocity at which it exits the nozzle, which characterize its dynamical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='14 This understanding has also a primary application, namely to make available helium drops with the size and velocity required by the experiments, together with size and velocity distributions as narrow as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This has led to recent experimental studies on the disintegration of liq- uid helium jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='15,16 Besides, a liquid thread with finite length (“filament” in the following) with no external constraint is ex- pected to contract trying to minimize its surface energy and eventually reach a spherical liquid drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' However, the out- come of the process is not always that simple, as an ample body of experiments and theoretical work on classical fluids has shown in the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We notice that liquid 4He filaments are regularly found in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='14–16 Liquid jets and filaments and their dynamical instabilities are well established subjects of study in classical fluids dy- namics because of practical questions and applications on the one hand, and because jet dynamics probes many physical properties and theoretical approaches on the other hand, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 17–19 and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Most studies concen- trate on viscous fluids because of practical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The underlying theoretical and numerical challenge is to solve the Navier-Stokes (NS) equation subject to appropriate boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The effect of viscosity and surface tension is embodied in the Ohnesorge number Oh defined as Oh = µ/√mρ0γR0, where m is the atom mass, ρ0 the atom density of the fluid, γ the surface tension, µ the viscosity coefficient, and R0 the radius of the jet or filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Inviscid filaments have been ad- dressed in passing by Schulkes,20 but owing to computational challenges, he could not simulate extreme interfacial defor- mations arising in crucial moments of the dynamics, as dur- ing filament breaking and pinch-off, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' the formation of two isolated drops from the opposite tips of the filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' While it is naturally assumed that solving the NS equation for small enough viscosities the results should be nearly indistinguish- able from the inviscid limit, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 19 and 21, a de- scription of superfluid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' inviscid and irrotational) jets and filaments is not available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Such study may be of relevance in view of the aforementioned studies on super- fluid 4He droplets and it is the motivation of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Our goal is to describe, at the microscopic level, the dynam- ics of contraction and breaking of zero temperature superfluid 4He nanojets and nanofilaments in vacuum using the well- established 4He density functional (He-DFT) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='22–25 The He-DFT approach is similar, in the superfluid 4He phase, to the Gross-Pitaevskii approach which has successfully been applied to the description of cold gases in the superfluid Bose- Einstein condensate phase, in particular in the study of quan- tized vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='26–28 Helium density functional and time-dependent density functional (He-TDDFT) methods have proven to be very pow- erful tools to address superfluid helium samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Within the He-DFT approach, the finite range of the He-He van der Waals (vdW) interaction is explicitly incorporated in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' As a consequence, the liquid-vacuum interface has a non- zero surface width, which is important in the description of nanoscopic 4He systems like the jets and filaments studied in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' It also takes into account the finite com- pressibility of the fluid and therefore the possibility of having arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='13699v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='mes-hall] 31 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Density profile in the radial direction of a cylinder of radius R0 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5 Å representing a 4He nanojet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' density excitations (ripplons, phonons and rotons) is naturally incorporated into the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' It also considers the pos- sibility of atom evaporation from the He sample during the real-time dynamics,29 which however has been found to be a negligible effect in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The He-DFT approach adds to the classical viscous fluids or molecular dynamics descriptions the possibility of disclos- ing purely superfluid effects in the dynamics, in particular quantized vortex nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' It has been recognized that a retracting viscous liquid filament may escape from pinch-off through the creation of vortex rings for Ohnesorge numbers in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='002 < Oh < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='1 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='21 Here we show that the same happens in the zero viscosity, irrotational superfluid case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Due to the computational burden associated with fully three-dimensional He-DFT simulations as the ones discussed here, we address jets and filaments of nanoscopic size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Stud- ies on the breakup of liquid nanojets are available in the litera- ture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' atomistic molecular dynamics simulations on the forma- tion, stability and breakup of viscous fluids have been carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='30 To our knowledge, no simulations of breakup of super- fluid nanojets and filaments have been published so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' II we briefly present the He-DFT approach used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In Sect III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='A we discuss the results for the dynamics of 4He jets, focusing on the conditions leading to fragmentation, and in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='B we study the contraction and possible break-up of 4He fila- ments with finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' A summary is presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In addition to the main text, we provide in the supplementary material the real-time dynamics of the 4He jets and filaments addressed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This multimedia material constitutes an important part of this work, since it helps capture physical details which would otherwise escape the written account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' THEORETICAL APPROACH Density functional theory for liquid helium is a phe- nomenological approach which constitutes a good compro- mise between accuracy and feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The parameters of the functional have been adjusted to reproduce various properties of the bulk superfluid such as equilibrium density, energy per atom and compressibility, as well as the main features of the dispersion relation of the elementary excitations of superfluid 4He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='22 A detailed description of the method can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 23–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Within He-DFT, the energy of a N-atom sample is written as a functional of the 4He atom density ρ(r) as E[ρ] = T[ρ]+Ec[ρ] = ¯h2 2m � dr|∇Ψ(r)|2 + � drEc[ρ] (1) where the first term is the kinetic energy, m is the mass of the 4He atom and Ψ(r) is the effective wave function (or or- der parameter) of the superfluid such that ρ(r) = |Ψ(r)|2 with � dr|Ψ(r)|2 = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The functional Ec(ρ) we have used con- tains the He-He interaction term within the Hartree approxi- mation and additional terms describing non-local correlation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='31 The equilibrium configuration of the system is obtained by solving, using an imaginary-time relaxation method,25 the Euler-Lagrange equation � − ¯h2 2m∇2 + δEc δρ � Ψ ≡ H [ρ]Ψ = ζ Ψ (2) where ζ is the 4He chemical potential corresponding to the number of He atoms in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Minimizing the action associated to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (1) leads to the He-TDDFT equation i¯h∂Ψ ∂t = � − ¯h2 2m∇2 + δEc δρ � Ψ ≡ H [ρ]Ψ (3) from which one can simulate the real-time evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The above equations have been solved using the 4He-DFT- BCN-TLS computing package,32 see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 24 and 25 and ref- erences therein for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Briefly, we work in cartesian coordinates, with the effective wave function Ψ(r,t) defined at the nodes of a 3D grid inside a calculation box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Pe- riodic boundary conditions (PBC) are imposed which allow to use the Fast Fourier Transform33 to efficiently compute the convolutions needed to obtain the DFT mean field H [ρ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The differential operators in H [ρ] are approximated by 13-point formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (2-3) have been solved using a space-step of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='2 Å, and the time-dependent Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (3) has been numerically integrated using a Hamming predictor-modifier-corrector ini- tiated by a fourth-order Runge-Kutta-Gill algorithm34 with a time-step of 2 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This time-step has been found to keep the energy of the jet and filaments properly conserved during the dynamics, as it corresponds to non-dissipative processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We have also checked that the jet configurations obtained in the course of the dynamics are robust against reasonable changes of the chosen space-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='015 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='005 0 0 5 10 15 20 25 30 r(A)3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Breaking dynamics of a cylinder subject to an axial perturbation of wavelength λ = 2πR0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The color bar shows the atom density in units of Å−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' RESULTS We have considered jets and filaments of sharp radius R0 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5 Å, defined as the radius at which the density equals ρ0/2, ρ0 being the liquid 4He atom density at zero temperature and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 4He nanoscopic jet The physics of liquid jets has been reviewed by Eggers and Villermaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='17 The thinning and breakup of a liquid jet is mainly determined by surface tension effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The stabil- ity of an infinite fluid cylinder of radius R0 was studied by Plateau,35 showing that it exists in an unstable equilibrium, and any perturbation with wavelength λ greater than 2πR0 is unstable and allows the surface tension to break up the cylin- der into droplets, thus decreasing the surface energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Lord Rayleigh later showed36 that for an inviscid liq- uid the fastest growing mode occurs when the wavelength of the axial undulation that ultimately leads to the fragmenta- tion of the jet into droplets is equal to λc = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01R0 (Rayleigh- Plateau instability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' When the jet breaks up, one or more small satellite drops -resulting from the necks breaking- may form between the larger droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The characteristic times for jet instability and breakup are set by the capillary time τc defined as τc = � mρ0R3 0 γ (4) with γ being the surface tension of the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In the case of 4He we have m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='325 × 1013 K, ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='021836 Å−3 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='274 K Å−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Hence, τc(R0 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5) = 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='7ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' It is customary to define the aspect ratio as Γ = ˜L/R0 where ˜L = L/2 is the half-length of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Here L coincides with the length of the simulation cell in the jet direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' From the linearized fluid dynamics equations for an inviscid and in- compressible fluid, a critical value Γc is predicted to trigger jet fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='36 It corresponds to the mode with wavelength λc = 2π/k, where k is such that ω(k) is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Here17 ω2(k) = � ξ I1(ξ) I0(ξ)(1−ξ 2) � 1 τ2c (5) where I0(x) and I1(x) = dI0(x)/dx are modified Bessel func- tions of the first kind and ξ ≡ kR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' From the maximum of ω one finds kR0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='697 and thus Γc = π/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='697 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='505 (6) t=0ps t=400 ps t=630ps t=700ps t=750ps t=800 ps 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='02 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='015 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='005 80 0 100 0 100 100 0 100 x (A) x (A)4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Neck shrinking as a function of time, shown on a log-scale (see the text for the definitions of δ and δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The points are the numerical values obtained from the simulation, whereas the dashed line shows the prediction of linear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In correspondence with it one has ωmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='343 � γ/(mρ0R3 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='343 τc (7) Similarly to what occurs in classical liquid jets, we have shown that the He-TDDFT approach yields the Rayleigh- Plateau instability for the superfluid nanojet when it is subject to a perturbation with the right wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Since the evolu- tion takes place in vacuum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' in the absence of ambient gas embedding the jet, the velocity of the jet itself is not playing any role and therefore we perform our simulations in a refer- ence frame where the jet is at rest (comoving frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' To this end, we simulate the jet by a cylindrical filament in a simula- tion box subject to PBC along the cylinder axis (the x-axis in the following).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Its equilibrium structure has been obtained by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' a plot of the jet density profile in the trans- verse direction is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The jet displays a bulk region of fairly constant density (slightly higher than the bulk 4He density ρ0 due to the compressive effect exerted by the surface tension on the lateral surface of the cylinder), delim- ited by a surface with a finite width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' As mentioned above, the radius of the cylinder R0 is defined as the distance from the symmetry axis of the point where ρ(r) = ρ0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We have verified by imaginary-time dynamics that the cylinder is indeed unstable against a small initial axial per- turbation with the proper wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' To do so, we consider a (periodically repeated) cylinder made of N = 12076 4He atoms and length L = 387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='2 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We have first found the equilib- rium geometry with a resulting radius R0 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Therefore the aspect ratio of the cylinder is Γ = ˜L/R0 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=', twice the critical aspect ratio Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' in this way the axial undu- lation caused by the mode with the Rayleigh-Plateau wave- length λc will produce two necks along the jet inducing the fragmentation into two droplets, as shown in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Next, we performed an imaginary-time dynamics starting from a configuration corresponding to a a slightly perturbed axially symmetric cylinder of radius R0, where the initial den- sity profile is given by: ρ(r) = ρ0 exp{[ � y2 +z2 −R(x)]/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5}+1 (8) with R(x) = R0[1−ε cos(4πx/L)] (9) and ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' With our choice of the length L and radius R0, the wavelength of the resulting density modulation is precisely equal to λc = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (9) ensures that the perturbed density is normalized so to have the same number of atoms as the unperturbed cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' If δ0 = ε R0, the maximum excursion of the radius along the cylinder axis is thus R = R0 ±δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The total energy of the axially perturbed cylinder turns out to be lower than that of the unperturbed cylinder, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' the sys- tem is energetically unstable toward a deformation leading to fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Starting from this configuration, we have per- formed an imaginary-time relaxation during which two necks develop eventually leading, as they shrink to zero, to two iden- tical spherical droplets as the lowest energy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Next, we have studied, by solving the He-TDDFT Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (3), the actual real-time dynamics of the fragmentation process, starting from the axially perturbed cylindrical jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 37, the perturbation is applied both to the density, as de- scribed above, and to the axial velocity of the jet as well, using the linearized solution of the Rayleigh-Plateau instability v = v0 sin(4πx/L) (10) where v0 = 2δ0 vmax/R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Notice that the radial perturbation is symmetric in the origin, whereas the velocity fluctuation is anti-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Here vmax = ωmaxR0/ξmax is calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (5) using ξ = ξmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='697, giving vmax ∼ 17 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Our starting value for the perturbation amplitude is δ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='452 Å, corresponding to the choice ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='021 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In order to apply this velocity field to the superfluid jet, we multiply the initial axially perturbed cylinder wave function Ψ(r) = ρ1/2(r) by the phase eiφ with φ = −2 δ0 R0 vmax �L/2 2π � cos(4πx/L) (11) and proceed with the real-time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Figure 2 shows snapshots of the jet density during the real- time dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' It can be seen that, starting from the perturbed cylinder, undulations whose amplitude increases with time ap- pear along the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The instability is caused by the fact that the Laplace pressure increases in constricted regions, driv- ing out the fluid and hence reducing further the neck radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The jet evolves into density bulges connected by thin threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Threads eventually break up and isolated drops appear in- stead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Figure 2 also shows that the threads between drops contract developing small end droplets displacing against each other whose collision yields a peak density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Not surprisingly, 100 10 0 100 200 300 400 500 600 700 t(ps)5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Breaking dynamics of a cylinder subject to multiple wavelength axial perturbations as explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The color bar shows the atom density in units of Å−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Contraction of a filament with aspect ratio Γ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The color bar shows the atom density in units of Å−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' t=0 ps t=460ps t=660ps t=718 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='02 t=760 ps t=866ps 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='015 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01 0 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='005 80 0 280 140 0 140 280-280 140 0 140 280 x (A) x (A)t=0 ps t=110 ps t=200ps t=254ps t=300ps t=400 ps 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='025 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='005 80 0 200 100 0 100 200 200 100 0 100 200 x (A) x (A)6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Contraction of a filament with aspect ratio Γ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The color bar shows the atom density in units of Å−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' t=0 ps t=150 ps t=260ps t=315ps t=380ps t=450 ps 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='025 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='005 80 0 200 100 0 100 200 200 100 0 100 200 x (A) x (A)7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Contraction of a filament with aspect ratio Γ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The color bar shows the atom density in units of Å−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' threads behave as the filaments described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' A similar pattern of alternating droplets and threads was ob- served in the study of the breakup of inviscid and irrotational capillary jets discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The lack of dissipation makes droplets and threads oscillate during the time elapsed by the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Once formed, they execute a series of vibrations, being alternately compressed and elongated in the jet direction with an expected frequency of the order of ω = � 8γ/mρ0R3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='39 It has been pointed out that no obvious effects due to superfluidity have been ob- served on the breakup behavior of a liquid He jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='14 Yet, Ko- latzki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='16 have found that He droplets undergo shape os- cillations that persist for much longer times than in the case of viscous drops, a signature of the superfluid character of these droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We would like to mention that if only the cylinder density is perturbed and no axial velocity field is applied to it, we find that jet breaking proceeds as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 2, the only difference being that it takes more time for the instability to fully develop and eventually lead to jet fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The actual time taken for the jet to break into droplets de- pends upon the amplitude of the initial density perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' It is defined as the time τb it takes for the wave amplitude with the largest frequency to grow up to R021,40 R0 = δ0eωmaxτb (12) where ωmax is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' With our choice for the initial perturbation amplitude δ0 we have τb = [ln(R0/δ0)]/ωmax = 695ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We have computed the dynamics of neck shrinking by mon- itoring during the real-time evolution the quantity δ(t) = (Rmax −Rmin)/2, where the radii Rmax and Rmin are measured at the two positions x = L/4 and x = L/2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The calculated values for δ(t)/δ0 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 3 on a loga- rithmic scale as a function of time, and are compared with the quantity eωmax t predicted from linear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' It is remarkable the good agreement between both for the whole duration of the breaking process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Finally, we have also investigated another scenario when the jet is subject to a more general perturbation on the equilib- rium density, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' we have started the real-time dynamics with the cylinder simultaneously perturbed by several axisymmet- ric perturbations of different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In order to accom- modate a reasonable number of modes with different wave- lengths compatible with the PBC used here, we perform the simulation in a cell longer than the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 2, with L equal to three times the critical wavelength associ- ated with the fastest mode, λc = 2πR0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We therefore consider an axially symmetric density perturbation given by a linear combination of six modes with small random ampli- tudes ε = δ0/R0 in the (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='03,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='03) range, and wavelengths λc,3λc,3λc/2,λc/2,λc/4, and 3λc/4, and perform a real-time t=0 ps t=150ps t=218 ps t=260 ps t=300ps t=376ps 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='025 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='015 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='005 80 0 200 100 0 100 200 200 100 0 100 200 x (A) x (A)8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Contraction of a filament with aspect ratio Γ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The color bar shows the atom density in units of Å−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Superfluid streamlines corresponding to the configurations Γ = 8 at t = 290 ps (top), and Γ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5 at t = 462 ps (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The color bar shows the atom density in units of Å−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' simulation starting from such initial state (t = 0 panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 4 some snapshots taken during the real- time evolution of this system, where it appears that among the various modes, the one eventually dominating in the course of time is indeed the critical one, dictated by λc, which leads to the formation of three necks, eventually resulting in the fragmentation into three droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' However, at variance with the case where the critical mode is the only one present (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 2) the jet does not break up into equal-size droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' For much longer filaments than the one investigated here, one might expect a distribution of slightly different drop sizes, some drops coming from the crests of the primary waves and others from the ligaments linking them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Determining the disturbance frequencies for jet breaking leading to the produc- tion of uniformly sized equidistant He drops has been one of the main concerns of a recent work16 in view of their exper- imental use in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' coherent diffraction imaging at x-ray free electron lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We have thus seen that the He-DFT approach is able to ad- dress jet breaking yielding results in agreement with linear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This is a needed first step before carrying out the study of contracting He filaments which we address in the follow- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' t=0 ps t=150 ps t=250 ps t=290ps t=320ps t=400 ps 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='025 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='005 80 0 200 100 0 100 200 200 100 0 100 200 x (A) x (A)40 t=290 ps 20 0 20 F= 8 40 100 50 0 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='024 (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='020 40 t=462 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='016 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='012 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='008 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='004 40 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5 100 50 0 50 100 c(A)9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Contraction of a filament with aspect ratio Γ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The color bar shows the atom density in units of Å−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Contraction of a filament with aspect ratio Γ = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The color bar shows the atom density in units of Å−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' t=0ps t=250 ps t=300ps t=400 ps t=440ps t=480ps 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='025 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='015 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='005 80 0 200 100 0 100 200 200 100 0 100 200 x (A) x (A)t=ops t=240ps t=290 ps t=400 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='025 t=480 ps t=600ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='02 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='015 40 0 00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='01 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='005 80 0 360 240-120 0 120240360-360-240-120 0 120 240 360 x (A) x (A)10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Contraction of filaments with different aspect ratios as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The slope of the solid line is the theoretical con- traction velocity R0/τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Contraction and fragmentation of free-standing filaments As with classical fluids, He jet breaking may lead not only to droplets but also to filaments, as observed in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='14–16 We model here these filaments as cylinders of radius R0 delimited by two hemispherical caps19 and study, using the He-TDDFT approach, their contraction due to the effect of the surface tension for different values of the aspect ratio Γ = ˜L/R0,18 where ˜L is the half-length of the filament from end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The configuration from which the real-time dynamics is initiated is a free-standing ideal filament (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' no density perturbation is applied), as usually done in numerical simulations of the contraction of viscous fluid filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='19–21 We have investigated filaments with different values of the aspect ratio, namely Γ = 4,5,6,8,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5, and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Some of these values coincide with those studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 19 at Oh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='001, which is considered to correspond to the inviscid regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Our goal is to study how the initial aspect ratio Γ determines the fate of the filament, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' either contraction into a single liq- uid body (stable state) or breaking into two or more droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Experimentally,18 it has been found for classical fluids that there is a critical initial aspect ratio Γ = 6 ± 1 below which a liquid filament is stable irrespective of the Oh value, and above which the filaments tend to break into separate droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In the following we describe the most salient features found during the real-time evolution of superfluid He filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' All simulations discussed below are displayed as movies in the supplementary material accompanying this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' These movies last for longer times than those reported in the follow- ing figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We do not discuss the filament appearance for such long times because undamped excitations and especially the annihilation of vortex rings, as discussed in the following, tend to produce turbulence41,42 whose description is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Filament with Γ = 4 This is the shortest filament that we have investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Its time evolution is similar to that predicted for short filaments by classical calculations and experiments,18,19 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' the filament contracts and oscillates back and forth without breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In the presence of some viscosity, the final configuration would be a single spherical droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' As shown by the temporal sequences in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 5, a blood- cell shape develops in the transverse direction (y-z plane) at around t = 240 ps, which develops an almost empty hole in the center at t = 254 ps (toroidal shape) before becoming compact again (frame at t = 300 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' After recovering the peanut-like shape along the filament axis (frame at t = 400 ps), the filament extends transversally at later times (t = 756 ps, not shown), and it is drawn again into a compact droplet, originating a high density spot at the touching region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The high density spot relaxes and launches a series of density waves propagating inside the filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This effect has been observed in previous simulations of the merging of two 4He nanodroplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='41,42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Filament with Γ = 5 According to classical calculations and experiments, a fila- ment with this value of Γ is also expected to display a stable dynamics, oscillating back and forth without breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='18,19 In- terestingly, simulation of this filament has disclosed the nucle- ation of quantized vortex rings, which appear in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 6 as dark spots in the snapshots at t = 315 ps and t = 380 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' For sym- metry reasons, only pairs of quantized vortex-antivortex rings (vortex ring pairs with opposite circulation) can be nucleated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' No such rings have been found in classical simulations carried out in the range of Ohnesorge numbers corresponding to the inviscid regime (below ∼ 2×10−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Yet, Hoepffner and Paré have found classical vortex rings for Ohnesorge numbers in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='002 < Oh < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='1 range but surprisingly enough not in the inviscid regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='21 We will highlight the role of these vortices for the longer filaments discussed in the following, where they become effective in preventing the filament breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In the case displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 6, vortices are nucleated dur- ing the contraction dynamics at surface indentations appearing between the end droplets or blobs and the rest of the cylindric filament (see the panel at t = 315 ps);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' this requires some in- ertia which can only be acquired when the filament is larger △ = 4 80 0 口 T=6 T=8 + T = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5 60 I= 15 Taylor-Culick A(A) 40 20 20 0 50 100 150 200 250 t(ps)11 than a critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Since this is the calculated filament with smallest Γ value for which we see vortex rings nucleation, one should expect the appearance of vortex rings for filaments with Γ ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Once nucleated, vortices move to the bulk of the filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The filament end caps collapse (panel at t = 315 ps) and launch additional vortex rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' One may see multiple vortex- antivortex ring pairs in a small volume which eventually anni- hilate, yielding an intense burst of density waves at later times (panel at t = 380 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Eventually, the filament oscillates be- tween the longitudinal and transverse directions, filled with density waves propagating inside the formed droplet (panel at t = 450 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We find similarities with the L0 = 5 case in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 19, but at variance with that reference, where breakup appears by complex oscillations at t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='2τc, in our simulation the end caps are reabsorbed in the bulk of the resulting stable droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Filament with Γ = 6 As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 7, the filament retracts and two end drops appear at the tips, clearly visible at around t = 150 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Drops grow in size and the filament between them contracts and shrinks into a thread, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' the configuration at t = 218 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This thread collapses at t = 236 ps, and the two droplets are temporarily apart, as shown in the panel for t = 260 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' However, due to the kinetic energy gained during the previous contraction stage, the two highly deformed fragments collide immediately after and merge again at t ∼ 280 ps to produce a single deformed droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The collision of the fragments produces a high density spot at the contact region (between t = 262 ps and t = 272 ps, see movie in the supplementary material) which expands yielding density waves propagating inside the filament, see the t = 300 ps frame in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The merged drop presents surface indenta- tions as those appearing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' at t = 300 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' These indentations act as nucleation sites for quantized vortex rings, which re- main close to the droplet surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The cores of some of these vortices are clearly visible in the frame at t = 376 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Notice that these vortices do not contribute to the escape from pinch- off since the thread connecting the end drops has collapsed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The density is no longer smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' rather, it is strongly perturbed by the presence of density waves produced by the merging of the two fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The evolution of this filament is similar to the L0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='0 fil- ament shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 5 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' For superfluid 4He we have found that the filament temporarily breaks into two deformed drops at t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='8τc, similar to the value one can read in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 5 of that reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' However, in our case drops collide and merge again, whereas in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 19 they seem to remain sep- arated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Another difference between classical and superfluid filaments is the appearance of quantized vortex rings and their subsequent annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Filament with Γ = 8 The dynamical evolution of this filament is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' As for the previous case, end drops develop, clearly vis- ible already after t ∼ 100 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The main filament connecting the end drops shrinks and a thin neck develops at the drop- filament contact region, which start pinching off the filament with two necks that reach their smallest radius at t = 254 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Before they completely shrink, vortex rings nucleate close to the necks at about t = 258 ps, being clearly formed at t = 270 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The streamlines of the superflow are drawn in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 9 for the configuration at t = 290 ps, clearly showing the characteristic pattern of lines wrapping the vortex core po- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' These vortex rings prevent necks from pinching, as they re- open immediately after their appearance (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' the frame at t = 290 ps), similarly to the mechanism discussed by Hoepffner and Paré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='21 A flow through the neck develops be- cause of the retraction and, according to these authors, this flow may detach into the jet downstream of the neck when fluid viscosity exceeds a threshold (Oh >∼ 2 × 10−3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='21 this sudden detachment creates a vortex ring which strongly mod- ifies the flow pressure: fluid is transported back into the neck which in turn reopens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' It is remarkable that the same happens in the case of superfluid 4He in spite of the lack of viscos- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' At t = 330 ps, another pair of vortex rings is nucleated at the droplet-filament indentation preventing pinching again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Finally, vortex-antivortex rings annihilate and disappear from the system producing as a result a burst of density waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The movie in the supplementary information shows the ap- pearance of surface protrusions at t = 452 ps which act as vortex nucleation sites, and their collapse yields a high den- sity spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Eventually, the contracted filament is permeated by a large number of vortex rings at t = 488 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This is at variance with the classical, inviscid fluid description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The evolution of this filament can be compared to that cor- responding to L0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='0 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 5 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Besides the vortex rings phenomenology, which is absent in the simula- tions of that reference, in our case end-pinching strictly never happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The closer the 4He filament gets to it is at t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='1τc, whereas the time for the filament breakup by end-pinching read from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 5 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 19 is t ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='6τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Filament with Γ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5 Similarly to the previous cases, end drops develop as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' A more violent approach is expected because the filament is longer and end drops have more time to acceler- ate under the traction exerted by surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The filament connecting the end drops contracts and necks appear at the drop-filament contact region, as shown at t = 250 ps, which start pinching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The neck shrinks to a minimum at t = 256 ps, escaping from pinch-off again because vortex rings are nucle- ated at t ∼ 260 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Vortex rings detach from the neck and move towards the bulk of the end drops (frame at t = 300 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The remaining filament develops bulges, which evolve to a more complex 12 structure (frame at t = 400 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The snapshot at t = 440 ps shows an almost complete frag- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' However, due to the opposite velocities acquired during the early stages of the contraction, the three fragments merge again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Other vortex rings are created in the process, nucleated at the necks during the re-merging, as shown in the frame at t = 480 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The streamlines of the superflow are drawn in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 9 for the configuration at t = 462 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Vortex ring annihilation at later times (see movie in the sup- plementary material) produces density waves arising from the collapse of their cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This is a phenomenon that we have not observed in the merging of He droplets,41,42 nor the shrink- ing of a vortex ring up to it collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' It is interesting to see that these small vortex rings travel towards the tips of the fil- ament, evaporating from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Eventually, vortex rings dis- appear and the contracted filament enters a complex dynamic regime, hosting plenty of density waves until the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The evolution of this filament should be similar to that cor- responding to L0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='0 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 5 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Be- sides the vortex rings phenomenology and wave dynamics, in our case end-pinching strictly never occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The filament gets close to it at t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='1τc (254 ps) and especially at t = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='0τc (432 ps), whereas the breakup time by end pinching read from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 5 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 19 is t ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='8τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Filament with Γ = 15 This is the largest filament we have investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In clas- sical simulations of sufficiently long filaments (like the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 11) and small Oh numbers, as the filament contracts it will succumb to end pinching18,20,43 even in cases where the Rayleigh-Plateau instability is expected to develop, subsequently resulting in the filament to break up into several drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' However, this instability does not occur, suggesting that the timescale for the Rayleigh-Plateau instability to grow is much larger than the timescale for the filament to fully con- tract even for long filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In the case of superfluid 4He, the sequence is similar to the Γ = 8 and Γ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5 cases, except that the number of necks has increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Well developed end drops appear at t = 100 ps, with a well developed necks at t = 160 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Figure 11 shows that end drops nearly pinch-off at t = 248 ps, but at t = 264 ps one may see vortex rings appearing at the necks, hindering pinch-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The vortex rings detach from the neck and move towards the bulk of the end drops and bulges ap- pear in the filament close to the end drops (panel at t = 290 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Bulges evolve to bulbs and, similarly to the Γ = 8 and Γ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='5 cases, intermediate drops develop during the time evolution whose number increases with the length of the fil- ament, as also observed in the simulations of classical low viscosity (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='003 ≤ Oh ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='02) filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='44 The evolution of this filament should be compared to that corresponding to L0 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='0 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 5 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Be- sides the phenomenology of vortex rings proliferation, also in this case end-pinching never occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' End drops are close to detach at t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='02τc (247 ps) but escape pinch off because of vortex ring nucleation, whereas the filament breakup time read from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 5 of that reference is t ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='8τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Finally, we have computed the contraction velocity for all the investigated filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We have defined the position of the tip of the filament as the location of its sharp surface (that at which the density equals ρ0/2) on the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Figure 12 shows the displacement of the tip position as a function of time for the studied filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' It appears that all curves collapse onto the same curve up to t ∼ 170 ps (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='76 τc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Consequently, within this range of time the retracting velocity is independent from the aspect ratio Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' For times in the 50ps ≤ t ≤ 170ps range, all filaments accurately fol- low the line with the slope equal to the Taylor-Culick velocity v = R0/τc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='348 Å/ps, which is the relevant velocity scale expected for the retraction process, originally proposed45,46 as the steady-state velocity of a capillary-driven retracting in- viscid planar liquid where inertia effects balance the capil- lary forces acting on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' For longer times the be- havior changes because there are either filament oscillations, changes in the tip shape, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The shorter the filament, the earlier these deviations start to show up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' The retracting velocity of liquid filaments has been studied for Ohnesorge numbers Oh ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='1,47 finding that the tip dynamics is charac- terized by an oscillating velocity whose mean value is close to the Taylor-Culick prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' These oscillations have also been found for Oh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='05 in the Γ = 20 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='47 In superfluid helium, though, we do not observe any oscillation with time of the tip retraction velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' SUMMARY We have studied the instability and breakup of nanoscopic superfluid 4He jets and filaments within He-DFT at zero tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We find that the fragmentation of long cylindrical jets closely follows the predictions of linear theory for inviscid fluids, resulting in the formation of larger droplets intercalated with smaller satellite droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' While some of our results for the contraction of free- standing filaments are consistent with those obtained in the inviscid regime which corresponds to Ohnesorge numbers smaller than 2 × 10−3,19 the novelty with respect to previous calculations for classical inviscid filaments is the appearance of quantized vortex rings in filaments with aspect ratio Γ > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Non-quantized vortex ring nucleation in the region connect- ing the end drops with the rest of the filament plays a central role in escaping filament breakup in the low-to-intermediate viscosity regime characterized by Ohnesorge numbers in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='002 < Oh < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='1 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='21 Our simulations show that a similar mechanism, associated with quantized vortex rings, is active in the superfluid regime at zero temperature, mostly prevent- ing the droplet formation through end-pinching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Vortices are also nucleated at surface protrusions appearing in the course of filament oscillations, similar to those found in the merging of He droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' As a result, filaments are permeated by vortex- antivortex ring pairs whose annihilation yields phonon/roton bursts which may leave the filament in a turbulent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='41,42 13 A key question is why vortex rings, which have appeared in the solution of the Navier-Stokes equation in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='002 < Oh < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='1 regime, cease to appear in the inviscid regime19,21 whereas we have found them in the superfluid regime within the He-DFT approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' It is known that the Gross-Pitaevskii and He-TDDFT equations, appropriated for superfluids, do not reduce to the zero-viscosity limit of the Navier-Stokes equation (Euler equation) for a barotropic fluid in irrotational flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='27 In the superfluid case, an extra term appears involving the gradient of the expression Q = ¯h2 2m ∇2ρ1/2 ρ1/2 (13) the so-called quantum pressure term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This term, which is missing in any classical approach, plays an important role when the density is highly inhomogeneous, as it happens near the core of a quantized vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' At variance, it is an ingredient naturally included in the Schrödinger He-TDDFT Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' We have thus seen that the He-DFT approach, which is a suitable method to describe pure and doped superfluid He nan- odroplets, can also address superfluid 4He jet breaking and the contraction of superfluid 4He filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Yet, we have found that upon filament breaking, the resulting fragments have a tendency to merge again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Two effects combine to favor this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' On the one hand, fragments, which are nanoscopic, have a non-zero surface width that helps recombination due to the overlap of the densities tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' On the other hand, the con- traction velocity acquired by the filament in the early stages of the contraction tends to push together the two highly de- formed drops even if they are temporarily apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' One should also consider the role of long-range van der Waals attractive interaction between separated fragments, which may also con- tribute to their merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' For the much larger sizes in the ex- periments, however, the vdW forces are expected to be neg- ligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' In fact, the force between two spherical particles of diameter D made of q atoms per unit volume interacting via the two-body vdW interaction λ/r6 is48 F ∝ − ˜F(x)/D, where x = d/D, d being the distance of closest approach between the spheres surfaces and ˜F(x) ∼ −1/(24x2) (x ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Therefore, for the sizes encountered in experiments the vdW attraction between fragments will be much reduced if not negligible, meaning that once a filament breaks into two fragments, re- combination into a single droplet due to the vdW attraction is unlike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' SUPPLEMENTARY MATERIAL See supplementary material for the video files showing the real time evolution of the processes discussed in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Rico Tanyag for useful exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' This work has been performed under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' PID2020-114626GB- I00 from the MICIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Author Contributions All authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request 1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Lehmann and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Scoles, Science 279, 2065 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Choi, G.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Ernst, in Handbook of High Resolution Spectroscopy, vol 3 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 1551, Whiley, New York (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 4 Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Sindzingre, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Klein, and D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Tanyag, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Jones, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Schorb, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Anielski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Belkacem, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Bernando, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Boll, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Bozek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Carron, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Delmas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Weise, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Zwart, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Bostedt, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Gessner, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Vilesov, Science 345, 906 (2014).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Cucini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Drabbels, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Finetti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Di Fraia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Gian- nessi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Grazioli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Iablonskyi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' LaForge, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Nishiyama, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Oliver Álvarez de Lara, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Piseri, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Plekan, K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Rupp, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Möller, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 121, 255301 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 9 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFST4oBgHgl3EQfAjil/content/2301.13699v1.pdf'} +page_content=' 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b/btE5T4oBgHgl3EQffA_l/content/tmp_files/2301.05624v1.pdf.txt @@ -0,0 +1,696 @@ +Layout-guided Indoor Panorama Inpainting with +Plane-aware Normalization +Chao-Chen Gao1[0000−0002−5325−3592], Cheng-Hsiu Chen1[0000−0002−8652−5830], +Jheng-Wei Su1[0000−0003−3148−002X], and Hung-Kuo Chu1[0000−0001−7153−4411] +1National Tsing Hua University, Taiwan +hkchu@cs.nthu.edu.tw +(a) Synthetic empty scene (b) Synthetic furnished scene (c) Real-world empty scene (d) Real-world furnished scene +Fig. 1: Indoor panorama inpainting. We present a learning-based indoor panorama +inpainting method that is capable of generating plausible results for the tasks of hole +filling (a)(c) and furniture removal (b)(d) in both synthetic (a)(b) and real-world (c)(d) +scenes. +Abstract. We present an end-to-end deep learning framework for indoor panoramic +image inpainting. Although previous inpainting methods have shown impressive +performance on natural perspective images, most fail to handle panoramic im- +ages, particularly indoor scenes, which usually contain complex structure and +texture content. To achieve better inpainting quality, we propose to exploit both +the global and local context of indoor panorama during the inpainting process. +Specifically, we take the low-level layout edges estimated from the input panorama +as a prior to guide the inpainting model for recovering the global indoor structure. +A plane-aware normalization module is employed to embed plane-wise style fea- +tures derived from the layout into the generator, encouraging local texture restora- +tion from adjacent room structures (i.e., ceiling, floor, and walls). Experimental +results show that our work outperforms the current state-of-the-art methods on +a public panoramic dataset in both qualitative and quantitative evaluations. Our +code is available online1. +1 https://ericsujw.github.io/LGPN-net/ +arXiv:2301.05624v1 [cs.CV] 13 Jan 2023 + +2 +Gao et al. +1 +Introduction +Image inpainting is a widely investigated topic in computer graphics and vision com- +munities, which aims at filling in missing regions of an image with photorealistic and +fine detailed content. It plays a crucial step toward many practical applications, such +as image restoration, object removal, etc. With the rapid development of deep learning, +image inpainting has been revisited and improved significantly in the past few years. +A considerable body of researches has been explored to generate impressive results on +perspective datasets. +In this work, we address the image inpainting problem in the context of indoor +panoramas. Indoor panoramas provide excellent media for the holistic scene under- +standing [40] that would further benefit several applications such as object detection, +depth estimation, furniture rearrangement, etc. In particular, removing foreground ob- +jects and filling the missing regions in an indoor panorama is essential for the interior +redesign task. However, the complex structures and textures presented in the indoor +scenes make the inpainting problem non-trivial and challenging for previous methods. +As shown in Figure 2(EC), results generated by a state-of-the-art deep learning method +fail to align the image structure along the layout boundaries and produce inconsistent +blurry image contents. +Recently, Gkitsas et al. [9] introduced PanoDR, a diminished reality-oriented in- +painting model for indoor panorama. The main idea is to translate a furnished indoor +panorama into its empty counterpart via a network that leverages both a generator and +an image-to-image translation module. The inpainting result is then obtained by com- +positing the predicted empty panorama and input panorama using the object mask. +However, there are still obvious artifacts near the boundaries of masked regions as +shown in Figure 2. +To achieve better inpainting quality, we present an end-to-end deep generative ad- +versarial framework that exploits both the global and local context of indoor panoramas +to guide the inpainting process. Specifically, we take the low-level layout boundaries +estimated from input panorama as a conditional input to guide the inpainting model, +encouraging the preservation of sharp boundaries in the filled image. A plane-aware +normalization module is then employed to embed local plane-wise style features de- +rived from the layout into the image decoder, encouraging local texture restoration from +adjacent room structures (i.e., ceiling, floor, and individual walls). We train and eval- +uate our model on a public indoor panorama dataset, Structured3D [41]. Experimental +results show that our method produces results superior to several state-of-the-art meth- +ods (see Figure 1, Figure 2 and Figure 5). The main contributions are summarized as +follows: +– We present an end-to-end generative adversarial network that incorporates both the +global and local context of indoor panoramas to guide the inpainting process. +– We introduce a plane-aware normalization module that guides the image decoder +with spatially varying normalization parameters per structural plane (i.e., ceiling, +floor, and individual walls). +– Our method achieves state-of-the-art performance and visual quality on synthetic +and real-world datasets. + +LGPN-net +3 +Input +EC [28] +PanoDR [9] +Ours +Fig. 2: Limitations of existing methods. EC [28] and PanoDR [9] fail to align the im- +age structure along the layout boundaries and produce inconsistent blurry image con- +tents in the inpainted regions (red mask). +2 +Related Work +Traditional image inpainting. There are two main genres among traditional image +inpainting works: diffusion-based methods and patch-based methods. Diffusion-based +methods [6,1,2,4,23,36] propagate pixels from neighboring regions to the missing ones +to synthesize the image content. On the other hand, patch-based methods [3,32,19,11,27,26,7] +fill the missing regions by searching for and copying similar image patches from the rest +of the image or existing image datasets. Without a high-level understanding of the im- +age contents, these methods easily fail on images with complex structures. +Learning-based image inpainting. With the rapid development of deep learning, sev- +eral image inpainting techniques based on convolutional neural networks (CNN) have +been proposed. These methods aim to learn a generator from a large dataset to produce +photorealistic image contents in the missing regions effectively. Context Encoders [30] +pioneers CNN-based image inpainting by proposing an adversarial network with an +encoder-decoder architecture. However, due to the information bottleneck layer of the +autoencoder, the results are often blurry, incoherent, and can not work on irregular +masks. Yu et al. [39] proposed a coarse-to-fine network and a context-aware mecha- +nism to reduce blurriness. Iizuka et al. [14] adopted local and global discriminators and +used dilated convolutions to increase the model’s receptive field and enhance coherence. +Liu et al. [25] proposed partial convolutions, which only consider valid pixels during +convolution, to handle irregular masks. Yu et al. [38] further extends the partial convo- +lutions by introducing a dynamic feature gating mechanism, named gated convolutions, +to deal with free-from masks. Both Liu et al. [25] and Yu et al. [38] adopt PatchGAN +discriminator [15] to improve the coherence further. Recently, several models were pro- + +000000 +OOODOODOOCOO +OOODOODOOOGO +OODOODOOCOO +OODOO4 +Gao et al. +posed to significantly improve the image painting quality by incorporating the structure +knowledge in a different context, including image edges [28,22], object contours [37], +smooth edge-preserving map [31], and gradient map [17]. Nazeri et al. [28] introduced +a two-stage network named EdgeConnect, which firstly recovers the missing edges in +the masked regions, followed by a generator conditioned on the reconstructed edge +map. The authors prove that the structure-to-content approach can effectively preserve +the structure information in the inpainting results. However, EdgeConnect uses canny +edges to represent structure features, which might be suitable for natural images but may +lead to complex local edges in indoor scenes. In contrast, our work exploits the Horizon- +Net [34] to estimate layout edges, representing the global room structure, which is suit- +able for our indoor inpainting task. In addition, our model is an end-to-end architecture +instead of a two-stage network. Yang et al. [17] developed a multi-task learning frame- +work to jointly learn the completion of image contents and structure map (edges and +gradient). A structure embedding scheme is employed to embed the learned structure +features while inpainting explicitly. The model further learns to exploit the recurrent +structures and contents via an attention mechanism. While demonstrating impressive +performance in generating realistic results, these structure-aware methods still fail to +model long-range structure correspondence such as the layout in the indoor scenes. On +the other hand, some works have successfully recovered a single partially occluded ob- +ject [5,20]. However, their architecture does not handle multiple object instances of the +same class and is thus not suitable for our context where the plane-wise segmentation +consists of different numbers of wall planes. +Image-to-image translation. The image inpainting is essentially a constrained image- +to-image translation problem. Significant efforts have been made to tackle various prob- +lems based on image-to-image translation architectures [15,42,18]. Here we focus on +the ones that are closely related to our work. Park et al. [29] introduced SPADE, which +utilizes a spatial adaptive normalization layer for synthesizing photorealistic images +given an input semantic segmentation map. Specifically, a spatially-adaptive learned +transform modulates the activation layer with a semantic segmentation map and ef- +fectively propagates the semantic information throughout the network. In contrast to +SPADE, which uses only one style code to control the image synthesis, Zhu et al. [43] +presents SEAN by extending the SPADE architecture with per-region style encoding. +By embedding one style code for individual semantic classes, SEAN shows signifi- +cant improvement over SPADE and generates the highest quality results. In the con- +text of indoor scenes, Gkitsas et al. [9] introduce PanoDR that combines image-to- +image translation with a generator to constrain the image inpainting with the underly- +ing scene structure. Percisely, to convert a furnished indoor panorama into its empty +counterpart, PanoDR exploits a generator for synthesizing photorealistic image con- +tents where the global layout structure is preserved via an image-to-image translation +module. The empty indoor panorama is then used to complete the masked regions in +the input panorama via a simple copy-and-paste process. Gkitsas et al. [10] extend the +architecture of PanoDR to make the model end-to-end trainable. However, the quan- +titative evaluation indicates that the performance improvement is marginal compared +with PanoDR. Our system also combines a generator with image-to-image translation +as PanoDR does. However, we obtain superior results than PanoDR by exploiting the + +LGPN-net +5 +Fig. 3: Architecture overview. Our network architecture follows the conventional gen- +erative adversarial network with an encoder-decoder scheme supervised by low- and +high-level loss functions and a discriminator. Given a masked indoor panoramic image +Iin with a corresponding mask M, our system uses an off-the-shelf layout prediction +network to predicts a layout map. The low-level boundary lines in Lm serve as a con- +ditional input to our network to assist the inpainting. Then, we compute two semantic +segmentation maps from the layout map Lm, declared L3−class and Lp−wise, where +the latter is used to generate plane-wise style codes for ceiling, floor, and individual +walls. Finally, these per plane style codes, together with L3−class, are fed to a struc- +tural plane-aware normalization module to constrain the inpainting. +global layout edges as a prior and adapting SEAN blocks in a local plane-wise man- +ner to guide the inpainting. Moreover, in contrast to PanoDR performs the inpainting +task via an indirect way, our system performs the inpainting task in an end-to-end fash- +ion, directly completing the mask areas instead of hallucinating an empty scene, thus +resulting in better visual quality and consistency. +3 +Overview +Figure 3 illustrates an overview of our architecture. Our system takes a masked panoramic +image Iin and the corresponding binary mask M as inputs and generates the inpainted +panoramic image Iout. The masked panoramic image is generated by Iin = Igt ⊙ (1 − M), +where Igt represents the ground-truth panoramic image and ⊙ denotes the Hadamard +product. Our system first utilizes an off-the-shelf model to estimate the room layout +Lm from input masked panoramic image. This layout map is then concatenated with +Iin and M to obtain a five-channel input map fed into the generator G. We further +derive two semantic segmentation maps L3−class and Lp−wise using the layout map +for the subsequent normalization module (Section 4.1). The image generation model +follows the conventional generative adversarial architecture with one content encoder +and one image decoder with one discriminator. (Section 4.2). To impose structure in- +formation during inpainting, we introduce a plane-aware normalization that modifies + +Horizon +Net +VGG-196 +Gao et al. +Fig. 4: Plane-aware normalization. Given an incomplete indoor panoramic image Iin +with mask M, we first predict two normalization values β and γ through several partial +convolution [25] blocks and a plane-wise average pooling based on the plane-wise seg- +mentation map Lp−wise. Second, we predict another set of normalization values β′ and +γ′ through several vanilla convolution blocks based on the 3-class segmentation map +L3−class. The final normalization values are thus computed using the weighted sum +weighted by learnable parameters αβ and αγ. +the SEAN [43] block with two semantic segmentation maps to guide the decoder with +spatially varying normalization parameters per structural plane (i.e., ceiling, floor, and +individual walls). Such a plane-aware normalization provides useful guidance for global +structure preservation as well as consistent local image content generation (Section 4.3). +Finally, common loss functions in image inpainting, including the reconstruction loss, +the perceptual loss, the style loss, and the adversarial loss are employed to train our +model (Section 4.4). +4 +Method +4.1 +Layout Guidance Map +We employ an off-the-shelf model, HorizonNet [34], to estimate a layout map from +input masked panorama. Through a recurrent neural network, the HorizonNet predicts +a 3-dimensional vector representing ceiling, ground, and corner location. We further +process the output vector to generate a layout map Lm comprising low-level boundary +lines. This layout map serves as a conditional input to encourage the preservation of +global layout structure while inpainting. Moreover, we extract two semantic segmen- +tation maps from the layout map that depict (i) the segmentation mask L3−class with +three semantic labels of indoor scene, i.e., ceiling, floor, and wall; and (ii) a plane-wise +segmentation mask Lp−wise where pixels are indexed in a per structural plane basis + +LGPN-net +7 +(i.e., ceiling, floor, or individual walls). These semantic segmentation maps are gener- +ated using conventional image processing operations (i.e, flood-fill) and will be used in +the later normalization module. +4.2 +Image Inpainting Backbone +As shown in Figure 3, our network architecture consists of one generator and one dis- +criminator. The generator G follows a conventional scheme with one content encoder +and one image decoder. The content encoder consists of two down-sampling convo- +lution blocks followed by eight residual blocks using dilated convolution [12]. The +image decoder uses a cascade of our proposed plane-aware residual blocks and two +up-sampling blocks. Motivated by EdgeConnect [28], we use PatchGAN [16] as our +discriminator to determine the real or fake sample by dividing the input image into sev- +eral patches. In the following sections, we will elaborate plane-aware residual block, +loss functions, and discriminator in more detail. +4.3 +Plane-aware Normalization +Considering the different styles among wall planes is very common in real-world indoor +scenes. We follow the architecture of SEAN [43] and propose leveraging two kinds of +segmentation maps Lp−wise and L3−class to establish our plane-aware normalization +(see Figure 4). Our plane-aware normalization consists of one style encoder and two +style blocks, which enhance the global style semantics and local style consistency of +the generated results. The inputs of the style encoder include masked panoramic image +Iin and mask image M. We use partial convolution blocks in style encoder instead of +vanilla convolution to make feature extraction conditioned only valid pixels. We first +adopt the plane-wise average pooling on the output features to generate style codes for +each plane based on Lp−wise. Second, we spatially broadcast each style code on the +corresponding area and output the local style block. On the other side, we predict the +global style block by passing the 3-class segmentation map L3−class through several +convolution layers. Finally, the remaining part of our plane-aware normalization follows +the same architecture of SEAN [43], and combines global and local style blocks into +the downstream β and γ parameters of the final batch normalization. +4.4 +Loss Functions +Here we elaborate on the low- and high-level loss functions and the discrimination used +for training our image generator. +Reconstruction loss measures the low-level pixel-based loss between the predicted and +ground-truth images. To encourage the generator to pay more attention to the missing +regions, we additionally calculate the L1 loss in the missing regions. The reconstruction +loss Lrec is defined as follows: +Lrec = ∥M ⊙ Igt − M ⊙ Iout∥1 + ∥Igt − Iout∥1 , +(1) + +8 +Gao et al. +where Igt and Iout represent the ground-truth image and the generator’s output, respec- +tively, and M is a binary mask. +Perceptual loss +encourages the predicted and ground-truth images to have similar +representation in high-level feature space extracted via a pre-trained VGG-19 [33], and +is defined as follows: +Lperc = +� +i +∥φi (Igt) − φi (Iout)∥1 , +(2) +where φi is the activation map of the ith layer of the pre-trained feature extraction +network. +Style loss calculates the co-variance difference between the activation maps. For the +activation map φi of size Ci × Hi × Wi, the style loss is defined as follows: +Lsty = +���Gφ +i (Igt) − Gφ +i (Iout) +��� +1 , +(3) +where Gφ +i is a Ci × Ci gram matrix [8] constructed by the activation map φi. +Adversarial loss is implemented with the patch-based discriminator [16], which out- +puts the feature map divided into several feature patches and uses hinge loss [24] to +optimize the generator G and the discriminator D. The adversarial loss for generator G +and discriminator D are defined as follows: +LG = −D (Iout) , +(4) +LD = λD (max (0, 1 + D (Iout)) + max (0, 1 − D (Igt))) ; +(5) +The overall loss function used in the generator G is defined as follows: +Ltotal = λrecLrec + λpercLperc + λstyLsty + λGLG, +(6) +where λrec, λperc, λsty, λG, and λD are the hyperparameters for weighting the loss +functions. +5 +Experiments +In this section, we evaluate the performance of our model by comparing it with several +state-of-the-art image inpainting approaches and conducting ablation studies to verify +the necessity of individual components in the proposed architecture. Please refer to our +online webpage for other experiments and more results2. +5.1 +Experimental Settings +Dataset and baselines. +We compare our model with the following state-of-the-art +structure-aware image inpainting models: +2 https://ericsujw.github.io/LGPN-net/ + +LGPN-net +9 +Input / Layout +EC [28] +LISK [17] +PanoDR [9] +Ours +GT +Fig. 5: Qualitative comparisons with state-of-the-arts. Top 8 rows: the inpainting re- +sults of the empty indoor scenes. Bottom 8 rows: the inpainting results of the furnished +indoor scenes. Our method produces superior results in generating image contents that +align the layout structure well and are consistent with the surrounding of the masked +regions. + +米米米米米米米10 +Gao et al. +– EC [28]: a two-stage adversarial network that comprises an edge completion model +followed by a generator. +– LISK [17]: a multi-task learning framework that exploits image structure embed- +ding and an attention mechanism in the generator. +– PanoDR [9]: a deep learning framework that combines image-to-image translation +with generator to condition the inpainting on the indoor scene structure. +The experiments were conducted on a public indoor panorama dataset, Structured3D [41], +which contains 21,835 indoor panoramas. The official data split is adopted for train- +ing(18,362), validation(1,776), and testing(1,697). We follow the same procedure as +PanoDR to generate mask images using contours of foreground furniture (see Section +3.1). We use the officially released implementation of baselines for training from scratch +and testing. Note that each indoor panorama in Structured3D has two representations of +the same scene (i.e., empty and furnished). Therefore, the experiments were conducted +in two phases to evaluate our model and baselines in different application scenarios +(i.e., structural inpainting vs. furniture removal). +Evaluation metrics. We take several commonly used image quality assessment met- +rics in previous inpainting approaches for quantitative evaluation. Specifically, we used +the low-level feature metrics, including Mean Absolute Error (MAE), Peak Signal-to- +Noise (PSNR), Structural Similarity Index (SSIM) [35], and Fr´echet Inception Distance +(FID) [13]. +Implementation details. We implement our model in PyTorch and conduct the exper- +iments on a single NVIDIA V100 with 32G VRAM. The resolution of the panoramic +images is resized to 512 × 256. We use Adam [21] optimizer in the training process +with the hyper-parameters setting of b1 = 0.0 and b2 = 0.9, a learning rate of 0.0001, +and a batch size of 8. We empirically set λrec = 1, λperc = 0.1, λsty = 250, λG = 0.1, +and λD = 0.5 in the total loss function (Equation 6). For HorizonNet [34], we use the +official pre-trained model for layout estimation. +5.2 +Evaluation on the Empty Scenes +In this experiment, we evaluate both the qualitative and quantitative performance of +our model on the image inpainting task by comparing it with baselines. The qualitative +comparisons are shown in Figure 5 (top 8 rows). In contrast to EC and LISK, which +fail to restore image structures in the masked regions, our method faithfully gener- +ates image contents adhering to the underlying layout structure. While PanoDR shows +slightly better structure preservation than EC and LISK, it fails to generate image con- +tents consistent with the surrounding of masked regions as our method does. Therefore, +our method achieves the best performance against all the baselines across all evaluation +metrics as shown in Table 1 (top). +5.3 +Evaluation on the Furnished Scenes +Furniture of irregular shape will more or less obscure the layout of the indoor scene, +making it more challenging to restore the regular structure in the missing area. There- +fore, in this experiment, we would like to evaluate how well our model learned from + +LGPN-net +11 +Table 1: Quantitative comparisons with state-of-the-arts. The top and bottom tables +summarize the performance of our model and baselines on the empty and furnished +scenes, respectively. +Dataset +Method +PSNR↑ SSIM↑ MAE↓ FID↓ +Empty scene +EC [28] +38.6936 0.9892 0.0039 3.9480 +LISK [17] +41.3761 0.9895 0.0055 4.1660 +PanoDR [9] 37.2431 0.9884 0.0040 4.3591 +Ours +41.8444 0.9919 0.0030 2.5265 +Furnished scene +EC [28] +31.4439 0.9493 0.0076 11.9955 +LISK [17] +34.7325 0.9553 0.0068 14.2676 +PanoDR [9] 34.3340 0.9641 0.0051 7.8399 +Ours +35.3923 0.9672 0.0047 7.2328 +Table 2: Quantitative results of the ablation study. We evaluate the effectiveness of +our design choices by gradually adding the individual components into the architecture. +PSNR ↑ SSIM ↑ MAE ↓ FID ↓ +Backbone +40.6449 0.9911 0.0034 3.3915 +Layout map only 41.2884 0.9916 0.0033 2.8105 +Full model +41.8444 0.9919 0.0030 2.5265 +empty scenes can generalize to the furnished scenes. Since the inpainting task setup +here exactly matches the one defined in the PanoDR, we use the pre-trained model +of PanoDR in this experiment for a fair comparison. As shown in Figure 5 (bottom 8 +rows), our method still clearly outperforms baselines in generating image contents that +align the layout structure well and are consistent with the surrounding of the masked +regions. The quantitative results are shown in Table 1 (bottom). It is worth noting that +the way PanoDR performs image completion via compositing the predicted empty im- +age and input image using the object mask will lead to severe artifacts where occlusion +occurred between foreground objects (see Figure 2(PanoDR)). +5.4 +Ablation Study +Here, we conduct ablation studies to validate our model from different perspectives. +First, we evaluate the necessity of individual design choices in our architecture. Then, +we conduct two experiments to evaluate how sensitive our model is to the size of input +masks and the quality of input layout maps. +Ablation on network architecture. In this experiment, we start with the backbone +model (Backbone) as the baseline, then progressively adding only layout guidance +map (Layout map only), and our plane-aware normalization (Full model). As shown +in Table 2, we obtain the best performance with the full model on all the metrics. The +qualitative comparisons shown in Figure 6 indicate that adding layout guidance map +generates clear structure boundaries in the final result (2nd and 3rd columns), while our + +12 +Gao et al. +Input +Backbone +Layout map only +Full model +GT +Fig. 6: Qualitative results of the ablation study. Side-by-side comparisons of inpaint- +ing results generated using our method by gradually adding individual components. +From left to right, input images and masks, our baseline model (Backbone), adding +the layout guidance map (Layout map only), full model with our plane-aware nor- +malization (Full model), and ground truth images. +full model with plane-aware normalization can constrain the image generation to the +adjacent structural planes and obtain visually consistent results (3rd and 4th columns). +Sensitivity to the mask size. In this experiment, we analyze the testing dataset and +classify the images into different categories according to the area proportions of input +masks. Table 3 shows the inpainting performance for each category. We can tell that +the inpainting quality degrades with the increasing mask size. A significant drop occurs +where the ratio of input mask is greater than 30%. +Sensitivity to the layout estimation. In order to explore the effect of the accuracy +of layout estimation on the inpainting quality, we first devise a mechanism to generate +layout maps with different levels of accuracy. Specifically, we feed masked images of +different mask sizes into HorizonNet. We start by generating randomly located rect- +angle masks of 5% image size and increase the mask ratio to 10%, 30% and 50% to +deliberately produce layout structures with decreasing quality. Then we take these lay- +out maps as conditional inputs of our model and compare the inpainting performance +empty-room testing dataset. As shown in Table 4, our model degrades marginally when +the quality of estimated layouts decreases from 0.96 to 0.84, indicating our model is +robust to the varying input layout maps. + +LGPN-net +13 +Table 3: Mask size vs. inpainting quality. +Mask Size(%) Count +Content +PSNR ↑ SSIM ↑ MAE ↓ +0-10 +1045 +44.3921 0.9967 0.0011 +10-20 +163 +34.2823 0.9841 0.0055 +20-30 +48 +30.4371 0.9726 0.0111 +30-40 +39 +25.0731 0.9386 0.0266 +40+ +13 +24.2958 0.9345 0.0305 +total +1308 +41.8444 0.9919 0.0030 +Table 4: Accuracy of layout estimation vs. inpainting quality. +Structure +Content +mIOU ↑ +PSNR ↑ SSIM ↑ MAE ↓ FID ↓ +0.9603 +42.3212 0.9925 0.0028 2.4322 +0.9561 +42.2871 0.9925 0.0028 2.4441 +0.9175 +42.0682 0.9923 0.0029 2.5624 +0.8489 +41.7300 0.9919 0.0030 2.8455 +5.5 +Qualitative Results on Real-world Scene +Real-world scenes have complex lighting and layout structure. However, the amount of +data in the real-world scene dataset and the quality of furniture category annotations +are insufficient for training our model, so we choose to train on the synthetic dataset +Structured3D [41]. Nevertheless, we still compare our results with PanoDR [9], which +also implements the furniture removal task, on the real-world scene dataset. Since the +real-world scene dataset does not contain paired data (i.e., scenes before and after furni- +ture removal), quantitative evaluation is infeasible and we can only provide qualitative +comparisons here. Figure 7 shows that our inpainted results have a higher quality of +structural maintenance and color restoration. Moreover, compared with PanoDR, we +can still exert more stable performance in real-world scenes. Please refer to our online +webpage for more results3. +6 +Conclusions +We proposed an end-to-end structural inpainting network for the indoor scene. We in- +troduce layout boundary line conditions the output structure and utilize the plane-aware +normalization to enhance planar style consistency. Experiment results show the out- +standing performance of our model in both structural inpainting and furniture removal +on the indoor scene. +3 https://ericsujw.github.io/LGPN-net/ + +14 +Gao et al. +Input +EC [28] +LISK [17] +PanoDR [9] +Ours +Fig. 7: Qualitative comparisons with state-of-the-arts on real-world scenes. Our +model clearly outperforms baselines by preserving layout boundary and restoring lo- +cal texture from adjacent room structures (i.e., floor and walls). +Input +PanoDR [9] +Ours +Fig. 8: Limitation. Both the state-of-the-art method and our model produce visual arti- +facts in the scenes presenting strong shading effect surrounding the removed furniture. +Limitations. In the real-world application of furniture removal, we can often see resid- +uals of shading effect caused by the removed furniture. These residuals are hard to +segment and even harder to model. As shown in Figure 8, our model is slightly affected +by these residuals but still produces more realistic results than PanoDR [9]. +Future work. We plan to adopt a more reasonable segmentation mask of the indoor +scene inpainting which can cover the shading area and thus improve our results in those +shaded scenes. +Acknowledgements. The project was funded in part by the National Science and Tech- +nology Council of Taiwan (110-2221-E-007-060-MY3, 110-2221-E-007-061-MY3). +References +1. Ashikhmin, M.: Synthesizing natural textures. In: Proceedings of the 2001 Symposium +on Interactive 3D Graphics. p. 217–226. I3D ’01, Association for Computing Machin- + +LGPN-net +15 +ery, New York, NY, USA (2001). https://doi.org/10.1145/364338.364405, https://doi. +org/10.1145/364338.364405 +2. Ballester, C., Bertalmio, M., Caselles, V., Sapiro, G., Verdera, J.: Filling-in by joint inter- +polation of vector fields and gray levels. 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In: IEEE/CVF Conference on Computer Vision and Pattern Recognition +(CVPR) (2020) + diff --git a/btE5T4oBgHgl3EQffA_l/content/tmp_files/load_file.txt b/btE5T4oBgHgl3EQffA_l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f19d1ff2871c3e9452ed176ccf7261a408950d3 --- /dev/null +++ b/btE5T4oBgHgl3EQffA_l/content/tmp_files/load_file.txt @@ -0,0 +1,773 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf,len=772 +page_content='Layout-guided Indoor Panorama Inpainting with Plane-aware Normalization Chao-Chen Gao1[0000−0002−5325−3592], Cheng-Hsiu Chen1[0000−0002−8652−5830], Jheng-Wei Su1[0000−0003−3148−002X], and Hung-Kuo Chu1[0000−0001−7153−4411] 1National Tsing Hua University, Taiwan hkchu@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='nthu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='tw (a) Synthetic empty scene (b) Synthetic furnished scene (c) Real-world empty scene (d) Real-world furnished scene Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 1: Indoor panorama inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We present a learning-based indoor panorama inpainting method that is capable of generating plausible results for the tasks of hole filling (a)(c) and furniture removal (b)(d) in both synthetic (a)(b) and real-world (c)(d) scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We present an end-to-end deep learning framework for indoor panoramic image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Although previous inpainting methods have shown impressive performance on natural perspective images, most fail to handle panoramic im- ages, particularly indoor scenes, which usually contain complex structure and texture content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' To achieve better inpainting quality, we propose to exploit both the global and local context of indoor panorama during the inpainting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Specifically, we take the low-level layout edges estimated from the input panorama as a prior to guide the inpainting model for recovering the global indoor structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' A plane-aware normalization module is employed to embed plane-wise style fea- tures derived from the layout into the generator, encouraging local texture restora- tion from adjacent room structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', ceiling, floor, and walls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Experimental results show that our work outperforms the current state-of-the-art methods on a public panoramic dataset in both qualitative and quantitative evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Our code is available online1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 1 https://ericsujw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='io/LGPN-net/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='05624v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='CV] 13 Jan 2023 2 Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 1 Introduction Image inpainting is a widely investigated topic in computer graphics and vision com- munities, which aims at filling in missing regions of an image with photorealistic and fine detailed content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' It plays a crucial step toward many practical applications, such as image restoration, object removal, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' With the rapid development of deep learning, image inpainting has been revisited and improved significantly in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' A considerable body of researches has been explored to generate impressive results on perspective datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In this work, we address the image inpainting problem in the context of indoor panoramas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Indoor panoramas provide excellent media for the holistic scene under- standing [40] that would further benefit several applications such as object detection, depth estimation, furniture rearrangement, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In particular, removing foreground ob- jects and filling the missing regions in an indoor panorama is essential for the interior redesign task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' However, the complex structures and textures presented in the indoor scenes make the inpainting problem non-trivial and challenging for previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' As shown in Figure 2(EC), results generated by a state-of-the-art deep learning method fail to align the image structure along the layout boundaries and produce inconsistent blurry image contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Recently, Gkitsas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [9] introduced PanoDR, a diminished reality-oriented in- painting model for indoor panorama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The main idea is to translate a furnished indoor panorama into its empty counterpart via a network that leverages both a generator and an image-to-image translation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The inpainting result is then obtained by com- positing the predicted empty panorama and input panorama using the object mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' However, there are still obvious artifacts near the boundaries of masked regions as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' To achieve better inpainting quality, we present an end-to-end deep generative ad- versarial framework that exploits both the global and local context of indoor panoramas to guide the inpainting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Specifically, we take the low-level layout boundaries estimated from input panorama as a conditional input to guide the inpainting model, encouraging the preservation of sharp boundaries in the filled image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' A plane-aware normalization module is then employed to embed local plane-wise style features de- rived from the layout into the image decoder, encouraging local texture restoration from adjacent room structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', ceiling, floor, and individual walls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We train and eval- uate our model on a public indoor panorama dataset, Structured3D [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Experimental results show that our method produces results superior to several state-of-the-art meth- ods (see Figure 1, Figure 2 and Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The main contributions are summarized as follows: – We present an end-to-end generative adversarial network that incorporates both the global and local context of indoor panoramas to guide the inpainting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' – We introduce a plane-aware normalization module that guides the image decoder with spatially varying normalization parameters per structural plane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', ceiling, floor, and individual walls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' – Our method achieves state-of-the-art performance and visual quality on synthetic and real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' LGPN-net 3 Input EC [28] PanoDR [9] Ours Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 2: Limitations of existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' EC [28] and PanoDR [9] fail to align the im- age structure along the layout boundaries and produce inconsistent blurry image con- tents in the inpainted regions (red mask).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 2 Related Work Traditional image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' There are two main genres among traditional image inpainting works: diffusion-based methods and patch-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Diffusion-based methods [6,1,2,4,23,36] propagate pixels from neighboring regions to the missing ones to synthesize the image content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' On the other hand, patch-based methods [3,32,19,11,27,26,7] fill the missing regions by searching for and copying similar image patches from the rest of the image or existing image datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Without a high-level understanding of the im- age contents, these methods easily fail on images with complex structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Learning-based image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' With the rapid development of deep learning, sev- eral image inpainting techniques based on convolutional neural networks (CNN) have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' These methods aim to learn a generator from a large dataset to produce photorealistic image contents in the missing regions effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Context Encoders [30] pioneers CNN-based image inpainting by proposing an adversarial network with an encoder-decoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' However, due to the information bottleneck layer of the autoencoder, the results are often blurry, incoherent, and can not work on irregular masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [39] proposed a coarse-to-fine network and a context-aware mecha- nism to reduce blurriness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Iizuka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [14] adopted local and global discriminators and used dilated convolutions to increase the model’s receptive field and enhance coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [25] proposed partial convolutions, which only consider valid pixels during convolution, to handle irregular masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [38] further extends the partial convo- lutions by introducing a dynamic feature gating mechanism, named gated convolutions, to deal with free-from masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Both Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [25] and Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [38] adopt PatchGAN discriminator [15] to improve the coherence further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Recently, several models were pro- 000000 OOODOODOOCOO OOODOODOOOGO OODOODOOCOO OODOO4 Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' posed to significantly improve the image painting quality by incorporating the structure knowledge in a different context, including image edges [28,22], object contours [37], smooth edge-preserving map [31], and gradient map [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Nazeri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [28] introduced a two-stage network named EdgeConnect, which firstly recovers the missing edges in the masked regions, followed by a generator conditioned on the reconstructed edge map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The authors prove that the structure-to-content approach can effectively preserve the structure information in the inpainting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' However, EdgeConnect uses canny edges to represent structure features, which might be suitable for natural images but may lead to complex local edges in indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In contrast, our work exploits the Horizon- Net [34] to estimate layout edges, representing the global room structure, which is suit- able for our indoor inpainting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In addition, our model is an end-to-end architecture instead of a two-stage network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [17] developed a multi-task learning frame- work to jointly learn the completion of image contents and structure map (edges and gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' A structure embedding scheme is employed to embed the learned structure features while inpainting explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The model further learns to exploit the recurrent structures and contents via an attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' While demonstrating impressive performance in generating realistic results, these structure-aware methods still fail to model long-range structure correspondence such as the layout in the indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' On the other hand, some works have successfully recovered a single partially occluded ob- ject [5,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' However, their architecture does not handle multiple object instances of the same class and is thus not suitable for our context where the plane-wise segmentation consists of different numbers of wall planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Image-to-image translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The image inpainting is essentially a constrained image- to-image translation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Significant efforts have been made to tackle various prob- lems based on image-to-image translation architectures [15,42,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Here we focus on the ones that are closely related to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [29] introduced SPADE, which utilizes a spatial adaptive normalization layer for synthesizing photorealistic images given an input semantic segmentation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Specifically, a spatially-adaptive learned transform modulates the activation layer with a semantic segmentation map and ef- fectively propagates the semantic information throughout the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In contrast to SPADE, which uses only one style code to control the image synthesis, Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [43] presents SEAN by extending the SPADE architecture with per-region style encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' By embedding one style code for individual semantic classes, SEAN shows signifi- cant improvement over SPADE and generates the highest quality results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In the con- text of indoor scenes, Gkitsas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [9] introduce PanoDR that combines image-to- image translation with a generator to constrain the image inpainting with the underly- ing scene structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Percisely, to convert a furnished indoor panorama into its empty counterpart, PanoDR exploits a generator for synthesizing photorealistic image con- tents where the global layout structure is preserved via an image-to-image translation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The empty indoor panorama is then used to complete the masked regions in the input panorama via a simple copy-and-paste process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Gkitsas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' [10] extend the architecture of PanoDR to make the model end-to-end trainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' However, the quan- titative evaluation indicates that the performance improvement is marginal compared with PanoDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Our system also combines a generator with image-to-image translation as PanoDR does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' However, we obtain superior results than PanoDR by exploiting the LGPN-net 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 3: Architecture overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Our network architecture follows the conventional gen- erative adversarial network with an encoder-decoder scheme supervised by low- and high-level loss functions and a discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Given a masked indoor panoramic image Iin with a corresponding mask M, our system uses an off-the-shelf layout prediction network to predicts a layout map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The low-level boundary lines in Lm serve as a con- ditional input to our network to assist the inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Then, we compute two semantic segmentation maps from the layout map Lm, declared L3−class and Lp−wise, where the latter is used to generate plane-wise style codes for ceiling, floor, and individual walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Finally, these per plane style codes, together with L3−class, are fed to a struc- tural plane-aware normalization module to constrain the inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' global layout edges as a prior and adapting SEAN blocks in a local plane-wise man- ner to guide the inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Moreover, in contrast to PanoDR performs the inpainting task via an indirect way, our system performs the inpainting task in an end-to-end fash- ion, directly completing the mask areas instead of hallucinating an empty scene, thus resulting in better visual quality and consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 3 Overview Figure 3 illustrates an overview of our architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Our system takes a masked panoramic image Iin and the corresponding binary mask M as inputs and generates the inpainted panoramic image Iout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The masked panoramic image is generated by Iin = Igt ⊙ (1 − M), where Igt represents the ground-truth panoramic image and ⊙ denotes the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Our system first utilizes an off-the-shelf model to estimate the room layout Lm from input masked panoramic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' This layout map is then concatenated with Iin and M to obtain a five-channel input map fed into the generator G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We further derive two semantic segmentation maps L3−class and Lp−wise using the layout map for the subsequent normalization module (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The image generation model follows the conventional generative adversarial architecture with one content encoder and one image decoder with one discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' To impose structure in- formation during inpainting, we introduce a plane-aware normalization that modifies Horizon Net VGG-196 Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 4: Plane-aware normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Given an incomplete indoor panoramic image Iin with mask M, we first predict two normalization values β and γ through several partial convolution [25] blocks and a plane-wise average pooling based on the plane-wise seg- mentation map Lp−wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Second, we predict another set of normalization values β′ and γ′ through several vanilla convolution blocks based on the 3-class segmentation map L3−class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The final normalization values are thus computed using the weighted sum weighted by learnable parameters αβ and αγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' the SEAN [43] block with two semantic segmentation maps to guide the decoder with spatially varying normalization parameters per structural plane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', ceiling, floor, and individual walls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Such a plane-aware normalization provides useful guidance for global structure preservation as well as consistent local image content generation (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Finally, common loss functions in image inpainting, including the reconstruction loss, the perceptual loss, the style loss, and the adversarial loss are employed to train our model (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 4 Method 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='1 Layout Guidance Map We employ an off-the-shelf model, HorizonNet [34], to estimate a layout map from input masked panorama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Through a recurrent neural network, the HorizonNet predicts a 3-dimensional vector representing ceiling, ground, and corner location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We further process the output vector to generate a layout map Lm comprising low-level boundary lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' This layout map serves as a conditional input to encourage the preservation of global layout structure while inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Moreover, we extract two semantic segmen- tation maps from the layout map that depict (i) the segmentation mask L3−class with three semantic labels of indoor scene, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', ceiling, floor, and wall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' and (ii) a plane-wise segmentation mask Lp−wise where pixels are indexed in a per structural plane basis LGPN-net 7 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', ceiling, floor, or individual walls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' These semantic segmentation maps are gener- ated using conventional image processing operations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='e, flood-fill) and will be used in the later normalization module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='2 Image Inpainting Backbone As shown in Figure 3, our network architecture consists of one generator and one dis- criminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The generator G follows a conventional scheme with one content encoder and one image decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The content encoder consists of two down-sampling convo- lution blocks followed by eight residual blocks using dilated convolution [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The image decoder uses a cascade of our proposed plane-aware residual blocks and two up-sampling blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Motivated by EdgeConnect [28], we use PatchGAN [16] as our discriminator to determine the real or fake sample by dividing the input image into sev- eral patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In the following sections, we will elaborate plane-aware residual block, loss functions, and discriminator in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='3 Plane-aware Normalization Considering the different styles among wall planes is very common in real-world indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We follow the architecture of SEAN [43] and propose leveraging two kinds of segmentation maps Lp−wise and L3−class to establish our plane-aware normalization (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Our plane-aware normalization consists of one style encoder and two style blocks, which enhance the global style semantics and local style consistency of the generated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The inputs of the style encoder include masked panoramic image Iin and mask image M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We use partial convolution blocks in style encoder instead of vanilla convolution to make feature extraction conditioned only valid pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We first adopt the plane-wise average pooling on the output features to generate style codes for each plane based on Lp−wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Second, we spatially broadcast each style code on the corresponding area and output the local style block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' On the other side, we predict the global style block by passing the 3-class segmentation map L3−class through several convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Finally, the remaining part of our plane-aware normalization follows the same architecture of SEAN [43], and combines global and local style blocks into the downstream β and γ parameters of the final batch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='4 Loss Functions Here we elaborate on the low- and high-level loss functions and the discrimination used for training our image generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Reconstruction loss measures the low-level pixel-based loss between the predicted and ground-truth images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' To encourage the generator to pay more attention to the missing regions, we additionally calculate the L1 loss in the missing regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The reconstruction loss Lrec is defined as follows: Lrec = ∥M ⊙ Igt − M ⊙ Iout∥1 + ∥Igt − Iout∥1 , (1) 8 Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' where Igt and Iout represent the ground-truth image and the generator’s output, respec- tively, and M is a binary mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Perceptual loss encourages the predicted and ground-truth images to have similar representation in high-level feature space extracted via a pre-trained VGG-19 [33], and is defined as follows: Lperc = � i ∥φi (Igt) − φi (Iout)∥1 , (2) where φi is the activation map of the ith layer of the pre-trained feature extraction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Style loss calculates the co-variance difference between the activation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' For the activation map φi of size Ci × Hi × Wi, the style loss is defined as follows: Lsty = ���Gφ i (Igt) − Gφ i (Iout) ��� 1 , (3) where Gφ i is a Ci × Ci gram matrix [8] constructed by the activation map φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Adversarial loss is implemented with the patch-based discriminator [16], which out- puts the feature map divided into several feature patches and uses hinge loss [24] to optimize the generator G and the discriminator D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The adversarial loss for generator G and discriminator D are defined as follows: LG = −D (Iout) , (4) LD = λD (max (0, 1 + D (Iout)) + max (0, 1 − D (Igt))) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' (5) The overall loss function used in the generator G is defined as follows: Ltotal = λrecLrec + λpercLperc + λstyLsty + λGLG, (6) where λrec, λperc, λsty, λG, and λD are the hyperparameters for weighting the loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 5 Experiments In this section, we evaluate the performance of our model by comparing it with several state-of-the-art image inpainting approaches and conducting ablation studies to verify the necessity of individual components in the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Please refer to our online webpage for other experiments and more results2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='1 Experimental Settings Dataset and baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We compare our model with the following state-of-the-art structure-aware image inpainting models: 2 https://ericsujw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='io/LGPN-net/ LGPN-net 9 Input / Layout EC [28] LISK [17] PanoDR [9] Ours GT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 5: Qualitative comparisons with state-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Top 8 rows: the inpainting re- sults of the empty indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Bottom 8 rows: the inpainting results of the furnished indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Our method produces superior results in generating image contents that align the layout structure well and are consistent with the surrounding of the masked regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 米米米米米米米10 Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' – EC [28]: a two-stage adversarial network that comprises an edge completion model followed by a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' – LISK [17]: a multi-task learning framework that exploits image structure embed- ding and an attention mechanism in the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' – PanoDR [9]: a deep learning framework that combines image-to-image translation with generator to condition the inpainting on the indoor scene structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The experiments were conducted on a public indoor panorama dataset, Structured3D [41], which contains 21,835 indoor panoramas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The official data split is adopted for train- ing(18,362), validation(1,776), and testing(1,697).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We follow the same procedure as PanoDR to generate mask images using contours of foreground furniture (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We use the officially released implementation of baselines for training from scratch and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Note that each indoor panorama in Structured3D has two representations of the same scene (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', empty and furnished).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Therefore, the experiments were conducted in two phases to evaluate our model and baselines in different application scenarios (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', structural inpainting vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' furniture removal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We take several commonly used image quality assessment met- rics in previous inpainting approaches for quantitative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Specifically, we used the low-level feature metrics, including Mean Absolute Error (MAE), Peak Signal-to- Noise (PSNR), Structural Similarity Index (SSIM) [35], and Fr´echet Inception Distance (FID) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We implement our model in PyTorch and conduct the exper- iments on a single NVIDIA V100 with 32G VRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The resolution of the panoramic images is resized to 512 × 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We use Adam [21] optimizer in the training process with the hyper-parameters setting of b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0 and b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9, a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0001, and a batch size of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We empirically set λrec = 1, λperc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='1, λsty = 250, λG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='1, and λD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='5 in the total loss function (Equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' For HorizonNet [34], we use the official pre-trained model for layout estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='2 Evaluation on the Empty Scenes In this experiment, we evaluate both the qualitative and quantitative performance of our model on the image inpainting task by comparing it with baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The qualitative comparisons are shown in Figure 5 (top 8 rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In contrast to EC and LISK, which fail to restore image structures in the masked regions, our method faithfully gener- ates image contents adhering to the underlying layout structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' While PanoDR shows slightly better structure preservation than EC and LISK, it fails to generate image con- tents consistent with the surrounding of masked regions as our method does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Therefore, our method achieves the best performance against all the baselines across all evaluation metrics as shown in Table 1 (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='3 Evaluation on the Furnished Scenes Furniture of irregular shape will more or less obscure the layout of the indoor scene, making it more challenging to restore the regular structure in the missing area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' There- fore, in this experiment, we would like to evaluate how well our model learned from LGPN-net 11 Table 1: Quantitative comparisons with state-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The top and bottom tables summarize the performance of our model and baselines on the empty and furnished scenes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Dataset Method PSNR↑ SSIM↑ MAE↓ FID↓ Empty scene EC [28] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='6936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0039 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9480 LISK [17] 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='3761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0055 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='1660 PanoDR [9] 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='2431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0040 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='3591 Ours 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='8444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0030 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='5265 Furnished scene EC [28] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='4439 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0076 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9955 LISK [17] 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='7325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0068 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='2676 PanoDR [9] 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='3340 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9641 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0051 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='8399 Ours 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='3923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0047 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='2328 Table 2: Quantitative results of the ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We evaluate the effectiveness of our design choices by gradually adding the individual components into the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' PSNR ↑ SSIM ↑ MAE ↓ FID ↓ Backbone 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='6449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0034 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='3915 Layout map only 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='2884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0033 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='8105 Full model 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='8444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0030 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='5265 empty scenes can generalize to the furnished scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Since the inpainting task setup here exactly matches the one defined in the PanoDR, we use the pre-trained model of PanoDR in this experiment for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' As shown in Figure 5 (bottom 8 rows), our method still clearly outperforms baselines in generating image contents that align the layout structure well and are consistent with the surrounding of the masked regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The quantitative results are shown in Table 1 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' It is worth noting that the way PanoDR performs image completion via compositing the predicted empty im- age and input image using the object mask will lead to severe artifacts where occlusion occurred between foreground objects (see Figure 2(PanoDR)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='4 Ablation Study Here, we conduct ablation studies to validate our model from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' First, we evaluate the necessity of individual design choices in our architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Then, we conduct two experiments to evaluate how sensitive our model is to the size of input masks and the quality of input layout maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Ablation on network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In this experiment, we start with the backbone model (Backbone) as the baseline, then progressively adding only layout guidance map (Layout map only), and our plane-aware normalization (Full model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' As shown in Table 2, we obtain the best performance with the full model on all the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The qualitative comparisons shown in Figure 6 indicate that adding layout guidance map generates clear structure boundaries in the final result (2nd and 3rd columns), while our 12 Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Input Backbone Layout map only Full model GT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 6: Qualitative results of the ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Side-by-side comparisons of inpaint- ing results generated using our method by gradually adding individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' From left to right, input images and masks, our baseline model (Backbone), adding the layout guidance map (Layout map only), full model with our plane-aware nor- malization (Full model), and ground truth images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' full model with plane-aware normalization can constrain the image generation to the adjacent structural planes and obtain visually consistent results (3rd and 4th columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Sensitivity to the mask size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In this experiment, we analyze the testing dataset and classify the images into different categories according to the area proportions of input masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Table 3 shows the inpainting performance for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We can tell that the inpainting quality degrades with the increasing mask size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' A significant drop occurs where the ratio of input mask is greater than 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Sensitivity to the layout estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In order to explore the effect of the accuracy of layout estimation on the inpainting quality, we first devise a mechanism to generate layout maps with different levels of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Specifically, we feed masked images of different mask sizes into HorizonNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We start by generating randomly located rect- angle masks of 5% image size and increase the mask ratio to 10%, 30% and 50% to deliberately produce layout structures with decreasing quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Then we take these lay- out maps as conditional inputs of our model and compare the inpainting performance empty-room testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' As shown in Table 4, our model degrades marginally when the quality of estimated layouts decreases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='96 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='84, indicating our model is robust to the varying input layout maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' LGPN-net 13 Table 3: Mask size vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' inpainting quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Mask Size(%) Count Content PSNR ↑ SSIM ↑ MAE ↓ 0-10 1045 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='3921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0011 10-20 163 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='2823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0055 20-30 48 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='4371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0111 30-40 39 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0731 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0266 40+ 13 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='2958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9345 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0305 total 1308 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='8444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0030 Table 4: Accuracy of layout estimation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' inpainting quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Structure Content mIOU ↑ PSNR ↑ SSIM ↑ MAE ↓ FID ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9603 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='3212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0028 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='4322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9561 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='2871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0028 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='4441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9175 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0682 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0029 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='5624 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='8489 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='7300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='9919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='0030 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='8455 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='5 Qualitative Results on Real-world Scene Real-world scenes have complex lighting and layout structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' However, the amount of data in the real-world scene dataset and the quality of furniture category annotations are insufficient for training our model, so we choose to train on the synthetic dataset Structured3D [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Nevertheless, we still compare our results with PanoDR [9], which also implements the furniture removal task, on the real-world scene dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Since the real-world scene dataset does not contain paired data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', scenes before and after furni- ture removal), quantitative evaluation is infeasible and we can only provide qualitative comparisons here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Figure 7 shows that our inpainted results have a higher quality of structural maintenance and color restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Moreover, compared with PanoDR, we can still exert more stable performance in real-world scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Please refer to our online webpage for more results3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 6 Conclusions We proposed an end-to-end structural inpainting network for the indoor scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We in- troduce layout boundary line conditions the output structure and utilize the plane-aware normalization to enhance planar style consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Experiment results show the out- standing performance of our model in both structural inpainting and furniture removal on the indoor scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 3 https://ericsujw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='io/LGPN-net/ 14 Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Input EC [28] LISK [17] PanoDR [9] Ours Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 7: Qualitative comparisons with state-of-the-arts on real-world scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Our model clearly outperforms baselines by preserving layout boundary and restoring lo- cal texture from adjacent room structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', floor and walls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Input PanoDR [9] Ours Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 8: Limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Both the state-of-the-art method and our model produce visual arti- facts in the scenes presenting strong shading effect surrounding the removed furniture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In the real-world application of furniture removal, we can often see resid- uals of shading effect caused by the removed furniture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' These residuals are hard to segment and even harder to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' As shown in Figure 8, our model is slightly affected by these residuals but still produces more realistic results than PanoDR [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' We plan to adopt a more reasonable segmentation mask of the indoor scene inpainting which can cover the shading area and thus improve our results in those shaded scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' The project was funded in part by the National Science and Tech- nology Council of Taiwan (110-2221-E-007-060-MY3, 110-2221-E-007-061-MY3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' 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Tang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', Gao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=': Structured3d: A large photo-realistic dataset for structured 3d modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In: Proceedings of The European Conference on Com- puter Vision (ECCV) (2020) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', Park, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', Isola, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=', Efros, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=': Unpaired image-to-image translation using cycle- consistent adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} +page_content=' In: Computer Vision (ICCV), 2017 IEEE International Con- ference on (2017) 43.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQffA_l/content/2301.05624v1.pdf'} diff --git a/btFRT4oBgHgl3EQfSDcU/content/tmp_files/2301.13527v1.pdf.txt b/btFRT4oBgHgl3EQfSDcU/content/tmp_files/2301.13527v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c73382591d0d8c42bf854d2ed403e6dbfa7cf41 --- /dev/null +++ b/btFRT4oBgHgl3EQfSDcU/content/tmp_files/2301.13527v1.pdf.txt @@ -0,0 +1,818 @@ +Real-Time Outlier Detection with Dynamic Process +Limits +Marek Wadinger and Michal Kvasnica +Institute of Information Engineering, Automation and Mathematics +Slovak University of Technology in Bratislava +Bratislava, Slovakia +{marek.wadinger, michal.kvasnica}@stuba.sk +Abstract—Anomaly detection methods are part of the systems +where rare events may endanger an operation’s profitability, +safety, and environmental aspects. Although many state-of- +the-art anomaly detection methods were developed to date, +their deployment is limited to the operation conditions present +during the model training. Online anomaly detection brings +the capability to adapt to data drifts and change points that +may not be represented during model development resulting in +prolonged service life. This paper proposes an online anomaly +detection algorithm for existing real-time infrastructures where +low-latency detection is required and novel patterns in data occur +unpredictably. The online inverse cumulative distribution-based +approach is introduced to eliminate common problems of offline +anomaly detectors, meanwhile providing dynamic process limits +to normal operation. The benefit of the proposed method is the +ease of use, fast computation, and deployability as shown in two +case studies of real microgrid operation data. +Index Terms—anomaly detection, interpretable machine learn- +ing, online machine learning, real-time systems, streaming ana- +lytics +I. INTRODUCTION +The era of Industry 4.0 is ruled by data. Effective data- +based decision-making is driven by the quantity of collected +data. Internet of Things (IoT) devices made data acquisition +seamless and positively influenced a wide range of industries. +It is estimated that the annual economic impact of IoT will +further grow and reach up to $6.2 trillion by 2025 [1]. +Various data collection mechanisms are used to buffer and +store the data for future processing. However, the tremendous +increase in data availability and the desire to extract valuable +insight led to problems with the unbounded buffering and +storage capacity. Real-time evaluation of the data streams +became an acronym for smart data processing. +Streaming data analytics introduced mechanisms for online +extraction and transformation while loading to the storage +only a fraction of the former data load, which allowed the +storage of the vital information carried by the data more +comprehensively. However, the unstable quality of the data +appeared to have the most crucial importance over the quantity. +Anomaly detection, well studied in the last decades, was +reborn to the world of new challenges. Former studies were +mainly concerned with a domain-specific detection of various +anomalies while trained offline [2]. However, anomalies of +diverse sources, from fraudulent web activity and suspicious +financial transactions to sensor failure, malfunctioning of the +hardware, and performance drops, mutate over time, and the +model had to be updated. +Companies expanded their research activities on the creation +and integration of generic frameworks combining prediction, +detection, and alert mechanisms. One of the first projects, +open-sourced for the public, are EGADS by Yahoo [3] and +AnomalyDetection by Twitter [4]. The frameworks’ modular- +ity allowed the automation of the anomaly detection of time- +series data and created space for discussion. +Moving from domain-specific to generic methods posed +new problems connected to type I errors, i.e., a false-positive +classification of normal behavior as anomalous. Accurate se- +lection of forecaster, detector, and alerting mechanism allowed +to tackle the problem, nevertheless, introduced considerable +dependence on expert domain knowledge and fine-tuning. +Further work proved improvement in performance while +relieving the tight requirements on domain knowledge [5]. +However, strict demands on detection systems ranging from +lasting up times to continuous monitoring with stable perfor- +mance pointed to the challenge of data stationarity. Change +points and concept drifts troubled unsupervised models, which +led to service downtime due to the model retraining. +The era of adaptive machine learning introduced incre- +mental learning schemes as a solution. Multiple studies for +learning modes, adaptation methods, and model management +swept through the machine learning community. Pannu et al. +proposed an adaptive anomaly detection system [6]. However, +the method represented a supervised operator-in-the-loop solu- +tion. Zhang et al. introduced an adaptive kernel density-based +algorithm that uses an adaptive kernel width [7]. Nonetheless, +training the models on big data had limitations resulting from +the storage and unbounded buffering of data. Online learning +models relaxed the need for data availability during model +training [8]. On the contrary, it processed the data from a +bounded buffer sequentially as in [9] and [10]. +Anomaly detection in microgrids, however, called for low +latency detection which implied real-time training and pre- +diction processes [11]. Such adaptation of streamed modeling +took into consideration strict boundaries on computational +time. For work in this area see [12] and [13]. +Alerting mechanisms in process automation detect situations +where signal value deviates from constraints. An alert watch- +dog is triggered on threshold violation by individual signals. +arXiv:2301.13527v1 [cs.LG] 31 Jan 2023 + +The constraints, or process limits, are usually predefined and +fixed. Nevertheless, factors such as aging and environmental +changes call for dynamic process limits. Setting up a procedure +for an evergrowing number of signal measurements is time- +consuming. Besides, it is impossible for signals where no prior +information about a correct process range is known. Those are +subject to external factors that are unknown at setup time. +In this article, we suggest using existing process automation +infrastructure based on alerting (PLC, SCADA, among others) +and applying machine learning for dynamic process range +based on changing conditions. We propose an unsupervised +anomaly detection algorithm capable of online adaptation to +change points and concept drifts, which adds to a recently +developed body of research. The approach is evaluated on two +case studies of microgrid sensors. To the author’s knowledge, +there are no studies to date concerned with providing adaptive +operation constraints. +The main benefits of the proposed solution are that it: +• Keeps existing IT infrastructure, saving costs, and does +not require operator retraining +• Automates alerting thresholds setup for a high number of +signals +• Automates alerting for signals with no a priori knowledge +of process limits +• Assesses changing environmental conditions and device +aging +• Uses self-learning approach on streamed data +II. PRELIMINARIES +This section introduces the main concepts which are build- +ing pillars of the developed approach. Subsection II-A will +discuss a one-pass algorithm that allows for online adaptation. +The following Subsection II-B proposes the ability to invert +the solution in a two-pass implementation. The mathemati- +cal background of distribution modeling in Subsection II-C +provides a basis for the Gaussian anomaly detection model +conceptualized in the last Subsection II-D of Preliminaries. +A. Welford’s Method +Streaming data analytics, restrict the unbounded buffer or +storage of the data, i.e., limits the uncontrolled growth of +memory usage with the increasing amount o input data. In +such cases, it is desired to keep the data only for the period of +time required to perform computations. For the given purpose +serve one-pass algorithms. This category of methods allows +processing on-the-fly without the need to store the entire data +stream. +Definition 2.1 (One-pass algorithm): The algorithm with a +single access to the data items in the order of their occurrence, +i.e., x1, x2, x3, ... is called one-pass algorithm [14] +Welford’s method represents a numerically stable one-pass +solution for the online computation of mean and variance [15]. +Given xi where i = 1, ..., n is the sample index in given +population n, the corrected sum of squares is defined as +Sn = +n +� +i=1 +(xi − ¯xn)2, +(1) +where the running mean ¯xn is +¯xn = n − 1 +n +¯xn−1 + 1 +nxn = ¯xn−1 + xn − ¯xn−1 +n +. +(2) +The following identities to update the corrected sum of squares +hold true +Sn = Sn−1 + (xn − ¯xn−1)(xn − ¯xn), +(3) +and the corresponding variance is +s2 +n = +Sn +n − 1. +(4) +As we can see in (3), we do access only current data sample +xn and previous value of ¯xn−1 which is updated in (2) using +the same data sample and the size of seen population n. +B. Inverse Welford’s Method +Let the incoming stream of data be subject to the concept +drift. Such alternation in statistical properties has a negative +influence on prediction accuracy. Therefore, an adaptation of +any machine learning model is crucial for successful long-term +operation. +Definition 2.2 (Concept drift): Concept drift is a change +in the statistical properties that occur in a sub-region of the +feature space. +The previous Subsection II-A defined the main concept +of online statistical computation that allows reacting to such +changes. However, the further in time the shift occurs, the +slower the adjustment of the running mean is, resulting from +a negative relationship in (2) between population size n and +influence of the last sample in population xn on the updated +value of ¯xn. For this reason, we define the expiration period +te, over which the running statistics are computed. After the +expiration period, the data items are forgotten. Such reversal +results in a need to store all the data in the window in order to +revert their effect. Given te = n−1 we can revert the influence +of the first data sample on the running mean as +¯xn−1 = +n +n − 1 ¯xn − +1 +n − 1xn−te = ¯xn − xn−te − ¯xn +n − 1 +, +(5) +then reverting the sum of squares follows as +Sn−1 = Sn − (xn−te − ¯xn−1)(xn−te − ¯xn), +(6) +which allows the computation of variance +s2 +n−1 = Sn−1 +n − 2. +(7) +C. Modeling Distribution +Statistical distribution can be used to create a generalized +model of a normal system behavior based on observed mea- +surement. Specifically, if no change point is expected in a +given subset of samples, the Gaussian normal distribution can +be fitted. Parameters of the normal distribution are used to +compute standard score (8) for each new observation. +Definition 2.3 (Standard Score): Standard score or Z-score +is a number that specifies the number of sample standard + +deviations s2 +n by which observation x deviates from the sample +mean ¯xn of normal distribution +zn = xn − ¯xn +s2n +. +(8) +In order to define the general probability of z-score belong- +ing to anomaly we use probability computed using Cumulative +Distribution Function (CDF). However, the z-score must be +bounded using an error function into the interval from 0 to 1. +Definition 2.4 (Approximate Error Function): The approxi- +mate error function represents the approximate probability that +the random variable X lies in the range of [ −zn, zn] denoted +as +EA(zn) = zn +e−z2 +n +√π ( 2/1 + 4/3x2 + 8/15x4 + ...) . +(9) +Definition 2.5 (Cumulative Distribution Function (CDF)): +CDF represents the probability that the random variable X +takes a value less than or equal to xi. FX : R → [0, 1]. For +generic normal distribution with sample mean ¯xn and sample +deviation sn the cumulative distribution function FX(x) equals +to +FX(xi)n = 1 +2( 1 + EA( zn +√ +2) ). +(10) +Given the probability, we can also derive the value of x to +which it belongs using a percent point function to compute +inverse CDF (ICDF) denoted also as FX(xi)−1 +n . +Definition 2.6 (Percent-Point Function (PPF)): PPF returns +the threshold value for random variable X under which it takes +a value less than or equal to the value, for which FX(x) takes +probability lower than selected quantile q. QX : [0, 1] → R. +An algorithm that calculates the value of the PPF is reported +below as Algorithm 1. +Algorithm 1 Percent-Point Function for Normal Distribution +Input: quantile q, sample mean ¯xn (2), sample variance s2 +n +(4) +Output: threshold value xn,q +Initialisation : +1: f ← 10; l ← −f; r ← f; +LOOP Process +2: while FX(l) − q > 0 do +3: +r ← l; +4: +l ← lf; +5: end while +6: while FX(r) − q < 0 do +7: +l ← r; +8: +r ← rf; +9: end while +10: ˜xn,q = arg minz ∥FX(z) − q∥ s.t. l ≤ z ≤ r +11: return ˜xn,q +� +s2n + ¯xn +D. Gaussian Anomaly Detection +Anomalies come in various kinds and flavors. Commonly +denoted types are point (spatial), contextual, and collective +(temporal) anomalies [2]. Spatial anomalies take on a value +that particularly deviates from the sample mean ¯xn. From +a statistical viewpoint, spatial anomalies can be considered +values x that significantly differ from the data distribution. +In empirical fields, such as machine learning, the three- +sigma rule defines a region of distribution where normal values +are expected to occur with near certainty. This assumption +makes approximately 0.27% of values in the given distribution +considered anomalous. +Definition 2.7 (Three-Sigma Rule of Thumb (3σ rule)): 3σ +rule represents a probability, that any value xi of random +variable X will lie within a region of values of normal +distribution at the distance from the sample mean µn of at +most 3 sample standard deviations σn. +P{|xi − µn| < 3σn} = 0.99730 +(11) +Anomalous values occur on both tails of the distribution. In +order to discriminate the anomalies using the three-sigma rule +on both tails of the distribution, we define the anomaly score +as follows +yi = 2 +����FX(xi)n − 1 +2 +���� , +(12) +where +yi ∈ [0, P{|xi − µn| < 3σn}), +(13a) +applies for normal observations and +yi ∈ [P{|xi − µn| < 3σn}, 1], +(13b) +for anomalies. +Using pure statistics to model normal behavior lets us ask +the question about the threshold value x which corresponds +to the area under the curve of CDF equal to the given +probability. A such query can be answered using inversion +of (12). However, inversion of (12) would fail the horizontal +line test. Therefore, we restrict the applicability of the inverse +only to FX(x)i ∈ [0.5, 1] +xi = FX +�yi +2 + 1 +2 +�−1 +n +(14) +In order to derive a lower threshold, the Gaussian distribu- +tion is fitted to the negative value of the streamed data and +evaluated accordingly using the previously defined equations. +III. ICDF-BASED REAL-VALUED THRESHOLD SYSTEM +We suggest a novel approach to provide dynamic process +limits using an online outlier detection algorithm capable of +handling concept drifts in real-time. Our main contribution +is based on using an inverse cumulative distribution function +(ICDF) to supply a real-valued threshold for anomaly detec- +tion, i.e., to find the values of the signal which corresponds +to the alert-triggering process limits. Therefore, in the context +of machine learning, we are tackling an inverse problem, i.e., +calculating the input that produced the observation. To utilize + +an adaptive ICDF-based threshold system, the univariate Gaus- +sian distribution has to be fitted to the data in online training +and ICDF evaluated on the fly. This method is divided into +four parts and described in the following lines. For a simplified +representation of the method see Algorithm 2. +A. Model Initialization +The initial conditions of the model parameters are µ0 = x0 +for mean and s2 +0 = 1 for variance. The score threshold is +constant and set to q = 0.9973. Moreover, there are two +user-defined parameters: the expiration period te, and the time +constant of the system tc. The expiration period, which defines +the period over which the time-rolling computations are per- +formed, can be altered to change the proportion of expected +anomalies and allows relaxation (longer expiration period) or +tightening (shorter expiration period) of the thresholds. The +time constant of the system determines the speed of change +point adaptation as it influences the selection of anomalous +points that will be used to update the model for a window +of values Y = {yi−tc, ..., yi} if the following condition holds +true +� +y∈Y y +n(Y ) +> q. +(15) +The existence of two tunable and easy-to-interpret hyper- +parameters makes it very easy to adapt the solution to any +univariate anomaly detection problem. +B. Online training +Training of the model takes place in an online fashion, +i.e., the model learns one sample at a time at the moment +of its arrival. Learning updates the mean and variance of the +underlying Gaussian distribution. The computation of moving +mean (2) and variance (4) is handled by Welford’s method. +Each sample after the expiration period is forgotten and its +effect reverted in the second pass. First, the new mean is +computed using (5) which accesses the first value in the +bounded buffer. The value is dropped in the same pass. Second, +the new sample variance is reverted based on (7) using the +new mean and current mean that is overwritten afterward. For +details see Subsection II-B. +C. Online prediction +In the prediction phase, z-score (8) is computed and passed +through EA (9) in order to evaluate FX(xi) from (10). The al- +gorithm marks the incoming data points if their corresponding +anomaly score (12) is out of the range defined by threshold q. +In other words, marks signal value xi that is higher or equal +to the threshold, which bounds the three-sigma region. +D. Dynamic Process Limits +Normal process operation is constrained online using ICDF. +The constant value of q and parameters of the fitted distribution +are both passed through Algorithm 1 to obtain value, which +corresponds to the value of x that would trigger an upper +bound outlier alarm at the given time instance. To obtain a +lower bound of operation conditions the same procedure is +applied to the distribution fitted on negative values of input. +Algorithm 2 Online Anomaly Detection Workflow +Input: expiration period te, time constant tc +Output: score yi, threshold xi,q +Initialisation : +1: i ← 1; n ← 1; q ← 0.9973; ¯x ← x0; s2 ← 1; +2: compute FX(x0) using (8); +LOOP Process +3: loop +4: +xi ← RECEIVE(); +5: +yi ← PREDICT(xi) using (12); +6: +xi,q ← GET(q, ¯x, s2) using Algorithm 1; +7: +if (13a) or (15) then +8: +¯x, s2 ← UPDATE(xi, ¯x, s2, n) using (2), (4); +9: +n ← n + 1; +10: +for xi−te do +11: +¯x, s2 ← REVERT(xi−te, ¯x, s2, n) using (5), (7); +12: +n ← n − 1; +13: +end for +14: +end if +15: +i ← i + 1; +16: end loop +IV. CASE STUDY +In this section, we demonstrate the applicability of the +proposed ICDF-based approach in two case studies of the +microgrid operation. The properties and performance were +investigated using streamed signals from the IoT devices. The +successful deployment demonstrates that this approach is suit- +able for existing alerting mechanisms of process automation +infrastructure. +The case studies were realized using Python 3.10.1 on a +MAC with an M1 CPU and 8 GB RAM. The percent point +function was solved using an iterative root-finding algorithm, +Brent’s method. +A. Battery Energy Storage System (BESS) +First, we verify our proposed method on BESS. The BESS +reports measurements of State of Charge (SoC), supply/draw +energy events, inner temperature, outer temperature, Heating, +ventilation, and air conditioning (HVAC) state. Tight control +of the battery cell temperature is needed for the optimal +performance and maximum lifespan of the battery. Identifying +anomalous events and removal of corrupted data might yield +significant improvement on the process control level. +The sampling rate of the signal measurement is 1 minute. +However, network communication is prone to packet dropout, +which results in non-uniform sampling. To protect the sensitive +business value of the data, we normalize all signals to the range +[0, 1]. The goal was to mark anomalous events in the data and +provide adaptive process limits from the online self-learning +model. + +Mar 3, 2022 +Mar 4, 2022 +Mar 7, 2022 +Mar 8, 2022 +Mar 10, 2022 +Mar 12, 2022 +Mar 14, 2022 +Mar 15, 2022 +Mar 21, 2022 +Mar 22, 2022 +Mar 24, 2022 +0 +0.2 +0.4 +0.6 +0.8 +1 +Average Cell Temperature +Normalized Temperature +Fig. 1. Time Series of Average Battery Cell Temperature measurement (green +line). Non-uniform ticks on the x-axis mark days of interest (NOTE: some +marks are hidden due to the readability). The y-axis renders the normalized +temperature. +Fig. 1 renders measurement of average battery cell tempera- +ture from 21st February until 26th March. We can observe mul- +tiple anomalies of various sources given this span, for instance, +packet dropout, suspicious events, intermittent sensor failure, +and change point in data distribution. Dates of observation +given the listed events will be provided later in the paper. +The initial conditions of the model states are set based +on Subsection III-A. The user-defined parameters, were set +to 7 days for the expiration period and 5 hours for the time +constant. Anomalies found during the first day of the service +are ignored due to the initialization of the detector. In this case +study, the anomaly detection problem was approached by the +online model fitting based on Subsection III-B +Using the online prediction described in Subsection III-C +we tag the sample as the anomaly or normal data point. Fig. 2 +renders vertical rectangles over the regions from the start until +the end of the predicted anomalous event. +The results on Average Cell Temperature in Fig. 2 show +that the model could capture anomalous patterns of various +sources. Despite self-learning without supervision, the model- +classified anomalies were also confirmed by the data provider +after inspection. For instance, a rare event of manipulation +with BESS on 3rd, followed by peak on 4th March. BESS +relocation on 7th, led to a change point which was alerted and +the system adapted completely over the course of 1 day. Test +events resulted in peak values through 10th to 15th March, and +faulty measurements on the 12th March followed by a packet +loss on 21st March were alerted too. The system tagged the +next two tests of temperature control switch-offs. +Findings that favor the model’s ability to discriminate +anomalous behavior are important for the meaningful real- +ization of the dynamic process thresholding. The real-valued +threshold mechanism, defined in Subsection III-D, provided +Mar 3, 2022 +Mar 4, 2022 +Mar 7, 2022 +Mar 8, 2022 +Mar 10, 2022 +Mar 12, 2022 +Mar 14, 2022 +Mar 15, 2022 +Mar 21, 2022 +Mar 22, 2022 +Mar 24, 2022 +0 +0.2 +0.4 +0.6 +0.8 +1 +Average Cell Temperature +Normalized Temperature +Fig. 2. Time Series of Average Battery Cell Temperature measurement (green +line) and predicted anomalous events (red vertical rectangles). +up-to-date upper and lower bounds for the signal. As for the +validity of the dynamic process limits, each breakout of the +signal value from within the range was also marked by the +anomaly detection system. Fig. 3 points to the capability to +adapt to change point on 7th March and mitigate the influence +of intermittent effects of anomalies on distribution. The speed +of the change point adaptation as well as the mitigation of +the effect of anomalies on the tightness of limits are governed +by the user-defined expiration period and time constant of the +system. +Mar 3, 2022 +Mar 4, 2022 +Mar 7, 2022 +Mar 8, 2022 +Mar 10, 2022 +Mar 12, 2022 +Mar 14, 2022 +Mar 15, 2022 +Mar 21, 2022 +Mar 22, 2022 +Mar 24, 2022 +0 +0.2 +0.4 +0.6 +0.8 +1 +Average Cell Temperature +Threshold +Normalized Temperature +Fig. 3. Time Series of Average Battery Cell Temperature measurement (green +line) and predicted anomalous events (red vertical rectangles). The reddish fill +bonded by the red line represents an area of anomalous behavior as given by +the anomaly detector. + +B. Power Inverter +A second case study demonstrates the proposed method’s +applicability to the temperature of the power inverter. During +high load periods, inverters can heat up swiftly. Technical +documentation of every inverter provides details on continuous +output rating as a function of temperature that implies static +process limits. Normally, for high temperatures, the rating +drops rapidly. Nevertheless, the impact of aging and ambient +conditions may render conservative limits impractical. Thus +the alerting mechanism for the detection of abnormal heating +shall be developed. Providing a real-valued anomaly threshold +tightens the theoretical operating conditions and gives the +ability to track the performance and deviations. +Mar 21, 2022 +Mar 22, 2022 +Mar 23, 2022 +Mar 29, 2022 +Apr 4, 2022 +Apr 5, 2022 +Apr 7, 2022 +Apr 8, 2022 +Apr 11, 2022 +Apr 12, 2022 +Apr 13, 2022 +0 +0.2 +0.4 +0.6 +0.8 +1 +Inverter Temperature +Threshold +Normalized Temperature +Fig. 4. +Time Series of Inverter Temperature measurement (green line) and +predicted anomalous events (red vertical rectangles). The reddish fill bonded +by the red line represents an area of anomalous behavior. +Fig. 4 depicts one month of operation of the inverter from +16th March to 17th April 2022. After the packet loss before +21st March, rare temperature events occurred. Both events +fell out of the normal operating conditions given by the +dynamic process limit. Four faulty sensor readings follow +from 22nd, 23rd, 29th March and 4th April. The first two +are tagged as anomalies, though almost missed due to the +prolonged data loss. Given a shorter time from initialization +than te the influence of the edge between drop and raise had +a relaxing effect on limits. Former finding proposes a need +for grace period modification, which would alter self-learning +until the buffer given by te is not fully filled. The third faulty +reading was tagged without influencing the distribution and +operational boundaries due to the effect of tc. Oscillations, +that kept the boundaries relaxed vanished after 29th March, +which further tightened the process limit range. After the +fourth caught fault which was not used to update the model, +the detector deliberately adapted the range of normal operation +during the next day. Outliers during the sensors rescaling +period from 7th April were all tagged. However, the relaxed +operational conditions would probably lead to ignorance of +smaller anomalous oscillations in given period. +V. CONCLUSION +This paper proposes a novel approach to real-time anomaly +detection that provides a physical threshold that bounds normal +process operation. Such an approach has wide applicability in +all the process automation fields where low latency evaluation +and online adaptation are crucial. Moreover, adaptive operation +constraints provide less conservative process limits and govern +important insight into systems behavior. The plug-and-play +feature of the model makes it easily deployable as shown in +two case studies. +The first case study performed on BESS examined the +average battery cell temperature and demonstrated the ability +to capture anomalies as well as the capacity to restrict the +operational area by inversion of the cumulative distribution +function. Following our investigation of state-of-the-art online +anomaly detection described in Section I we conclude, that +although the robustness and performance of complex methods +may exceed the performance of the proposed method, the +ability to invert the prediction to depict real-time operational +restrictions and eschew using non-comprehensible parameters +makes it superior for a wide range of use cases. However, +the performance might be greatly afflicted when the time +constraints of the observed system are not known. This +restriction is much weaker than the restriction of the need +for data scientists skilled in the hyper-parameter tuning of +unsupervised models. Moreover, hyper-parameter tuning calls +for ground truth information about anomalies, which requires +an exhaustive collection and is not possible in real time. +Future works on the method will follow three practical +challenges: Firstly, the multivariate online anomaly detection +based on the developed method will be researched. The +multivariate implementation would allow the detection of +temporal anomalies and the use of features that render spatio- +temporal characteristics of the modeled system. This is the +common property of most of the online anomaly detection +methods that do not offer real-valued thresholds on operational +conditions. The multivariate clusters can reveal regions of +normal operation that would be otherwise detected incorrectly. +Secondly, the challenge of varying positive and negative +process limits thresholds will be examined. As depicted in +Fig 4 the positive and negative outliers, in many cases, result +from different mechanisms that caused them. The current +approach draws a range of normal operational conditions +centered around the moving mean value. +Thirdly, automated system identification using normal op- +eration data would further simplify the usage by removing +the requirement for system dynamics knowledge. The usage +of normal distribution makes the three-sigma rule constrain +the number of anomalies only theoretically. This allows the +number of anomalies in a given time window to vary greatly +and thus the performance is not very sensitive to the selection +of the threshold. On the contrary, the time window impacts +the model’s performance. + +REFERENCES +[1] J. Manyika, M. 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Available: https://www.tandfonline.com/doi/abs/ +10.1080/00401706.1962.10490022 + diff --git a/btFRT4oBgHgl3EQfSDcU/content/tmp_files/load_file.txt b/btFRT4oBgHgl3EQfSDcU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f434056b40d2ee1a3632f3eca0f3aaa7667ed7c8 --- /dev/null +++ b/btFRT4oBgHgl3EQfSDcU/content/tmp_files/load_file.txt @@ -0,0 +1,451 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf,len=450 +page_content='Real-Time Outlier Detection with Dynamic Process Limits Marek Wadinger and Michal Kvasnica Institute of Information Engineering, Automation and Mathematics Slovak University of Technology in Bratislava Bratislava, Slovakia {marek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='wadinger, michal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='kvasnica}@stuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='sk Abstract—Anomaly detection methods are part of the systems where rare events may endanger an operation’s profitability, safety, and environmental aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Although many state-of- the-art anomaly detection methods were developed to date, their deployment is limited to the operation conditions present during the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Online anomaly detection brings the capability to adapt to data drifts and change points that may not be represented during model development resulting in prolonged service life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' This paper proposes an online anomaly detection algorithm for existing real-time infrastructures where low-latency detection is required and novel patterns in data occur unpredictably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The online inverse cumulative distribution-based approach is introduced to eliminate common problems of offline anomaly detectors, meanwhile providing dynamic process limits to normal operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The benefit of the proposed method is the ease of use, fast computation, and deployability as shown in two case studies of real microgrid operation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Index Terms—anomaly detection, interpretable machine learn- ing, online machine learning, real-time systems, streaming ana- lytics I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' INTRODUCTION The era of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='0 is ruled by data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Effective data- based decision-making is driven by the quantity of collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Internet of Things (IoT) devices made data acquisition seamless and positively influenced a wide range of industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' It is estimated that the annual economic impact of IoT will further grow and reach up to $6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='2 trillion by 2025 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Various data collection mechanisms are used to buffer and store the data for future processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' However, the tremendous increase in data availability and the desire to extract valuable insight led to problems with the unbounded buffering and storage capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Real-time evaluation of the data streams became an acronym for smart data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Streaming data analytics introduced mechanisms for online extraction and transformation while loading to the storage only a fraction of the former data load, which allowed the storage of the vital information carried by the data more comprehensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' However, the unstable quality of the data appeared to have the most crucial importance over the quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Anomaly detection, well studied in the last decades, was reborn to the world of new challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Former studies were mainly concerned with a domain-specific detection of various anomalies while trained offline [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' However, anomalies of diverse sources, from fraudulent web activity and suspicious financial transactions to sensor failure, malfunctioning of the hardware, and performance drops, mutate over time, and the model had to be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Companies expanded their research activities on the creation and integration of generic frameworks combining prediction, detection, and alert mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' One of the first projects, open-sourced for the public, are EGADS by Yahoo [3] and AnomalyDetection by Twitter [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The frameworks’ modular- ity allowed the automation of the anomaly detection of time- series data and created space for discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Moving from domain-specific to generic methods posed new problems connected to type I errors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=', a false-positive classification of normal behavior as anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Accurate se- lection of forecaster, detector, and alerting mechanism allowed to tackle the problem, nevertheless, introduced considerable dependence on expert domain knowledge and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Further work proved improvement in performance while relieving the tight requirements on domain knowledge [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' However, strict demands on detection systems ranging from lasting up times to continuous monitoring with stable perfor- mance pointed to the challenge of data stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Change points and concept drifts troubled unsupervised models, which led to service downtime due to the model retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The era of adaptive machine learning introduced incre- mental learning schemes as a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Multiple studies for learning modes, adaptation methods, and model management swept through the machine learning community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Pannu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' proposed an adaptive anomaly detection system [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' However, the method represented a supervised operator-in-the-loop solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' introduced an adaptive kernel density-based algorithm that uses an adaptive kernel width [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Nonetheless, training the models on big data had limitations resulting from the storage and unbounded buffering of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Online learning models relaxed the need for data availability during model training [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' On the contrary, it processed the data from a bounded buffer sequentially as in [9] and [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Anomaly detection in microgrids, however, called for low latency detection which implied real-time training and pre- diction processes [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Such adaptation of streamed modeling took into consideration strict boundaries on computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' For work in this area see [12] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Alerting mechanisms in process automation detect situations where signal value deviates from constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' An alert watch- dog is triggered on threshold violation by individual signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='13527v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='LG] 31 Jan 2023 The constraints, or process limits, are usually predefined and fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Nevertheless, factors such as aging and environmental changes call for dynamic process limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Setting up a procedure for an evergrowing number of signal measurements is time- consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Besides, it is impossible for signals where no prior information about a correct process range is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Those are subject to external factors that are unknown at setup time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' In this article, we suggest using existing process automation infrastructure based on alerting (PLC, SCADA, among others) and applying machine learning for dynamic process range based on changing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' We propose an unsupervised anomaly detection algorithm capable of online adaptation to change points and concept drifts, which adds to a recently developed body of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The approach is evaluated on two case studies of microgrid sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' To the author’s knowledge, there are no studies to date concerned with providing adaptive operation constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The main benefits of the proposed solution are that it: Keeps existing IT infrastructure, saving costs, and does not require operator retraining Automates alerting thresholds setup for a high number of signals Automates alerting for signals with no a priori knowledge of process limits Assesses changing environmental conditions and device aging Uses self-learning approach on streamed data II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' PRELIMINARIES This section introduces the main concepts which are build- ing pillars of the developed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Subsection II-A will discuss a one-pass algorithm that allows for online adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The following Subsection II-B proposes the ability to invert the solution in a two-pass implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The mathemati- cal background of distribution modeling in Subsection II-C provides a basis for the Gaussian anomaly detection model conceptualized in the last Subsection II-D of Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Welford’s Method Streaming data analytics, restrict the unbounded buffer or storage of the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=', limits the uncontrolled growth of memory usage with the increasing amount o input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' In such cases, it is desired to keep the data only for the period of time required to perform computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' For the given purpose serve one-pass algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' This category of methods allows processing on-the-fly without the need to store the entire data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='1 (One-pass algorithm): The algorithm with a single access to the data items in the order of their occurrence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=', x1, x2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' is called one-pass algorithm [14] Welford’s method represents a numerically stable one-pass solution for the online computation of mean and variance [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Given xi where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=', n is the sample index in given population n, the corrected sum of squares is defined as Sn = n � i=1 (xi − ¯xn)2, (1) where the running mean ¯xn is ¯xn = n − 1 n ¯xn−1 + 1 nxn = ¯xn−1 + xn − ¯xn−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' (2) The following identities to update the corrected sum of squares hold true Sn = Sn−1 + (xn − ¯xn−1)(xn − ¯xn), (3) and the corresponding variance is s2 n = Sn n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' (4) As we can see in (3), we do access only current data sample xn and previous value of ¯xn−1 which is updated in (2) using the same data sample and the size of seen population n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Inverse Welford’s Method Let the incoming stream of data be subject to the concept drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Such alternation in statistical properties has a negative influence on prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Therefore, an adaptation of any machine learning model is crucial for successful long-term operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='2 (Concept drift): Concept drift is a change in the statistical properties that occur in a sub-region of the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The previous Subsection II-A defined the main concept of online statistical computation that allows reacting to such changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' However, the further in time the shift occurs, the slower the adjustment of the running mean is, resulting from a negative relationship in (2) between population size n and influence of the last sample in population xn on the updated value of ¯xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' For this reason, we define the expiration period te, over which the running statistics are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' After the expiration period, the data items are forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Such reversal results in a need to store all the data in the window in order to revert their effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Given te = n−1 we can revert the influence of the first data sample on the running mean as ¯xn−1 = n n − 1 ¯xn − 1 n − 1xn−te = ¯xn − xn−te − ¯xn n − 1 , (5) then reverting the sum of squares follows as Sn−1 = Sn − (xn−te − ¯xn−1)(xn−te − ¯xn), (6) which allows the computation of variance s2 n−1 = Sn−1 n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' (7) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Modeling Distribution Statistical distribution can be used to create a generalized model of a normal system behavior based on observed mea- surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Specifically, if no change point is expected in a given subset of samples, the Gaussian normal distribution can be fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Parameters of the normal distribution are used to compute standard score (8) for each new observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='3 (Standard Score): Standard score or Z-score is a number that specifies the number of sample standard deviations s2 n by which observation x deviates from the sample mean ¯xn of normal distribution zn = xn − ¯xn s2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' (8) In order to define the general probability of z-score belong- ing to anomaly we use probability computed using Cumulative Distribution Function (CDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' However, the z-score must be bounded using an error function into the interval from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='4 (Approximate Error Function): The approxi- mate error function represents the approximate probability that the random variable X lies in the range of [ −zn, zn] denoted as EA(zn) = zn e−z2 n √π ( 2/1 + 4/3x2 + 8/15x4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' (9) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='5 (Cumulative Distribution Function (CDF)): CDF represents the probability that the random variable X takes a value less than or equal to xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' FX : R → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' For generic normal distribution with sample mean ¯xn and sample deviation sn the cumulative distribution function FX(x) equals to FX(xi)n = 1 2( 1 + EA( zn √ 2) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' (10) Given the probability, we can also derive the value of x to which it belongs using a percent point function to compute inverse CDF (ICDF) denoted also as FX(xi)−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='6 (Percent-Point Function (PPF)): PPF returns the threshold value for random variable X under which it takes a value less than or equal to the value, for which FX(x) takes probability lower than selected quantile q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' QX : [0, 1] → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' An algorithm that calculates the value of the PPF is reported below as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Algorithm 1 Percent-Point Function for Normal Distribution Input: quantile q, sample mean ¯xn (2), sample variance s2 n (4) Output: threshold value xn,q Initialisation : 1: f ← 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' l ← −f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' r ← f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' LOOP Process 2: while FX(l) − q > 0 do 3: r ← l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 4: l ← lf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 5: end while 6: while FX(r) − q < 0 do 7: l ← r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 8: r ← rf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 9: end while 10: ˜xn,q = arg minz ∥FX(z) − q∥ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' l ≤ z ≤ r 11: return ˜xn,q � s2n + ¯xn D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Gaussian Anomaly Detection Anomalies come in various kinds and flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Commonly denoted types are point (spatial), contextual, and collective (temporal) anomalies [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Spatial anomalies take on a value that particularly deviates from the sample mean ¯xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' From a statistical viewpoint, spatial anomalies can be considered values x that significantly differ from the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' In empirical fields, such as machine learning, the three- sigma rule defines a region of distribution where normal values are expected to occur with near certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' This assumption makes approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='27% of values in the given distribution considered anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='7 (Three-Sigma Rule of Thumb (3σ rule)): 3σ rule represents a probability, that any value xi of random variable X will lie within a region of values of normal distribution at the distance from the sample mean µn of at most 3 sample standard deviations σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' P{|xi − µn| < 3σn} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='99730 (11) Anomalous values occur on both tails of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' In order to discriminate the anomalies using the three-sigma rule on both tails of the distribution, we define the anomaly score as follows yi = 2 ����FX(xi)n − 1 2 ���� , (12) where yi ∈ [0, P{|xi − µn| < 3σn}), (13a) applies for normal observations and yi ∈ [P{|xi − µn| < 3σn}, 1], (13b) for anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Using pure statistics to model normal behavior lets us ask the question about the threshold value x which corresponds to the area under the curve of CDF equal to the given probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' A such query can be answered using inversion of (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' However, inversion of (12) would fail the horizontal line test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Therefore, we restrict the applicability of the inverse only to FX(x)i ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='5, 1] xi = FX �yi 2 + 1 2 �−1 n (14) In order to derive a lower threshold, the Gaussian distribu- tion is fitted to the negative value of the streamed data and evaluated accordingly using the previously defined equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' ICDF-BASED REAL-VALUED THRESHOLD SYSTEM We suggest a novel approach to provide dynamic process limits using an online outlier detection algorithm capable of handling concept drifts in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Our main contribution is based on using an inverse cumulative distribution function (ICDF) to supply a real-valued threshold for anomaly detec- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=', to find the values of the signal which corresponds to the alert-triggering process limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Therefore, in the context of machine learning, we are tackling an inverse problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=', calculating the input that produced the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' To utilize an adaptive ICDF-based threshold system, the univariate Gaus- sian distribution has to be fitted to the data in online training and ICDF evaluated on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' This method is divided into four parts and described in the following lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' For a simplified representation of the method see Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Model Initialization The initial conditions of the model parameters are µ0 = x0 for mean and s2 0 = 1 for variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The score threshold is constant and set to q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='9973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Moreover, there are two user-defined parameters: the expiration period te, and the time constant of the system tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The expiration period, which defines the period over which the time-rolling computations are per- formed, can be altered to change the proportion of expected anomalies and allows relaxation (longer expiration period) or tightening (shorter expiration period) of the thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The time constant of the system determines the speed of change point adaptation as it influences the selection of anomalous points that will be used to update the model for a window of values Y = {yi−tc, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=', yi} if the following condition holds true � y∈Y y n(Y ) > q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' (15) The existence of two tunable and easy-to-interpret hyper- parameters makes it very easy to adapt the solution to any univariate anomaly detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Online training Training of the model takes place in an online fashion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=', the model learns one sample at a time at the moment of its arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Learning updates the mean and variance of the underlying Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The computation of moving mean (2) and variance (4) is handled by Welford’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Each sample after the expiration period is forgotten and its effect reverted in the second pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' First, the new mean is computed using (5) which accesses the first value in the bounded buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The value is dropped in the same pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Second, the new sample variance is reverted based on (7) using the new mean and current mean that is overwritten afterward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' For details see Subsection II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Online prediction In the prediction phase, z-score (8) is computed and passed through EA (9) in order to evaluate FX(xi) from (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The al- gorithm marks the incoming data points if their corresponding anomaly score (12) is out of the range defined by threshold q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' In other words, marks signal value xi that is higher or equal to the threshold, which bounds the three-sigma region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Dynamic Process Limits Normal process operation is constrained online using ICDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The constant value of q and parameters of the fitted distribution are both passed through Algorithm 1 to obtain value, which corresponds to the value of x that would trigger an upper bound outlier alarm at the given time instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' To obtain a lower bound of operation conditions the same procedure is applied to the distribution fitted on negative values of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Algorithm 2 Online Anomaly Detection Workflow Input: expiration period te, time constant tc Output: score yi, threshold xi,q Initialisation : 1: i ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' n ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' q ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='9973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' ¯x ← x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' s2 ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 2: compute FX(x0) using (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' LOOP Process 3: loop 4: xi ← RECEIVE();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 5: yi ← PREDICT(xi) using (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 6: xi,q ← GET(q, ¯x, s2) using Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 7: if (13a) or (15) then 8: ¯x, s2 ← UPDATE(xi, ¯x, s2, n) using (2), (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 9: n ← n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 10: for xi−te do 11: ¯x, s2 ← REVERT(xi−te, ¯x, s2, n) using (5), (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 12: n ← n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 13: end for 14: end if 15: i ← i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 16: end loop IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' CASE STUDY In this section, we demonstrate the applicability of the proposed ICDF-based approach in two case studies of the microgrid operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The properties and performance were investigated using streamed signals from the IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The successful deployment demonstrates that this approach is suit- able for existing alerting mechanisms of process automation infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The case studies were realized using Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='1 on a MAC with an M1 CPU and 8 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The percent point function was solved using an iterative root-finding algorithm, Brent’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Battery Energy Storage System (BESS) First, we verify our proposed method on BESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The BESS reports measurements of State of Charge (SoC), supply/draw energy events, inner temperature, outer temperature, Heating, ventilation, and air conditioning (HVAC) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Tight control of the battery cell temperature is needed for the optimal performance and maximum lifespan of the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Identifying anomalous events and removal of corrupted data might yield significant improvement on the process control level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The sampling rate of the signal measurement is 1 minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' However, network communication is prone to packet dropout, which results in non-uniform sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' To protect the sensitive business value of the data, we normalize all signals to the range [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The goal was to mark anomalous events in the data and provide adaptive process limits from the online self-learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Mar 3, 2022 Mar 4, 2022 Mar 7, 2022 Mar 8, 2022 Mar 10, 2022 Mar 12, 2022 Mar 14, 2022 Mar 15, 2022 Mar 21, 2022 Mar 22, 2022 Mar 24, 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='8 1 Average Cell Temperature Normalized Temperature Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Time Series of Average Battery Cell Temperature measurement (green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Non-uniform ticks on the x-axis mark days of interest (NOTE: some marks are hidden due to the readability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The y-axis renders the normalized temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 1 renders measurement of average battery cell tempera- ture from 21st February until 26th March.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' We can observe mul- tiple anomalies of various sources given this span, for instance, packet dropout, suspicious events, intermittent sensor failure, and change point in data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Dates of observation given the listed events will be provided later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The initial conditions of the model states are set based on Subsection III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The user-defined parameters, were set to 7 days for the expiration period and 5 hours for the time constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Anomalies found during the first day of the service are ignored due to the initialization of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' In this case study, the anomaly detection problem was approached by the online model fitting based on Subsection III-B Using the online prediction described in Subsection III-C we tag the sample as the anomaly or normal data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 2 renders vertical rectangles over the regions from the start until the end of the predicted anomalous event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The results on Average Cell Temperature in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 2 show that the model could capture anomalous patterns of various sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Despite self-learning without supervision, the model- classified anomalies were also confirmed by the data provider after inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' For instance, a rare event of manipulation with BESS on 3rd, followed by peak on 4th March.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' BESS relocation on 7th, led to a change point which was alerted and the system adapted completely over the course of 1 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Test events resulted in peak values through 10th to 15th March, and faulty measurements on the 12th March followed by a packet loss on 21st March were alerted too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The system tagged the next two tests of temperature control switch-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Findings that favor the model’s ability to discriminate anomalous behavior are important for the meaningful real- ization of the dynamic process thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The real-valued threshold mechanism, defined in Subsection III-D, provided Mar 3, 2022 Mar 4, 2022 Mar 7, 2022 Mar 8, 2022 Mar 10, 2022 Mar 12, 2022 Mar 14, 2022 Mar 15, 2022 Mar 21, 2022 Mar 22, 2022 Mar 24, 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='8 1 Average Cell Temperature Normalized Temperature Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Time Series of Average Battery Cell Temperature measurement (green line) and predicted anomalous events (red vertical rectangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' up-to-date upper and lower bounds for the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' As for the validity of the dynamic process limits, each breakout of the signal value from within the range was also marked by the anomaly detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 3 points to the capability to adapt to change point on 7th March and mitigate the influence of intermittent effects of anomalies on distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The speed of the change point adaptation as well as the mitigation of the effect of anomalies on the tightness of limits are governed by the user-defined expiration period and time constant of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Mar 3, 2022 Mar 4, 2022 Mar 7, 2022 Mar 8, 2022 Mar 10, 2022 Mar 12, 2022 Mar 14, 2022 Mar 15, 2022 Mar 21, 2022 Mar 22, 2022 Mar 24, 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='8 1 Average Cell Temperature Threshold Normalized Temperature Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Time Series of Average Battery Cell Temperature measurement (green line) and predicted anomalous events (red vertical rectangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The reddish fill bonded by the red line represents an area of anomalous behavior as given by the anomaly detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Power Inverter A second case study demonstrates the proposed method’s applicability to the temperature of the power inverter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' During high load periods, inverters can heat up swiftly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Technical documentation of every inverter provides details on continuous output rating as a function of temperature that implies static process limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Normally, for high temperatures, the rating drops rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Nevertheless, the impact of aging and ambient conditions may render conservative limits impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Thus the alerting mechanism for the detection of abnormal heating shall be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Providing a real-valued anomaly threshold tightens the theoretical operating conditions and gives the ability to track the performance and deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Mar 21, 2022 Mar 22, 2022 Mar 23, 2022 Mar 29, 2022 Apr 4, 2022 Apr 5, 2022 Apr 7, 2022 Apr 8, 2022 Apr 11, 2022 Apr 12, 2022 Apr 13, 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content='8 1 Inverter Temperature Threshold Normalized Temperature Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Time Series of Inverter Temperature measurement (green line) and predicted anomalous events (red vertical rectangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The reddish fill bonded by the red line represents an area of anomalous behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' 4 depicts one month of operation of the inverter from 16th March to 17th April 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' After the packet loss before 21st March, rare temperature events occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Both events fell out of the normal operating conditions given by the dynamic process limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Four faulty sensor readings follow from 22nd, 23rd, 29th March and 4th April.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The first two are tagged as anomalies, though almost missed due to the prolonged data loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Given a shorter time from initialization than te the influence of the edge between drop and raise had a relaxing effect on limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Former finding proposes a need for grace period modification, which would alter self-learning until the buffer given by te is not fully filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The third faulty reading was tagged without influencing the distribution and operational boundaries due to the effect of tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Oscillations, that kept the boundaries relaxed vanished after 29th March, which further tightened the process limit range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' After the fourth caught fault which was not used to update the model, the detector deliberately adapted the range of normal operation during the next day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Outliers during the sensors rescaling period from 7th April were all tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' However, the relaxed operational conditions would probably lead to ignorance of smaller anomalous oscillations in given period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' CONCLUSION This paper proposes a novel approach to real-time anomaly detection that provides a physical threshold that bounds normal process operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Such an approach has wide applicability in all the process automation fields where low latency evaluation and online adaptation are crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Moreover, adaptive operation constraints provide less conservative process limits and govern important insight into systems behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The plug-and-play feature of the model makes it easily deployable as shown in two case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The first case study performed on BESS examined the average battery cell temperature and demonstrated the ability to capture anomalies as well as the capacity to restrict the operational area by inversion of the cumulative distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Following our investigation of state-of-the-art online anomaly detection described in Section I we conclude, that although the robustness and performance of complex methods may exceed the performance of the proposed method, the ability to invert the prediction to depict real-time operational restrictions and eschew using non-comprehensible parameters makes it superior for a wide range of use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' However, the performance might be greatly afflicted when the time constraints of the observed system are not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' This restriction is much weaker than the restriction of the need for data scientists skilled in the hyper-parameter tuning of unsupervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Moreover, hyper-parameter tuning calls for ground truth information about anomalies, which requires an exhaustive collection and is not possible in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Future works on the method will follow three practical challenges: Firstly, the multivariate online anomaly detection based on the developed method will be researched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The multivariate implementation would allow the detection of temporal anomalies and the use of features that render spatio- temporal characteristics of the modeled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' This is the common property of most of the online anomaly detection methods that do not offer real-valued thresholds on operational conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The multivariate clusters can reveal regions of normal operation that would be otherwise detected incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Secondly, the challenge of varying positive and negative process limits thresholds will be examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' As depicted in Fig 4 the positive and negative outliers, in many cases, result from different mechanisms that caused them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The current approach draws a range of normal operational conditions centered around the moving mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' Thirdly, automated system identification using normal op- eration data would further simplify the usage by removing the requirement for system dynamics knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' The usage of normal distribution makes the three-sigma rule constrain the number of anomalies only theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' This allows the number of anomalies in a given time window to vary greatly and thus the performance is not very sensitive to the selection of the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' On the contrary, the time window impacts the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFRT4oBgHgl3EQfSDcU/content/2301.13527v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': 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Layout-Aware Summarization +Laura Nguyen1,3 +Thomas Scialom2∗ +Benjamin Piwowarski3 +Jacopo Staiano4∗ +1reciTAL, Paris, France +2Meta AI, Paris, France +3Sorbonne Université, CNRS, ISIR, F-75005 Paris, France +4University of Trento, Italy +laura@recital.ai +tscialom@fb.com +benjamin.piwowarski@cnrs.fr +jacopo.staiano@unitn.it +Abstract +Text Summarization is a popular task and an +active area of research for the Natural Lan- +guage Processing community. It requires ac- +counting for long input texts, a characteristic +which poses computational challenges for neu- +ral models. Moreover, real-world documents +come in a variety of complex, visually-rich, +layouts. This information is of great relevance, +whether to highlight salient content or to en- +code long-range interactions between textual +passages. Yet, all publicly available summa- +rization datasets only provide plain text con- +tent. To facilitate research on how to exploit vi- +sual/layout information to better capture long- +range dependencies in summarization models, +we present LoRaLay, a collection of datasets +for long-range summarization with accompa- +nying visual/layout information. +We extend +existing and popular English datasets (arXiv +and PubMed) with visual/layout information +and propose four novel datasets – consistently +built from scholar resources – covering French, +Spanish, Portuguese, and Korean languages. +Further, we propose new baselines merging +layout-aware and long-range models – two or- +thogonal approaches – and obtain state-of-the- +art results, showing the importance of combin- +ing both lines of research. +1 +Introduction +Deep learning techniques have enabled remarkable +progress in Natural Language Processing (NLP) +in recent years (Devlin et al., 2018; Raffel et al., +2019; Brown et al., 2020). However, the majority +of models, benchmarks, and tasks have been de- +signed for unimodal approaches, i.e. focusing ex- +clusively on a single source of information, namely +plain text. While it can be argued that for specific +NLP tasks, such as textual entailment or machine +translation, plain text is all that is needed, there +exist several tasks for which disregarding the vi- +sual appearance of text is clearly sub-optimal: in +*Work partially done while at reciTAL. +a real-world context (business documentation, sci- +entific articles, etc.), text does not naturally come +as a sequence of characters, but is rather displayed +in a bi-dimensional space containing rich visual +information. The layout of e.g. this very paper +provides valuable semantics to the reader: in which +section are we right now? At the blink of an eye, +this information is readily accessible via the salient +section title (formatted differently and placed to +highlight its role) preceding these words. Just to +emphasize this point, imagine having to scroll this +content in plain text to access such information. +In the last couple of years, the research commu- +nity has shown a growing interest in addressing +these limitations. Several approaches have been +proposed to deal with visually-rich documents and +integrate layout information into language mod- +els, with direct applications to Document Under- +standing tasks. Joint multi-modal pretraining (Xu +et al., 2021; Powalski et al., 2021; Appalaraju et al., +2021) has been key to reach state-of-the-art per- +formance on several benchmarks (Jaume et al., +2019; Grali´nski et al., 2020; Mathew et al., 2021). +Nonetheless, a remaining limitation is that these +(transformer-based) approaches are not suitable for +processing long documents, the quadratic complex- +ity of self-attention constraining their use to short +sequences. Such models are hence unable to en- +code global context (e.g. long-range dependencies +among text blocks). +Focusing on compressing the most relevant infor- +mation from long texts to short summaries, the Text +Summarization task naturally lends itself to benefit +from such global context. Notice that, in practice, +the limitations linked to sequence length are also +amplified by the lack of visual/layout information +in the existing datasets. Therefore, in this work, +we aim at spurring further research on how to in- +corporate multimodal information to better capture +long-range dependencies. +Our contributions can be summarized as follows: +arXiv:2301.11312v1 [cs.CL] 26 Jan 2023 + +• We extend two popular datasets for long-range +summarization, arXiv and PubMed (Cohan +et al., 2018), by including visual and layout +information – thus allowing direct comparison +with previous works; +• We release 4 additional layout-aware summa- +rization datasets (128K documents), covering +French, Spanish, Portuguese, and Korean lan- +guages; +• We provide baselines including adapted archi- +tectures for multi-modal long-range summa- +rization, and report results showing that (1) +performance is far from being optimal; and +(2) layout provides valuable information. +All the datasets are available on HuggingFace.1 +2 +Related Work +2.1 +Layout/Visually-rich Datasets +Document Understanding covers problems that in- +volve reading and interpreting visually-rich docu- +ments (in contrast to plain texts), requiring com- +prehending the conveyed multimodal information. +Hence, several tasks with a central layout aspect +have been proposed by the document understand- +ing community. Key Information Extraction tasks +consist in extracting the values of a given set of +keys, e.g., the total amount in a receipt or the date +in a form. In such tasks, documents have a layout +structure that is crucial for their interpretation. No- +table datasets include FUNSD (Jaume et al., 2019) +for form understanding in scanned documents, and +SROIE (Huang et al., 2019), as well as CORD +(Park et al., 2019), for information extraction from +receipts. Grali´nski et al. (2020) elicit progress on +deeper and more complex Key Information Extrac- +tion by introducing the Kleister datasets, a collec- +tion of business documents with varying lengths, +released as PDF files. However, the documents +in Kleister often contain single-column layouts, +which are simpler than the various multi-column +layouts considered in LoRaLay. Document VQA +is another popular document understanding task +that requires processing multimodal information +(e.g., text, layout, font style, images) conveyed by +a document to be able to answer questions about a +1https://hf.co/datasets/nglaura/arxivlay-summarization, +https://hf.co/datasets/nglaura/pubmedlay-summarization, +https://hf.co/datasets/nglaura/hal-summarization, +https://hf.co/datasets/nglaura/scielo-summarization, +https://hf.co/datasets/nglaura/koreascience-summarization +visually rich document (e.g., What is the date given +at the top left of the form?, Whose picture is given +in this figure?). The DocVQA dataset (Mathew +et al., 2021) and InfographicsVQA (Mathew et al., +2022) are commonly-used VQA datasets that re- +spectively provide industry documents and info- +graphic images, encouraging research on under- +standing documents with complex interplay of text, +layout and graphical elements. Finally, to foster +research on visually-rich document understanding, +Borchmann et al. (2021) introduce the Document +Understanding Evaluation (DUE) benchmark, a +unified benchmark for end-to-end document under- +standing, created by combining several datasets. +DUE includes several available and transformed +datasets for VQA, Key Information Extraction and +Machine Reading Comprehension tasks. +2.2 +Existing Summarization Datasets +Several large-scale summarization datasets have +been proposed to boost research on text summa- +rization systems. Hermann et al. (2015) proposed +the CNN/DailyMail dataset, a collection of English +articles extracted from the CNN and The Daily +Mail portals. Each news article is associated with +multi-sentence highlights which serve as reference +summaries. Scialom et al. (2020) bridge the gap be- +tween English and non-English resources for text +summarization by introducing MLSum, a large- +scale multilingual summarization corpus providing +news articles written in French, German, Spanish, +Turkish and Russian. Going toward more challeng- +ing scenarios involving significantly longer doc- +uments, the arXiv and PubMed datasets (Cohan +et al., 2018) consist of scientific articles collected +from academic repositories, wherein the paper ab- +stracts are used as summaries. To encourage a shift +towards building more abstractive summarization +models with global content understanding, Sharma +et al. (2019) introduce BIGPATENT, a large-scale +dataset made of U.S. patent filings. Here, invention +descriptions serve as reference summaries. +The vast majority of summarization datasets only +deal with plain text documents. As opposed to +other Document Understanding tasks (e.g., form +understanding, visual QA) in which the placement +of text on the page and/or visual components are +the main source of information needed to find the +desired data (Borchmann et al., 2021), text plays +a predominant role in document summarization. +However, guidelines for summarizing texts – espe- + +cially long ones – often recommend roughly pre- +viewing them to break them down into their major +sections (Toprak and Almacio˘glu, 2009; Luo et al., +2019). This suggests that NLP systems might lever- +age multimodal information in documents. Miculi- +cich and Han (2022) propose a two-stage method +which detects text segments and incorporates this +information in an extractive summarization model. +Cao and Wang (2022) collect a new dataset for +long and structure-aware document summarization, +consisting of 21k documents written in English and +extracted from WikiProject Biography. +Although not all documents are explicitly or- +ganized into clearly defined sections, the great +majority contains layout and visual clues (e.g., a +physical organization into paragraphs, bigger head- +ings/subheadings) which help structure their textual +contents and facilitate reading. Thus, we argue that +layout is crucial to summarize long documents. We +propose a corpus of more than 345K long docu- +ments with layout information. Furthermore, to +address the need for multilingual training data (Chi +et al., 2020), we include not only English docu- +ments, but also French, Spanish, Portuguese and +Korean ones. +3 +Datasets Construction +Inspired by the way the arXiv and PubMed datasets +were built (Cohan et al., 2018), we construct our +corpus from research papers, with abstracts as +ground-truth summaries. As the PDF format allows +simultaneous access to textual, visual and layout +information, we collect PDF files to construct our +datasets, and provide their URLs.2 +For each language, we select a repository that +contains a high number of academic articles (in the +order of hundreds of thousands) and provides easy +access to abstracts. More precisely, we chose the +following repositories: +• Archives Ouverte HAL (French),3 an open +archive of scholarly documents from all aca- +demic fields. As HAL is primarily directed +towards French academics, a great proportion +of articles are written in French; +• SciELO (Spanish and Portuguese),4 an open +access database of academic articles published +in journal collections from Latin America, +2We make the corpus-construction code publicly available at https:// +github.com/recitalAI/loralay-datasets. +3https://hal.archives-ouvertes.fr/ +4https://www.scielo.org/ +Iberian Peninsula and South Africa, and cov- +ering a broad range of topics (e.g. agricultural +sciences, engineering, health sciences, letters +and arts). Languages include English, Span- +ish, and Portuguese. +• KoreaScience (Korean),5 an open archive of +Korean scholarly publications in the fields of +natural sciences, life sciences, engineering, +and humanities and social sciences. Articles +are written in English or Korean. +Further, we provide enhanced versions of the +arXiv and PubMed datasets, respectively denoted +as arXiv-Lay and PubMed-Lay, for which layout +information is provided. +3.1 +Collecting the Data +Extended Datasets +The arXiv and PubMed +datasets (Cohan et al., 2018) contain long scien- +tific research papers extracted from the arXiv and +PubMed repositories. We augment them by provid- +ing their PDFs, allowing access to layout and visual +information. As the abstracts contained in the orig- +inal datasets are all lowercased, we do not reuse +them, but rather extract the raw abstracts using the +corresponding APIs. +Note that we were unable to retrieve all the orig- +inal documents. For the most part, we failed to +retrieve the corresponding abstracts, as they did not +necessarily match the ones contained in the PDF +files (due to e.g. PDF-parsing errors). We also +found that some PDF files were unavailable, while +others were corrupted or scanned documents.6 In +total, about 39% (35%) of the original documents +in arXiv (PubMed) were lost. +arXiv-Lay +The original arXiv dataset (Cohan +et al., 2018) was constructed by converting the +LATEX files to plain text. To be consistent with +the other datasets – for which LATEX files are not +available – we instead use the PDF files to extract +both text and layout elements. For each document +contained in the original dataset, we fetch (when +possible) the corresponding PDF file using Google +Cloud Storage buckets. As opposed to the original +procedure, we do not remove tables nor discard +sections that follow the conclusion. We retrieve +the corresponding abstracts from a metadata file +provided by Kaggle.7 +5http://www.koreascience.or.kr +6For more details on this, see Section A.1 in the Appendix. +7https://www.kaggle.com/Cornell-University/arxiv + +PubMed-Lay +For PubMed, we use the PMC +OAI Service8 to retrieve abstracts and PDF files. +HAL +We use the HAL API9 to download re- +search papers written in French. To avoid exces- +sively long (e.g. theses) or short (e.g. posters) +documents, extraction is restricted to journal and +conference papers. +SciELO +Using Scrapy,10 we crawl the following +SciELO collections: Ecuador, Colombia, Paraguay, +Uruguay, Bolivia, Peru, Portugal, Spain and Brazil. +We download documents written either in Spanish +or Portuguese, according to the metadata, obtaining +two distinct datasets: SciELO-ES (Spanish) and +SciELO-PT (Portuguese). +KoreaScience +Similarly, we scrape the Korea- +Science website to extract research papers. We +limit search results to documents whose publishers’ +names contain the word Korean. This rule was de- +signed after sampling documents in the repository, +and is the simplest way to get a good proportion +of papers written in Korean.11 Further, search is +restricted to papers published between 2012 and +2021, as recent publications are more likely to have +digital-born, searchable PDFs. Finally, we down- +load the PDF files of documents that contain an +abstract written in Korean. +3.2 +Data Pre-processing +For each corpus, we use the 95th percentile of the +page distribution as an upper bound to filter out +documents with too many pages, while the 5th (1st +for HAL and SciELO) percentile of the summary +length distribution is used as a minimum thresh- +old to remove documents whose abstracts are too +short. As our baselines do not consider visual in- +formation, we only extract text and layout from +the PDF files. Layout is incorporated by provid- +ing the spatial position of each word in a docu- +ment page image, represented by its bounding box +(x0, y0, x1, y1), where (x0, y0) and (x1, y1) respec- +tively denote the coordinates of the top-left and +bottom-right corners. Using the PDF rendering li- +brary Poppler12, text and word bounding boxes are +extracted from each PDF, and the sequence order is +recovered based on heuristics around the document +layout (e.g., tables, columns). Abstracts are then +8https://www.ncbi.nlm.nih.gov/pmc/tools/oai/ +9https://api.archives-ouvertes.fr/docs/search +10https://scrapy.org/ +11For further details, see Section A.2 in the Appendix. +12https://poppler.freedesktop.org/ +removed by searching for exact matches; when no +exact match is found, we use fuzzysearch13 +and regex14 to find near matches.15 For the non- +English datasets, documents might contain several +abstracts, written in different languages. To avoid +information leakage, we retrieve the abstract of +each document in every language available – ac- +cording to the API for HAL or the websites for +SciELO and KoreaScience – and remove them us- +ing the same strategy as for the main language. In +the case an abstract cannot be found, we discard +the document to prevent any unforeseen leakage. +The dataset construction process is illustrated in +Section A in the Appendix. +3.3 +Datasets Statistics +The statistics of our proposed datasets, along with +those computed on existing summarization datasets +of long documents (Cohan et al., 2018; Sharma +et al., 2019) are reported in Table 1. We see that +document lengths are comparable or greater than +for the arXiv, PubMed and BigPatent datasets. +For arXiv-Lay and PubMed-Lay, we retain the +original train/validation/splits and try to reconstruct +them as faithfully to the originals as possible. For +the new datasets, we order documents based on +their publication dates and provide splits following +a chronological ordering. For HAL and Korea- +Science, we retain 3% of the articles as validation +data, 3% as test, and the remaining as training data. +To match the number of validation/test documents +in HAL and KoreaScience, we split the data into +90% for training, 5% for validation and 5% for test, +for both SciELO datasets. +4 +Experiments +4.1 +Models +For reproducibility purposes, we make the mod- +els implementation, along with the fine-tuning and +evaluation scripts, publicly available.16 +We do not explore the use of visual information +in long document summarization, as the focus is on +evaluating baseline performance using state-of-the- +art summarization models augmented with layout +information. While visual features might provide +a better understanding of structures such as tables +and figures, we do not expect substantial gains with +13https://pypi.org/project/fuzzysearch/ +14https://pypi.org/project/regex/ +15We use a maximum Levenshtein distance of 20 with fuzzysearch, and a +maximum number of errors of 3 with regex. +16https://github.com/recitalAI/loralay-modeling + +Dataset +# Docs +Mean +Mean +Article +Summary +Length +Length +arXiv (Cohan et al., 2018) +215,913 +3,016 +203 +PubMed (Cohan et al., 2018) +133,215 +4,938 +220 +BigPatent (Sharma et al., 2019) +1,341,362 +3,572 +117 +arXiv-Lay +130,919 +7,084 +125 +PubMed-Lay +86,668 +4,038 +144 +HAL +46,148 +4,543 +134 +SciELO-ES +23,170 +4,977 +172 +SciELO-PT +21,563 +6,853 +162 +KoreaScience +37,498 +3,192 +95 +Table 1: +Datasets statistics. +Article and summary +lengths are computed in words. +For KoreaScience, +words are obtained via white-space tokenization. Dif- +ference between arXiv and arXiv-Lay is due to the fact +that we retain the whole document, while Cohan et al. +(2018) truncate it after the conclusion. +respect to layout-aware models. Indeed, the infor- +mation provided in figures (i.e., information that +cannot be captured by layout or text) are commonly +described in the caption or related paragraphs. +Text-only models with standard input size +We +use Pegasus (Zhang et al., 2020) as a text-only base- +line for arXiv-Lay and PubMed-Lay. Pegasus is +an encoder-decoder model pre-trained using gap- +sentences generation, making it a state-of-the-art +model for abstractive summarization. For the non- +English datasets, we rely on a finetuned MBART as +our baseline. MBART (Liu et al., 2020) is a multi- +lingual sequence-to-sequence model pretrained on +large-scale monolingual corpora in many languages +using the BART objective (Lewis et al., 2019). We +use its extension, MBART-50 (Tang et al., 2020),17 +which is created from the original MBART by ex- +tending its embeddings layers and pre-training it on +a total of 50 languages. Both Pegasus and MBART +are limited to a maximum sequence length of 1,024 +tokens, which is well below the median length of +each dataset. +Layout-aware models with standard input size +We introduce layout-aware extensions of Pega- +sus and MBART, respectively denoted as Pe- +gasus+Layout and MBART+Layout. Following +LayoutLM (Xu et al., 2020), which is state-of- +the-art on several document understanding tasks +(Jaume et al., 2019; Huang et al., 2019; Harley +et al., 2015), each token bounding box coordinates +(x0, y0, x1, y1) is normalized into an integer in the +range [0, 1000]. Spatial positions are encoded us- +ing four embedding tables, namely two for the co- +ordinate axes (x and y), and the other two for the +17For the sake of clarity, we refer to MBART-50 as MBART. +bounding box size (width and height). The layout +representation of a token is formed by summing +the resulting embedding representations The final +representation of a token is then obtained through +point-wise summation of its textual, 1D-positional +and layout embeddings. +Long-range, +text-only +models +To +process +longer sequences, we leverage BigBird (Zaheer +et al., 2020), a sparse-attention based Transformer +which reduces the quadratic dependency to a linear +one. For arXiv-Lay and PubMed-Lay, we initialize +BigBird from Pegasus (Zaheer et al., 2020) and for +the non-English datasets, we use the weights of +MBART. The resulting models are referred to as +BigBird-Pegasus and BigBird-MBART. For both +models, BigBird sparse attention is used only in +the encoder. Both models can handle up to 4,096 +inputs tokens, which is greater than the median +length in PubMed-Lay, HAL and KoreaScience. +Long-range, layout-aware models +We also in- +clude layout information in long-range text-only +models. Similarly to layout-aware models with +standard input size, we integrate layout informa- +tion into our long-range models by encoding each +token’s spatial position in the page. The resulting +models are denoted as BigBird-Pegasus+Layout +and BigBird-MBART+Layout. +Additional State-of-the-Art Baselines +We fur- +ther consider additional state-of-the-art baselines +for summarization: i) the text-only T5 (Raffel et al., +2019) with standard input size, ii) the long-range +Longformer-Encoder-Decoder (LED) (Beltagy +et al., 2020), and iii) the layout-aware, long-range +LED+Layout, which we implement similarly to +the previous layout-aware models. +4.2 +Implementation Details +We initialize our Pegasus-based and MBART-based +models with, respectively, the google/pegasus-large +and facebook/mbart-large-50 checkpoints shared +through the Hugging Face Model Hub. As for T5 +and LED, we use the weights from t5-base and +allenai/led-base-16384, respectively.18 +Following Zhang et al. (2020) and Zaheer et al. +(2020), we fine-tune our models up to 74k (100k) +steps on arXiv-Lay (PubMed-Lay). On HAL, the +total number of steps is set to 100k, while it is de- +18The large versions of T5 and LED did not fit into GPU due to their size. + +Dataset +Instances +Input Length +Output Length +Train +Dev +Test +Median +90%-ile +Median +90%-ile +arXiv (Cohan et al., 2018) +203,037 +6,436 +6,440 +6,151 +14,405 +171 +352 +PubMed (Cohan et al., 2018) +119,924 +6,633 +6,658 +2,715 +6,101 +212 +318 +arXiv-Lay +122,189 +4,374 +4,356 +6,225 +12,541 +150 +249 +PubMed-Lay +78,234 +4,084 +4,350 +3,761 +7,109 +182 +296 +HAL +43,379 +1,384 +1,385 +4,074 +8,761 +179 +351 +SciELO-ES +20,853 +1,158 +1,159 +4,859 +8,519 +226 +382 +SciELO-PT +19,407 +1,078 +1,078 +6,090 +9,655 +239 +374 +KoreaScience +35,248 +1,125 +1,125 +2,916 +5,094 +219 +340 +Table 2: Datasets splits and statistics. Input and output lengths are computed in tokens, obtained using Pegasus +and MBART-50’s tokenizers for the English and non-English datasets, respectively. +creased to 50k for the other non-English datasets.19 +For each model, we select the checkpoint with +the best validation loss. For Pegasus and MBART +models, inputs are truncated at 1,024 tokens. For +BigBird-Pegasus models, we follow Zaheer et al. +(2020) and set the maximum input length at 3,072 +tokens. As the median input length is much greater +in almost every non-English dataset, we increase +the maximum input length to 4,096 tokens for +BigBird-MBART models. +Output length is re- +stricted to 256 tokens for all models, which is +enough to fully capture at least 50% of the sum- +maries in each dataset. +For evaluation, we use beam search and report a +single run for each model and dataset. Following +Zhang et al. (2020); Zaheer et al. (2020), we set the +number of beams to 8 for Pegasus-based models, +and 5 for BigBird-Pegasus-based models. For the +non-English datasets, we set it to 5 for all models, +for fair comparison. For all experiments, we use +a length penalty of 0.8. For more implementation +details, see Section B.1 in the Appendix. +5 +Results and Discussion +5.1 +General Results +In Table 3, we report the ROUGE-L scores ob- +tained on arXiv and PubMed datasets (reported by +Zaheer et al. (2020)), as well as on the correspond- +ing layout-augmented counterparts we release. 20 +On arXiv-Lay and PubMed-Lay, we observe that, +while the addition of layout to Pegasus does not +improve the ROUGE-L scores, there are gains in in- +tegrating layout information into BigBird-Pegasus. +To assess whether these gains are significant, we +perform significance analysis at the 0.05 level us- +ing bootstrap, and estimate a ROUGE-L thresh- +19We tested different values for the number of steps (10k, 25k, 50k, 100k) +and chose the one that gave the best validation scores for MBART. +20For detailed results, please refer to Section C.1 in the Appendix. +old that predicts when improvements are signifi- +cant. ROUGE-L improvements between each pair +of models are reported in Table 11 in the appendix. +On arXiv-Lay, we compute a threshold of 1.48 +ROUGE-L, showing that BigBird-Pegasus+Layout +significantly outperforms all Pegasus-based mod- +els. In particular, we find a 1.56 ROUGE-L im- +provement between BigBird-Pegasus and its layout- +augmented counterpart, demonstrating that the ad- +dition of layout to long-range modeling signifi- +cantly improves summarization. On PubMed-Lay, +we compute a threshold of 1.77. Hence, the 0.96 +ROUGE-L improvement from BigBird-Pegasus to +its layout-augmented counterpart is not significant. +However, the variance in font sizes in PubMed-Lay +is much smaller compared to arXiv-Lay (see Ta- +ble 12 in the appendix), reflecting an overall more +simplistic layout. Therefore, we argue that lay- +out integration has a lesser impact in PubMed-Lay, +which can explain the non-significance of results. +In addition, we find that BigBird-Pegasus signifi- +cantly outperforms Pegasus and Pegasus+Layout +only when augmented with layout, with an im- +provement of, respectively, 2.3 and 2.2 points. This +demonstrates the importance of combining layout +and long-range modeling. +While T5 and LED obtain competitive results, +we find that the gain in adding layout to LED is +minor. However, the models we consider have all +been pre-trained only on plain text. As a result, +the layout representations are learnt from scratch +during fine-tuning. Similarly to us, Borchmann +et al. (2021) show that their layout-augmented T5 +does not necessarily improve the scores, and that +performance is significantly enhanced only when +the model has been pre-trained on layout-rich data. +Further, we observe, for both Pegasus and +BigBird-Pegasus, a drop in performance w.r.t. the +scores obtained on the original datasets. This can +be explained by two factors. First, our extended + +Model +# Params +arXiv/ +arXiv-Lay +PubMed/ +PubMed-Lay +Pegasus (Zhang et al., 2020) +568M +38.83 +41.34 +BigBird-Pegasus (Zaheer et al., 2020) +576M +41.77 +42.33 +T5 (Raffel et al., 2019) +223M +37.90 +39.23 +LED (Beltagy et al., 2020) +161M +40.74 +41.54 +LED+Layout +165M +40.96 +41.83 +Pegasus +568M +39.07 +39.75 +Pegasus+Layout +572M +39.25 +39.85 +BigBird-Pegasus +576M +39.59 +41.09 +BigBird-Pegasus+Layout +581M +41.15 +42.05 +Table 3: ROUGE-L scores on arXiv-Lay and PubMed-Lay. Reported results obtained by Pegasus and BigBird- +Pegasus on the original arXiv and PubMed are reported with a gray background. The best results obtained on +arXiv-Lay and PubMed-Lay are denoted in bold. +Model +# Params +HAL +(fr) +SciELO-ES +(es) +SciELO-PT +(pt) +KoreaScience +(ko) +MBART +610M +42.00 +36.55 +36.42 +16.94 +MBART+Layout +615M +41.67 +37.47 +34.37 +14.98 +BigBird-MBART +617M +45.04 +37.76 +39.63 +18.55 +BigBird-MBART+Layout +621M +45.20 +40.71 +40.51 +19.95 +Table 4: ROUGE-L scores on the non-English datasets. The best results for each dataset are reported in bold. +Dataset +Train +Validation +Test +HAL (fr) +90.72 +90.54 +85.84 +SciELO-ES (es) +84.86 +84.28 +84.90 +SciELO-PT (pt) +90.95 +90.58 +91.96 +KoreaScience (ko) +73.53 +70.26 +68.78 +Table 5: Percent confidence obtained for the main lan- +guage, for each dataset split. +datasets contain less training data due to the inabil- +ity to process all original documents. Secondly, +the settings are different: while the original arXiv +and PubMed datasets contain clear discourse in- +formation (e.g., each section is delimited by mark- +ers) obtained from LATEX files, documents in our +extended versions are built by parsing raw PDF +files. Therefore, the task is more challenging for +text-only baselines, as they have no access to the +discourse structure of documents, which further +underlines the importance of taking the structural +information, brought by visual cues, into account. +Table 4 presents the ROUGE-L scores reported +on the non-English datasets. On HAL, we note +that BigBird-MBART does not benefit from lay- +out. After investigation, we hypothesize that this is +due to the larger presence of single-column and +simple layouts, which makes layout integration +less needed. On both SciELO datasets, we notice +that combining layout with long-range modeling +brings substantial improvements over MBART. Fur- +ther, we find that the plain-text BigBird models do +not improve over the layout-aware Pegasus and +MBART on arXiv-Lay and SciELO-ES, demon- +strating that simply capturing more context does +not always suffice. Regarding performance on Ko- +reaScience, we can see a significant drop in perfor- +mance for every model w.r.t the other non-English +datasets. At first glance, we notice a high amount +of English segments (e.g., tables, figure captions, +scientific concepts) in documents in KoreaScience. +To investigate this, we use the cld2 library21 to de- +tect the language in each non-English document. +We consider the percent confidence of the top-1 +matching language as an indicator of the presence +of the main language (i.e., French, Spanish, Por- +tuguese or Korean) in a document, and average +the results to obtain a score for the whole dataset. +Table 5 reports the average percent confidence ob- +tained on each split, for each dataset. We find +that the percentage of text written in the main lan- +guage in KoreaScience (i.e., Korean) is smaller +than in other datasets. As the MBART-based mod- +els expect only one language in a document (the +information is encoded using a special token), we +claim the strong presence of non-Korean segments +in KoreaScience causes them to suffer from inter- +ference problems. Therefore, we highlight that +KoreaScience is a more challenging dataset, and +21https://github.com/GregBowyer/cld2-cffi + +n < Q1 +Q1 ≤ n < Q2 +Q2 ≤ n < Q3 +Q3 ≤ n +0 +0.5 +1 +1.5 +2 +2.5 +Difference in ROUGE-L +(a) Article length +m < Q1 +Q1 ≤ m < Q2 +Q2 ≤ m < Q3 +Q3 ≤ m +0 +0.5 +1 +1.5 +2 +Difference in ROUGE-L +(b) Summary length +σ < Q1 +Q1 ≤ σ < Q2 +Q2 ≤ σ < Q3 +Q3 ≤ σ +0 +0.5 +1 +1.5 +2 +2.5 +Difference in ROUGE-L +(c) σ of bounding box height +Figure 1: Benefit of using layout on arXiv-Lay (blue) and PubMed-Lay (red), defined as the difference in ROUGE- +L scores between BigBird-Pegasus+Layout and BigBird-Pegasus. For each dataset, quartiles are calculated from +the distributions of article lengths (a), summary lengths (b) and variance in the height of the bounding boxes (c). +ROUGE-L scores are then computed per quartile range, and averaged over each range. +we hope our work will boost research on better +long-range, multimodal and multilingual models. +Overall, results show a clear benefit of integrat- +ing layout information for long document summa- +rization. +5.2 +Human Evaluation +Metric +BigBird +BigBird+Layout +Precision % +35.15 (0.81) +37.51 (0.70) +Recall % +28.07 (0.73) +33.59 (0.86) +Coherence +3.80 (0.38) +3.75 (0.62) +Fluency +4.48 (0.03) +4.34 (0.16) +Overlap % +8.77 (0.24) +7.49 (0.36) +Flow % +30.75 (0.68) +33.02 (0.71) +Table 6: Average human judgement scores obtained by +comparing gold-truth abstracts and summaries gener- +ated by BigBird and BigBird+Layout from 50 docu- +ments sampled from arXiv-Lay and HAL. Inter-rater +agreement is computed using Krippendorff’s alpha co- +efficient, and enclosed between parentheses. +To gain more insight into the effect of docu- +ment layout for summarizing long textual content, +we conduct a human evaluation of summaries gen- +erated by BigBird-Pegasus/BigBird-MBART and +their layout-aware counterparts. We choose the +BigBird-based models over the LED ones, as the +gain in augmenting BigBird with layout is much +more apparent. We evenly sample 50 documents +from arXiv-Lay and HAL test sets, filtering docu- +ments by their topics (computer science) to match +the judgment capabilities of the three human an- +notators. We design an evaluation interface (see +Section C.2 in the appendix). For each sentence si +in the generated summary, we ask the annotators +to highlight the relevant tokens in si, along with +the equivalent parts in the ground-truth abstract (de- +noted hi). Further, we ask them to rate the summary +in terms of coherence and fluency, on a scale of 0 +to 5, following the DUC quality guidelines (Dang, +2005). Finally, annotators are asked to penalize +summaries with hallucinated facts. The highlight- +ing process allows us to compute precision and +recall as the percentage of highlighted information +in the generated summary and the ground-truth ab- +stract, respectively. Moreover, we can compute an +overlap ratio as the percentage of highlighted infor- +mation that appears several times in the generated +summary. Lastly, we calculate a flow percentage +that evaluates how well the order of the ground- +truth information is preserved by computing the +percentage of times where the highlighted text hi +in the gold summary for one generated sentence +si follows the highlighted text hi−1 for the previ- +ous sentence si−1 (i.e. where any token from hi +occurs after a token in hi−1). Table 6 reports the +scores for each metric and model, averaged over all +50 documents, along with inter-rater agreements, +computed using Krippendorff’s alpha coefficient. +We find that adding layout to the models signifi- +cantly improves precision and recall, results in less +overlap (repetition), and is more in line with the +ground truth order. Further, annotators did not en- +counter any hallucinated fact in the 50 generated +summaries. To conclude, reported results show that +human annotators strongly agree that adding lay- +out generates better summaries, further validating +our claim that layout provides vital information for +summarization tasks. +5.3 +Case Studies +To have a better understanding of the previous re- +sults, we focus on uncovering the cases in which +layout is most helpful. To this end, we identify fea- + +tures that relate to the necessity of having layout: 1) +article length, as longer texts are intuitively easier +to understand with layout, 2) summary length, as +longer summaries are likely to cover more salient +information, and 3) variance in font sizes (using +the height of the bounding boxes), and, as such, +the complexity of the layout. The benefit of using +layout is measured as the difference in ROUGE- +L scores between BigBird-Pegasus+Layout and +its purely textual counterpart, on arXiv-Lay and +PubMed-Lay. We compute quartiles from the dis- +tributions of article lengths, ground-truth summary +lengths, and variance in the height of bounding +boxes.22 Based on the aforementioned factors, the +scores obtained by each model are then grouped +by quartile range, and averaged over each range, +see Figure 1. On arXiv-Lay, we find that layout +brings most improvement when dealing with the +25% longest documents and summaries, while, for +both datasets, layout is least beneficial for the short- +est documents and summaries. These results cor- +roborate our claim that layout can bring important +information about long-range context. Concerning +the third factor, we see, on PubMed-Lay, that layout +is most helpful for documents that have the widest +ranges of font sizes, showcasing the advantage of +using layout to capture salient information. +6 +Limitations and Risks +The proposed corpus is limited to a single domain, +that of scientific literature. Such limitation arguably +extends also to the layout diversity of documents. +In terms of risks, we acknowledge the presence +of Personally Identifiable Information such as au- +thor names and affiliations; nonetheless, such infor- +mation is already voluntarily made public by the +authors themselves. +7 +Conclusion +We have presented LoRaLay, a set of large-scale +datasets for long-range and layout-aware text sum- +marization. LoRaLay provides the research com- +munity with 4 novel multimodal corpora cover- +ing French, Spanish, Portuguese, and Korean lan- +guages, built from scientific articles. Furthermore, +it includes additional layout and visual informa- +tion for existing long-range summarization datasets +(arXiv and PubMed). We provide adapted architec- +tures merging layout-aware and long-range models, +22The quartiles are provided in Appendix C.3. +and show the importance of layout information in +capturing long-range dependencies. +8 +Acknowledgements +We thank the reviewers for their insightful com- +ments. This work is supported by the Associa- +tion Nationale de la Recherche et de la Technolo- +gie (ANRT) under CIFRE grant N2020/0916. It +was partially performed using HPC resources from +GENCI-IDRIS (Grant 2021-AD011011841). +References +Srikar Appalaraju, Bhavan Jasani, Bhargava Urala +Kota, Yusheng Xie, and R Manmatha. 2021. Doc- +former: End-to-end transformer for document under- +standing. arXiv preprint arXiv:2106.11539. +Iz Beltagy, Matthew E Peters, and Arman Cohan. +2020. Longformer: The long-document transformer. +arXiv preprint arXiv:2004.05150. +Łukasz Borchmann, Michał Pietruszka, Tomasz Stanis- +lawek, Dawid Jurkiewicz, Michał Turski, Karolina +Szyndler, and Filip Grali´nski. 2021. 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PMLR. + +LoRaLay: A Multilingual and +Multimodal Dataset for Long +Range and Layout-Aware +Summarization – Appendix +A +Datasets Construction +w1 bbox1 +w2 bbox2 +w3 bbox3 +w4 bbox4 +w5 bbox5 +… +w1 bbox1 +w2 bbox2 +w3 bbox3 +w4 bbox4 +w5 bbox5 +… +Data Repository +w1 bbox1 +w2 bbox2 +w3 bbox3 +w4 bbox4 +w5 bbox5 +… +w1 bbox1 +w2 bbox2 +w3 bbox3 +w4 bbox4 +w5 bbox5 +… +w1 bbox1 +w2 bbox2 +w3 bbox3 +w4 bbox4 +w5 bbox5 +… +w1 bbox1 +w2 bbox2 +w3 bbox3 +w4 bbox4 +w5 bbox5 +… +(1) PDF Extraction +(2) Filtering +(3) Text Extraction +(4) Abstract Removal +Figure 2: Dataset Construction Process. +A.1 +Extended Datasets – Lost Documents +Figure 3 provides details on the amount of original +documents lost in the process of augmenting arXiv +and PubMed with layout/visual information. We +observe four types of failures, and provide numbers +for each type: +• The link to the document’s PDF file is not +provided (Unavailable PDF); +• The PDF file is corrupted (i.e., cannot be +opened) (Corrupted PDF); +• The document is not digital-born, making it +impossible to parse it with PDF parsing tools +( Scanned PDF); +• The document’s abstract cannot be found in +the PDF (Irretrievable Abstract). +Figure 3: Distribution of failure types in arXiv-Lay +(top) and PubMed-Lay (bottom). +A.2 +KoreaScience – Extraction Rule +Korean documents in KoreaScience are extracted +by restricting search results to documents contain- +ing the word "Korean" in the publisher’s name. We +show that this rule does not bias the sample to- +wards a specific research area. We compute the +distribution of topics covered by all publishers, and +compare it to the distribution of topics covered by +publishers whose name contains the word Korean. +Figure 4 shows that the distribution obtained using +our rule remains roughly the same as the original. +Nature +Life +Artificial +Human +Society +Human Science and Technology +0 +10 +20 +30 +40 +Publishers with `Korean` in name +All publishers +Figure 4: Distribution of topics covered by all publish- +ers (red) vs distribution of topics covered by publishers +whose name contains the word Korean (blue). +A.3 +Samples +We provide samples of documents from each +dataset in Figure 5. + +e +>>>>>>>>>> +(P2) +Figure 1: A sketch of the deep-inelastic electron-photon scattering process +process the structure of the quasi-real photon, , radiated off an electron from one beam is +probed by the virtual photon, *. The * is radiated off an electron from the other beam such +that this electron is deflected into the detector. +The detailed formalism for the scattering of photons of arbitrary virtualities can be found +in [1]. For deep-inelastic electron-photon scattering on quasi-real photons the equation reduces +to the well known formula: +doe→ex = 2 +2元Q2 +Q2 +ddQ2 +zM +O+d +The absolute values of the four momentum squared of the virtual and quasi-real photons are +denoted Q2 and p2, with p2 < Q?. The symbols α and y denote the usual dimensionless +variables of deep-inelastic scattering, W denotes the invariant mass of the final state excluding +the electrons, and α is the fine structure constant. The fux of the incoming photons, f(z, P2), +where z is the fraction of the electron energy carried by the photon, is usually taken from the +equivalent photon approximation,EPA.At leading order, the structure function F2(c,Q2)is +proportional to the parton content, fa/, of the photon, and therefore reveals the structure of +the photon. In the region of small y studied, y < l, the contribution of the term containing +FL(, Q?) is small, and is usually neglected. +2.1 +QED structure +photon scattering events in which a pair of muons is produced by the * system. Figure 2 +shows the present world data on this measurement. An update is expected when the ongoing +L3 analysis [4] is finalized. The data span a range of about two orders of magnitude in Q? and +have a precision down to about 5%. With this precision, the treatment of the small but non- +zero virtuality of the quasi-real photon is important, as are electroweak radiative corrections +is different for the various experiments, see [1] for details. +In addition to the measurements of F2,QE further structure functions [5] have been obtained +by analyzing the azimuthal correlation between the scattering plane of the deep inelastically +scattered electron and the plane spanned by the muon pair. Good agreement between data +and predictions has been found. Also the scattering of two highly virtual photons has been +2 +PHOTON09SSN1225-0740 +Journal of the Korean Regional Science Association Vol.37, No.2(2021) +https://doi.org/10.22669/krsa.2021.37.2.063 +* +**·***，**** +Changes in Spatial Distribution of Manufacturing Startup +Activities in the Capital Region, Korea: +A Spatial Markov Chain Approach* +Changhyun Song* . Soonbeom Ahn** . Up Lim*** +Abstract:This study aims to explore how manufacturing start-up activities from 2000 to 2018 have changed spatially +and to predict changes in distribution patterns of future start-up activities. For the analysis, the Census on Establishments +microdata from 2000 to 2018 were used, and the manufacturing industry was classified into four detailed industrial +mailchanghyunsongyonsei.ac.kr) +***mail：uplimyonsei.ac.kr)HAD +ISSN-L:2530-5115 +DOl:http://doi.org/10.22585/hospdomic.v5i4.148 +Tendencias temporales de los patrones de busqueda +sobre Servicios de Atencion de Salud a Domicilio +antes y después del COVID-19 +Temporal trends in Home Care Services search patterns +before and after COVID-19 +RubénPalomo-Llinaresl0000-0002-1890-4337 +JuliaSanchez-Tormo20000-0001-9341-8737 +BenjaminPalomo-Llinares30000-0002-3892-3551 +1.Universidad Miguel Hernandez,Departamento de Salud Publica e Historia de la Ciencia,Sant Joan d'Alacant,Alicante +Espana +2.Centro Internacional Virtual de Investigacion en Nutricion (CiViIN),Alicante, Espana. +3.Universitat Miguel Hernandez d'Elx,Elche,Espana +Correspondencia/Correspondence +Financiacion/Funding +Ruben Palomo-Llinares +Este trabajo no ha recibido ninguna financiacion. +palomo.rub@gmail.com +Contribuciones de autoria/Author contributions +Recibido/Received +Todos los autores han contribuido por igual en la realiczacion +29.09.2021 +de este trabajo. +Aceptado/Accepted +12.10.2021 +Conflicto de Intereses/Competing interest +Los autores no presentan conflicto de intereses +COMO CITAR ESTE TRABAJO I HOW TO CITE THIS PAPER +Palomo-Llinares R, Sanchez-Tormo J,Palomo-Llinares B.Tendenc +as temporales de los patrones de busqueda sobre +Servicios de Atencion de Salud a Domicilio antes y despues del COVID-19. Hosp Domic. 2021;5(4):187-95. +Hosp Domic. 2021;5(4):187-195 +1870.111% +IrretrievableAbstracts +UnavailablePDFs +99.9%UnavailablePDFs +IrretrievableAbstracts +Corrupted PDFs +40.8% +Scanned PDFs +49.7% +9% +0.521%A.4 +Datasets Statistics +The distribution of research areas in arXiv-Lay and +HAL are provided in Figures 6 and 7, respectively. +Such distributions are not available for the other +datasets, as we did not have access to topic infor- +mation during extraction. +Figure 6: Distribution of research areas in arXiv-Lay. +Figure 7: Distribution of research areas in HAL. +B +Experiments +B.1 +Implementation Details +Models were implemented in Python using Py- +Torch (Paszke et al., 2017) and Hugging Face (Wolf +et al., 2019) librairies. In all experiments, we use +Adafactor (Shazeer and Stern, 2018), a stochastic +optimization method based on Adam (Kingma and +Ba, 2014) that reduces memory usage while retain- +ing the empirical benefits of adaptivity. We set +a learning rate warmup over the first 10% steps – +except on arXiv-Lay where it is set to 10k consis- +tently with Zaheer et al. (2020), and use a square +root decay of the learning rate. All our experiments +have been run on four Nvidia V100 with 32GB +each. +C +Results +C.1 +Detailed Results +Model +R-1 +R-2 +R-L +MBART +47.05 +22.23 +42.00 +MBART+Layout +46.65 +21.96 +41.67 +BigBird-MBART +49.85 +25.71 +45.04 +BigBird-MBART+Layout +49.99 +25.20 +45.20 +Table 8: ROUGE scores on HAL. Best results are re- +ported in bold. +Model +R-1 +R-2 +R-L +MBART +17.33 +7.70 +16.94 +MBART+Layout +15.43 +6.69 +14.98 +BigBird-MBART +18.96 +8.01 +18.55 +BigBird-MBART+Layout +20.36 +9.49 +19.95 +Table 10: ROUGE scores on KoreaScience. The best +results are reported in bold. +C.2 +Human Evaluation +Using the Streamlit23 framework, we design and +develop an interface to aid human evaluation of +summarization models.24 +23https://streamlit.io/ +24The +code +is +publicly +available +at +https://anonymous.4open.science/r/ +loralay-eval-interface-C20D. + +CondensedMatter +Astrophysics +Physics +26.6% +27.8% +Mathematics +QuantumPhysics +ComputerScience +2.03% +High Energy Physics - Phenomenology +10.3% +-2.28% +NuclearTheory +6.53% +-2.53% +.58% +9.28% +General Relativity and Quantum Cosmology +NonlinearSciences +6HumanitiesandSocialSciences +17.2% +Physics +ComputerScience +10.9% +EngineeringSciences +47.5% +LifeSciences +EnvironmentalSciences +9.73% +SciencesoftheUniverse +CognitiveScience +Mathematics +3.23% +0.013% +1.45% +0.117% +0.182% +L0.219% +L0.312% +0.939% +L1.1%Model +arXiv / arXiv-Lay +PubMed / PubMed-Lay +R-1 +R-2 +R-L +R-1 +R-2 +R-L +Pegasus (Zhang et al., 2020) +44.21 +16.95 +38.83 +45.97 +20.15 +41.34 +BigBird-Pegasus (Zaheer et al., 2020) +46.63 +19.02 +41.77 +46.32 +20.65 +42.33 +T5 (Raffel et al., 2019) +42.79 +15.98 +37.90 +42.88 +17.58 +39.23 +LED (Beltagy et al., 2020) +45.41 +18.14 +40.74 +45.28 +19.86 +41.54 +LED+Layout +45.51 +18.55 +40.96 +45.41 +19.74 +41.83 +MBART +37.64 +13.29 +33.49 +41.19 +16.04 +37.47 +Pegasus +43.81 +17.27 +39.07 +43.52 +17.96 +39.75 +Pegasus+Layout +44.10 +17.01 +39.25 +43.59 +18.24 +39.85 +BigBird-Pegasus +44.43 +17.74 +39.59 +44.80 +19.32 +41.09 +BigBird-Pegasus+Layout +46.02 +18.95 +41.15 +45.69 +20.38 +42.05 +Table 7: ROUGE scores on arXiv-Lay and PubMed-Lay. Reported results obtained by Pegasus and BigBird- +Pegasus on the original arXiv and PubMed are reported with a gray background. The best results obtained on +arXiv-Lay and PubMed-Lay are denoted in bold. +Model +SciELO-ES +SciELO-PT +R-1 +R-2 +R-L +R-1 +R-2 +R-L +MBART +41.04 +15.65 +36.55 +41.18 +15.53 +36.42 +MBART+Layout +42.27 +15.73 +37.47 +39.45 +14.17 +34.37 +BigBird-MBART +42.64 +16.60 +37.76 +44.85 +18.70 +39.63 +BigBird-MBART+Layout +45.64 +19.33 +40.71 +45.47 +20.40 +40.51 +Table 9: ROUGE scores on the SciELO datasets. The best results are reported in bold. +Figure 8: LoRaLay evaluation interface. +C.3 +Analysis of the Impact of Layout +Table 12 lists the quartiles computed from the dis- +tributions of article lengths, summary lengths, and +variation in the height of bounding boxes, for arXiv- +Lay and PubMed-Lay. + +LoRaLayEvaluationInterface +Evaluation guidelines +Document0806.3537(1/50) +Statistical LearningofArbitraryComputableClassifiers +Linktofulldocument +Model A +ModelB +Ground-truthabstract +Statistical learningtheory chiefly studiesrestricted hypothesis classes,particularlythosewith +finiteVapnik-Chervonenkis(VC)dimension.Thefundamentalquantityofinterest isthesample +complexity:thenumberofsamplesrequiredto learntoaspecified level ofaccuracy.Herewe +considerlearningoverthesetofallcomputablelabelingfunctions.SincetheVC-dimensionis +infiniteandapriori (uniform)boundsonthenumberofsamplesare impossible,weletthe +learningalgorithmdecidewhenithasseensuficientsamplestohavelearned.Wefirstshowthat +learning inthis setting is indeedpossible,and developalearningalgorithm.Wethenshow, +however,thatboundingsamplecomplexity independentlyof thedistribution isimpossible.Notably +this impossibilityis entirelydueto therequirementthat the learningalgorithmbecomputable,and +notdue to thestatistical nature of theproblem. +You selectedthefollowingsentencegeneratedbyModel B.Highlightthepartsinthesentencethat +canbefoundintheground-truthabstract. +Conventionalstatisticallearningtheoryattemptstoboundthenumberof +samplesneededtolearntoaspecifiedlevelofaccuracyforeachoftheabove +models (e.g.neural networks,supportvectormachines) +Next sentenceSummarygeneratedbyModelB +Sentence +Precision (%) +V +Conventional statistical learningtheoryattempts toboundthenumber +40.62 +ofsamplesneededtolearntoaspecifiedlevelofaccuracyforeachof +the abovemodels (e.g.neural networks, support vectormachines) +0 +However,ifweallowourselvestochangethemodel, thentheC +16.67 +dimensionoftheoverall learningalgorithm isnotfinite,andmuchof +statistical learningtheorydoesnotdirectlyapply +0 +In contrast, weprovethatdistribution-independentboundsdonotexist +72.22 +altogetherforcomputable learningalgorithms in oursetting. +Ourresults imply thatcomputable learningalgorithms in theuniversal +0.0 +settingmust"wastesamples"inthesenseofrequiringmoresamples +thanisnecessaryforstatisticalreasonsalone +Recall(%) +30.15 +Coherence +00 +01 +02 +O3 +. +4 +O5 +Fluency +00 +01 +O2 +O3 +O4 +lamunabletoevaluatethisdocument. +NextLED +LED+Layout +Pegasus +Pegasus+Layout +BigBird-Pegasus +BigBird-Pegasus+Layout +T5 +2.84 / 2.31 +3.06 / 2.60 +1.17 / 0.52 +1.35 / 0.62 +1.69 / 1.86 +3.25 / 2.82 +LED +– +0.22 / 0.29 +1.67 / 1.79 +1.49 / 1.69 +1.15 / 0.45 +0.41 / 0.51 +LED+Layout +– +– +1.89 / 2.08 +1.71 / 1.98 +1.38 / 0.74 +0.19 / 0.22 +Pegasus +– +– +– +0.34 / 0.10 +0.52 / 1.34 +2.08 / 2.30 +Pegasus+Layout +– +– +– +– +0.34 / 1.24 +1.90 / 2.20 +BigBird-Pegasus +– +– +– +– +– +1.56 / 0.96 +Table 11: Absolute ROUGE-L score differences between each pair of models, on arXiv-Lay/PubMed-Lay. +Distribution +Q1 +Q2 +Q3 +arXiv-Lay +PubMed-Lay +arXiv-Lay +PubMed-Lay +arXiv-Lay +PubMed-Lay +Article Length +6,226 +3,513 +9,142 +5,557 +13,190 +8,036 +Summary Length +119 +130 +159 +182 +202 +247 +σ of bounding box height +3.37 +1.34 +3.98 +1.73 +4.70 +2.28 +Table 12: Quartiles calculated from the distributions of article lengths, summary lengths, and variation in the height +of bounding boxes, for arXiv-Lay and PubMed-Lay. + +arXiv:0907.2782v1 [hep-ex] 16 Jul 2009 +Experimental Review of Photon Structure Func- +tion Data +Richard Nisius +Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut), F¨ohringer Ring 6, D-80805 M¨un- +chen, Germany, E-mail: Richard.Nisius@mpp.mpg.de∗ +DOI: will be assigned +MPP-2009-131 +The present knowledge of the structure of the photon is presented based on results obtained +by measurements of photon structure functions at e+e− collider. Results are presented both +for the QED structure of the photon as well as for the hadronic structure, where the data +are also compared to recent parametrisations of the hadronic structure function F γ +2 (x, Q2). +Prospects of future photon structure function measurements, especially at an International +Linear Collider are outlined. +1 +Introduction +The measurements of photon structure functions have a long tradition since the first of such +measurements was performed by the PLUTO Collaboration in 1981. The investigations concern +the QED structure of the photon as well as the hadronic structure. For the hadronic structure +function F γ +2 (x, Q2) the main areas of interest are the behavior at low values of x and the +evolution with the momentum scale Q2, which is predicted by QCD to be logarithmic. The +experimental information is dominated by the results from the four LEP experiments. +This review is based on earlier work [1, 2] and as an extension provides a number of updated +figures, together with a comparison of the experimental data with new parametrisations of +F γ +2 (x, Q2) that became available since then. Only results on the structure of quasi-real photons +are discussed here. The structure of virtual photons and the corresponding measurements of +effective structure functions are detailed in [3]. +2 +Structure function measurements +The photon can fluctuate into a fermion–anti-fermion state consistent with the quantum num- +bers of the photon and within the limitations set by the Heisenberg uncertainty principle. These +fluctuations are favored, i.e. have the longest lifetimes, for high energetic photons of low virtu- +ality. If such a fluctuation of the photon is probed, the photon reveals its structure. Using this +feature, measurements of photon structure functions are obtained from the differential cross- +section of the deep-inelastic electron-photon scattering1 process sketched in Figure 1. In this +∗Invited talk presented at the Photon09 Conference in Hamburg on May 12, 2009. +1In this paper, the term electron encompasses positrons throughout. +PHOTON09 +1 +(a) arXiv-Lay +© 2004 Hindawi Publishing Corporation +Journal of Biomedicine and Biotechnology • 2004:5 (2004) 306–313 • PII. S111072430440401X • http://jbb.hindawi.com +MINIREVIEW ARTICLE +Anthocyanins and Human Health: An In Vitro +Investigative Approach +Mary Ann Lila∗ +Department of Natural Resources & Environmental Sciences, College of Agricultural Consumer and Environmental Sciences, +University of Illinois, Urbana, IL 61801, USA +Received 2 April 2004; revised 10 May 2004; accepted 12 May 2004 +Anthocyanin pigments and associated flavonoids have demonstrated ability to protect against a myriad of human diseases, yet they +have been notoriously difficult to study with regard to human health. Anthocyanins frequently interact with other phytochemicals +to potentiate biological effects, thus contributions from individual components are difficult to decipher. The complex, multicompo- +nent structure of compounds in a bioactive mixture and the degradation of flavonoids during harsh extraction procedures obscure +the precise assignment of bioactivity to individual pigments. Extensive metabolic breakdown after ingestion complicates tracking of +anthocyanins to assess absorption, bioavailability, and accumulation in various organs. Anthocyanin pigments and other flavonoids +that are uniformly, predictably produced in rigorously controlled plant cell culture systems can be a great advantage for health and +nutrition research because they are quickly, easily isolated, lack interferences found in whole fruits, can be elicited to provoke rapid +and prolific accumulation, and are amenable to biolabeling so that metabolic fate can be investigated after ingestion. +ANTHOCYANINS AND BIOMEDICINAL PROPERTIES +Anthocyanins are members of the flavonoid group +of phytochemicals, a group predominant in teas, honey, +wines, fruits, vegetables, nuts, olive oil, cocoa, and cereals. +The flavonoids, perhaps the most important single group +of phenolics in foods, comprise a group of over 4000 +C15 aromatic plant compounds with multiple substitution +patterns (www.nal.usda.gov/fnic/foodcomp/index.html). +The primary players in this group include the an- +thocyanins (eg, cyanidin, pelargonidin, petunidin), the +flavonols (quercetin, kaempferol), flavones (luteolin, +apigenin), flavanones (myricetin, naringin, hesperetin, +naringenin), flavan-3-ols (catechin, epicatechin, gallocat- +echin), and, although sometimes classified separately, the +isoflavones (genistein, daidzein). Phytochemicals in this +class are frequently referred to as bioflavonoids due to +their multifaceted roles in human health maintenance, +and anthocyanins in food are typically ingested as com- +ponents of complex mixtures of flavonoid components. +Daily intake is estimated from 500 mg to 1 g, but can be +several g/d if an individual is consuming flavonoid supple- +ments (grape seed extract, ginkgo biloba, or pycnogenol; +see, eg, [1]). +The colorful anthocyanins are the most recognized, +visible members of the bioflavonoid phytochemicals. The +free-radical scavenging and antioxidant capacities of an- +thocyanin pigments are the most highly publicized of the +modus operandi used by these pigments to intervene with +human therapeutic targets, but, in fact, research clearly +suggests that other mechanisms of action are also respon- +sible for observed health benefits [2, 3, 4, 5]. Anthocyanin +isolates and anthocyanin-rich mixtures of bioflavonoids +may provide protection from DNA cleavage, estrogenic +activity (altering development of hormone-dependent +disease symptoms), enzyme inhibition, boosting produc- +tion of cytokines (thus regulating immune responses), +anti-inflammatory activity, lipid peroxidation, decreas- +ing capillary permeability and fragility, and membrane +strengthening [6, 7, 8, 9, 10]. The chemical structure (po- +sition, number, and types of substitutions) of the indi- +vidual anthocyanin molecule also has a bearing on the +degree to which anthocyanins exert their bioactive prop- +erties [11, 12] and the structure/function relationships +also influence the intracellular localization of the pig- +ments [7]. The anthocyanin literature includes some con- +troversy over the relative contributions of glycosylated an- +thocyanins versus aglycones in terms of bioavailability +and bioactive potential [7, 13, 14, 15, 16]. Originally, it +was assumed that only aglycones could enter the circu- +lation circuit, however, absorption and metabolism of an- +thocyanin glycosides has now been demonstrated. The na- +ture of the sugar conjugate and the aglycone are important +determinants of anthocyanin absorption and excretion in +both humans and rats [15]. +The roles of anthocyanin pigments as medicinal +agents have been well-accepted dogma in folk medicine +throughout the world, and, in fact, these pigments are +linked to an amazingly broad-based range of health ben- +efits. For example, anthocyanins from Hibiscus sp have +(b) PubMed-Lay +1 + +Les représentations des enseignants de ZEP sur la relation école/famille à +travers le prisme des élèves en grande réussite scolaire + +Publié dans la revue Cahier E&D 2017 Cahier N° 28 Familles, Parents, Ecole +Lien vers le site Education & Devenir +Lien vers le site Les cahiers pédagogiques +Résumé +Les familles sont des partenaires essentiels de l’école. Pourtant, la relation école/famille est souvent décrite +comme problématique. Quelles représentations les enseignants ont de cette relation et de l’influence du +milieu familial sur la réussite de leurs élèves ? Nous avons réalisé une enquête nationale auprès de 1790 +professeurs des écoles (PE) en zone d’éducation prioritaire (ZEP) puis des entretiens avec dix d’entre eux. Le +prisme des élèves en grande réussite scolaire (EGRS) dans les ZEP a été choisi pour étudier la différence de +perceptions des enseignants en fonction de la réussite de l’élève. Les PE décrivent le profil idéal des parents +d’élèves. Ils souhaitent davantage d’implication de la part des familles et voudraient mettre en place une +réelle coéducation qu’ils jugent indispensable à la réussite des élèves. +Mots clefs +Représentations – enseignants – coéducation – grande réussite scolaire - éducation prioritaire +Caroline HACHE – ADEF – AMU +(caroline.hache@univ-amu.fr) + +Introduction +Lorsqu’ils étudient la proportion d’élèves de milieu populaire ayant obtenu le baccalauréat général sans +redoubler, Ould Ferhat et Terrail (2005) indique qu’un désir fort de la part des parents peut faire la différence +entre les élèves qui réussissent et ceux qui échouent. On retrouve dans la littérature (Lorcerie, 2015) une +catégorisation des conduites des élèves lorsqu’ils font face aux apprentissages en fonction de l’attitude de +leurs parents. Les textes officielsi encouragent une relation positive école/famille, car la famille est +considérée comme un partenaire de l’école avec une place importante dans la scolarité de l’élève (Houssaye, +2001). Que pensent les enseignants de ces déclarations ? Quelles sont les représentations des enseignants +concernant l’influence des familles populaires sur la réussite scolaire de leur enfant ? +Notre étude se propose, dans une première enquête, d’interroger par questionnaire 1790 enseignants +d’école élémentaire, toutes en ZEPii, autour de leur quotidien dans les classes et, dans une deuxième +enquête, de réaliser des entretiens avec dix d’entre eux. Le sujet des parents d’élèves a pris une place +importante dans les entretiens de tous les enseignants, comme ceux interrogés par Moisan et Simon (1997, +p. 68) qui ont plus parlé « des parents que des élèves ». +Le choix du prisme des élèves en grande réussite scolaire (EGRS) et donc ici, des parents de ceux-ci, a été pris +pour étudier l’avis des enseignants sur un profil particulier, celui des familles dont les élèves réussissent +(Hache, 2016) alors que l’on s’attendrait à ce qu’ils soient en difficulté scolaire. En effet, Charlot (2001, p. 7) +les appelle les « réussites paradoxales » car ils réussissent dans un milieu qualifié de défavorable pour la +réussite scolaire. Cela a permis aux enseignants de s’exprimer sur la différence ou l’absence de différence +entre les parents des EGRS et les autres. +(c) HAL +187 +Hosp Domic. 2021;5(4):187-195 +ISSN-L: 2530-5115 +DOI: http://doi.org/10.22585/hospdomic.v5i4.148 +Correspondencia/Correspondence +Rubén Palomo-Llinares +palomo.rub@gmail.com +Recibido/Received +29.09.2021 +Aceptado/Accepted +12.10.2021 +Conflicto de Intereses/Competing interest +Los autores no presentan conflicto de intereses. +Financiación/Funding +Este trabajo no ha recibido ninguna financiación. +Contribuciones de autoría/Author contributions +Todos los autores han contribuido por igual en la realiczación +de este trabajo. +cómo citar este trabajo | how to cite this paper +Palomo-Llinares R, Sanchez-Tormo J, Palomo-Llinares B. Tendencias temporales de los patrones de búsqueda sobre +Servicios de Atención de Salud a Domicilio antes y después del COVID-19. Hosp Domic. 2021;5(4):187-95. +Tendencias temporales de los patrones de búsqueda +sobre Servicios de Atención de Salud a Domicilio +antes y después del COVID-19 +Temporal trends in Home Care Services search patterns +before and after COVID-19 +Rubén Palomo-Llinares1 ORCID 0000-0002-1890-4337 +Julia Sanchez-Tormo2 ORCID 0000-0001-9341-8737 +Benjamín Palomo-Llinares3 ORCID 0000-0002-3892-3551 +1. Universidad Miguel Hernández, Departamento de Salud Pública e Historia de la Ciencia, Sant Joan d´Alacant, Alicante, +España. +2. Centro Internacional Virtual de Investigación en Nutrición (CIVIN), Alicante, España. +3. Universitat Miguel Hernández d’Elx, Elche, España. +(d) SciELO-ES +1 +Gilberto da Silva Guizelin +Uma luz sobre as relações Brasil-Moçambique no oitocentos: +a Missão Consular de João Luiz Airoza (1827-1828) +rev. hist. (São Paulo), n.178, a03318, 2019 +http://dx.doi.org/10.11606/issn.2316-9141.rh.2019.144021 +ARTIGO +UMA LUZ SOBRE AS +RELAÇÕES BRASIL- +MOÇAMBIQUE NO +OITOCENTOS: A +MISSÃO CONSULAR +DE JOÃO LUIZ AIROZA +(1827-1828)* +Gilberto da Silva Guizelin** +Universidade de São Paulo +São Paulo – São Paulo – Brasil +Resumo +Após a assinatura da Convenção de 1826 com a Grã-Bretanha, pela qual o go- +verno de D. Pedro I concordou, em troca do reconhecimento britânico, coibir o +tráfico transatlântico de africanos para o Império a partir de 1830, foram criadas +representações consulares brasileiras na África Portuguesa com a explícita fina- +lidade de proteger a atuação de negreiros brasileiros nos últimos anos de legali- +dade do comércio de escravos sob a bandeira imperial. Neste sentido, o presente +artigo investiga a atuação de João Luiz Airoza, cônsul do Brasil em Moçambique, +entre 1827 e 1828, na defesa do circuito negro entre o Brasil e a África Oriental. +Para tanto, o texto aqui apresentado priorizou como fonte de estudo a documen- +tação consular produzida por Airoza e dirigida à antiga Secretaria de Estado dos +Negócios Estrangeiros. +Palavras-chave +Relações internacionais – Relações Brasil-Moçambique – Missão consular – +Tráfico de escravos – África Oriental. +Contato +Rua Belo Horizonte, 433, apto. 603 +86020-060 – Londrina – Paraná – Brasil +guizelin.gs@gmail.com +* Todas as obras e todos os documentos utilizados na pesquisa e na elaboração do artigo são +citados nas notas e na bibliografia. +** Doutor em História pela Faculdade de Ciências Humanas e Sociais de Franca, da Universidade +Estadual Paulista Júlio de Mesquita Filho (Unesp). Pós-doutorando em História pela Faculdade de +Filosofia, Letras e Ciências Humanas, da Universidade de São Paulo (FFLCH/USP). Bolsista pós- +doc processo nº 2018/07798-1, Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP). +(e) SciELO-PT +63 +수도권 제조업 창업 활동의 공간적 분포 변화 +- 공간 마르코프 체인의 응용 -* +송창현**·안순범***·임업**** +Changes in Spatial Distribution of Manufacturing Startup +Activities in the Capital Region, Korea: +A Spatial Markov Chain Approach* +Changhyun Song* · Soonbeom Ahn** · Up Lim*** +국문요약 본 연구는 2000년부터 2018년까지를 분석의 시간적 범위로 설정하여 제조업 창업 활동이 공간적으로 +어떠한 변화를 보여왔는지를 탐색적으로 분석하고, 향후 창업 활동의 분포 패턴 변화를 예측하는 것을 목적으로 한 +다. 분석을 위해 2000년부터 2018년까지의 「전국사업체조사」 마이크로데이터 제조업 사업체 자료를 활용하였다. +한국산업연구원의 ISTANS 분류체계에서 제시하는 40대 제조업 기준에 따라 제조업을 4개의 세부 산업군으로 구 +분한 후, 수도권 행정구역 읍면동 수준에서 공간자기상관 분석 및 공간 마르코프 체인 분석을 수행하였다. 분석 결 +과에 따르면, 고위기술산업군 및 중고위기술산업군의 창업 활동은 시간이 흐름에 따라 경기도 남부를 중심으로 집 +중되고 있는 것으로 나타났으며, 중저위기술산업군 및 저위기술산업군 창업 활동의 집중은 수도권 외곽으로 분산 +되고 있는 것으로 나타났다. 2000년부터 2018년까지의 추세를 연장하여 2036년까지의 분포 변화를 예측하였을 +때, 창업 활동이 활발히 발생하는 지역 및 그와 인접하고 있는 지역의 경우 향후 분위 상승의 가능성이 높은 것으로 +나타나 긍정적인 공간 효과가 존재하는 것으로 확인되었다. 본 연구는 일자리 창출의 주요 원천이 되는 제조업 창 +업 활동의 분포 패턴 변화를 동태적으로 분석함으로써 창업 육성 및 일자리 창출과 관련한 지역 정책에의 시사점을 +제공하고자 하였다. +주제어 제조업, 창업, 탐색적공간자료 분석, 공간마르코프 체인 +Abstract: This study aims to explore how manufacturing start-up activities from 2000 to 2018 have changed spatially +and to predict changes in distribution patterns of future start-up activities. For the analysis, the Census on Establishments +microdata from 2000 to 2018 were used, and the manufacturing industry was classified into four detailed industrial +* 이 논문은 2019년 대한민국 교육부와 한국연구재단의 인문사회분야 중견연구자지원사업의 지원을 받아 수행된 연구임(NRF-2019 +S1A5A2A01045590). 본 연구는 2020년 한국지역학회 후기학술대회에서 우수논문상을 수상한 연구임. +** 연세대학교 대학원 도시공학과 석박사통합과정(주저자, E-mail: changhyunsong@yonsei.ac.kr) +*** 연세대학교 대학원 도시공학과 석사과정(공동저자, E-mail: soonbeomahn@yonsei.ac.kr) +**** 연세대학교 도시공학과 교수(교신저자, E-mail: uplim@yonsei.ac.kr) +ISSN 1225-0740 +https://doi.org/10.22669/krsa.2021.37.2.063 +Journal of the Korean Regional Science Association Vol.37, No.2(2021) +한국지역학회지 『지역연구』 제37권 제2호 pp.63-82 +(f) KoreaScience +Figure 5: Samples from each dataset. + +cC +000cC +BY +NCDOSSIE +Mocambique em +perspectiva: +historias conectadas +interdisciplinaridade +enovos +sujeitos historicos +Matias Ntundo,2013,xilogravura (detalhe) \ No newline at end of file diff --git a/cNFIT4oBgHgl3EQfnCvP/content/tmp_files/load_file.txt b/cNFIT4oBgHgl3EQfnCvP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..40c0f5bbdae46a0249c748f7e204dbcb2655622e --- /dev/null +++ b/cNFIT4oBgHgl3EQfnCvP/content/tmp_files/load_file.txt @@ -0,0 +1,1193 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf,len=1192 +page_content='LoRaLay: A Multilingual and Multimodal Dataset for Long Range and Layout-Aware Summarization Laura Nguyen1,3 Thomas Scialom2∗ Benjamin Piwowarski3 Jacopo Staiano4∗ 1reciTAL, Paris, France 2Meta AI, Paris, France 3Sorbonne Université, CNRS, ISIR, F-75005 Paris, France 4University of Trento, Italy laura@recital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='ai tscialom@fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='com benjamin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='piwowarski@cnrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='fr jacopo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='staiano@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='it Abstract Text Summarization is a popular task and an active area of research for the Natural Lan- guage Processing community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' It requires ac- counting for long input texts, a characteristic which poses computational challenges for neu- ral models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Moreover, real-world documents come in a variety of complex, visually-rich, layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' This information is of great relevance, whether to highlight salient content or to en- code long-range interactions between textual passages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Yet, all publicly available summa- rization datasets only provide plain text con- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' To facilitate research on how to exploit vi- sual/layout information to better capture long- range dependencies in summarization models, we present LoRaLay, a collection of datasets for long-range summarization with accompa- nying visual/layout information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We extend existing and popular English datasets (arXiv and PubMed) with visual/layout information and propose four novel datasets – consistently built from scholar resources – covering French, Spanish, Portuguese, and Korean languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Further, we propose new baselines merging layout-aware and long-range models – two or- thogonal approaches – and obtain state-of-the- art results, showing the importance of combin- ing both lines of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 1 Introduction Deep learning techniques have enabled remarkable progress in Natural Language Processing (NLP) in recent years (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' However, the majority of models, benchmarks, and tasks have been de- signed for unimodal approaches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' focusing ex- clusively on a single source of information, namely plain text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' While it can be argued that for specific NLP tasks, such as textual entailment or machine translation, plain text is all that is needed, there exist several tasks for which disregarding the vi- sual appearance of text is clearly sub-optimal: in Work partially done while at reciTAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' a real-world context (business documentation, sci- entific articles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' ), text does not naturally come as a sequence of characters, but is rather displayed in a bi-dimensional space containing rich visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The layout of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' this very paper provides valuable semantics to the reader: in which section are we right now?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' At the blink of an eye, this information is readily accessible via the salient section title (formatted differently and placed to highlight its role) preceding these words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Just to emphasize this point, imagine having to scroll this content in plain text to access such information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' In the last couple of years, the research commu- nity has shown a growing interest in addressing these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Several approaches have been proposed to deal with visually-rich documents and integrate layout information into language mod- els, with direct applications to Document Under- standing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Joint multi-modal pretraining (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Powalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Appalaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2021) has been key to reach state-of-the-art per- formance on several benchmarks (Jaume et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Grali´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Nonetheless, a remaining limitation is that these (transformer-based) approaches are not suitable for processing long documents, the quadratic complex- ity of self-attention constraining their use to short sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Such models are hence unable to en- code global context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' long-range dependencies among text blocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Focusing on compressing the most relevant infor- mation from long texts to short summaries, the Text Summarization task naturally lends itself to benefit from such global context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Notice that, in practice, the limitations linked to sequence length are also amplified by the lack of visual/layout information in the existing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Therefore, in this work, we aim at spurring further research on how to in- corporate multimodal information to better capture long-range dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Our contributions can be summarized as follows: arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='11312v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='CL] 26 Jan 2023 We extend two popular datasets for long-range summarization, arXiv and PubMed (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2018), by including visual and layout information – thus allowing direct comparison with previous works;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We release 4 additional layout-aware summa- rization datasets (128K documents), covering French, Spanish, Portuguese, and Korean lan- guages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We provide baselines including adapted archi- tectures for multi-modal long-range summa- rization, and report results showing that (1) performance is far from being optimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' and (2) layout provides valuable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' All the datasets are available on HuggingFace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 Layout/Visually-rich Datasets Document Understanding covers problems that in- volve reading and interpreting visually-rich docu- ments (in contrast to plain texts), requiring com- prehending the conveyed multimodal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Hence, several tasks with a central layout aspect have been proposed by the document understand- ing community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Key Information Extraction tasks consist in extracting the values of a given set of keys, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', the total amount in a receipt or the date in a form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' In such tasks, documents have a layout structure that is crucial for their interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' No- table datasets include FUNSD (Jaume et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019) for form understanding in scanned documents, and SROIE (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019), as well as CORD (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019), for information extraction from receipts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Grali´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2020) elicit progress on deeper and more complex Key Information Extrac- tion by introducing the Kleister datasets, a collec- tion of business documents with varying lengths, released as PDF files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' However, the documents in Kleister often contain single-column layouts, which are simpler than the various multi-column layouts considered in LoRaLay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Document VQA is another popular document understanding task that requires processing multimodal information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', text, layout, font style, images) conveyed by a document to be able to answer questions about a 1https://hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='co/datasets/nglaura/arxivlay-summarization, https://hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='co/datasets/nglaura/pubmedlay-summarization, https://hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='co/datasets/nglaura/hal-summarization, https://hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='co/datasets/nglaura/scielo-summarization, https://hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='co/datasets/nglaura/koreascience-summarization visually rich document (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', What is the date given at the top left of the form?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', Whose picture is given in this figure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The DocVQA dataset (Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2021) and InfographicsVQA (Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2022) are commonly-used VQA datasets that re- spectively provide industry documents and info- graphic images, encouraging research on under- standing documents with complex interplay of text, layout and graphical elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Finally, to foster research on visually-rich document understanding, Borchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2021) introduce the Document Understanding Evaluation (DUE) benchmark, a unified benchmark for end-to-end document under- standing, created by combining several datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' DUE includes several available and transformed datasets for VQA, Key Information Extraction and Machine Reading Comprehension tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2 Existing Summarization Datasets Several large-scale summarization datasets have been proposed to boost research on text summa- rization systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Hermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2015) proposed the CNN/DailyMail dataset, a collection of English articles extracted from the CNN and The Daily Mail portals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Each news article is associated with multi-sentence highlights which serve as reference summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Scialom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2020) bridge the gap be- tween English and non-English resources for text summarization by introducing MLSum, a large- scale multilingual summarization corpus providing news articles written in French, German, Spanish, Turkish and Russian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Going toward more challeng- ing scenarios involving significantly longer doc- uments, the arXiv and PubMed datasets (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2018) consist of scientific articles collected from academic repositories, wherein the paper ab- stracts are used as summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' To encourage a shift towards building more abstractive summarization models with global content understanding, Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2019) introduce BIGPATENT, a large-scale dataset made of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' patent filings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Here, invention descriptions serve as reference summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The vast majority of summarization datasets only deal with plain text documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' As opposed to other Document Understanding tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', form understanding, visual QA) in which the placement of text on the page and/or visual components are the main source of information needed to find the desired data (Borchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2021), text plays a predominant role in document summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' However, guidelines for summarizing texts – espe- cially long ones – often recommend roughly pre- viewing them to break them down into their major sections (Toprak and Almacio˘glu, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' This suggests that NLP systems might lever- age multimodal information in documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Miculi- cich and Han (2022) propose a two-stage method which detects text segments and incorporates this information in an extractive summarization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Cao and Wang (2022) collect a new dataset for long and structure-aware document summarization, consisting of 21k documents written in English and extracted from WikiProject Biography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Although not all documents are explicitly or- ganized into clearly defined sections, the great majority contains layout and visual clues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', a physical organization into paragraphs, bigger head- ings/subheadings) which help structure their textual contents and facilitate reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Thus, we argue that layout is crucial to summarize long documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We propose a corpus of more than 345K long docu- ments with layout information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Furthermore, to address the need for multilingual training data (Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020), we include not only English docu- ments, but also French, Spanish, Portuguese and Korean ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 3 Datasets Construction Inspired by the way the arXiv and PubMed datasets were built (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2018), we construct our corpus from research papers, with abstracts as ground-truth summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' As the PDF format allows simultaneous access to textual, visual and layout information, we collect PDF files to construct our datasets, and provide their URLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2 For each language, we select a repository that contains a high number of academic articles (in the order of hundreds of thousands) and provides easy access to abstracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' More precisely, we chose the following repositories: Archives Ouverte HAL (French),3 an open archive of scholarly documents from all aca- demic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' As HAL is primarily directed towards French academics, a great proportion of articles are written in French;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' SciELO (Spanish and Portuguese),4 an open access database of academic articles published in journal collections from Latin America, 2We make the corpus-construction code publicly available at https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='com/recitalAI/loralay-datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 3https://hal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='archives-ouvertes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='fr/ 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='scielo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='org/ Iberian Peninsula and South Africa, and cov- ering a broad range of topics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' agricultural sciences, engineering, health sciences, letters and arts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Languages include English, Span- ish, and Portuguese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' KoreaScience (Korean),5 an open archive of Korean scholarly publications in the fields of natural sciences, life sciences, engineering, and humanities and social sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Articles are written in English or Korean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Further, we provide enhanced versions of the arXiv and PubMed datasets, respectively denoted as arXiv-Lay and PubMed-Lay, for which layout information is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 Collecting the Data Extended Datasets The arXiv and PubMed datasets (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2018) contain long scien- tific research papers extracted from the arXiv and PubMed repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We augment them by provid- ing their PDFs, allowing access to layout and visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' As the abstracts contained in the orig- inal datasets are all lowercased, we do not reuse them, but rather extract the raw abstracts using the corresponding APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Note that we were unable to retrieve all the orig- inal documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For the most part, we failed to retrieve the corresponding abstracts, as they did not necessarily match the ones contained in the PDF files (due to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' PDF-parsing errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We also found that some PDF files were unavailable, while others were corrupted or scanned documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='6 In total, about 39% (35%) of the original documents in arXiv (PubMed) were lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' arXiv-Lay The original arXiv dataset (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2018) was constructed by converting the LATEX files to plain text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' To be consistent with the other datasets – for which LATEX files are not available – we instead use the PDF files to extract both text and layout elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For each document contained in the original dataset, we fetch (when possible) the corresponding PDF file using Google Cloud Storage buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' As opposed to the original procedure, we do not remove tables nor discard sections that follow the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We retrieve the corresponding abstracts from a metadata file provided by Kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='7 5http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='koreascience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='or.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='kr 6For more details on this, see Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 7https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='com/Cornell-University/arxiv PubMed-Lay For PubMed, we use the PMC OAI Service8 to retrieve abstracts and PDF files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' HAL We use the HAL API9 to download re- search papers written in French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' To avoid exces- sively long (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' theses) or short (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' posters) documents, extraction is restricted to journal and conference papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' SciELO Using Scrapy,10 we crawl the following SciELO collections: Ecuador, Colombia, Paraguay, Uruguay, Bolivia, Peru, Portugal, Spain and Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We download documents written either in Spanish or Portuguese, according to the metadata, obtaining two distinct datasets: SciELO-ES (Spanish) and SciELO-PT (Portuguese).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' KoreaScience Similarly, we scrape the Korea- Science website to extract research papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We limit search results to documents whose publishers’ names contain the word Korean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' This rule was de- signed after sampling documents in the repository, and is the simplest way to get a good proportion of papers written in Korean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='11 Further, search is restricted to papers published between 2012 and 2021, as recent publications are more likely to have digital-born, searchable PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Finally, we down- load the PDF files of documents that contain an abstract written in Korean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2 Data Pre-processing For each corpus, we use the 95th percentile of the page distribution as an upper bound to filter out documents with too many pages, while the 5th (1st for HAL and SciELO) percentile of the summary length distribution is used as a minimum thresh- old to remove documents whose abstracts are too short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' As our baselines do not consider visual in- formation, we only extract text and layout from the PDF files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Layout is incorporated by provid- ing the spatial position of each word in a docu- ment page image, represented by its bounding box (x0, y0, x1, y1), where (x0, y0) and (x1, y1) respec- tively denote the coordinates of the top-left and bottom-right corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Using the PDF rendering li- brary Poppler12, text and word bounding boxes are extracted from each PDF, and the sequence order is recovered based on heuristics around the document layout (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', tables, columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Abstracts are then 8https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='ncbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='nlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='gov/pmc/tools/oai/ 9https://api.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='archives-ouvertes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='fr/docs/search 10https://scrapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='org/ 11For further details, see Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 12https://poppler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='freedesktop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='org/ removed by searching for exact matches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' when no exact match is found, we use fuzzysearch13 and regex14 to find near matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='15 For the non- English datasets, documents might contain several abstracts, written in different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' To avoid information leakage, we retrieve the abstract of each document in every language available – ac- cording to the API for HAL or the websites for SciELO and KoreaScience – and remove them us- ing the same strategy as for the main language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' In the case an abstract cannot be found, we discard the document to prevent any unforeseen leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The dataset construction process is illustrated in Section A in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='3 Datasets Statistics The statistics of our proposed datasets, along with those computed on existing summarization datasets of long documents (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019) are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We see that document lengths are comparable or greater than for the arXiv, PubMed and BigPatent datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For arXiv-Lay and PubMed-Lay, we retain the original train/validation/splits and try to reconstruct them as faithfully to the originals as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For the new datasets, we order documents based on their publication dates and provide splits following a chronological ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For HAL and Korea- Science, we retain 3% of the articles as validation data, 3% as test, and the remaining as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' To match the number of validation/test documents in HAL and KoreaScience, we split the data into 90% for training, 5% for validation and 5% for test, for both SciELO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 Models For reproducibility purposes, we make the mod- els implementation, along with the fine-tuning and evaluation scripts, publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='16 We do not explore the use of visual information in long document summarization, as the focus is on evaluating baseline performance using state-of-the- art summarization models augmented with layout information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' While visual features might provide a better understanding of structures such as tables and figures, we do not expect substantial gains with 13https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='org/project/fuzzysearch/ 14https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='org/project/regex/ 15We use a maximum Levenshtein distance of 20 with fuzzysearch, and a maximum number of errors of 3 with regex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 16https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='com/recitalAI/loralay-modeling Dataset # Docs Mean Mean Article Summary Length Length arXiv (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2018) 215,913 3,016 203 PubMed (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2018) 133,215 4,938 220 BigPatent (Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019) 1,341,362 3,572 117 arXiv-Lay 130,919 7,084 125 PubMed-Lay 86,668 4,038 144 HAL 46,148 4,543 134 SciELO-ES 23,170 4,977 172 SciELO-PT 21,563 6,853 162 KoreaScience 37,498 3,192 95 Table 1: Datasets statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Article and summary lengths are computed in words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For KoreaScience, words are obtained via white-space tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Dif- ference between arXiv and arXiv-Lay is due to the fact that we retain the whole document, while Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2018) truncate it after the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' respect to layout-aware models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Indeed, the infor- mation provided in figures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', information that cannot be captured by layout or text) are commonly described in the caption or related paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Text-only models with standard input size We use Pegasus (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020) as a text-only base- line for arXiv-Lay and PubMed-Lay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Pegasus is an encoder-decoder model pre-trained using gap- sentences generation, making it a state-of-the-art model for abstractive summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For the non- English datasets, we rely on a finetuned MBART as our baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' MBART (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020) is a multi- lingual sequence-to-sequence model pretrained on large-scale monolingual corpora in many languages using the BART objective (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We use its extension, MBART-50 (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020),17 which is created from the original MBART by ex- tending its embeddings layers and pre-training it on a total of 50 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Both Pegasus and MBART are limited to a maximum sequence length of 1,024 tokens, which is well below the median length of each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Layout-aware models with standard input size We introduce layout-aware extensions of Pega- sus and MBART, respectively denoted as Pe- gasus+Layout and MBART+Layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Following LayoutLM (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020), which is state-of- the-art on several document understanding tasks (Jaume et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Harley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2015), each token bounding box coordinates (x0, y0, x1, y1) is normalized into an integer in the range [0, 1000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Spatial positions are encoded us- ing four embedding tables, namely two for the co- ordinate axes (x and y), and the other two for the 17For the sake of clarity, we refer to MBART-50 as MBART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' bounding box size (width and height).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The layout representation of a token is formed by summing the resulting embedding representations The final representation of a token is then obtained through point-wise summation of its textual, 1D-positional and layout embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Long-range, text-only models To process longer sequences, we leverage BigBird (Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020), a sparse-attention based Transformer which reduces the quadratic dependency to a linear one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For arXiv-Lay and PubMed-Lay, we initialize BigBird from Pegasus (Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020) and for the non-English datasets, we use the weights of MBART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The resulting models are referred to as BigBird-Pegasus and BigBird-MBART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For both models, BigBird sparse attention is used only in the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Both models can handle up to 4,096 inputs tokens, which is greater than the median length in PubMed-Lay, HAL and KoreaScience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Long-range, layout-aware models We also in- clude layout information in long-range text-only models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Similarly to layout-aware models with standard input size, we integrate layout informa- tion into our long-range models by encoding each token’s spatial position in the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The resulting models are denoted as BigBird-Pegasus+Layout and BigBird-MBART+Layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Additional State-of-the-Art Baselines We fur- ther consider additional state-of-the-art baselines for summarization: i) the text-only T5 (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019) with standard input size, ii) the long-range Longformer-Encoder-Decoder (LED) (Beltagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020), and iii) the layout-aware, long-range LED+Layout, which we implement similarly to the previous layout-aware models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2 Implementation Details We initialize our Pegasus-based and MBART-based models with, respectively, the google/pegasus-large and facebook/mbart-large-50 checkpoints shared through the Hugging Face Model Hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' As for T5 and LED, we use the weights from t5-base and allenai/led-base-16384, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='18 Following Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2020) and Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2020), we fine-tune our models up to 74k (100k) steps on arXiv-Lay (PubMed-Lay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' On HAL, the total number of steps is set to 100k, while it is de- 18The large versions of T5 and LED did not fit into GPU due to their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Dataset Instances Input Length Output Length Train Dev Test Median 90%-ile Median 90%-ile arXiv (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2018) 203,037 6,436 6,440 6,151 14,405 171 352 PubMed (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2018) 119,924 6,633 6,658 2,715 6,101 212 318 arXiv-Lay 122,189 4,374 4,356 6,225 12,541 150 249 PubMed-Lay 78,234 4,084 4,350 3,761 7,109 182 296 HAL 43,379 1,384 1,385 4,074 8,761 179 351 SciELO-ES 20,853 1,158 1,159 4,859 8,519 226 382 SciELO-PT 19,407 1,078 1,078 6,090 9,655 239 374 KoreaScience 35,248 1,125 1,125 2,916 5,094 219 340 Table 2: Datasets splits and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Input and output lengths are computed in tokens, obtained using Pegasus and MBART-50’s tokenizers for the English and non-English datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' creased to 50k for the other non-English datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='19 For each model, we select the checkpoint with the best validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For Pegasus and MBART models, inputs are truncated at 1,024 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For BigBird-Pegasus models, we follow Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2020) and set the maximum input length at 3,072 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' As the median input length is much greater in almost every non-English dataset, we increase the maximum input length to 4,096 tokens for BigBird-MBART models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Output length is re- stricted to 256 tokens for all models, which is enough to fully capture at least 50% of the sum- maries in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For evaluation, we use beam search and report a single run for each model and dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Following Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2020), we set the number of beams to 8 for Pegasus-based models, and 5 for BigBird-Pegasus-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For the non-English datasets, we set it to 5 for all models, for fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For all experiments, we use a length penalty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For more implementation details, see Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 5 Results and Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 General Results In Table 3, we report the ROUGE-L scores ob- tained on arXiv and PubMed datasets (reported by Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2020)), as well as on the correspond- ing layout-augmented counterparts we release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 20 On arXiv-Lay and PubMed-Lay, we observe that, while the addition of layout to Pegasus does not improve the ROUGE-L scores, there are gains in in- tegrating layout information into BigBird-Pegasus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' To assess whether these gains are significant, we perform significance analysis at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='05 level us- ing bootstrap, and estimate a ROUGE-L thresh- 19We tested different values for the number of steps (10k, 25k, 50k, 100k) and chose the one that gave the best validation scores for MBART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 20For detailed results, please refer to Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' old that predicts when improvements are signifi- cant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' ROUGE-L improvements between each pair of models are reported in Table 11 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' On arXiv-Lay, we compute a threshold of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='48 ROUGE-L, showing that BigBird-Pegasus+Layout significantly outperforms all Pegasus-based mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' In particular, we find a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='56 ROUGE-L im- provement between BigBird-Pegasus and its layout- augmented counterpart, demonstrating that the ad- dition of layout to long-range modeling signifi- cantly improves summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' On PubMed-Lay, we compute a threshold of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Hence, the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='96 ROUGE-L improvement from BigBird-Pegasus to its layout-augmented counterpart is not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' However, the variance in font sizes in PubMed-Lay is much smaller compared to arXiv-Lay (see Ta- ble 12 in the appendix), reflecting an overall more simplistic layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Therefore, we argue that lay- out integration has a lesser impact in PubMed-Lay, which can explain the non-significance of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' In addition, we find that BigBird-Pegasus signifi- cantly outperforms Pegasus and Pegasus+Layout only when augmented with layout, with an im- provement of, respectively, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' This demonstrates the importance of combining layout and long-range modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' While T5 and LED obtain competitive results, we find that the gain in adding layout to LED is minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' However, the models we consider have all been pre-trained only on plain text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' As a result, the layout representations are learnt from scratch during fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Similarly to us, Borchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2021) show that their layout-augmented T5 does not necessarily improve the scores, and that performance is significantly enhanced only when the model has been pre-trained on layout-rich data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Further, we observe, for both Pegasus and BigBird-Pegasus, a drop in performance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' the scores obtained on the original datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' This can be explained by two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' First, our extended Model # Params arXiv/ arXiv-Lay PubMed/ PubMed-Lay Pegasus (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020) 568M 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='83 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='34 BigBird-Pegasus (Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020) 576M 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='77 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='33 T5 (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019) 223M 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='90 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='23 LED (Beltagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2020) 161M 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='74 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='54 LED+Layout 165M 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='96 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='83 Pegasus 568M 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='07 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='75 Pegasus+Layout 572M 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='25 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='85 BigBird-Pegasus 576M 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='59 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='09 BigBird-Pegasus+Layout 581M 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='15 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='05 Table 3: ROUGE-L scores on arXiv-Lay and PubMed-Lay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Reported results obtained by Pegasus and BigBird- Pegasus on the original arXiv and PubMed are reported with a gray background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The best results obtained on arXiv-Lay and PubMed-Lay are denoted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Model # Params HAL (fr) SciELO-ES (es) SciELO-PT (pt) KoreaScience (ko) MBART 610M 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='00 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='55 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='42 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='94 MBART+Layout 615M 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='67 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='47 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='37 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='98 BigBird-MBART 617M 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='04 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='76 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='63 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='55 BigBird-MBART+Layout 621M 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='20 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='71 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='51 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='95 Table 4: ROUGE-L scores on the non-English datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The best results for each dataset are reported in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Dataset Train Validation Test HAL (fr) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='72 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='54 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='84 SciELO-ES (es) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='86 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='28 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='90 SciELO-PT (pt) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='95 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='58 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='96 KoreaScience (ko) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='53 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='26 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='78 Table 5: Percent confidence obtained for the main lan- guage, for each dataset split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' datasets contain less training data due to the inabil- ity to process all original documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Secondly, the settings are different: while the original arXiv and PubMed datasets contain clear discourse in- formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', each section is delimited by mark- ers) obtained from LATEX files, documents in our extended versions are built by parsing raw PDF files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Therefore, the task is more challenging for text-only baselines, as they have no access to the discourse structure of documents, which further underlines the importance of taking the structural information, brought by visual cues, into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Table 4 presents the ROUGE-L scores reported on the non-English datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' On HAL, we note that BigBird-MBART does not benefit from lay- out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' After investigation, we hypothesize that this is due to the larger presence of single-column and simple layouts, which makes layout integration less needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' On both SciELO datasets, we notice that combining layout with long-range modeling brings substantial improvements over MBART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Fur- ther, we find that the plain-text BigBird models do not improve over the layout-aware Pegasus and MBART on arXiv-Lay and SciELO-ES, demon- strating that simply capturing more context does not always suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Regarding performance on Ko- reaScience, we can see a significant drop in perfor- mance for every model w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='t the other non-English datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' At first glance, we notice a high amount of English segments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', tables, figure captions, scientific concepts) in documents in KoreaScience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' To investigate this, we use the cld2 library21 to de- tect the language in each non-English document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We consider the percent confidence of the top-1 matching language as an indicator of the presence of the main language (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', French, Spanish, Por- tuguese or Korean) in a document, and average the results to obtain a score for the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Table 5 reports the average percent confidence ob- tained on each split, for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We find that the percentage of text written in the main lan- guage in KoreaScience (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', Korean) is smaller than in other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' As the MBART-based mod- els expect only one language in a document (the information is encoded using a special token), we claim the strong presence of non-Korean segments in KoreaScience causes them to suffer from inter- ference problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Therefore, we highlight that KoreaScience is a more challenging dataset, and 21https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='com/GregBowyer/cld2-cffi n < Q1 Q1 ≤ n < Q2 Q2 ≤ n < Q3 Q3 ≤ n 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5 Difference in ROUGE-L (a) Article length m < Q1 Q1 ≤ m < Q2 Q2 ≤ m < Q3 Q3 ≤ m 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5 2 Difference in ROUGE-L (b) Summary length σ < Q1 Q1 ≤ σ < Q2 Q2 ≤ σ < Q3 Q3 ≤ σ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5 Difference in ROUGE-L (c) σ of bounding box height Figure 1: Benefit of using layout on arXiv-Lay (blue) and PubMed-Lay (red), defined as the difference in ROUGE- L scores between BigBird-Pegasus+Layout and BigBird-Pegasus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For each dataset, quartiles are calculated from the distributions of article lengths (a), summary lengths (b) and variance in the height of the bounding boxes (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' ROUGE-L scores are then computed per quartile range, and averaged over each range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' we hope our work will boost research on better long-range, multimodal and multilingual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Overall, results show a clear benefit of integrat- ing layout information for long document summa- rization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2 Human Evaluation Metric BigBird BigBird+Layout Precision % 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='15 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='81) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='51 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='70) Recall % 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='07 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='73) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='59 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='86) Coherence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='38) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='75 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='62) Fluency 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='48 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='03) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='34 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='16) Overlap % 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='24) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='36) Flow % 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='75 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='68) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='71) Table 6: Average human judgement scores obtained by comparing gold-truth abstracts and summaries gener- ated by BigBird and BigBird+Layout from 50 docu- ments sampled from arXiv-Lay and HAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Inter-rater agreement is computed using Krippendorff’s alpha co- efficient, and enclosed between parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' To gain more insight into the effect of docu- ment layout for summarizing long textual content, we conduct a human evaluation of summaries gen- erated by BigBird-Pegasus/BigBird-MBART and their layout-aware counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We choose the BigBird-based models over the LED ones, as the gain in augmenting BigBird with layout is much more apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We evenly sample 50 documents from arXiv-Lay and HAL test sets, filtering docu- ments by their topics (computer science) to match the judgment capabilities of the three human an- notators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We design an evaluation interface (see Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2 in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For each sentence si in the generated summary, we ask the annotators to highlight the relevant tokens in si, along with the equivalent parts in the ground-truth abstract (de- noted hi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Further, we ask them to rate the summary in terms of coherence and fluency, on a scale of 0 to 5, following the DUC quality guidelines (Dang, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Finally, annotators are asked to penalize summaries with hallucinated facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The highlight- ing process allows us to compute precision and recall as the percentage of highlighted information in the generated summary and the ground-truth ab- stract, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Moreover, we can compute an overlap ratio as the percentage of highlighted infor- mation that appears several times in the generated summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Lastly, we calculate a flow percentage that evaluates how well the order of the ground- truth information is preserved by computing the percentage of times where the highlighted text hi in the gold summary for one generated sentence si follows the highlighted text hi−1 for the previ- ous sentence si−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' where any token from hi occurs after a token in hi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Table 6 reports the scores for each metric and model, averaged over all 50 documents, along with inter-rater agreements, computed using Krippendorff’s alpha coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We find that adding layout to the models signifi- cantly improves precision and recall, results in less overlap (repetition), and is more in line with the ground truth order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Further, annotators did not en- counter any hallucinated fact in the 50 generated summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' To conclude, reported results show that human annotators strongly agree that adding lay- out generates better summaries, further validating our claim that layout provides vital information for summarization tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='3 Case Studies To have a better understanding of the previous re- sults, we focus on uncovering the cases in which layout is most helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' To this end, we identify fea- tures that relate to the necessity of having layout: 1) article length, as longer texts are intuitively easier to understand with layout, 2) summary length, as longer summaries are likely to cover more salient information, and 3) variance in font sizes (using the height of the bounding boxes), and, as such, the complexity of the layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The benefit of using layout is measured as the difference in ROUGE- L scores between BigBird-Pegasus+Layout and its purely textual counterpart, on arXiv-Lay and PubMed-Lay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We compute quartiles from the dis- tributions of article lengths, ground-truth summary lengths, and variance in the height of bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='22 Based on the aforementioned factors, the scores obtained by each model are then grouped by quartile range, and averaged over each range, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' On arXiv-Lay, we find that layout brings most improvement when dealing with the 25% longest documents and summaries, while, for both datasets, layout is least beneficial for the short- est documents and summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' These results cor- roborate our claim that layout can bring important information about long-range context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Concerning the third factor, we see, on PubMed-Lay, that layout is most helpful for documents that have the widest ranges of font sizes, showcasing the advantage of using layout to capture salient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 6 Limitations and Risks The proposed corpus is limited to a single domain, that of scientific literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Such limitation arguably extends also to the layout diversity of documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' In terms of risks, we acknowledge the presence of Personally Identifiable Information such as au- thor names and affiliations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' nonetheless, such infor- mation is already voluntarily made public by the authors themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 7 Conclusion We have presented LoRaLay, a set of large-scale datasets for long-range and layout-aware text sum- marization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' LoRaLay provides the research com- munity with 4 novel multimodal corpora cover- ing French, Spanish, Portuguese, and Korean lan- guages, built from scientific articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Furthermore, it includes additional layout and visual informa- tion for existing long-range summarization datasets (arXiv and PubMed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We provide adapted architec- tures merging layout-aware and long-range models, 22The quartiles are provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' and show the importance of layout information in capturing long-range dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 8 Acknowledgements We thank the reviewers for their insightful com- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' This work is supported by the Associa- tion Nationale de la Recherche et de la Technolo- gie (ANRT) under CIFRE grant N2020/0916.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' In In- ternational Conference on Machine Learning, pages 11328–11339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='LoRaLay: A Multilingual and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Multimodal Dataset for Long ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Range and Layout-Aware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Summarization – Appendix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Datasets Construction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='w1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='bbox1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='w2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='bbox2 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='PDF Extraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Filtering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Text Extraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Abstract Removal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Figure 2: Dataset Construction Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 Extended Datasets – Lost Documents Figure 3 provides details on the amount of original documents lost in the process of augmenting arXiv and PubMed with layout/visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We observe four types of failures, and provide numbers for each type: The link to the document’s PDF file is not provided (Unavailable PDF);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The PDF file is corrupted (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', cannot be opened) (Corrupted PDF);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The document is not digital-born, making it impossible to parse it with PDF parsing tools ( Scanned PDF);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The document’s abstract cannot be found in the PDF (Irretrievable Abstract).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Figure 3: Distribution of failure types in arXiv-Lay (top) and PubMed-Lay (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2 KoreaScience – Extraction Rule Korean documents in KoreaScience are extracted by restricting search results to documents contain- ing the word "Korean" in the publisher’s name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We show that this rule does not bias the sample to- wards a specific research area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We compute the distribution of topics covered by all publishers, and compare it to the distribution of topics covered by publishers whose name contains the word Korean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Figure 4 shows that the distribution obtained using our rule remains roughly the same as the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Nature Life Artificial Human Society Human Science and Technology 0 10 20 30 40 Publishers with `Korean` in name All publishers Figure 4: Distribution of topics covered by all publish- ers (red) vs distribution of topics covered by publishers whose name contains the word Korean (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='3 Samples We provide samples of documents from each dataset in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' e >>>>>>>>>> (P2) Figure 1: A sketch of the deep-inelastic electron-photon scattering process process the structure of the quasi-real photon, , radiated off an electron from one beam is probed by the virtual photon, *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The * is radiated off an electron from the other beam such that this electron is deflected into the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The detailed formalism for the scattering of photons of arbitrary virtualities can be found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For deep-inelastic electron-photon scattering on quasi-real photons the equation reduces to the well known formula: doe→ex = 2 2元Q2 Q2 ddQ2 zM +O+d The absolute values of the four momentum squared of the virtual and quasi-real photons are denoted Q2 and p2, with p2 < Q?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='. The symbols α and y denote the usual dimensionless variables of deep-inelastic scattering, W denotes the invariant mass of the final state excluding the electrons, and α is the fine structure constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The fux of the incoming photons, f(z, P2), where z is the fraction of the electron energy carried by the photon, is usually taken from the equivalent photon approximation,EPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='At leading order, the structure function F2(c,Q2)is proportional to the parton content, fa/, of the photon, and therefore reveals the structure of the photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' In the region of small y studied, y < l, the contribution of the term containing FL(, Q?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=') is small, and is usually neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 QED structure photon scattering events in which a pair of muons is produced by the * system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Figure 2 shows the present world data on this measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' An update is expected when the ongoing L3 analysis [4] is finalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The data span a range of about two orders of magnitude in Q?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' and have a precision down to about 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' With this precision, the treatment of the small but non- zero virtuality of the quasi-real photon is important, as are electroweak radiative corrections is different for the various experiments, see [1] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' In addition to the measurements of F2,QE further structure functions [5] have been obtained by analyzing the azimuthal correlation between the scattering plane of the deep inelastically scattered electron and the plane spanned by the muon pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Good agreement between data and predictions has been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Also the scattering of two highly virtual photons has been 2 PHOTON09SSN1225-0740 Journal of the Korean Regional Science Association Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2(2021) https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='22669/krsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='063 **·***，**** Changes in Spatial Distribution of Manufacturing Startup Activities in the Capital Region, Korea: A Spatial Markov Chain Approach* Changhyun Song* .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Soonbeom Ahn** .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Up Lim*** Abstract:This study aims to explore how manufacturing start-up activities from 2000 to 2018 have changed spatially and to predict changes in distribution patterns of future start-up activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For the analysis, the Census on Establishments microdata from 2000 to 2018 were used, and the manufacturing industry was classified into four detailed industrial mailchanghyunsongyonsei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='kr) ***mail：uplimyonsei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='kr)HAD ISSN-L:2530-5115 DOl:http://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='22585/hospdomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='v5i4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='148 Tendencias temporales de los patrones de busqueda sobre Servicios de Atencion de Salud a Domicilio antes y después del COVID-19 Temporal trends in Home Care Services search patterns before and after COVID-19 RubénPalomo-Llinaresl0000-0002-1890-4337 JuliaSanchez-Tormo20000-0001-9341-8737 BenjaminPalomo-Llinares30000-0002-3892-3551 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content="Universidad Miguel Hernandez,Departamento de Salud Publica e Historia de la Ciencia,Sant Joan d'Alacant,Alicante Espana 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Centro Internacional Virtual de Investigacion en Nutricion (CiViIN),Alicante, Espana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content="Universitat Miguel Hernandez d'Elx,Elche,Espana Correspondencia/Correspondence Financiacion/Funding Ruben Palomo-Llinares Este trabajo no ha recibido ninguna financiacion." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' palomo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='rub@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='com Contribuciones de autoria/Author contributions Recibido/Received Todos los autores han contribuido por igual en la realiczacion 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2021 de este trabajo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Aceptado/Accepted 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2021 Conflicto de Intereses/Competing interest Los autores no presentan conflicto de intereses COMO CITAR ESTE TRABAJO I HOW TO CITE THIS PAPER Palomo-Llinares R, Sanchez-Tormo J,Palomo-Llinares B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Tendenc as temporales de los patrones de busqueda sobre Servicios de Atencion de Salud a Domicilio antes y despues del COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Hosp Domic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5(4):187-95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Hosp Domic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5(4):187-195 1870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='111% IrretrievableAbstracts UnavailablePDFs 99.' metadata={'source': 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Such distributions are not available for the other datasets, as we did not have access to topic infor- mation during extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Figure 6: Distribution of research areas in arXiv-Lay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Figure 7: Distribution of research areas in HAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' B Experiments B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 Implementation Details Models were implemented in Python using Py- Torch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2017) and Hugging Face (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=', 2019) librairies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' In all experiments, we use Adafactor (Shazeer and Stern, 2018), a stochastic optimization method based on Adam (Kingma and Ba, 2014) that reduces memory usage while retain- ing the empirical benefits of adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' We set a learning rate warmup over the first 10% steps – except on arXiv-Lay where it is set to 10k consis- tently with Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (2020), and use a square root decay of the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' All our experiments have been run on four Nvidia V100 with 32GB each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' C Results C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='1 Detailed Results Model R-1 R-2 R-L MBART 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='05 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='23 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='00 MBART+Layout 46.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='20 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='20 Table 8: ROUGE scores on HAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Best results are re- ported in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Model R-1 R-2 R-L MBART 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='33 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='70 16.' metadata={'source': 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20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='36 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='49 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='95 Table 10: ROUGE scores on KoreaScience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The best results are reported in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2 Human Evaluation Using the Streamlit23 framework, we design and develop an interface to aid human evaluation of summarization models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='24 23https://streamlit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='io/ 24The code is publicly available at https://anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='4open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='science/r/ loralay-eval-interface-C20D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' CondensedMatter Astrophysics Physics 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='04 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='47 Pegasus 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='81 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='27 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='07 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='52 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='96 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='75 Pegasus+Layout 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='10 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='01 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='25 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='59 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='24 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='85 BigBird-Pegasus 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='43 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='74 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='59 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='80 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='32 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='09 BigBird-Pegasus+Layout 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='02 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='95 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='15 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='69 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='38 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='05 Table 7: ROUGE scores on arXiv-Lay and PubMed-Lay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Reported results obtained by Pegasus and BigBird- Pegasus on the original arXiv and PubMed are reported with a gray background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The best results obtained on arXiv-Lay and PubMed-Lay are denoted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Model SciELO-ES SciELO-PT R-1 R-2 R-L R-1 R-2 R-L MBART 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='04 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='65 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='55 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='18 15.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='37 BigBird-MBART 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='64 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='60 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='76 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='85 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='70 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='63 BigBird-MBART+Layout 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='64 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='33 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='71 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='47 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='40 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='51 Table 9: ROUGE scores on the SciELO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The best results are reported in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Figure 8: LoRaLay evaluation interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='3 Analysis of the Impact of Layout Table 12 lists the quartiles computed from the dis- tributions of article lengths, summary lengths, and variation in the height of bounding boxes, for arXiv- Lay and PubMed-Lay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' LoRaLayEvaluationInterface Evaluation guidelines Document0806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='3537(1/50) Statistical LearningofArbitraryComputableClassifiers Linktofulldocument Model A ModelB Ground-truthabstract Statistical learningtheory chiefly studiesrestricted hypothesis classes,particularlythosewith finiteVapnik-Chervonenkis(VC)dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Thefundamentalquantityofinterest isthesample complexity:thenumberofsamplesrequiredto learntoaspecified level ofaccuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Herewe considerlearningoverthesetofallcomputablelabelingfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='SincetheVC-dimensionis infiniteandapriori (uniform)boundsonthenumberofsamplesare impossible,weletthe learningalgorithmdecidewhenithasseensuficientsamplestohavelearned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Wefirstshowthat learning inthis setting is indeedpossible,and developalearningalgorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Wethenshow, however,thatboundingsamplecomplexity independentlyof thedistribution isimpossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Notably this impossibilityis entirelydueto therequirementthat the learningalgorithmbecomputable,and notdue to thestatistical nature of theproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' You selectedthefollowingsentencegeneratedbyModel B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Highlightthepartsinthesentencethat canbefoundintheground-truthabstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Conventionalstatisticallearningtheoryattemptstoboundthenumberof samplesneededtolearntoaspecifiedlevelofaccuracyforeachoftheabove models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='neural networks,supportvectormachines) Next sentenceSummarygeneratedbyModelB Sentence Precision (%) V Conventional statistical learningtheoryattempts toboundthenumber 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='62 ofsamplesneededtolearntoaspecifiedlevelofaccuracyforeachof the abovemodels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='neural networks, support vectormachines) 0 However,ifweallowourselvestochangethemodel, thentheC 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='67 dimensionoftheoverall learningalgorithm isnotfinite,andmuchof statistical learningtheorydoesnotdirectlyapply 0 In contrast, weprovethatdistribution-independentboundsdonotexist 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='22 altogetherforcomputable learningalgorithms in oursetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Ourresults imply thatcomputable learningalgorithms in theuniversal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='0 settingmust"wastesamples"inthesenseofrequiringmoresamples thanisnecessaryforstatisticalreasonsalone Recall(%) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='15 Coherence 00 01 02 O3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 4 O5 Fluency 00 01 O2 O3 O4 lamunabletoevaluatethisdocument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' NextLED LED+Layout Pegasus Pegasus+Layout BigBird-Pegasus BigBird-Pegasus+Layout T5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='84 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='06 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='17 / 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='08 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='30 Pegasus+Layout – – – – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='34 / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='90 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='20 BigBird-Pegasus – – – – – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='56 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='96 Table 11: Absolute ROUGE-L score differences between each pair of models, on arXiv-Lay/PubMed-Lay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Distribution Q1 Q2 Q3 arXiv-Lay PubMed-Lay arXiv-Lay PubMed-Lay arXiv-Lay PubMed-Lay Article Length 6,226 3,513 9,142 5,557 13,190 8,036 Summary Length 119 130 159 182 202 247 σ of bounding box height 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='73 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='28 Table 12: Quartiles calculated from the distributions of article lengths, summary lengths, and variation in the height of bounding boxes, for arXiv-Lay and PubMed-Lay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2782v1 [hep-ex] 16 Jul 2009 Experimental Review of Photon Structure Func- tion Data Richard Nisius Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut), F¨ohringer Ring 6, D-80805 M¨un- chen, Germany, E-mail: Richard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='Nisius@mpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='de∗ DOI: will be assigned MPP-2009-131 The present knowledge of the structure of the photon is presented based on results obtained by measurements of photon structure functions at e+e− collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Results are presented both for the QED structure of the photon as well as for the hadronic structure, where the data are also compared to recent parametrisations of the hadronic structure function F γ 2 (x, Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Prospects of future photon structure function measurements, especially at an International Linear Collider are outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 1 Introduction The measurements of photon structure functions have a long tradition since the first of such measurements was performed by the PLUTO Collaboration in 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The investigations concern the QED structure of the photon as well as the hadronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For the hadronic structure function F γ 2 (x, Q2) the main areas of interest are the behavior at low values of x and the evolution with the momentum scale Q2, which is predicted by QCD to be logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The experimental information is dominated by the results from the four LEP experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' This review is based on earlier work [1, 2] and as an extension provides a number of updated figures, together with a comparison of the experimental data with new parametrisations of F γ 2 (x, Q2) that became available since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Only results on the structure of quasi-real photons are discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The structure of virtual photons and the corresponding measurements of effective structure functions are detailed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 2 Structure function measurements The photon can fluctuate into a fermion–anti-fermion state consistent with the quantum num- bers of the photon and within the limitations set by the Heisenberg uncertainty principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' These fluctuations are favored, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' have the longest lifetimes, for high energetic photons of low virtu- ality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' If such a fluctuation of the photon is probed, the photon reveals its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Using this feature, measurements of photon structure functions are obtained from the differential cross- section of the deep-inelastic electron-photon scattering1 process sketched in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' In this ∗Invited talk presented at the Photon09 Conference in Hamburg on May 12, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 1In this paper, the term electron encompasses positrons throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' PHOTON09 1 (a) arXiv-Lay © 2004 Hindawi Publishing Corporation Journal of Biomedicine and Biotechnology • 2004:5 (2004) 306–313 • PII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' S111072430440401X • http://jbb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='hindawi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='com MINIREVIEW ARTICLE Anthocyanins and Human Health: An In Vitro Investigative Approach Mary Ann Lila∗ Department of Natural Resources & Environmental Sciences, College of Agricultural Consumer and Environmental Sciences, University of Illinois, Urbana, IL 61801, USA Received 2 April 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' revised 10 May 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' accepted 12 May 2004 Anthocyanin pigments and associated flavonoids have demonstrated ability to protect against a myriad of human diseases, yet they have been notoriously difficult to study with regard to human health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Anthocyanins frequently interact with other phytochemicals to potentiate biological effects, thus contributions from individual components are difficult to decipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The complex, multicompo- nent structure of compounds in a bioactive mixture and the degradation of flavonoids during harsh extraction procedures obscure the precise assignment of bioactivity to individual pigments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Extensive metabolic breakdown after ingestion complicates tracking of anthocyanins to assess absorption, bioavailability, and accumulation in various organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Anthocyanin pigments and other flavonoids that are uniformly, predictably produced in rigorously controlled plant cell culture systems can be a great advantage for health and nutrition research because they are quickly, easily isolated, lack interferences found in whole fruits, can be elicited to provoke rapid and prolific accumulation, and are amenable to biolabeling so that metabolic fate can be investigated after ingestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' ANTHOCYANINS AND BIOMEDICINAL PROPERTIES Anthocyanins are members of the flavonoid group of phytochemicals, a group predominant in teas, honey, wines, fruits, vegetables, nuts, olive oil, cocoa, and cereals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The flavonoids, perhaps the most important single group of phenolics in foods, comprise a group of over 4000 C15 aromatic plant compounds with multiple substitution patterns (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='usda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='gov/fnic/foodcomp/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='html).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The primary players in this group include the an- thocyanins (eg, cyanidin, pelargonidin, petunidin), the flavonols (quercetin, kaempferol), flavones (luteolin, apigenin), flavanones (myricetin, naringin, hesperetin, naringenin), flavan-3-ols (catechin, epicatechin, gallocat- echin), and, although sometimes classified separately, the isoflavones (genistein, daidzein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Phytochemicals in this class are frequently referred to as bioflavonoids due to their multifaceted roles in human health maintenance, and anthocyanins in food are typically ingested as com- ponents of complex mixtures of flavonoid components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Daily intake is estimated from 500 mg to 1 g, but can be several g/d if an individual is consuming flavonoid supple- ments (grape seed extract, ginkgo biloba, or pycnogenol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' see, eg, [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The colorful anthocyanins are the most recognized, visible members of the bioflavonoid phytochemicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The free-radical scavenging and antioxidant capacities of an- thocyanin pigments are the most highly publicized of the modus operandi used by these pigments to intervene with human therapeutic targets, but, in fact, research clearly suggests that other mechanisms of action are also respon- sible for observed health benefits [2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Anthocyanin isolates and anthocyanin-rich mixtures of bioflavonoids may provide protection from DNA cleavage, estrogenic activity (altering development of hormone-dependent disease symptoms), enzyme inhibition, boosting produc- tion of cytokines (thus regulating immune responses), anti-inflammatory activity, lipid peroxidation, decreas- ing capillary permeability and fragility, and membrane strengthening [6, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The chemical structure (po- sition, number, and types of substitutions) of the indi- vidual anthocyanin molecule also has a bearing on the degree to which anthocyanins exert their bioactive prop- erties [11, 12] and the structure/function relationships also influence the intracellular localization of the pig- ments [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The anthocyanin literature includes some con- troversy over the relative contributions of glycosylated an- thocyanins versus aglycones in terms of bioavailability and bioactive potential [7, 13, 14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Originally, it was assumed that only aglycones could enter the circu- lation circuit, however, absorption and metabolism of an- thocyanin glycosides has now been demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The na- ture of the sugar conjugate and the aglycone are important determinants of anthocyanin absorption and excretion in both humans and rats [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' The roles of anthocyanin pigments as medicinal agents have been well-accepted dogma in folk medicine throughout the world, and, in fact, these pigments are linked to an amazingly broad-based range of health ben- efits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For example, anthocyanins from Hibiscus sp have (b) PubMed-Lay 1 Les représentations des enseignants de ZEP sur la relation école/famille à travers le prisme des élèves en grande réussite scolaire Publié dans la revue Cahier E&D 2017 Cahier N° 28 Familles, Parents, Ecole Lien vers le site Education & Devenir Lien vers le site Les cahiers pédagogiques Résumé Les familles sont des partenaires essentiels de l’école.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Pourtant, la relation école/famille est souvent décrite comme problématique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Quelles représentations les enseignants ont de cette relation et de l’influence du milieu familial sur la réussite de leurs élèves ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Nous avons réalisé une enquête nationale auprès de 1790 professeurs des écoles (PE) en zone d’éducation prioritaire (ZEP) puis des entretiens avec dix d’entre eux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Le prisme des élèves en grande réussite scolaire (EGRS) dans les ZEP a été choisi pour étudier la différence de perceptions des enseignants en fonction de la réussite de l’élève.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Les PE décrivent le profil idéal des parents d’élèves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Ils souhaitent davantage d’implication de la part des familles et voudraient mettre en place une réelle coéducation qu’ils jugent indispensable à la réussite des élèves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Mots clefs Représentations – enseignants – coéducation – grande réussite scolaire - éducation prioritaire Caroline HACHE – ADEF – AMU (caroline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='hache@univ-amu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='fr) Introduction Lorsqu’ils étudient la proportion d’élèves de milieu populaire ayant obtenu le baccalauréat général sans redoubler, Ould Ferhat et Terrail (2005) indique qu’un désir fort de la part des parents peut faire la différence entre les élèves qui réussissent et ceux qui échouent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' On retrouve dans la littérature (Lorcerie, 2015) une catégorisation des conduites des élèves lorsqu’ils font face aux apprentissages en fonction de l’attitude de leurs parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Les textes officielsi encouragent une relation positive école/famille, car la famille est considérée comme un partenaire de l’école avec une place importante dans la scolarité de l’élève (Houssaye, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Que pensent les enseignants de ces déclarations ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Quelles sont les représentations des enseignants concernant l’influence des familles populaires sur la réussite scolaire de leur enfant ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Notre étude se propose, dans une première enquête, d’interroger par questionnaire 1790 enseignants d’école élémentaire, toutes en ZEPii, autour de leur quotidien dans les classes et, dans une deuxième enquête, de réaliser des entretiens avec dix d’entre eux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Le sujet des parents d’élèves a pris une place importante dans les entretiens de tous les enseignants, comme ceux interrogés par Moisan et Simon (1997, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 68) qui ont plus parlé « des parents que des élèves ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Le choix du prisme des élèves en grande réussite scolaire (EGRS) et donc ici, des parents de ceux-ci, a été pris pour étudier l’avis des enseignants sur un profil particulier, celui des familles dont les élèves réussissent (Hache, 2016) alors que l’on s’attendrait à ce qu’ils soient en difficulté scolaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' En effet, Charlot (2001, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 7) les appelle les « réussites paradoxales » car ils réussissent dans un milieu qualifié de défavorable pour la réussite scolaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Cela a permis aux enseignants de s’exprimer sur la différence ou l’absence de différence entre les parents des EGRS et les autres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (c) HAL 187 Hosp Domic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5(4):187-195 ISSN-L: 2530-5115 DOI: http://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='22585/hospdomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='v5i4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='148 Correspondencia/Correspondence Rubén Palomo-Llinares palomo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='rub@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='com Recibido/Received 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2021 Aceptado/Accepted 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2021 Conflicto de Intereses/Competing interest Los autores no presentan conflicto de intereses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Financiación/Funding Este trabajo no ha recibido ninguna financiación.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Contribuciones de autoría/Author contributions Todos los autores han contribuido por igual en la realiczación de este trabajo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' cómo citar este trabajo | how to cite this paper Palomo-Llinares R, Sanchez-Tormo J, Palomo-Llinares B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Tendencias temporales de los patrones de búsqueda sobre Servicios de Atención de Salud a Domicilio antes y después del COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Hosp Domic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='5(4):187-95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Tendencias temporales de los patrones de búsqueda sobre Servicios de Atención de Salud a Domicilio antes y después del COVID-19 Temporal trends in Home Care Services search patterns before and after COVID-19 Rubén Palomo-Llinares1 ORCID 0000-0002-1890-4337 Julia Sanchez-Tormo2 ORCID 0000-0001-9341-8737 Benjamín Palomo-Llinares3 ORCID 0000-0002-3892-3551 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Universidad Miguel Hernández, Departamento de Salud Pública e Historia de la Ciencia, Sant Joan d´Alacant, Alicante, España.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Centro Internacional Virtual de Investigación en Nutrición (CIVIN), Alicante, España.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Universitat Miguel Hernández d’Elx, Elche, España.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (d) SciELO-ES 1 Gilberto da Silva Guizelin Uma luz sobre as relações Brasil-Moçambique no oitocentos: a Missão Consular de João Luiz Airoza (1827-1828) rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' hist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (São Paulo), n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='178, a03318, 2019 http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='11606/issn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2316-9141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='144021 ARTIGO UMA LUZ SOBRE AS RELAÇÕES BRASIL- MOÇAMBIQUE NO OITOCENTOS: A MISSÃO CONSULAR DE JOÃO LUIZ AIROZA (1827-1828)* Gilberto da Silva Guizelin** Universidade de São Paulo São Paulo – São Paulo – Brasil Resumo Após a assinatura da Convenção de 1826 com a Grã-Bretanha, pela qual o go- verno de D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Pedro I concordou, em troca do reconhecimento britânico, coibir o tráfico transatlântico de africanos para o Império a partir de 1830, foram criadas representações consulares brasileiras na África Portuguesa com a explícita fina- lidade de proteger a atuação de negreiros brasileiros nos últimos anos de legali- dade do comércio de escravos sob a bandeira imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Neste sentido, o presente artigo investiga a atuação de João Luiz Airoza, cônsul do Brasil em Moçambique, entre 1827 e 1828, na defesa do circuito negro entre o Brasil e a África Oriental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Para tanto, o texto aqui apresentado priorizou como fonte de estudo a documen- tação consular produzida por Airoza e dirigida à antiga Secretaria de Estado dos Negócios Estrangeiros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Palavras-chave Relações internacionais – Relações Brasil-Moçambique – Missão consular – Tráfico de escravos – África Oriental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Contato Rua Belo Horizonte, 433, apto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 603 86020-060 – Londrina – Paraná – Brasil guizelin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='gs@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='com Todas as obras e todos os documentos utilizados na pesquisa e na elaboração do artigo são citados nas notas e na bibliografia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' ** Doutor em História pela Faculdade de Ciências Humanas e Sociais de Franca, da Universidade Estadual Paulista Júlio de Mesquita Filho (Unesp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Pós-doutorando em História pela Faculdade de Filosofia, Letras e Ciências Humanas, da Universidade de São Paulo (FFLCH/USP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' Bolsista pós- doc processo nº 2018/07798-1, Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' (e) SciELO-PT 63 수도권 제조업 창업 활동의 공간적 분포 변화 공간 마르코프 체인의 응용 -* 송창현**·안순범***·임업**** Changes in Spatial Distribution of Manufacturing Startup Activities in the Capital Region, Korea: A Spatial Markov Chain Approach* Changhyun Song* · Soonbeom Ahn** · Up Lim*** 국문요약 본 연구는 2000년부터 2018년까지를 분석의 시간적 범위로 설정하여 제조업 창업 활동이 공간적으로 어떠한 변화를 보여왔는지를 탐색적으로 분석하고, 향후 창업 활동의 분포 패턴 변화를 예측하는 것을 목적으로 한 다.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 분석을 위해 2000년부터 2018년까지의 「전국사업체조사」 마이크로데이터 제조업 사업체 자료를 활용하였다.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 한국산업연구원의 ISTANS 분류체계에서 제시하는 40대 제조업 기준에 따라 제조업을 4개의 세부 산업군으로 구 분한 후, 수도권 행정구역 읍면동 수준에서 공간자기상관 분석 및 공간 마르코프 체인 분석을 수행하였다.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 분석 결 과에 따르면, 고위기술산업군 및 중고위기술산업군의 창업 활동은 시간이 흐름에 따라 경기도 남부를 중심으로 집 중되고 있는 것으로 나타났으며, 중저위기술산업군 및 저위기술산업군 창업 활동의 집중은 수도권 외곽으로 분산 되고 있는 것으로 나타났다.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 2000년부터 2018년까지의 추세를 연장하여 2036년까지의 분포 변화를 예측하였을 때, 창업 활동이 활발히 발생하는 지역 및 그와 인접하고 있는 지역의 경우 향후 분위 상승의 가능성이 높은 것으로 나타나 긍정적인 공간 효과가 존재하는 것으로 확인되었다.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 본 연구는 일자리 창출의 주요 원천이 되는 제조업 창 업 활동의 분포 패턴 변화를 동태적으로 분석함으로써 창업 육성 및 일자리 창출과 관련한 지역 정책에의 시사점을 제공하고자 하였다.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 주제어 제조업, 창업, 탐색적공간자료 분석, 공간마르코프 체인 Abstract: This study aims to explore how manufacturing start-up activities from 2000 to 2018 have changed spatially and to predict changes in distribution patterns of future start-up activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' For the analysis, the Census on Establishments microdata from 2000 to 2018 were used, and the manufacturing industry was classified into four detailed industrial 이 논문은 2019년 대한민국 교육부와 한국연구재단의 인문사회분야 중견연구자지원사업의 지원을 받아 수행된 연구임(NRF-2019 S1A5A2A01045590).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' 본 연구는 2020년 한국지역학회 후기학술대회에서 우수논문상을 수상한 연구임.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' ** 연세대학교 대학원 도시공학과 석박사통합과정(주저자, E-mail: changhyunsong@yonsei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='kr) *** 연세대학교 대학원 도시공학과 석사과정(공동저자, E-mail: soonbeomahn@yonsei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='kr) **** 연세대학교 도시공학과 교수(교신저자, E-mail: uplim@yonsei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='kr) ISSN 1225-0740 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='22669/krsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='063 Journal of the Korean Regional Science Association Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='37, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='2(2021) 한국지역학회지 『지역연구』 제37권 제2호 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content='63-82 (f) KoreaScience Figure 5: Samples from each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} +page_content=' cC 000cC BY NCDOSSIE Mocambique em perspectiva: historias conectadas interdisciplinaridade enovos sujeitos historicos Matias Ntundo,2013,xilogravura (detalhe)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf'} diff --git a/d9E2T4oBgHgl3EQfGQZm/content/tmp_files/2301.03655v1.pdf.txt b/d9E2T4oBgHgl3EQfGQZm/content/tmp_files/2301.03655v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f4bb0062d784c776124359b7f7617e33353ee500 --- /dev/null +++ b/d9E2T4oBgHgl3EQfGQZm/content/tmp_files/2301.03655v1.pdf.txt @@ -0,0 +1,1511 @@ +Submitted to the Annals of Applied Statistics +BAYESIAN ADDITIVE MAIN EFFECTS AND MULTIPLICATIVE +INTERACTION MODELS USING TENSOR REGRESSION FOR +MULTI-ENVIRONMENTAL TRIALS +BY ANTÔNIA A. L. DOS SANTOS1, DANILO A. SARTI1, RAFAEL A. MORAL1, ANDREW +C. PARNELL1,2 +1Hamilton Institute, Department of Mathematics and Statistics, Maynooth University, Ireland +2Insight Centre for Data Analytics, Maynooth University, Ireland +We propose a Bayesian tensor regression model to accommodate the ef- +fect of multiple factors on phenotype prediction. We adopt a set of prior dis- +tributions that resolve identifiability issues that may arise between the pa- +rameters in the model. Simulation experiments show that our method out- +performs previous related models and machine learning algorithms under +different sample sizes and degrees of complexity. We further explore the ap- +plicability of our model by analysing real-world data related to wheat pro- +duction across Ireland from 2010 to 2019. Our model performs competitively +and overcomes key limitations found in other analogous approaches. Finally, +we adapt a set of visualisations for the posterior distribution of the tensor +effects that facilitate the identification of optimal interactions between the +tensor variables whilst accounting for the uncertainty in the posterior distri- +bution. +1. Introduction. +The phenotypic performance of a cultivar is associated with many po- +tentially interacting variables (Hara, Piekutowska and Niedbała, 2021). These may include, +but are not limited to: genetic factors; environment exposure; soil type; climatic conditions; +and season. Any combinations of these factors may contribute, either positively or negatively, +to the variability of the production of the crop of interest (Kross et al., 2020). Statistical mod- +elling of the effect of these variables, both singly and jointly, is an important decision-making +tool for farmers and those in the agricultural sector for predicting e.g. yield (Adisa et al., +2019). +One of the main interactions that is believed to impact most the production of a crop +is the one between genotype and environment. For notational convenience, we denote such +interactions as G × E. This sort of interaction is characterised by cultivars that do not behave +consistently in differing environments. Therefore, it is necessary to estimate the amount of +variation in crop yield that is caused by the interaction. Many models have been proposed to +estimate G × E (Gauch Jr, Piepho and Annicchiarico, 2008; Crossa, Vargas and Joshi, 2010; +Gauch Jr, 2013). The most popular is perhaps the additive main effects and multiplicative +interaction (AMMI) model (Gauch Jr, 1988), which consists of two components. The first +term is the additive component, which contains the main effects of categorically structured +genotype and environmental factors. The second term involves a sum of a multiplication of +parameters, which are constrained to an orthonormal space and represent how strong/weak +the interactions between the genotypes and environments are. To date, the models in this area +have mostly been restricted to using these two sole (but important) covariates. +Our approach allows for more components beyond genotype and environment to be +included in the AMMI model. We follow the Bayesian tensor regression technique of +Guhaniyogi, Qamar and Dunson (2017) to allow for any number of interacting categori- +cal factors. Tensors are algebraic structures that generalise matrices and provide a generic +Keywords and phrases: BAMMIT model, Tensors, Bayesian Inference. +1 +arXiv:2301.03655v1 [stat.ML] 9 Jan 2023 + +2 +way of describing multidimensional arrays on a given number of axes. Tensor decomposition +methods have the advantage of capturing the information in the data with a multi-linear struc- +ture and bring a unique representation without the requirement for additional constraints like +sparsity or statistical independence (Jørgensen et al., 2018). The two main tensor decomposi- +tions are the PARAFAC (Carroll and Chang, 1970; Harshman et al., 1970) and Tucker models +(Tucker, 1963). Tensors have been used in many fields of study, including physics (Gaillac, +Pullumbi and Coudert, 2016), chemistry (Facelli, 2011), medicine (Peyrat et al., 2007), and +data mining (Mørup, 2011). Guhaniyogi, Qamar and Dunson (2017) propose a tensor-based +Bayesian regression model where vector/tensor covariates are used to estimate a univariate +response through a class of multiway shrinkage priors. They illustrate the model on real- +world data from the brain connectome as well as providing theoretical results concerning +the speed at which the posterior distribution converges to the true posterior (i.e., contraction +rate). Similarly, Papadogeorgou, Zhang and Dunson (2021) propose a soft tensor regression +to investigate the connection between human traits and brain structural connectomics. +In this paper, we propose the Bayesian additive main effects and multiplicative interaction +tensor model (BAMMIT), which generalises the AMMI model to contain a tensor of inter- +acting terms. We extend the standard AMMI model to include new parameters to the additive +and multiplicative terms of the model, taking into account factors other than genotype and +environment on the phenotype of a given cultivar. Common extra factors might include soil +types, replications, time points, or growth stages. We present our new model in a Bayesian +hierarchical format where we place prior distributions on the main and tensor product terms +so as to guarantee the model’s identifiability and impose orthonormality constraints, which +are an essential part of both the original AMMI and our BAMMIT models. Our model as +proposed is easily extendable to more complex dependence structures, and we explore how +one such extension (time dependence) might be used in our case study. +We evaluate our proposed approach through a set of simulation experiments. Our interest +is to investigate the model’s performance when the complexity increases, that is, when other +variables besides genotype and environment are included and there are different sample sizes. +We compare the prediction of our model with other machine learning models in terms of the +root mean squared error (RMSE) and the coefficient of determination (R2). We explore the +proposed model in a real-world application where we analyse wheat data gathered across +Ireland from 2010 to 2019. Again, our model demonstrates competitive performance when +compared to previous approaches. Finally, we show through a new set of visualisations how +the posterior distribution of the components of the BAMMIT model can be better assessed in +order to quickly identify optimal interactions as well as the uncertainty associated with them. +Our paper is structured as follows. In Section 2.1, we review the AMMI model and present +the constraints imposed on its two components. In Section 2.2, we introduce our BAMMIT +model with its extended additive and multiplicative terms. We outline the interpretability and +identifiability constraints, as well as the priors considered for the parameters and a description +of obtaining the posteriors. In Section 3, we compare the results from BAMMIT with other +relevant models based on synthetic data. In Section 4, we analyse a real-world application +involving wheat production in Ireland. Finally, we review and discuss the findings of the +work in Section 5. +2. A Bayesian AMMI Tensor (BAMMIT) Model. +In this section, we review the vanilla +AMMI model and define terminology and notation. We then introduce the BAMMIT model +detailing the necessary constraints to ensure identifiability as well as the prior distributions +and inferential scheme. + +AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION +3 +2.1. The AMMI model. +The traditional AMMI model takes into account only two cate- +gorical factors, genotype and environment, and is given by a combination of two parts, one +additive and one multiplicative. Let yij be the outcome variable. We write the model as: +yij = µ + b(1) +i ++ b(2) +j ++ +Q +� +q=1 +λqβ(1) +iq β(2) +jq + εij, εij ∼ N(0,σ2), +(1) +where b(1) +i +and b(2) +j +represent the marginal effect of the ith genotype and jth environment, +respectively, i = 1,...,B1 and j = 1,...,B2. The bilinear term (i.e. the summation) is com- +posed of Q components, each of which having a variable λq and the scores β(1) +iq and β(2) +jq . The +parameter λq measures the interaction strength of the qth component and is ordered such that +λ1 ≥ λ2 ≥ ··· ≥ λQ. The scores β(1) +iq and β(2) +jq represent the importance of the ith genotype +and the jth environment in the interaction. To ensure identifiability, the bilinear term is con- +strained so that � +i β(1) +iq β(1) +iq′ = � +j β(2) +jq β(2) +jq′ = 0, for q ̸= q′ and � +i(β(1) +iq )2 = � +j(β(2) +jq )2 = 1. +There are a range of approaches to estimating the parameters of the AMMI model. In the +frequentist paradigm, the additive term of Equation (1) is estimated by ordinary least squares +ignoring the interaction term, and subsequently a singular value decomposition (SVD) on +the matrix of residuals is used to estimate the multiplicative terms (Gabriel, 1978). Within +the Bayesian context, Viele and Srinivasan (2000) proposed the use of Markov chain Monte +Carlo (MCMC) to estimate the parameters of the AMMI model ensuring that the inherent +constraints of the model were not violated. Liu (2001) formulated a more stable and compu- +tationally faster Gibbs sampler. Crossa et al. (2011) and Perez-Elizalde, Jarquin and Crossa +(2012) proposed a Gibbs sampler such that the algorithm was stabilised and incorporated sta- +tistical inference in the visualisation of biplots (Gabriel, 1971), drawing credibility regions +for the interaction effects. By contrast, Josse et al. (2014) introduced an approach to deal +with the overparametrization issue of the model by defining priors for the complete set of pa- +rameters ignoring the constraints, then applying a postprocessing on the posterior samples of +each parameter. Sarti et al. (2021) used Bayesian additive regression trees (BART) in which +a ‘double-grow’ BART is responsible for capturing the interaction term. +The number of terms in the summation, Q, is usually fixed. It is assumed that Q ≤ +min(B1 − 1,B2 − 1). The total variability measured by the principal components is linked to +the number Q, such that by setting Q = min(B1 − 1,B2 − 1) the model can capture all the +variance in the interaction. In practice, Q is commonly an integer between 1 and 3 as this al- +lows for easier interpretation and visualisation of the interaction effects via biplots. However, +many approaches can be applied to determine the value of Q. Examples include Cornelius +(1993) who applied parametric significance tests; other authors who employed cross valida- +tion techniques (dos S. Dias and Krzanowski, 2003; Gabriel, 2002; Hadasch, Forkman and +Piepho, 2017), or those using resampling techniques (Malik et al., 2018; Malik, Forkman and +Piepho, 2019). Examples in the Bayesian field include Perez-Elizalde, Jarquin and Crossa +(2012) and da Silva et al. (2015) where the prior choice and Bayes factor deal with deter- +mining the number of components of the multiplicative term. The non-parametric Bayesian +approach of Sarti et al. (2021) bypasses the need to provide Q completely but, like many +BART models, suffers from interpretability problems due to the complexity of the regression +trees. +One of the reasons for the popularity of the AMMI model is its strong predictive perfor- +mance (Gauch Jr, 2006; Gauch Jr, Piepho and Annicchiarico, 2008), accuracy (Gauch and +Moran, 2019) and its stability evaluation system (Gauch Jr, 1988; Yue et al., 2022). Given its +desirable properties, many extensions can be found in the literature, as highlighted above. In +this work, we aim to maintain the structure of the AMMI model and add the effects of other +categorical factors that are commonly available in real-world METs. + +4 +2.2. The BAMMIT model. +The model in (1) can be extended to include the effect of many +factors apart from genotype and environment. Let yij...v be an outcome variable, in a setting +with a total of N observations and V predictors. We define the BAMMIT model as: +yij...v = µ + b(1) +i ++ b(2) +j ++ ··· + b(V ) +v ++ +Q +� +q=1 +λq +� +β(1) +iq β(2) +jq × ··· × β(V ) +vq +� ++ εij...v. +(2) +This is similar to the AMMI model described in (1), however now we have V factors instead +of only two. Alternatively, we can rewrite the coefficients of the additive and multiplicative +terms of (2) in tensor notation. Let b(v) = (b(v) +1 ,...,b(v) +Bv)⊤ be a Bv-dimensional vector of +parameters of the vth predictor and β(v) +q += (β(v) +1q ,...,β(v) +Bvq)⊤ be a Bv-dimensional vector of +singular values, with q = 1,...,Q. Binding the column vectors β(v) +q , we get β(v), a matrix of +dimension Bv × Q. We define N = +��V +v=1 Bv +� +as the total number of observations (though, +for example, replication may increase N without any need for extra parameters). +For notational convenience, we define a cumulative direct sum and a cumulative Kro- +necker product resulting in an N-dimensional vector as +V +v=1 b(v) = b(1) +··· +b(V ) and +�V +v=1 β(v) +q += β(1) +q +⊗ ··· ⊗ β(V ) +q +, respectively. The direct sum operation is defined such that +for vectors a = (a1,a2)⊤ and b = (b1,b2,b3)⊤, for example, +a +b = (a1 + b1,a1 + b2,a1 + b3,a2 + b1,a2 + b2,a2 + b3)⊤ . +Following the tensor notation presented, the BAMMIT model can be written more com- +pactly as: +y = µ + +V +v=1 +b(v) + +Q +� +q=1 +λq +� V +� +v=1 +β(v) +q +� ++ ε, +(3) +where y is an N-dimensional vector, as before µ is the grand mean, λq is the strength +of the qth component, and ε is a noise vector such that each entry εn ∼ N(0,σ2), with +n = 1,...,N. Note that each vector b(1),b(2),...,b(V ) consists of B1,B2,...,BV values, +respectively, each of which representing the levels of a factor (e.g., 8 genotypes, 10 environ- +ments and 4 soil types would yield B1 = 8,B2 = 10,B3 = 4, with V = 3). The cumulative +direct sum operator +then ensures sums of main effects representing all possible combi- +nations between levels, each corresponding to one observation in the data set. The additive +term represents the individual effect of each predictor, while the summation captures via Q +components the interactions between the individual effects. In the case where there is only +the effect of two variables, the model in Equation 3 is reduced to the AMMI model. The sum- +mation term provides a regularisation on the complexity of the model, with larger Q yielding +a more complex set of interactions. +The model in the form presented in Equation (3) allows for the inclusion and study of +multiple categorical predictors beyond the standard G × E pair used in AMMI models, and +the understanding of their effects in two parts, individually and when interacting. As in the +traditional AMMI model, Q is fixed and represents how many multiplicative terms are in- +cluded in the model. Common extra predictors that might be added to the model include soil +type, time, or growth stages, amongst many others. Being able to tractably estimate the effect +of each of these on a phenotype would be extremely useful for practitioners, whilst retaining +the simple interpretation of the parameters in the AMMI model. + +AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION +5 +2.3. Prior distributions in the BAMMIT model. +In order to ensure the tractability of the +coefficients in the model, it is necessary to establish restrictions on both the additive and in- +teraction terms. For the main effects term, the only constraint to be made is that the covariates +are centered. For the interaction term, we note that it is not trivial to ensure the identifiability +of each parameter individually, only the entire product term (Guhaniyogi, Qamar and Dun- +son, 2017). The constrains we use are: +1. 1′ +B1b(1) = 1′ +B2b(2) = ··· = 1′ +BV b(V ) = 0. +2. 1′ +B1β(1) +q += 1′ +B2β(2) +q += ··· = 1′ +BV β(V ) +q += 0. +3. β(1)′β(1) = β(2)′β(2) = ··· = β(V )′β(V ) = 1. +4. λ1 ≥ λ2 ≥ ··· ≥ λQ ≥ 0. +In the Bayesian context, these constraints are ensured from the definition at the prior level. +For example, in the Bayesian AMMI model proposed byPerez-Elizalde, Jarquin and Crossa +(2012), the von Mises-Fisher distribution is considered for the coefficients of the multiplica- +tive term. In the tensor field, Guhaniyogi, Qamar and Dunson (2017) introduce multiway +shrinkage priors in their tensor regression model. In our approach, we provide a new method +by which the constraints are met by applying the restrictions above through parameter trans- +formations which we describe next. +Formally, we frame a hierarchical model in which prior distributions of the grand mean, +main additive effects and variance parameters are +µ ∼ N(µµ,σ2 +µ), +b(v) ∼ N(0,σ2 +b(v)), +λq ∼ N+(0,σ2 +λ), +σ−2 +y +∼ G(a0,a1), +σb(v) ∼ t+(0,a2), +σλ ∼ t+(0,a3), +where N, N+, G, and t+ are the Normal, truncated Normal, Gamma, and truncated t-Student +distributions, respectively. The hyperparameters of the grand mean µµ and σ2 +µ are fixed as are +all ak terms, with k = 1, ..., 3. We treat the additive effects as random and so estimate σb(v), +though a ‘fixed effects’ version could also be obtained. We express the prior knowledge on +the standard deviations of the additive term parameters and the λ parameter using a truncated +t distribution. Additionally, we impose that the estimated λ vector values are in descending +order. +For the product parameters in the interaction term, we use a transformation to ensure the +constraints are met. Specifically, we generate an auxiliary variable θβ +(v) +iq +from a standard +N(0,1) distribution (the transformation is invariant to the scale of this distribution), with +i = 1,...,Bv, v = 1,...,V , q = 1,,...,Q. Then, we centre by the mean µβ +(v) +q +of the vector +β(v) +q , that is, for each vector β(v) +q +we calculate its mean and then subtract it from the auxiliary +variable θβ +(v) +iq for the respective value of q. Finally, we get the final β(v) +iq value via: +β(v) +iq = +� +θβ +(v) +iq − µβ +(v) +q +��� +i +� +θβ +(v) +iq − µβ +(v) +q +�2 +�−1/2 +. +Applying this procedure to the parameters of the matrix β(v) guarantees that the identifia- +bility constraints (2) and (3) of the model are met in the inferential process. + +6 +3. Simulation. +To evaluate the performance of the BAMMIT model, we simulate data +from Equation (3) over a grid of V = 2,3,4, where Bv is constructed to allow for differences +in the interaction structures. We set up the simulation experiments as follows: +(i) V = 2 and N = 120, with B1 = 12, B2 = 10; +(ii) V = 3 and N = 480, with B1 = 12, B2 = 10, B3 = 4; +(iii) V = 4 and N = 960, with B1 = 12, B2 = 10, B3 = 4, B4 = 2; +(iv) V = 3 and N = 5000, with B1 = 100, B2 = 10 and B3 = 5. +Our goal with the above scenarios is to explore the performance of the BAMMIT model +in situations where the AMMI model can be applied (case i) and situations where the number +of genotypes/environments is small, medium and large. Scenarios (ii) and (iii) present a chal- +lenge to the classic AMMI model because it cannot be applied directly. Together, scenarios +(i), (ii) and (iii) evaluate our model’s performance when the number of predictors increases. +Finally, scenario (iv) presents a computationally challenging scenario for BAMMIT as it in- +volves a large number of observations. +For each of these scenarios we set the real number of terms Qsim = {1,2,3} taking +λ = {{10},{8,10},{8,10,12}}. We simulated 12 training and 12 test data sets. In other +words, in scenario (i) there are two predictors and 120 observations, setting 12 values for +the first predictor and 10 for the second. Given these number of observations and variables, +we generate three data sets, one where the value of Qsim = 1 and λ = 10, another where +Qsim = 2 and λ is defined as 8 and 10, and finally, a data set in which Qsim = 3 and λ takes +the values 8, 10 and 12. The same understanding extends to the other cases. In all scenarios, +we set µ = 100, and σ2 = 1. +To fit the BAMMIT model, we run a Markov chain Monte Carlo (MCMC) algorithm +through the probabilistic programming language Just Another Gibbs Sampler (JAGS; Plum- +mer et al., 2003) and the R package R2jags (Su and Yajima, 2021). We set Q = {1,2,3} as +the true number of components, µµ = 100, σ2 +µ = 10, a0 = a1 = 0.1, and a2 = a3 = 1. We use +three chains, 4000 iterations per chain, discarding the first 2000 as burn-in, and a thinning +rate of two. Regarding computational time, a data set with three predictors (V = 3), N = 100 +and Q = 1 takes on average one minute to run, whilst to run a data set with N = 1000 takes +30 minutes, and with N = 5000, it takes on average 6 hours. We discuss computational issues +further in Section 5. All experiments were implemented in R, and the code used is available +at https://github.com/Alessandra23/bammit. +To assess the performance of the model when V > 2, we compare BAMMIT with two +models extensively employed for prediction purposes, namely Random Forests (RF) and +eXtreme Gradient Boosting (XGB). We also compare with the traditional AMMI and AM- +BARTI model (Sarti et al., 2021), though these are unavoidably restricted to using only the +first two variables. For the RF model, we use the package randomForest (Liaw et al., +2002) selecting the default settings, mtry= 2 and 500 trees. For the XGB model, we use +the package xgboost (Chen et al., 2019) setting 50 iterations. For the AMBARTI model +we use the package AMBARTI 1 setting 50 trees, 500 as burn-in and 1000 iterations as post +burn-in. All the models were fitted to the training data. We checked the accuracy, using the +test data, by comparing the posterior mean estimates with the true parameter values used in +the simulations. We use the root mean squared error (RMSE) to measure predictive power +(how close ˆy is to the true y) and R2 to assess the proportion of explained variability. +1The code is available at https://github.com/ebprado/AMBARTI. + +AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION +7 +3.1. Simulation results. +The scatterplot in Figure 1 shows this comparison of the additive +portion of Equation (3), taking V = 4, Qsim = 2, N = 960 when the true value of λ = +{8,10}. Each point is an estimated level of the parameter and the error bars are the 90% +credible intervals. By visual inspection, the estimates of the effects of the four main predictors +are close to the true values, with narrower intervals for predictors with a greater number of +levels. +b(1) +b(2) +b(3) +b(4) +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +-3 +-2 +-1 +0 +1 +2 +True +Estimated +Fig 1: Scatterplots of true versus estimated additive term for simulation scenario (iii), setting +Qsim = 2, λ = {8,10}. The bars represent the 90% credible interval. +In Figure 2, we compare the estimates against the true values in the case where the number +of predictors varies. Each point represents an interaction term estimate in a total of 120 (V = +2), 480 (V = 3) and 960 (V = 4) points, and the bars, again, represent the 90% credible +intervals. We observe that when V = 4, the dispersion is smaller and the interaction estimates +are more concentrated around zero. This can be explained because as more predictors are +added to the additive term of the model, the greater the approximation of the response by the +predictors and the smaller the amount approximated by the interaction term, despite inserting +more variables in both terms of the model. Also, note that the interaction is comprised of +all the new variables together, and that this interaction may not be that strong. For example, +suppose we are looking at the genotype × environment × soil type × growth stage interaction. +In this case, the interaction of the four factors together is not as strong as if we were looking +only at subsets of these interactions, such as genotype × environment × growth stage. +V = 2 +V = 3 +V = 4 +-2 +0 +2 +4 +-2 +0 +2 +4 +-2 +0 +2 +4 +-5.0 +-2.5 +0.0 +2.5 +5.0 +True +Estimated +. +Fig 2: Scatterplots of true versus estimated interaction terms for simulations scenarios (i), (ii) +and (iii) setting Qsim = 1 and λ = 10. The bars represent the 90% credible intervals. + +8 +To investigate how much the estimation of the interaction term is influenced by the choice +of the value of Q, we study the case when the data are simulated with Qsim = 2 and V = +3, but the number of components in the model fit is Q = {1,2,3}. The fixed value in the +simulation was determined because, in real-world applications, the true number of terms in +the interaction is not known. Thus we wanted to compare the behaviour of the model in +a situation where we know the true value of Q in the data, though it is of course fixed in +the model. Figure 3 shows the 90% credible error bars solely for the interaction term. As +expected, the performance of the model setting Q = 3 is better since there is an increase in +the complexity of the model fit to match the data. When Q is set too small, as in the left +panel, we see the model being unable to capture the interaction terms. However, when the +value of Q used in the model fit is at least as big as the value used in the simulation, we obtain +superior results. +In terms of predictions, Table 1 shows the prediction RMSE and the R2 considering the +cases where we have three and four predictors in the models (simulation scenarios (ii) and +(iii)). To fit BAMMIT and AMMI models we used Q = 2. As stated above, the AMBARTI +and AMMI models were fitted disregarding the effects of the other variables. Specifically, in +scenario (iii), for example, there were three predictors, but the two aforementioned models +disregarded the effect of the third predictor. The BAMMIT model clearly performed better +than the other two models. In addition to the prediction advantage, our model stands out from +RF and XGB as it can accommodate the interaction between variables, while at the same time +providing estimates based on posterior distributions. Another highlight is that BAMMIT is +able to satisfactorily explain the variability of the response variable, since the R2 obtained +in all scenarios was above 75%. In a real world scenario where the data were not simulated +from the BAMMIT model we might expect that the machine learning approaches would be +more competitive in terms of their performance. However they would still not allow for clear +interpretation of the interaction effects. +V = 3 +V = 4 +BAMMIT +AMBARTI +AMMI +RF +XGB +BAMMIT +AMBARTI +AMMI +RF +XGB +RMSE +0.92 +2.54 +2.52 +1.68 +1.26 +0.96 +2.74 +2.71 +1.74 +1.15 +R2 +0.78 +0.02 +0.02 +0.58 +0.69 +0.81 +0.01 +0.01 +0.68 +0.78 +TABLE 1 +RMSE and R2 for ˆy on out-of-sample data for scenarios (ii) and (iii). +4. Case Study. +In this section, we investigate the performance of the model on a real +data set. The data was collected over ten years (2010 – 2019) and concerns the production +of a common species of wheat (Triticum aestivum L.) in Ireland, with the response being the +yield of wheat measured in tonnes per hectare (t/ha). The data comes from the Horizon2020 +EU InnoVar project (www.h2020innovar.eu) and was supplied by the Irish Department +of Agriculture, Food and the Marine. The experiments were conducted using a randomised +complete block design with four replicates. The data set contains 85 genotypes and 17 envi- +ronments, all anonymised and named as g1,...,g85 and e1,...,e17, respectively. Owing to +not all genotypes being observed in each location in all seasons, the total number of observa- +tions genotype × location × year × block is 6,368, rather than 14,450. An advantage of the +BAMMIT model is that we are able to impute the missing combinations as part of the model +fit. +A subsample of this data was previously explored by Sarti et al. (2021), considering only +two factors: genotype and environment. However, in our work, we include the additional +variables year and block, present in the Irish data set, as a third and fourth effect in the + +AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION +9 +Q = 1 +Q = 2 +Q = 3 +-2 +0 +2 +-2 +0 +2 +-2 +0 +2 +-4 +-2 +0 +2 +4 +True +Estimated +Fig 3: Scatterplots of true versus estimated interaction term for a simulation scenario setting +N = 480, Qsim = 2, λ = {8,10}. Each plot is a result of the Bayesian fit for three possible +values of Q (1,2,3). The bars represent the 90% credible interval. +BAMMIT model. We expect to detect if the year is an important predictor in the models, as +affirmed by Hara, Piekutowska and Niedbała (2021) that the ability to predict the yield in a +certain year can be useful for making decisions, such as cultivation planning and storage. +We are interested in answering some specific questions. Initially, when fixing the year and +block effect, we would like to know which genotype has the best performance, in which +environment, and also which environment provides the highest yield. When considering all +the variables, we investigated which year and block had the best performance. One of the +main tasks is to understand how accurate our predictions are and to examine the uncertainty +associated with the answers from the previous questions. As an extension to the model, we +include additional structure to the time component of the model by adding autoregressive +terms in year. +4.1. Autoregressive structure on the time predictor (AR-BAMMIT). +In this particular ap- +plication, where one of the variables in the BAMMIT model is the year of production, we +have the option of extending the model by applying a different structure to the time predictor +in both terms, that is, additive and multiplicative. The model is now: +yijtr = µ + b(1) +i ++ b(2) +j ++ b(3) +t ++ b(4) +r ++ +Q +� +q=1 +λqβ(1) +iq β(2) +jiqβ(3) +tq β(4) +rq + ϵijtr, +(4) +b(3) +t += αb + φbb(3) +t−1 + ηt, +(5) +θ(3) +tq = αθ + φθθ(3) +(t−1)q + ωt, +(6) +β(3) +tq = +� +θ(3) +tq − µθ +(3) +q +��� +t +(θ(3) +tq − µθ +(3) +q )2 +�−1/2 +, +(7) +where the indexes i,j,t and r are associated to genotypic, environmental, time and block +effects, respectively. The priors for the entire model follow the same structure as before, such +that Equation (7) is defined to meet the conditions of identifiability of the model. To estimate +the autoregressive parameters φb and φθ we use uniform priors in the interval [−1,1]. A +normal distribution with mean zero and variance 100 was considered for the parameters αb + +10 +and αθ. The priors for ηt and ωt were a truncated t-Student distribution with parameters zero +and one. More details about the model structure are given in Appendix B. +4.2. Results. +To fit the BAMMIT model we selected two replicates randomly for training +and the last two for validation, in a total of 3,184 observations in each data set. Therefore, +we have V = 4, B1 = 85 genotypes, B2 = 17 environments, B3 = 10 years and B4 = 2 +block effects. Assuming there is little information to establish the hyperparameters for our +priors, we set µµ = 10, σ2 +µ = 1, obtained from the mean and the standard deviation of the +data, a0 = a1 = 0.1, and a2 = a3 = 1. To input the number of Q terms, we run the model +with Q = 1,2,3. The model is fitted with three Markov chains, 4,000 iterations per chain, +discarding 2,000 and a thinning rate of two. +We compare our BAMMIT approach with traditional AMMI and AMBARTI models. In +addition, we fit the AR-BAMMIT model, following Equations (5) to (7). For the AMBARTI +model, we considered 50 trees, 1000 iterations as burn-in and 1000 iterations as post burn-in. +To assess the performance of the predictions ˆy, we display the RMSE and the R2 in Table 2 +for each model. As the AMBARTI and classic AMMI models can only handle the genotype +and environment variables, we performed two procedures to treat the training and test data +sets. The first takes into account all rows in the data set and ignores the year and block +variables, so that rows corresponding to the same genotype and the same environment are +treated as different (columns AMBARTI ALL YEARS and AMMI ALL YEARS in Table 2). +In the second approach, the data set are separated by years, so that for each year we run the +model. Then, we regroup the estimates obtained for each group of years, having a single data +set, and calculate the metrics (AMBARTI BY YEAR and AMMI BY YEAR columns in the +table). The results presented in the table were observed considering Q = 1 for the BAMMIT, +AR-BAMMIT and AMMI models (ALL YEARS and BY YEAR). +BAMMIT +AR-BAMMIT +AMBARTI +BY YEAR +AMBARTI +ALL YEARS +AMMI +BY YEAR +AMMI +ALL YEARS +RMSE +0.68 +0.68 +0.59 +1.66 +0.66 +1.60 +R2 +0.88 +0.89 +0.91 +0.34 +0.88 +0.38 +TABLE 2 +Metrics for out-of-sample y for different models. ALL YEARS refers to the fact that the model took into account +all rows of the data set and ignored the year and the block variables effect. BY YEAR means that the data set was +separated by years and the model was run for each group. +The results presented in Table 2 show that the AMBARTI and AMMI models were su- +perior to BAMMIT in terms of prediction when the data set is separated by year. However, +when all the years are together, these models performed poorly. For the AMBARTI, these +results can be explained by the high numbers of genotypes and environments. As mentioned +above, considering all the 10 years, there are 85 genotypes and 17 environments, and this +case the AMBARTI model is not efficient in the generation of the 2B1−1 − 1 and 2B2−1 − 1 +2-partition combinations for genotypes and environments. Thus, due to the high numbers +of possible combinations, the interaction component of the AMBARTI model is not able to +estimate the interactions between genotypes and environments efficiently and the model per- +forms poorly. Also, the inferior performance of classic AMMI model and AMBARTI was +expected, since the adjusted models do not take into account the effects of the other vari- +ables. It is Interesting to note that, for the BAMMIT model, the best RMSE was obtained by +setting Q = 1 in the model fit, since when Q = 2 and Q = 3 the RMSE was 0.69 and 0.71 +respectively. + +AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION +11 +4.3. Posterior visualisation. +A common way to visualise the genotype and environment +interactions in an AMMI model is through biplots (Gabriel, 1971). However, Sarti et al. +(2021) used a heatmap to visualise the predictions and interactions. In this visualisation ap- +proach, it is possible to identify in a more immediate way the best interactions between +genotypes and environments. A shortcoming of their approach concerns the quantification of +uncertainty, which cannot be observed directly on the graph. To address this particular issue, +in this work we show the prediction for interactions through a heatmap as in Sarti et al. (2021) +and the uncertainty is showed as Value-suppressing uncertainty palettes (VSUP) as presented +by Inglis, Parnell and Hurley (2022). +First introduced by Correll, Moritz and Heer (2018) value-suppressing uncertainty palettes +are bivariate colour palettes that represent a measure or value and its uncertainty. The outputs +for each combination of value and uncertainty in traditional bivariate palettes are often shown +as a 2D square (for example see, Robertson and O’Callaghan (1986)), However, VSUP plots +combine cells in the palette using a tree structure to suppress the measure or value at higher +levels of uncertainty. In VSUP, when the uncertainty is low, more bins are allocated to the +colour space. When increasing in uncertainty, the values are suppressed into fewer bins that +blend together their colour value. By doing this, the values will become more distinct as the +level of uncertainty reduces, with the intention of making it easier to detect the difference +between low and high uncertainty. +In Figure 4 we display an ordered heatmap and VSUP legend for the Irish data selecting +the year 2015 in block three from the test data set. This year corresponds to the best average +wheat production observed in the data. The remaining years are shown in Appendix A. For +these plots we use the standard deviation of the predictions as our measure of uncertainty and +the value shown is the median prediction for each variable pair. The plots are ordered so that +generally high predicted yield values are pushed to the top left of the plot and descend to the +bottom right. The environments e2, e9, e11 and e16 were the worst environments observed, +having a small median value compared to the others. On the other hand, environment e1 was +the best and most stable, presenting a median yield value higher than the others and a lower +uncertainty. The environment e6 had a middling production across some genotypes, but with +a higher uncertainty when compared to the others. Applying the same interpretation to the +genotypes, we observed that the genotypes g3, g26 and g85 had the best performance. The +genotypes g17 and g20 presented a good production of predicted wheat on environment e1, e5 +and e7, however with a higher standard deviation than the other genotypes, which also pro- +duced around 14 t/ha in this environment. The worst genotype × environment combinations +observed were e9 × g81, e2 × g81, e2 × g71, e2 × g62, e2 × g48 and e2 × g15, while the best +were e1 × g85, e1 × g30, e1 × g26 and e1 × g3. The results are comparable to those of Sarti +et al. (2021). +In Figure 5 we plot the median yield by year. We can see that the best production occurred +in the years 2015 to 2017, with the first having a smaller variation than the others. Another +important factor to observe is the effect of interaction, especially related to the genotype +effect. Yan et al. (2000) affirms that in multi-environment trials (MET), the primary source +of interest in the evaluation of the genotypes is via the genotype effect plus g × e interaction. +This enables plant breeders to determine not only those genotypes with optimal performance +but also evaluate variability across environments. In Figure 6 we present the interaction effect +added to the individual genotype effect. As in Figure 4, the graph is oriented downwards, that +is, high values of the sum (genotype + genotype × environment) at the top and low values +at the bottom. We observe that genotype g3 is optimal in having a generally high performance +score whilst also being stable across environments. . + +12 +g81 +g17 +g15 +g71 +g62 +g48 +g38 +g60 +g5 +g13 +g37 +g36 +g20 +g10 +g30 +g26 +g3 +g85 +e1 +e7 +e5 +e6 +e4 +e11 +e16 +e9 +e2 +Environment +Genotype +ŷ +11 +12 +13 +14 +15 +sd +0.48 +0.49 +0.50 +0.51 +0.52 +Fig 4: Predicted yields from the BAMMIT model for the data set in 2015. Production this year +was high, between 11 and 15 tonnes per hectare, with positive emphasis on the environment +e1, the genotype g26 and on the combination g26 × e7. +8 +12 +16 +2010 +2011 +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +Year +ŷ +Fig 5: Median yield by year. The bars are the ˆy plus two times the standard deviation. + +AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION +13 +g81 +g17 +g15 +g71 +g62 +g48 +g38 +g60 +g5 +g13 +g37 +g36 +g20 +g10 +g30 +g26 +g85 +g3 +e5 +e16 +e6 +e11 +e9 +e2 +e4 +e1 +e7 +Environment +Genotype +g^ + int +-0.3 +-0.2 -0.1 0.0 +0.1 +sd +0.06 +0.08 +0.10 +0.12 +0.14 +Fig 6: Predicted addiction of genotype and interaction from the BAMMIT model for the data +set in 2015. +5. Discussion. +In this paper, we propose a generalisation of the AMMI model which +uses the tensor regression approach of Guhaniyogi, Qamar and Dunson (2017) and Papado- +georgou, Zhang and Dunson (2021) to extend the model to allow for multiple interacting +categorical variables. The main idea is to allow for more realistic understanding of pheno- +typic effects beyond the usually considered pair of genotype and environment. We envisage +that in the future such models may be used to further indicate interactions between season, +soil, weather conditions, growth stage and other potential predictors. The priors we use on +the hierarchical model were built in order to meet the restrictions imposed on the model and +to ensure identifiability. For each simulated data set and for the estimates obtained from the +Bayesian fit, the constraints were checked. +In the simulation results, the model displayed promising performance, in the sense that +the estimates corresponded with the true values of the parameters from the simulation. As +expected, when the established number of terms in the interaction is three, the Bayesian +model had a better fit, regardless of the true number used in the simulation. The choice of the +number of components Q is still arbitrary, taking into account only the typical values already +mentioned in the literature. Nonetheless, a more sophisticated approach to choosing the rank +Q can be applied, as shown by Guhaniyogi, Qamar and Dunson (2017). +When the model was applied to the real data, it was possible to establish which geno- +types, environments, blocks and years had the highest wheat production. Also, we could +visualise just the interaction effect and so it was possible to determine the optimal interac- +tions. The purpose of showing the results in the visualisations presented in this paper is to +aid a researcher’s ability to interpret the results and improve recommendations. Although the +AMBARTI model had a superior result in relation to BAMMIT, it performed well, even with +a small number of interaction terms. When compared with the Bayesian AMMI model, both +BAMMIT and AR-BAMMIT were superior in terms of prediction. + +14 +In relation to the computational time, the cost for the method was high, especially when the +number of components Q increased; when setting Q = 3 it took eight hours to complete the +fit. This drawback was true in general when the size of the data set was large (around 5,000 +total observations or more) with the model taking hours to form a valid posterior distribution. +In future work, we could be use faster computational methods such as variational inference +(Blei, Kucukelbir and McAuliffe, 2017; Dos Santos et al., 2022) or those as discussed in +Papadogeorgou, Zhang and Dunson (2021) and Zhang et al. (2020). +As we have seen, there is an obvious extension of these models through the prior distribu- +tions. New structures can be added to certain predictors, as was done here for the year variable +in the data set. Any temporal and spatial components could have their inherent characteris- +tics inserted in the model. Another important point is the insertion of continuous variables, +or latent representations of them, since the current structure does not allow for this type of +variable. +Acknowledgments. +Antônia A. L. dos Santos, Andrew Parnell, and Danilo Sarti re- +ceived funding for their work from the European Union’s Horizon 2020 research and in- +novation programme under grant agreement No 818144. In addition Andrew Parnell’s work +was supported by: a Science Foundation Ireland Career Development Award (17/CDA/4695); +an investigator award (16/IA/4520); a Marine Research Programme funded by the Irish +Government, co-financed by the European Regional Development Fund (Grant-Aid Agree- +ment No. PBA/CC/18/01); SFI Centre for Research Training in Foundations of Data Sci- +ence 18/CRT/6049, and SFI Research Centre awards I-Form 16/RC/3872 and Insight +12/RC/2289_P2. For the purpose of Open Access, the author has applied a CC BY public +copyright licence to any Author Accepted Manuscript version arising from this submission. +APPENDIX A: ADDITIONAL RESULTS +In this section, we complement the results presented in the Section 4. Figure 7 presents +the graphs for the predicted yields for the BAMMIT model applied to the Irish time series +data set (except year 2015). Analysing only the legend of the figures and looking at the value +scale it is possible to see that the forecast of wheat yield for the year 2015 (presented in Figure +4) was higher than that for all the other years. In order to clearly observe the behaviour of +genotype and environment predictors in each year, we do not scale the estimated value and +the uncertainty legend to be equal across plots. Thus, we prevent years with high production +and low uncertainty having an intense colour while years with contrary behaviour have a +washed-out colour. +g74 +g14 +g22 +g2 +g78 +g9 +g27 +g11 +g25 +g15 +g79 +g16 +g75 +g83 +g24 +g12 +g1 +g59 +g67 +g26 +g3 +e6 +e12 +e2 +e1 +e4 +e11 +e7 +e8 +Environment +Genotype +ŷ +5 +7 +9 +11 +13 +sd +0.4 +0.6 +0.8 +1.0 +1.2 +(a) 2010 +g22 +g32 +g78 +g27 +g9 +g43 +g84 +g15 +g79 +g83 +g24 +g75 +g12 +g59 +g38 +g34 +g26 +g3 +e7 +e6 +e11 +e2 +e13 +e1 +e4 +e3 +e8 +Environment +Genotype +ŷ +8 +10 +12 +14 +16 +sd +0.40 +0.45 +0.50 +0.55 +0.60 +(b) 2011 +Fig 7: Predicted yields from the BAMMIT model for the Irish data set. + +AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION +15 +g64 +g22 +g78 +g19 +g9 +g31 +g27 +g57 +g15 +g79 +g24 +g75 +g59 +g18 +g38 +g13 +g26 +g3 +e1 +e8 +e7 +e14 +e6 +e3 +e4 +e11 +e2 +Environment +Genotype +ŷ +4 +6 +8 +10 +12 +sd +0.4 +0.5 +0.6 +0.7 +0.8 +(c) 2012 +g70 +g31 +g9 +g39 +g63 +g15 +g79 +g44 +g59 +g18 +g62 +g48 +g38 +g13 +g26 +g3 +g85 +g76 +e11 +e6 +e3 +e15 +e10 +e7 +e2 +e4 +Environment +Genotype +ŷ +8 +9 +10 +11 +12 +sd +0.4 +0.5 +0.6 +0.7 +0.8 +(d) 2013 +g9 +g81 +g6 +g15 +g79 +g71 +g62 +g48 +g18 +g38 +g13 +g37 +g36 +g49 +g20 +g26 +g3 +g85 +e7 +e2 +e6 +e9 +e17 +e5 +e1 +e4 +e11 +Environment +Genotype +ŷ +7 +8 +9 +10 +11 +sd +0.48 +0.49 +0.50 +0.51 +0.52 +(e) 2014 +g50 +g51 +g71 +g62 +g38 +g60 +g5 +g54 +g41 +g37 +g35 +g20 +g10 +g30 +g26 +g55 +g28 +g3 +g82 +g4 +e5 +e1 +e6 +e2 +e9 +e4 +e7 +e11 +Environment +Genotype +ŷ +8 +9 +10 +11 +12 +sd +0.48 +0.49 +0.50 +0.51 +0.52 +(f) 2016 +g56 +g50 +g71 +g38 +g69 +g42 +g37 +g45 +g35 +g20 +g10 +g26 +g55 +g29 +g28 +g3 +g73 +g82 +g4 +g21 +e1 +e4 +e7 +e2 +e9 +e11 +e5 +e6 +Environment +Genotype +ŷ +8 +10 +12 +14 +16 +sd +0.48 +0.51 +0.54 +0.57 +0.60 +(g) 2017 +g38 +g65 +g7 +g37 +g20 +g10 +g26 +g29 +g61 +g47 +g3 +g77 +g40 +g68 +g82 +g4 +g52 +g80 +g23 +e2 +e9 +e5 +e7 +e11 +e1 +e4 +e6 +Environment +Genotype +ŷ +7 +8 +9 +10 +11 +sd +0.48 +0.49 +0.50 +0.51 +0.52 +(h) 2018 +g66 +g58 +g20 +g7 +g72 +g10 +g46 +g26 +g29 +g77 +g53 +g8 +g40 +g68 +g82 +g4 +g33 +g52 +g80 +g23 +e11 +e4 +e7 +e1 +e5 +e9 +e2 +e6 +Environment +Genotype +ŷ +9 +10 +11 +12 +13 +sd +0.50 +0.75 +1.00 +1.25 +1.50 +(i) 2019 +Fig 7: Predicted yields from the BAMMIT model for the Irish data set on block three. + +16 +APPENDIX B: STUDY OF AR-BAMMIT MODEL +The construction of the BAMMIT model allows new structures to be applied to the pa- +rameters, such as spatial or temporal. However, the insertion of these new structures brings +greater complexity to the model, since it is necessary that the restrictions continue to be +guaranteed. In Section 4.1, a temporal architecture is applied to the parameters associated +with years in the Equation 3. In this section, we intend to explore the aforementioned AR- +BAMMIT method (Section 4.1) by carrying out a small simulation study. To ensure the +needed restrictions, we apply the transformation made earlier in Section 2.3 to the param- +eters of the interaction term also to the parameters of the new model structure. +The simulation scenarios considered were scenarios (ii) and (iii) of Section 3, where only +one of the variables in the additive term and one variable in the multiplicative term have +the autoregressive structure. In Figure 8, we present the scatterplot of the true and estimated +main effects values for the simulation scenario (ii). Figure 9 shows the posterior density of +the precision parameter and a scatterplot of the interaction term in this scenario. +b(1) +b(2) +b(3) +-1 +0 +1 +-1 +0 +1 +-1 +0 +1 +-2 +-1 +0 +1 +2 +True +Estimated +Fig 8: Posterior density for the years and for the first four genotypes and environments, setting +V = 3, N = 480, B1 = 8, B2 = 4, B3 = 10, Qsim = 1, λ = 12. +REFERENCES +ADISA, O. M., BOTAI, J. O., ADEOLA, A. M., HASSEN, A., BOTAI, C. M., DARKEY, D. and +TESFAMARIAM, E. (2019). Application of artificial neural network for predicting maize production in South +Africa. Sustainability 11 1145. +BLEI, D. M., KUCUKELBIR, A. and MCAULIFFE, J. D. (2017). Variational inference: A review for statisticians. +Journal of the American statistical Association 112 859–877. +CARROLL, J. D. and CHANG, J.-J. (1970). Analysis of individual differences in multidimensional scaling via an +N-way generalization of “Eckart-Young” decomposition. 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SIAM journal on mathematics of data science 2 444–479. + diff --git a/d9E2T4oBgHgl3EQfGQZm/content/tmp_files/load_file.txt b/d9E2T4oBgHgl3EQfGQZm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..862687ae21bb70254c89843147d3d556af552e5d --- /dev/null +++ b/d9E2T4oBgHgl3EQfGQZm/content/tmp_files/load_file.txt @@ -0,0 +1,907 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf,len=906 +page_content='Submitted to the Annals of Applied Statistics BAYESIAN ADDITIVE MAIN EFFECTS AND MULTIPLICATIVE INTERACTION MODELS USING TENSOR REGRESSION FOR MULTI-ENVIRONMENTAL TRIALS BY ANTÔNIA A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' DOS SANTOS1, DANILO A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' SARTI1, RAFAEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' MORAL1, ANDREW C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' PARNELL1,2 1Hamilton Institute, Department of Mathematics and Statistics, Maynooth University, Ireland 2Insight Centre for Data Analytics, Maynooth University, Ireland We propose a Bayesian tensor regression model to accommodate the ef- fect of multiple factors on phenotype prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We adopt a set of prior dis- tributions that resolve identifiability issues that may arise between the pa- rameters in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Simulation experiments show that our method out- performs previous related models and machine learning algorithms under different sample sizes and degrees of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We further explore the ap- plicability of our model by analysing real-world data related to wheat pro- duction across Ireland from 2010 to 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Our model performs competitively and overcomes key limitations found in other analogous approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Finally, we adapt a set of visualisations for the posterior distribution of the tensor effects that facilitate the identification of optimal interactions between the tensor variables whilst accounting for the uncertainty in the posterior distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The phenotypic performance of a cultivar is associated with many po- tentially interacting variables (Hara, Piekutowska and Niedbała, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' These may include, but are not limited to: genetic factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' environment exposure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' soil type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' climatic conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' and season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Any combinations of these factors may contribute, either positively or negatively, to the variability of the production of the crop of interest (Kross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Statistical mod- elling of the effect of these variables, both singly and jointly, is an important decision-making tool for farmers and those in the agricultural sector for predicting e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' yield (Adisa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' One of the main interactions that is believed to impact most the production of a crop is the one between genotype and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For notational convenience, we denote such interactions as G × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' This sort of interaction is characterised by cultivars that do not behave consistently in differing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Therefore, it is necessary to estimate the amount of variation in crop yield that is caused by the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Many models have been proposed to estimate G × E (Gauch Jr, Piepho and Annicchiarico, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Crossa, Vargas and Joshi, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Gauch Jr, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The most popular is perhaps the additive main effects and multiplicative interaction (AMMI) model (Gauch Jr, 1988), which consists of two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The first term is the additive component, which contains the main effects of categorically structured genotype and environmental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The second term involves a sum of a multiplication of parameters, which are constrained to an orthonormal space and represent how strong/weak the interactions between the genotypes and environments are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To date, the models in this area have mostly been restricted to using these two sole (but important) covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Our approach allows for more components beyond genotype and environment to be included in the AMMI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We follow the Bayesian tensor regression technique of Guhaniyogi, Qamar and Dunson (2017) to allow for any number of interacting categori- cal factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Tensors are algebraic structures that generalise matrices and provide a generic Keywords and phrases: BAMMIT model, Tensors, Bayesian Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='03655v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='ML] 9 Jan 2023 2 way of describing multidimensional arrays on a given number of axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Tensor decomposition methods have the advantage of capturing the information in the data with a multi-linear struc- ture and bring a unique representation without the requirement for additional constraints like sparsity or statistical independence (Jørgensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The two main tensor decomposi- tions are the PARAFAC (Carroll and Chang, 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Harshman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 1970) and Tucker models (Tucker, 1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Tensors have been used in many fields of study, including physics (Gaillac, Pullumbi and Coudert, 2016), chemistry (Facelli, 2011), medicine (Peyrat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 2007), and data mining (Mørup, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Guhaniyogi, Qamar and Dunson (2017) propose a tensor-based Bayesian regression model where vector/tensor covariates are used to estimate a univariate response through a class of multiway shrinkage priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' They illustrate the model on real- world data from the brain connectome as well as providing theoretical results concerning the speed at which the posterior distribution converges to the true posterior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', contraction rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Similarly, Papadogeorgou, Zhang and Dunson (2021) propose a soft tensor regression to investigate the connection between human traits and brain structural connectomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In this paper, we propose the Bayesian additive main effects and multiplicative interaction tensor model (BAMMIT), which generalises the AMMI model to contain a tensor of inter- acting terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We extend the standard AMMI model to include new parameters to the additive and multiplicative terms of the model, taking into account factors other than genotype and environment on the phenotype of a given cultivar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Common extra factors might include soil types, replications, time points, or growth stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We present our new model in a Bayesian hierarchical format where we place prior distributions on the main and tensor product terms so as to guarantee the model’s identifiability and impose orthonormality constraints, which are an essential part of both the original AMMI and our BAMMIT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Our model as proposed is easily extendable to more complex dependence structures, and we explore how one such extension (time dependence) might be used in our case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We evaluate our proposed approach through a set of simulation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Our interest is to investigate the model’s performance when the complexity increases, that is, when other variables besides genotype and environment are included and there are different sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We compare the prediction of our model with other machine learning models in terms of the root mean squared error (RMSE) and the coefficient of determination (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We explore the proposed model in a real-world application where we analyse wheat data gathered across Ireland from 2010 to 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Again, our model demonstrates competitive performance when compared to previous approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Finally, we show through a new set of visualisations how the posterior distribution of the components of the BAMMIT model can be better assessed in order to quickly identify optimal interactions as well as the uncertainty associated with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Our paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='1, we review the AMMI model and present the constraints imposed on its two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='2, we introduce our BAMMIT model with its extended additive and multiplicative terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We outline the interpretability and identifiability constraints, as well as the priors considered for the parameters and a description of obtaining the posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In Section 3, we compare the results from BAMMIT with other relevant models based on synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In Section 4, we analyse a real-world application involving wheat production in Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Finally, we review and discuss the findings of the work in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' A Bayesian AMMI Tensor (BAMMIT) Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In this section, we review the vanilla AMMI model and define terminology and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We then introduce the BAMMIT model detailing the necessary constraints to ensure identifiability as well as the prior distributions and inferential scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The AMMI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The traditional AMMI model takes into account only two cate- gorical factors, genotype and environment, and is given by a combination of two parts, one additive and one multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Let yij be the outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We write the model as: yij = µ + b(1) i + b(2) j + Q � q=1 λqβ(1) iq β(2) jq + εij, εij ∼ N(0,σ2), (1) where b(1) i and b(2) j represent the marginal effect of the ith genotype and jth environment, respectively, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',B1 and j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The bilinear term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' the summation) is com- posed of Q components, each of which having a variable λq and the scores β(1) iq and β(2) jq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The parameter λq measures the interaction strength of the qth component and is ordered such that λ1 ≥ λ2 ≥ ··· ≥ λQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The scores β(1) iq and β(2) jq represent the importance of the ith genotype and the jth environment in the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To ensure identifiability, the bilinear term is con- strained so that � i β(1) iq β(1) iq′ = � j β(2) jq β(2) jq′ = 0, for q ̸= q′ and � i(β(1) iq )2 = � j(β(2) jq )2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' There are a range of approaches to estimating the parameters of the AMMI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In the frequentist paradigm, the additive term of Equation (1) is estimated by ordinary least squares ignoring the interaction term, and subsequently a singular value decomposition (SVD) on the matrix of residuals is used to estimate the multiplicative terms (Gabriel, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Within the Bayesian context, Viele and Srinivasan (2000) proposed the use of Markov chain Monte Carlo (MCMC) to estimate the parameters of the AMMI model ensuring that the inherent constraints of the model were not violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Liu (2001) formulated a more stable and compu- tationally faster Gibbs sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Crossa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2011) and Perez-Elizalde, Jarquin and Crossa (2012) proposed a Gibbs sampler such that the algorithm was stabilised and incorporated sta- tistical inference in the visualisation of biplots (Gabriel, 1971), drawing credibility regions for the interaction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' By contrast, Josse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2014) introduced an approach to deal with the overparametrization issue of the model by defining priors for the complete set of pa- rameters ignoring the constraints, then applying a postprocessing on the posterior samples of each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Sarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2021) used Bayesian additive regression trees (BART) in which a ‘double-grow’ BART is responsible for capturing the interaction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The number of terms in the summation, Q, is usually fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' It is assumed that Q ≤ min(B1 − 1,B2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The total variability measured by the principal components is linked to the number Q, such that by setting Q = min(B1 − 1,B2 − 1) the model can capture all the variance in the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In practice, Q is commonly an integer between 1 and 3 as this al- lows for easier interpretation and visualisation of the interaction effects via biplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' However, many approaches can be applied to determine the value of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Examples include Cornelius (1993) who applied parametric significance tests;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' other authors who employed cross valida- tion techniques (dos S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Dias and Krzanowski, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Gabriel, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Hadasch, Forkman and Piepho, 2017), or those using resampling techniques (Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Malik, Forkman and Piepho, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Examples in the Bayesian field include Perez-Elizalde, Jarquin and Crossa (2012) and da Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2015) where the prior choice and Bayes factor deal with deter- mining the number of components of the multiplicative term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The non-parametric Bayesian approach of Sarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2021) bypasses the need to provide Q completely but, like many BART models, suffers from interpretability problems due to the complexity of the regression trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' One of the reasons for the popularity of the AMMI model is its strong predictive perfor- mance (Gauch Jr, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Gauch Jr, Piepho and Annicchiarico, 2008), accuracy (Gauch and Moran, 2019) and its stability evaluation system (Gauch Jr, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Yue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Given its desirable properties, many extensions can be found in the literature, as highlighted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In this work, we aim to maintain the structure of the AMMI model and add the effects of other categorical factors that are commonly available in real-world METs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The BAMMIT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The model in (1) can be extended to include the effect of many factors apart from genotype and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Let yij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='v be an outcome variable, in a setting with a total of N observations and V predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We define the BAMMIT model as: yij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='v = µ + b(1) i + b(2) j + ··· + b(V ) v + Q � q=1 λq � β(1) iq β(2) jq × ··· × β(V ) vq � + εij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2) This is similar to the AMMI model described in (1), however now we have V factors instead of only two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Alternatively, we can rewrite the coefficients of the additive and multiplicative terms of (2) in tensor notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Let b(v) = (b(v) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',b(v) Bv)⊤ be a Bv-dimensional vector of parameters of the vth predictor and β(v) q = (β(v) 1q ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',β(v) Bvq)⊤ be a Bv-dimensional vector of singular values, with q = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Binding the column vectors β(v) q , we get β(v), a matrix of dimension Bv × Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We define N = ��V v=1 Bv � as the total number of observations (though, for example, replication may increase N without any need for extra parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For notational convenience, we define a cumulative direct sum and a cumulative Kro- necker product resulting in an N-dimensional vector as V v=1 b(v) = b(1) ··· b(V ) and �V v=1 β(v) q = β(1) q ⊗ ··· ⊗ β(V ) q , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The direct sum operation is defined such that for vectors a = (a1,a2)⊤ and b = (b1,b2,b3)⊤, for example, a b = (a1 + b1,a1 + b2,a1 + b3,a2 + b1,a2 + b2,a2 + b3)⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Following the tensor notation presented, the BAMMIT model can be written more com- pactly as: y = µ + V v=1 b(v) + Q � q=1 λq � V � v=1 β(v) q � + ε, (3) where y is an N-dimensional vector, as before µ is the grand mean, λq is the strength of the qth component, and ε is a noise vector such that each entry εn ∼ N(0,σ2), with n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Note that each vector b(1),b(2),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',b(V ) consists of B1,B2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',BV values, respectively, each of which representing the levels of a factor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 8 genotypes, 10 environ- ments and 4 soil types would yield B1 = 8,B2 = 10,B3 = 4, with V = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The cumulative direct sum operator then ensures sums of main effects representing all possible combi- nations between levels, each corresponding to one observation in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The additive term represents the individual effect of each predictor, while the summation captures via Q components the interactions between the individual effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In the case where there is only the effect of two variables, the model in Equation 3 is reduced to the AMMI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The sum- mation term provides a regularisation on the complexity of the model, with larger Q yielding a more complex set of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The model in the form presented in Equation (3) allows for the inclusion and study of multiple categorical predictors beyond the standard G × E pair used in AMMI models, and the understanding of their effects in two parts, individually and when interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' As in the traditional AMMI model, Q is fixed and represents how many multiplicative terms are in- cluded in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Common extra predictors that might be added to the model include soil type, time, or growth stages, amongst many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Being able to tractably estimate the effect of each of these on a phenotype would be extremely useful for practitioners, whilst retaining the simple interpretation of the parameters in the AMMI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Prior distributions in the BAMMIT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In order to ensure the tractability of the coefficients in the model, it is necessary to establish restrictions on both the additive and in- teraction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For the main effects term, the only constraint to be made is that the covariates are centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For the interaction term, we note that it is not trivial to ensure the identifiability of each parameter individually, only the entire product term (Guhaniyogi, Qamar and Dun- son, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The constrains we use are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 1′ B1b(1) = 1′ B2b(2) = ··· = 1′ BV b(V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 1′ B1β(1) q = 1′ B2β(2) q = ··· = 1′ BV β(V ) q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' β(1)′β(1) = β(2)′β(2) = ··· = β(V )′β(V ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' λ1 ≥ λ2 ≥ ··· ≥ λQ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In the Bayesian context, these constraints are ensured from the definition at the prior level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For example, in the Bayesian AMMI model proposed byPerez-Elizalde, Jarquin and Crossa (2012), the von Mises-Fisher distribution is considered for the coefficients of the multiplica- tive term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In the tensor field, Guhaniyogi, Qamar and Dunson (2017) introduce multiway shrinkage priors in their tensor regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In our approach, we provide a new method by which the constraints are met by applying the restrictions above through parameter trans- formations which we describe next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Formally, we frame a hierarchical model in which prior distributions of the grand mean, main additive effects and variance parameters are µ ∼ N(µµ,σ2 µ), b(v) ∼ N(0,σ2 b(v)), λq ∼ N+(0,σ2 λ), σ−2 y ∼ G(a0,a1), σb(v) ∼ t+(0,a2), σλ ∼ t+(0,a3), where N, N+, G, and t+ are the Normal, truncated Normal, Gamma, and truncated t-Student distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The hyperparameters of the grand mean µµ and σ2 µ are fixed as are all ak terms, with k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We treat the additive effects as random and so estimate σb(v), though a ‘fixed effects’ version could also be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We express the prior knowledge on the standard deviations of the additive term parameters and the λ parameter using a truncated t distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Additionally, we impose that the estimated λ vector values are in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For the product parameters in the interaction term, we use a transformation to ensure the constraints are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Specifically, we generate an auxiliary variable θβ (v) iq from a standard N(0,1) distribution (the transformation is invariant to the scale of this distribution), with i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',Bv, v = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',V , q = 1,,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Then, we centre by the mean µβ (v) q of the vector β(v) q , that is, for each vector β(v) q we calculate its mean and then subtract it from the auxiliary variable θβ (v) iq for the respective value of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Finally, we get the final β(v) iq value via: β(v) iq = � θβ (v) iq − µβ (v) q ��� i � θβ (v) iq − µβ (v) q �2 �−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Applying this procedure to the parameters of the matrix β(v) guarantees that the identifia- bility constraints (2) and (3) of the model are met in the inferential process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To evaluate the performance of the BAMMIT model, we simulate data from Equation (3) over a grid of V = 2,3,4, where Bv is constructed to allow for differences in the interaction structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We set up the simulation experiments as follows: (i) V = 2 and N = 120, with B1 = 12, B2 = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (ii) V = 3 and N = 480, with B1 = 12, B2 = 10, B3 = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (iii) V = 4 and N = 960, with B1 = 12, B2 = 10, B3 = 4, B4 = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (iv) V = 3 and N = 5000, with B1 = 100, B2 = 10 and B3 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Our goal with the above scenarios is to explore the performance of the BAMMIT model in situations where the AMMI model can be applied (case i) and situations where the number of genotypes/environments is small, medium and large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Scenarios (ii) and (iii) present a chal- lenge to the classic AMMI model because it cannot be applied directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Together, scenarios (i), (ii) and (iii) evaluate our model’s performance when the number of predictors increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Finally, scenario (iv) presents a computationally challenging scenario for BAMMIT as it in- volves a large number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For each of these scenarios we set the real number of terms Qsim = {1,2,3} taking λ = {{10},{8,10},{8,10,12}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We simulated 12 training and 12 test data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In other words, in scenario (i) there are two predictors and 120 observations, setting 12 values for the first predictor and 10 for the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Given these number of observations and variables, we generate three data sets, one where the value of Qsim = 1 and λ = 10, another where Qsim = 2 and λ is defined as 8 and 10, and finally, a data set in which Qsim = 3 and λ takes the values 8, 10 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The same understanding extends to the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In all scenarios, we set µ = 100, and σ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To fit the BAMMIT model, we run a Markov chain Monte Carlo (MCMC) algorithm through the probabilistic programming language Just Another Gibbs Sampler (JAGS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Plum- mer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 2003) and the R package R2jags (Su and Yajima, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We set Q = {1,2,3} as the true number of components, µµ = 100, σ2 µ = 10, a0 = a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='1, and a2 = a3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We use three chains, 4000 iterations per chain, discarding the first 2000 as burn-in, and a thinning rate of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Regarding computational time, a data set with three predictors (V = 3), N = 100 and Q = 1 takes on average one minute to run, whilst to run a data set with N = 1000 takes 30 minutes, and with N = 5000, it takes on average 6 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We discuss computational issues further in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' All experiments were implemented in R, and the code used is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='com/Alessandra23/bammit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To assess the performance of the model when V > 2, we compare BAMMIT with two models extensively employed for prediction purposes, namely Random Forests (RF) and eXtreme Gradient Boosting (XGB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We also compare with the traditional AMMI and AM- BARTI model (Sarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 2021), though these are unavoidably restricted to using only the first two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For the RF model, we use the package randomForest (Liaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 2002) selecting the default settings, mtry= 2 and 500 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For the XGB model, we use the package xgboost (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 2019) setting 50 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For the AMBARTI model we use the package AMBARTI 1 setting 50 trees, 500 as burn-in and 1000 iterations as post burn-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' All the models were fitted to the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We checked the accuracy, using the test data, by comparing the posterior mean estimates with the true parameter values used in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We use the root mean squared error (RMSE) to measure predictive power (how close ˆy is to the true y) and R2 to assess the proportion of explained variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 1The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='com/ebprado/AMBARTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The scatterplot in Figure 1 shows this comparison of the additive portion of Equation (3), taking V = 4, Qsim = 2, N = 960 when the true value of λ = {8,10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Each point is an estimated level of the parameter and the error bars are the 90% credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' By visual inspection, the estimates of the effects of the four main predictors are close to the true values, with narrower intervals for predictors with a greater number of levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' b(1) b(2) b(3) b(4) 2 1 0 1 2 2 1 0 1 2 2 1 0 1 2 2 1 0 1 2 3 2 1 0 1 2 True Estimated Fig 1: Scatterplots of true versus estimated additive term for simulation scenario (iii), setting Qsim = 2, λ = {8,10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The bars represent the 90% credible interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In Figure 2, we compare the estimates against the true values in the case where the number of predictors varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Each point represents an interaction term estimate in a total of 120 (V = 2), 480 (V = 3) and 960 (V = 4) points, and the bars, again, represent the 90% credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We observe that when V = 4, the dispersion is smaller and the interaction estimates are more concentrated around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' This can be explained because as more predictors are added to the additive term of the model, the greater the approximation of the response by the predictors and the smaller the amount approximated by the interaction term, despite inserting more variables in both terms of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Also, note that the interaction is comprised of all the new variables together, and that this interaction may not be that strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For example, suppose we are looking at the genotype × environment × soil type × growth stage interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In this case, the interaction of the four factors together is not as strong as if we were looking only at subsets of these interactions, such as genotype × environment × growth stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' V = 2 V = 3 V = 4 2 0 2 4 2 0 2 4 2 0 2 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='0 True Estimated .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Fig 2: Scatterplots of true versus estimated interaction terms for simulations scenarios (i), (ii) and (iii) setting Qsim = 1 and λ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The bars represent the 90% credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 8 To investigate how much the estimation of the interaction term is influenced by the choice of the value of Q, we study the case when the data are simulated with Qsim = 2 and V = 3, but the number of components in the model fit is Q = {1,2,3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The fixed value in the simulation was determined because, in real-world applications, the true number of terms in the interaction is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Thus we wanted to compare the behaviour of the model in a situation where we know the true value of Q in the data, though it is of course fixed in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Figure 3 shows the 90% credible error bars solely for the interaction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' As expected, the performance of the model setting Q = 3 is better since there is an increase in the complexity of the model fit to match the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' When Q is set too small, as in the left panel, we see the model being unable to capture the interaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' However, when the value of Q used in the model fit is at least as big as the value used in the simulation, we obtain superior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In terms of predictions, Table 1 shows the prediction RMSE and the R2 considering the cases where we have three and four predictors in the models (simulation scenarios (ii) and (iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To fit BAMMIT and AMMI models we used Q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' As stated above, the AMBARTI and AMMI models were fitted disregarding the effects of the other variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Specifically, in scenario (iii), for example, there were three predictors, but the two aforementioned models disregarded the effect of the third predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The BAMMIT model clearly performed better than the other two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In addition to the prediction advantage, our model stands out from RF and XGB as it can accommodate the interaction between variables, while at the same time providing estimates based on posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Another highlight is that BAMMIT is able to satisfactorily explain the variability of the response variable, since the R2 obtained in all scenarios was above 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In a real world scenario where the data were not simulated from the BAMMIT model we might expect that the machine learning approaches would be more competitive in terms of their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' However they would still not allow for clear interpretation of the interaction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' V = 3 V = 4 BAMMIT AMBARTI AMMI RF XGB BAMMIT AMBARTI AMMI RF XGB RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='15 R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='78 TABLE 1 RMSE and R2 for ˆy on out-of-sample data for scenarios (ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Case Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In this section, we investigate the performance of the model on a real data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The data was collected over ten years (2010 – 2019) and concerns the production of a common species of wheat (Triticum aestivum L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=') in Ireland, with the response being the yield of wheat measured in tonnes per hectare (t/ha).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The data comes from the Horizon2020 EU InnoVar project (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='h2020innovar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='eu) and was supplied by the Irish Department of Agriculture, Food and the Marine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The experiments were conducted using a randomised complete block design with four replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The data set contains 85 genotypes and 17 envi- ronments, all anonymised and named as g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',g85 and e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=',e17, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Owing to not all genotypes being observed in each location in all seasons, the total number of observa- tions genotype × location × year × block is 6,368, rather than 14,450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' An advantage of the BAMMIT model is that we are able to impute the missing combinations as part of the model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' A subsample of this data was previously explored by Sarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2021), considering only two factors: genotype and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' However, in our work, we include the additional variables year and block, present in the Irish data set, as a third and fourth effect in the AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION 9 Q = 1 Q = 2 Q = 3 2 0 2 2 0 2 2 0 2 4 2 0 2 4 True Estimated Fig 3: Scatterplots of true versus estimated interaction term for a simulation scenario setting N = 480, Qsim = 2, λ = {8,10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Each plot is a result of the Bayesian fit for three possible values of Q (1,2,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The bars represent the 90% credible interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' BAMMIT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We expect to detect if the year is an important predictor in the models, as affirmed by Hara, Piekutowska and Niedbała (2021) that the ability to predict the yield in a certain year can be useful for making decisions, such as cultivation planning and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We are interested in answering some specific questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Initially, when fixing the year and block effect, we would like to know which genotype has the best performance, in which environment, and also which environment provides the highest yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' When considering all the variables, we investigated which year and block had the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' One of the main tasks is to understand how accurate our predictions are and to examine the uncertainty associated with the answers from the previous questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' As an extension to the model, we include additional structure to the time component of the model by adding autoregressive terms in year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Autoregressive structure on the time predictor (AR-BAMMIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In this particular ap- plication, where one of the variables in the BAMMIT model is the year of production, we have the option of extending the model by applying a different structure to the time predictor in both terms, that is, additive and multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The model is now: yijtr = µ + b(1) i + b(2) j + b(3) t + b(4) r + Q � q=1 λqβ(1) iq β(2) jiqβ(3) tq β(4) rq + ϵijtr, (4) b(3) t = αb + φbb(3) t−1 + ηt, (5) θ(3) tq = αθ + φθθ(3) (t−1)q + ωt, (6) β(3) tq = � θ(3) tq − µθ (3) q ��� t (θ(3) tq − µθ (3) q )2 �−1/2 , (7) where the indexes i,j,t and r are associated to genotypic, environmental, time and block effects, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The priors for the entire model follow the same structure as before, such that Equation (7) is defined to meet the conditions of identifiability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To estimate the autoregressive parameters φb and φθ we use uniform priors in the interval [−1,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' A normal distribution with mean zero and variance 100 was considered for the parameters αb 10 and αθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The priors for ηt and ωt were a truncated t-Student distribution with parameters zero and one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' More details about the model structure are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To fit the BAMMIT model we selected two replicates randomly for training and the last two for validation, in a total of 3,184 observations in each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Therefore, we have V = 4, B1 = 85 genotypes, B2 = 17 environments, B3 = 10 years and B4 = 2 block effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Assuming there is little information to establish the hyperparameters for our priors, we set µµ = 10, σ2 µ = 1, obtained from the mean and the standard deviation of the data, a0 = a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='1, and a2 = a3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To input the number of Q terms, we run the model with Q = 1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The model is fitted with three Markov chains, 4,000 iterations per chain, discarding 2,000 and a thinning rate of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We compare our BAMMIT approach with traditional AMMI and AMBARTI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In addition, we fit the AR-BAMMIT model, following Equations (5) to (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For the AMBARTI model, we considered 50 trees, 1000 iterations as burn-in and 1000 iterations as post burn-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To assess the performance of the predictions ˆy, we display the RMSE and the R2 in Table 2 for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' As the AMBARTI and classic AMMI models can only handle the genotype and environment variables, we performed two procedures to treat the training and test data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The first takes into account all rows in the data set and ignores the year and block variables, so that rows corresponding to the same genotype and the same environment are treated as different (columns AMBARTI ALL YEARS and AMMI ALL YEARS in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In the second approach, the data set are separated by years, so that for each year we run the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Then, we regroup the estimates obtained for each group of years, having a single data set, and calculate the metrics (AMBARTI BY YEAR and AMMI BY YEAR columns in the table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The results presented in the table were observed considering Q = 1 for the BAMMIT, AR-BAMMIT and AMMI models (ALL YEARS and BY YEAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' BAMMIT AR-BAMMIT AMBARTI BY YEAR AMBARTI ALL YEARS AMMI BY YEAR AMMI ALL YEARS RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='60 R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='38 TABLE 2 Metrics for out-of-sample y for different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' ALL YEARS refers to the fact that the model took into account all rows of the data set and ignored the year and the block variables effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' BY YEAR means that the data set was separated by years and the model was run for each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The results presented in Table 2 show that the AMBARTI and AMMI models were su- perior to BAMMIT in terms of prediction when the data set is separated by year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' However, when all the years are together, these models performed poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For the AMBARTI, these results can be explained by the high numbers of genotypes and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' As mentioned above, considering all the 10 years, there are 85 genotypes and 17 environments, and this case the AMBARTI model is not efficient in the generation of the 2B1−1 − 1 and 2B2−1 − 1 2-partition combinations for genotypes and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Thus, due to the high numbers of possible combinations, the interaction component of the AMBARTI model is not able to estimate the interactions between genotypes and environments efficiently and the model per- forms poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Also, the inferior performance of classic AMMI model and AMBARTI was expected, since the adjusted models do not take into account the effects of the other vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' It is Interesting to note that, for the BAMMIT model, the best RMSE was obtained by setting Q = 1 in the model fit, since when Q = 2 and Q = 3 the RMSE was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='69 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='71 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Posterior visualisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' A common way to visualise the genotype and environment interactions in an AMMI model is through biplots (Gabriel, 1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' However, Sarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2021) used a heatmap to visualise the predictions and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In this visualisation ap- proach, it is possible to identify in a more immediate way the best interactions between genotypes and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' A shortcoming of their approach concerns the quantification of uncertainty, which cannot be observed directly on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To address this particular issue, in this work we show the prediction for interactions through a heatmap as in Sarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2021) and the uncertainty is showed as Value-suppressing uncertainty palettes (VSUP) as presented by Inglis, Parnell and Hurley (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' First introduced by Correll, Moritz and Heer (2018) value-suppressing uncertainty palettes are bivariate colour palettes that represent a measure or value and its uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The outputs for each combination of value and uncertainty in traditional bivariate palettes are often shown as a 2D square (for example see, Robertson and O’Callaghan (1986)), However, VSUP plots combine cells in the palette using a tree structure to suppress the measure or value at higher levels of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In VSUP, when the uncertainty is low, more bins are allocated to the colour space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' When increasing in uncertainty, the values are suppressed into fewer bins that blend together their colour value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' By doing this, the values will become more distinct as the level of uncertainty reduces, with the intention of making it easier to detect the difference between low and high uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In Figure 4 we display an ordered heatmap and VSUP legend for the Irish data selecting the year 2015 in block three from the test data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' This year corresponds to the best average wheat production observed in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The remaining years are shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For these plots we use the standard deviation of the predictions as our measure of uncertainty and the value shown is the median prediction for each variable pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The plots are ordered so that generally high predicted yield values are pushed to the top left of the plot and descend to the bottom right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The environments e2, e9, e11 and e16 were the worst environments observed, having a small median value compared to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' On the other hand, environment e1 was the best and most stable, presenting a median yield value higher than the others and a lower uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The environment e6 had a middling production across some genotypes, but with a higher uncertainty when compared to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Applying the same interpretation to the genotypes, we observed that the genotypes g3, g26 and g85 had the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The genotypes g17 and g20 presented a good production of predicted wheat on environment e1, e5 and e7, however with a higher standard deviation than the other genotypes, which also pro- duced around 14 t/ha in this environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The worst genotype × environment combinations observed were e9 × g81, e2 × g81, e2 × g71, e2 × g62, e2 × g48 and e2 × g15, while the best were e1 × g85, e1 × g30, e1 × g26 and e1 × g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The results are comparable to those of Sarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In Figure 5 we plot the median yield by year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We can see that the best production occurred in the years 2015 to 2017, with the first having a smaller variation than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Another important factor to observe is the effect of interaction, especially related to the genotype effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2000) affirms that in multi-environment trials (MET), the primary source of interest in the evaluation of the genotypes is via the genotype effect plus g × e interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' This enables plant breeders to determine not only those genotypes with optimal performance but also evaluate variability across environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In Figure 6 we present the interaction effect added to the individual genotype effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' As in Figure 4, the graph is oriented downwards, that is, high values of the sum (genotype + genotype × environment) at the top and low values at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We observe that genotype g3 is optimal in having a generally high performance score whilst also being stable across environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 12 g81 g17 g15 g71 g62 g48 g38 g60 g5 g13 g37 g36 g20 g10 g30 g26 g3 g85 e1 e7 e5 e6 e4 e11 e16 e9 e2 Environment Genotype ŷ 11 12 13 14 15 sd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='52 Fig 4: Predicted yields from the BAMMIT model for the data set in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Production this year was high, between 11 and 15 tonnes per hectare, with positive emphasis on the environment e1, the genotype g26 and on the combination g26 × e7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 8 12 16 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Year ŷ Fig 5: Median yield by year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The bars are the ˆy plus two times the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' AN EXTENSION TO BAYESIAN AMMI MODELS USING TENSOR REGRESSION 13 g81 g17 g15 g71 g62 g48 g38 g60 g5 g13 g37 g36 g20 g10 g30 g26 g85 g3 e5 e16 e6 e11 e9 e2 e4 e1 e7 Environment Genotype g^ + int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='1 sd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='14 Fig 6: Predicted addiction of genotype and interaction from the BAMMIT model for the data set in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In this paper, we propose a generalisation of the AMMI model which uses the tensor regression approach of Guhaniyogi, Qamar and Dunson (2017) and Papado- georgou, Zhang and Dunson (2021) to extend the model to allow for multiple interacting categorical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The main idea is to allow for more realistic understanding of pheno- typic effects beyond the usually considered pair of genotype and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' We envisage that in the future such models may be used to further indicate interactions between season, soil, weather conditions, growth stage and other potential predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The priors we use on the hierarchical model were built in order to meet the restrictions imposed on the model and to ensure identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For each simulated data set and for the estimates obtained from the Bayesian fit, the constraints were checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In the simulation results, the model displayed promising performance, in the sense that the estimates corresponded with the true values of the parameters from the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' As expected, when the established number of terms in the interaction is three, the Bayesian model had a better fit, regardless of the true number used in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The choice of the number of components Q is still arbitrary, taking into account only the typical values already mentioned in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Nonetheless, a more sophisticated approach to choosing the rank Q can be applied, as shown by Guhaniyogi, Qamar and Dunson (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' When the model was applied to the real data, it was possible to establish which geno- types, environments, blocks and years had the highest wheat production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Also, we could visualise just the interaction effect and so it was possible to determine the optimal interac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The purpose of showing the results in the visualisations presented in this paper is to aid a researcher’s ability to interpret the results and improve recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Although the AMBARTI model had a superior result in relation to BAMMIT, it performed well, even with a small number of interaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' When compared with the Bayesian AMMI model, both BAMMIT and AR-BAMMIT were superior in terms of prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 14 In relation to the computational time, the cost for the method was high, especially when the number of components Q increased;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' when setting Q = 3 it took eight hours to complete the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' This drawback was true in general when the size of the data set was large (around 5,000 total observations or more) with the model taking hours to form a valid posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In future work, we could be use faster computational methods such as variational inference (Blei, Kucukelbir and McAuliffe, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Dos Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', 2022) or those as discussed in Papadogeorgou, Zhang and Dunson (2021) and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' As we have seen, there is an obvious extension of these models through the prior distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' New structures can be added to certain predictors, as was done here for the year variable in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Any temporal and spatial components could have their inherent characteris- tics inserted in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Another important point is the insertion of continuous variables, or latent representations of them, since the current structure does not allow for this type of variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Antônia A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' dos Santos, Andrew Parnell, and Danilo Sarti re- ceived funding for their work from the European Union’s Horizon 2020 research and in- novation programme under grant agreement No 818144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In addition Andrew Parnell’s work was supported by: a Science Foundation Ireland Career Development Award (17/CDA/4695);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' an investigator award (16/IA/4520);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' a Marine Research Programme funded by the Irish Government, co-financed by the European Regional Development Fund (Grant-Aid Agree- ment No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' PBA/CC/18/01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' SFI Centre for Research Training in Foundations of Data Sci- ence 18/CRT/6049, and SFI Research Centre awards I-Form 16/RC/3872 and Insight 12/RC/2289_P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' For the purpose of Open Access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' APPENDIX A: ADDITIONAL RESULTS In this section, we complement the results presented in the Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Figure 7 presents the graphs for the predicted yields for the BAMMIT model applied to the Irish time series data set (except year 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Analysing only the legend of the figures and looking at the value scale it is possible to see that the forecast of wheat yield for the year 2015 (presented in Figure 4) was higher than that for all the other years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In order to clearly observe the behaviour of genotype and environment predictors in each year, we do not scale the estimated value and the uncertainty legend to be equal across plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Thus, we prevent years with high production and low uncertainty having an intense colour while years with contrary behaviour have a washed-out colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' g74 g14 g22 g2 g78 g9 g27 g11 g25 g15 g79 g16 g75 g83 g24 g12 g1 g59 g67 g26 g3 e6 e12 e2 e1 e4 e11 e7 e8 Environment Genotype ŷ 5 7 9 11 13 sd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='50 (i) 2019 Fig 7: Predicted yields from the BAMMIT model for the Irish data set on block three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' 16 APPENDIX B: STUDY OF AR-BAMMIT MODEL The construction of the BAMMIT model allows new structures to be applied to the pa- rameters, such as spatial or temporal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' However, the insertion of these new structures brings greater complexity to the model, since it is necessary that the restrictions continue to be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='1, a temporal architecture is applied to the parameters associated with years in the Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In this section, we intend to explore the aforementioned AR- BAMMIT method (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='1) by carrying out a small simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' To ensure the needed restrictions, we apply the transformation made earlier in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content='3 to the param- eters of the interaction term also to the parameters of the new model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' The simulation scenarios considered were scenarios (ii) and (iii) of Section 3, where only one of the variables in the additive term and one variable in the multiplicative term have the autoregressive structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' In Figure 8, we present the scatterplot of the true and estimated main effects values for the simulation scenario (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Figure 9 shows the posterior density of the precision parameter and a scatterplot of the interaction term in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' b(1) b(2) b(3) 1 0 1 1 0 1 1 0 1 2 1 0 1 2 True Estimated Fig 8: Posterior density for the years and for the first four genotypes and environments, setting V = 3, N = 480, B1 = 8, B2 = 4, B3 = 10, Qsim = 1, λ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' REFERENCES ADISA, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', BOTAI, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' O.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Cultivar evaluation and mega-environment investigation based on the GGE biplot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Crop science 40 597–605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' YUE, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', GAUCH, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', WEI, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', XIE, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', CHEN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', PENG, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', BU, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' and JIANG, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Genotype by Environment Interaction Analysis for Grain Yield and Yield Components of Summer Maize Hybrids across the Huanghuaihai Region in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' Agriculture 12 602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' ZHANG, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', LUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=', RASKUTTI, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' and YUAN, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' ISLET: Fast and optimal low-rank tensor regression via importance sketching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} +page_content=' SIAM journal on mathematics of data science 2 444–479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9E2T4oBgHgl3EQfGQZm/content/2301.03655v1.pdf'} diff --git a/ddE4T4oBgHgl3EQfQQxz/content/tmp_files/2301.04980v1.pdf.txt b/ddE4T4oBgHgl3EQfQQxz/content/tmp_files/2301.04980v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4db9739f5742d5fc5114489cd90a582247770936 --- /dev/null +++ b/ddE4T4oBgHgl3EQfQQxz/content/tmp_files/2301.04980v1.pdf.txt @@ -0,0 +1,1768 @@ +Prepared for submission to JHEP +Bound on the central charge of CFTs in large dimension +Abhijit Gaddea, Mrunmay Jagadaleb, Shraiyance Jaina and Trakshu Sharmaa +a Department of Theoretical Physics, Tata Institute of Fundamental Research, +Mumbai 400005, India. +b Walter Burke Institute for Theoretical Physics, California Institute of Technology, +Pasadena, CA 91125, USA. +E-mail: abhijit@theory.tifr.res.in, mjagadal@caltech.edu, +shraiyance.jain@tifr.res.in, trakshu.sharma@tifr.res.in +Abstract: +In this paper, we use crossing symmetry and unitarity constraints to put a +lower bound on the central charge of conformal field theories in large space-time dimensions +D. +Specifically, we work with the four-point function of identical scalars φ with scaling +dimension ∆φ, and use a certain class of analytic functionals to show that the OPE coefficient +squared c2 +φφT µν must be exponentially small in D. For this to hold, we need to make a mild +assumption about the nature of the spectrum below 2∆φ. Our argument is robust and can be +applied to any OPE coefficient squared c2 +φφO with ∆O < 2∆φ. This suggests that conformal +field theories in large dimensions (if they exist) must be exponentially close to generalized +free field theories. +arXiv:2301.04980v1 [hep-th] 12 Jan 2023 + +Contents +1 +Introduction +1 +2 +Review of analytic functional bootstrap +2 +2.1 +Functionals for OPE coefficient maximization +3 +2.2 +Analytic functionals for z = ¯z +4 +2.3 +Computing the functional in large D +7 +3 +Bounds on Central Charge in Large D CFTs +9 +3.1 +Modifying the set P +12 +3.2 +Applicability of the bound +13 +4 +Corrections at large but finite D +15 +A Simplifying the functional +18 +B Conformal blocks in large D and saddle points +21 +1 +Introduction +The central charge of a conformal field theory is a measure of its degrees of freedom. For a +unit-normalized stress tensor T µν, the central charge is inversely proportional to the three- +point function coefficient squared c2 +φφT µν for any operator φ. Hence, it governs the strength +of the gravitational coupling in the dual theory in anti-de Sitter space. Therefore, bounding +the central charge is important from the point of view of charting not only the space of CFTs +but also the landscape of quantum gravity in AdS. +Lower bounds on the central charge of CFTs in two and four dimensions have been +computed using numerical bootstrap [1, 2].1 It is believed that no non-trivial conformal field +theory exists in dimensions greater than six2. Here, a non-trivial CFT means that it is (a) +unitary, (b) not free, and (c) contains the stress tensor in its spectrum. One can alternatively +formulate this conjecture as: +• In D > 6, a unitary CFT that is not free must have c2 +φφTµν = 0. +The advantage of this formulation is that it opens a way of addressing this problem in a +quantitative way. In this paper, we will show: +1See [3–6] and references therein for an introduction and review of the vast literature on the conformal +bootstrap program and [7] and references therein for a focus on analytic methods. +2See [8] for discussion regarding this point. +– 1 – + +• In large D, a unitary CFT - with a reasonable condition on the low lying spectrum (this +includes not being free) - must have c2 +φφTµν < Aφe−αφD. The exponent αφ is an O(1) +number given in equation (3.5). +The problem of constraining CFTs in large dimensions was considered in [8] by two of the +present authors. It was concluded that the unitary CFTs in large dimensions must be expo- +nentially close to generalized free field theories in a certain sense. Happily, our results in this +paper concur with the results of [8]. We will comment more on the connection towards the +end of section 3. +The exponential lower bound on the central charge may lead one to naively conclude that +there is an exponential hierarchy between the cosmological constant scale and the Planck scale +in large dimensions. This is not so. For a D-dimensional CFT, +MD−1 +Λ +GN ∼ c2 +φφTµν = Aφe−αφD +⇒ +MΛ +MP +∼ e−αφ. +(1.1) +We will use the crossing symmetry and unitarity of the four-point function of identical +scalars φ in a large D CFT to put upper bounds on the OPE coefficient c2 +φφTµν. This is +accomplished using analytic functional bootstrap [9–15]. In particular, we will use the analytic +functionals in [11] that were used to bound certain OPE coefficients in one-dimensional CFTs +in the large ∆φ limit. We will review these tools in section 2. Their application to CFTs in +large dimensions is made in section 2.3. The lower bound on the central charge for CFTs in +large dimensions is obtained in section 3. In section 4, we use the same functionals to obtain +approximate numerical bounds for CFTs in large but finite dimensions. The paper contains +two appendices that supplement the discussion in the bulk of the paper. +2 +Review of analytic functional bootstrap +Consider the four-point function of identical scalar primary operators φ of dimension ∆φ. +This four-point function is fixed by conformal symmetry up to a function of cross-ratios (u, v) +as, +⟨φ(x1)φ(x2)φ(x3)φ(x4)⟩ = +1 +|x12x34|2∆φ G(u, v) +(2.1) +where +u ≡ z¯z = x2 +12x2 +34 +x2 +13x2 +24 +, v ≡ (1 − z)(1 − ¯z) = x2 +14x2 +23 +x2 +13x2 +24 +. +The s-channel OPE expansion i.e. corresponding to x1 → x2, x3 → x4 is convergent in z, ¯z ∈ +C\[1, ∞). The t-channel OPE expansion x1 → x4, x2 → x3 is convergent in z, ¯z ∈ C\(−∞, 0]. +We will consider the correlator in the overlapping region of convergence (z, ¯z) ∈ R×R where +R = C \ {(−∞, 0] ∪ [1, ∞)}. The OPE expansions take the form, +G(u, v) =s +� +O∈φ×φ +c2 +φφOG∆O,ℓO(z, ¯z) +(2.2) +G(u, v) =t +(z¯z)∆φ +((1 − z)(1 − ¯z))∆φ +� +O∈φ×φ +c2 +φφOG∆O,ℓO(1 − z, 1 − ¯z) +(2.3) +– 2 – + +Due to conformal symmetry and permutation symmetry only operators with even spin ℓ +appear in these expansions. The equality of expansions (2.2) and (2.3) is called the crossing +equation. It is convenient to express the crossing equation in terms of an elegant sum rule, +� +O∈φ×φ +c2 +φφOF ∆φ +∆O,ℓO(z, ¯z) = 0 +(2.4) +where +F ∆φ +∆O,ℓO(z, ¯z) ≡ G∆O,ℓO(z, ¯z) +(z¯z)∆φ +− G∆O,ℓO(1 − z, 1 − ¯z) +((1 − z)(1 − ¯z))∆φ . +The functions F ∆φ +∆O,ℓO(z, ¯z) are holomorphic in R × R and obey +F(z, ¯z) = F(¯z, z) = −F(1 − z, 1 − ¯z). +(2.5) +Let us call the vector space of such functions V. Unitarity implies cφφO ∈ R and hence c2 +φφO ≥ +0. Therefore, the sum rule sets a positive linear combination of the vectors F ∆φ +∆O,ℓO(z, ¯z) to +zero. +Consider a linear functional ω, that is an element of the dual of the vector space. One +can act this functional ω on the sum rule (2.4) to get, +� +O∈φ×φ +c2 +φφO ω +� +F ∆φ +∆O,ℓO(z, ¯z) +� += 0. +(2.6) +Some simple example of such functionals include evaluation and taking derivatives at a point +in R×R. The numerical bootstrap typically uses ω to be derivatives at the crossing symmetric +point z = ¯z = 1 +2. Note that to get (2.6) from (2.4) we have swapped the action of functional +ω with an infinite sum over operators appearing in the OPE. Not all functionals satisfy this +property. Following [10] we call this property of ω, the swapping condition. Further we want +the functionals to be finite on F ∆φ +∆,ℓ(z, ¯z) with ∆, ℓ satisfying the unitarity bound i.e. ∆ ≥ d−2 +2 +for ℓ = 0 and ∆ ≥ ℓ + d − 2 for ℓ > 0. We will only consider functionals which satisfy the +swapping and finiteness conditions. +2.1 +Functionals for OPE coefficient maximization +In this paper, we will be concerned with obtaining an upper bound on the OPE coefficient +squared c2 +φφOb of a primary operator Ob ∈ φ × φ. Let P be the set of all (∆, ℓ) values where +the functional is non-negative and S be the set of all CFT operators except for identity 1 and +Ob. The sum rule constraining CFT data is, +F ∆φ +1 +(z, ¯z) + c2 +φφObF ∆φ +∆b,ℓb(z, ¯z) + +� +(∆,ℓ)∈S +c2 +φφOF ∆φ +∆,ℓ(z, ¯z) = 0 +(2.7) +The action of a functional ω on the sum rule is, +ω(1) + c2 +φφObω(∆b, ℓb) + +� +S +c2 +φφOω(∆, ℓ) = 0. +(2.8) +– 3 – + +0 +1 +2 +3 +4 +5 +6 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +Δ +ω +1 +2 +S +Figure 1: Extremal Functional example +Therefore, the OPE coefficient c2 +φφOb can be expressed as +c2 +φφOb = − +ω(1) +ω(∆b, ℓb) − +� +S c2 +φφOω(∆, ℓ) +ω(∆b, ℓb) +(2.9) +At this point, it is easy to see that we can obtain an upper bound on c2 +φφOb by constructing +a functional that satisfies, +ω(1) < 0, +ω(∆b, ℓb) > 0, +S ⊂ P. +(2.10) +Existence of such functional would give us the bound, +c2 +φφOb ≤ +−ω(1) +ω(∆b, ℓb). +(2.11) +Moreover this inequality is saturated when ω(∆, ℓ) = 0, ∀(∆, ℓ) ∈ S. Such functionals are +called extremal functionals. A cartoon of an extremal functional for one dimensional CFT is +given in figure 1, where ∆b is taken to lie between 1 and 2 and S is {∆ : ∆ > 2}. The set S +is precisely the set of double zeros of the functional. +2.2 +Analytic functionals for z = ¯z +In this section we will consider analytical functionals that act on the functions restricted to +the locus z = ¯z. These functionals were first constructed in [11] for one dimensional CFTs. +It is straightforward to repurpose them as functionals for CFTs in general dimensions but +acting only on the specialization z = ¯z. Let ˜V be the vector space of functions F(z) that is +holomorphic in R and obey F(z) = −F(1 − z). After specializing a function in V to z = ¯z, +we precisely get a function in ˜V. +– 4 – + +The authors of [11] consider a class of functionals acting on ˜V given by the integral of +the discontinuity along the branch cut [1, ∞) weighted by a kernel h(z). +ω(F) = +1 +2πi +� ∞ +1 +dz h(z)Disc[F(z)] = +1 +2πi +� 0 +−∞ +dz h(1 − z)Disc[F(z)]. +(2.12) +Here Disc[F(z)] = limϵ→0+ F(z + iϵ) − F(z − iϵ) is the discontinuity along the branch cut. +In the second equality we have done the change of variables from z → 1 − z and used +F(1 − z) = −F(z). The kernel h(z) is analytic on C \ (−∞, 1) with possible branch cuts at +(−∞, 0] and [0, 1]. Without loss of generality, we can assume h(z) ∈ R for z ∈ (1, ∞). This +implies h∗(z) = h(z∗). The kernel h(z) satisfies the properties, +1. h(z) is analytic away from possible poles or branch points at z = 0, 1 and ∞. +2. h(z) is bounded by A1|z|−1−ϵ1 for some A1, ϵ1 > 0 as z → ∞. +3. The discontinuity of h(z) around z = 1 is bounded by A2|z − 1|2∆φ−1+ϵ2 for some +A2, ϵ2 > 0 as z → 1. +The second and the third properties follow from finiteness and swapping conditions respec- +tively. See [10, 11] for details regarding this point. The action of the functional in (2.12) on +the function F ∆φ +∆,ℓ(z, z) is given by +ω(∆, ℓ) = +1 +2πi +� 0 +−∞ +dz h(1 − z)Disc +�G∆,ℓ(z) +z2∆φ +− G∆,ℓ(1 − z) +(1 − z)2∆φ +� +(2.13) +Here, we use G∆,ℓ(z) = G∆,ℓ(z, z) for the conformal block specialized to z = ¯z. After certain +contour manipulations detailed in appendix A, the functional reduces to, +ω(∆, ℓ) = g(∆, ℓ) − Im +� +eiπ(∆−2∆φ)f(∆, ℓ) +� +, +(2.14) +where, f and g are given by, +f(∆, ℓ) = +� 0 +−∞ +dz f(z) +ˆG∆,ℓ(z) +|z|2∆φ +(2.15) +g(∆, ℓ) = +� 0 +−∞ +dz (1 − z)2∆φ−2g +� +z +z − 1 +� ˆG∆,ℓ(z) +|z|2∆φ += +� 1 +0 +dz g(z)G∆,ℓ(z) +z2∆φ +. +(2.16) +Here we have introduced3 +ˆG∆,ℓ(z) = |G∆,ℓ(z + iϵ)| +for z ∈ (−∞, 0) +(2.17) +f(z) = h(z) − h(1 − z) +π +for Im(z) > 0, +(2.18) +g(z) = −Disc [h(z)] +2πi +for z ∈ (0, 1) +(2.19) +3Our definition of f(z) differs from the one used in [11] by a factor of i. That is the reason Im[f(z)] appears +in our gluing condition (2.20) rather than Re[f(z)]. +– 5 – + +The function g is analytically continued in the z variable and the function f is continued to +the region with Im(z) < 0 via f(z) ≡ −f(1 − z). With this analytic continuation, we have +f∗(z) = f(z∗). The definitions of the kernels f and g in terms of the kernel h imply that they +obey the following relation called the gluing condition, +Im[f(z)] + g(z) + g(1 − z) = 0 +for z ∈ (0, 1). +(2.20) +This manipulation is valid for ∆ > ∆c, where ∆c is some positive scaling dimension. This +is because although the functional ω(∆) is always finite by definition, the individual integrals +in equations (2.15) and (2.16) can diverge near z = 0. Since the conformal block goes like +|z|∆ as z → 0, the integrals (2.15) and (2.16) are convergent for ∆ greater than certain ∆c. +The precise value of ∆c depends on the behavior of the kernel h(z) near z = 0. Outside this +range i.e. for ∆ < ∆c, we need to resort to the manifestly finite expression of the functional +(2.13) for evaluation. +Since Disc[h(z)] is purely imaginary and G∆,ℓ(z) is real for z ∈ (0, 1), this means g(∆, ℓ) +is real. However h(z + iϵ) generically has both real and imaginary parts. So f(∆, ℓ) can +be written as f(∆, ℓ) = ir(∆, ℓ)eiπγ(∆,ℓ) with r(∆, ℓ) ∈ R+ and γ(∆, ℓ) ∈ R. With this, the +functional (2.14) becomes, +ω(∆, ℓ) = g(∆, ℓ) − r(∆, ℓ) cos(π(∆ − 2∆φ + γ(∆, ℓ))). +(2.21) +If the choice of h is such that g, r, γ are slowly varying compared to the oscillations of the +cosine in above equation and further if r(∆, ℓ) = g(∆, ℓ), then one gets an extremal functional +which has double zeros at +∆n = 2∆φ + 2n − γ(∆n, ℓ) +n ∈ Z≥0. +(2.22) +The functional, in particular the integrals f(∆, ℓ) and g(∆, ℓ) are difficult to compute +analytically in general. If the conformal dimensions ∆φ and ∆ both are taken to be large +then these integrals can be performed via saddle point approximation. So at this stage, we +will take the limit of large D. This will also set all the conformal dimensions to be large, +thanks to the unitarity bound. +If we want the saddle points of f(∆, ℓ) and g(∆, ℓ) to be universal and not depend on the +kernel f and g, and if we further want f(∆, ℓ) and g(∆, ℓ) to be of the same order (which is +necessary to get the double zeroes (2.22)) then we need to take +f(z) ∼ O(1) +for Im(z) > 0, +g(z) = (1 − z)2∆φ˜g(z), +with ˜g(z) ∼ O(1) +for z ∈ (0, 1). +(2.23) +In the limit of large ∆, ∆φ and with the scaling of kernels f(z) and g(z) given above, it is easy +to see that the integrals for f and g are convergent for ∆ below some ∆c. If f(z) = O(z−1+ϵ) +near z = 0 then ∆c = 2∆φ. +We will assume that f(z) obeys this property. +Therefore, +the computation of the functional ω(∆) can be divided into three regions, depending on the +method of computation, namely (I) ω(∆ > 2∆φ), (II) ω(0 < ∆ < 2∆φ) and (III) ω(1). +– 6 – + +2.3 +Computing the functional in large D +In this section we will compute the functional ω(∆, ℓ) in the large D limit. For this we will +need the conformal block in the large D limit. These blocks were first computed in [16]. They +gave an explicit expression in terms of the hypergeometric function, +G∆,ℓ(y+, y−) = +2∆ +√y− − y+ +A∆(y+)A1−ℓ(y−) +(2.24) +with, +Aβ(x) = x +β +2 2F 1 +�β +2 , β − 1 +2 +, β − D +2 + 1, x +� +, +y± = +z¯z +(1 ± +� +(1 − z)(1 − ¯z))2 . +(2.25) +One immediate observation is that on the z = ¯z slice, y− = 1. Hence, the spin dependance of +the block trivializes as we get A1−ℓ(1) = 1. We will drop the spin label from now on as the +functional ω(∆, ℓ) does not depend on ℓ in the large D limit. We are interested in scaling all +∆ = δD. In this limit the large D block simplifies even further on z = ¯z locus. We get +GδD(z) = +� +1 − +� +z +2 − z +�2 �− 1 +2 vδ(z) eDqδ(z). +(2.26) +The functions vδ(z) and qδ(z) are given in equation (B.6). The saddle point in integrals (2.15) +and (2.16) come from extremizing GδD(z)/z2δφD with respect to z in the large D limit. This +is computed in appendix B. We get, +z∗(δ) = (2δφ − δ)(2δφ + δ − 1) +δφ(4δφ − 1) +. +(2.27) +No we will evaluate the functional in the three regions mentioned earlier. +(I) δ > 2δφ: For this range of δ the saddle point z∗ ∈ (−∞, 0). The functional is +ω(∆ > 2∆φ) = µ(δφ, z∗) +� +(1 − z∗)−2˜g +� +z∗ +z∗ − 1 +� +− cos(π(∆ − 2∆φ + γ(z∗))) |f(z∗)| +� +(2.28) +where, µ(δφ, z∗) is a positive pre-factor given below. It is independent of f(z) and ˜g(z). +µ(δφ, z∗) = +ˆGδD(z∗) +|z∗|2δφD +� +2π +−δφDq′′ +δ(z∗). +(2.29) +The phase γ(z) is defined by f(z + i0+) = i|f(z)|eiπγ(z) and is of O(1). It is clear from the +expression that the functional ω(∆) for ∆ ≥ 2∆φ is oscillating because of the cosine factor. +Since we want positive functional for ∆ > 2∆φ, this means the first term in the square +brackets should be greater than or equal to the second, +(1 − z)−2˜g +� +z +z − 1 +� +≥ |f(z)| +for z ∈ (−∞, 0). +(2.30) +– 7 – + +To get an extremal functional we will require this inequality be saturated. Extremality along +with the gluing condition (2.20) yields the constraint on the kernel f(z), +Im(f(z)) = z2∆φ−2|f(1/z)| + (1 − z)2∆φ−2|f(1/(1 − z))| = 0 +z ∈ (0, 1). +(2.31) +In the second equality we use the fact that ∆φ is large, z ∈ (0, 1) and f(z) ∼ O(1). +The functional takes the form +ω(∆ > 2∆φ) = 2µ(δφ, z∗)|f(z∗)| sin2 �π +2 (∆ − 2∆φ + γ(z∗)) +� +. +(2.32) +It has double zeroes at ∆ = 2∆φ + 2n − γ(z∗). +(II) 0 < δ < 2δφ: For this range of δ the integrals (2.15) and (2.16) are divergent so we have +to use the original definition of the functional (2.13) to evaluate it. This is done in appendix +B. The result is, +ω(0 < ∆ < 2∆φ) = Re (f(z∗)) GδD(z∗) +z2δφD +∗ +� +2π +δφDq′′ +δ(z∗) +(2.33) +Here z∗ is the one defined in (2.27). Notice that the value of the functional scale exponentially +with D. +(III) Identity operator 1: This functional is also evaluated in appendix B. The result is, +ω(1) = − +� 0 +−∞ +dz|f(z)|. +(2.34) +Recall the conditions (2.10) on ω to get a bound on the OPE coefficient. It is clear from +(2.34) that ω(1) < 0 is already obeyed. To impose the condition ω(∆b, ℓ) > 0, we need to +impose Re(f(z)) > 0 for z ∈ (0, 1). As the functional is non-negative for δ > 2δφ, the set of +operators with quantum numbers {(∆, ℓ) : ∆ > 2∆φ} is definitely contained in P. If the rest +of the CFT operators also lie in the set P then the bound on the OPE coefficient squared for +∆b ≤ 2∆φ is +c2 +φφOb ≤ − ω(1) +ω(∆b) = +� 0 +−∞ dz|f(z)| +Re(f(zb)) +�� +δφD +2π +z2δφD +b +� +q′′ +δ(zb) +GδbD(zb) +, +� +zb = z∗(δb). +(2.35) +The expression inside the big parenthesis is independent of f(z). It is evaluated explicitly in +appendix B. Let us collect all the properties of f(z). +1. It is analytic in C \ (−∞, ∞) with possible singularities only at 0, 1 and ∞. +2. f(z) = −f(1 − z). This follows from the definition (2.18) of f(z). +3. Im[f(z)] = 0 for z ∈ (0, 1). This is derived in (2.31). +– 8 – + +4. f(z) = O +� +z−1+ϵ� +near z = 0. This is needed for ∆c = 2∆φ +5. f(z) = O +� +z−1−ϵ� +near z = ∞. This follows from decay property of h(z). +If we show the existence of f(z) satisfying above properties then the bound (2.35) holds. As +shown in appendix B, the bound takes the form c2 +φφOb < A(δb, δφ)e−α(δb,δφ)D with α(δb, δφ) > 0 +for δb < 2δφ. The explicit expressions for A(δb, δφ) and α(δb, δφ) are cumbersome and are give +in appendix B. Note that exponent α is robust and does not depend on f(z). However, the +set P of the quantum numbers (∆, ℓ) where the functional ω is non-negative depends crucially +on f(z). +3 +Bounds on Central Charge in Large D CFTs +In this section, we will apply the technology of section 2 to obtain lower bound on the central +charge CT of a CFT in large D. +When the stress tensor of the CFT is canonically normalized i.e. when it obeys the Ward +identity +∂µ⟨T µνφ(x1) . . . φ(xn)⟩ = − +� +i +δ(x − xi)⟨φ(x1) . . . ∂µφ(xi) . . . φ(xn)⟩, +(3.1) +the central charge governs its two point function. +⟨T µν(x)T λσ(0)⟩ = CT +S2 +d +1 +x2d +�1 +2(IµλIνσ + IµσIνλ) − 1 +Dδµνδλσ� +. +(3.2) +Here Iµν(x) ≡ δµν −2xµxν/x2 and Sd = 2πD/2/Γ(D/2) is the volume of the D−1 dimensional +sphere. With this definition of the central charge, its value for a single free degree of freedom +is +• Cscalar +T += D/(D − 1) +• Cfermion +T += D/2 +• C(D−2)/2 form +T += D2/2. +There is 1 degree of freedom for a free scalar field and this value for a free fermion field and +a free (D − 2)/2-form are 2D/2 and Γ(D − 2)/Γ2(D/2) respectively. +In order to apply the analytic bounds for the stress tensor we must first normalize its +two point function to unity. With the new normalization, CT appears in the OPE coefficient +squared c2 +φφT µν as +c2 +φφT µν = 1 +CT +� ∆φD +(D − 1) +�2 +. +(3.3) +– 9 – + +1 +2 +3 +4 +5 +6 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +δϕ +α(δϕ,1) +Figure 2: The function α(δφ, 1) is plotted against δφ. It is always greater than zero for +δφ > 1/2. +If we now apply the bound computed in section 2, we get +CT > +� ∆φD +(D − 1) +�2 +1 +A(δφ, 1)eα(δφ,1)D. +(3.4) +Here we have set δb = 1 because the role of Ob is played by T µν and because its conformal +dimension is D, δb = 1. For δb = 1, the expression for α simplifies. +α(δφ, 1) = log +� +2(2δφ − 1) +�4δφ − 1 +� 4δφ − 1 +2(2δφ − 1) +�2δφ� +. +(3.5) +It is easy to see that α > 0 for all values of δφ in the unitary range i.e. for δφ > 1/2. The +plot of α(δφ, 1) against δφ is given in figure 2. The function A(δφ, 1) depends on the details +of the function f(z). Recall that this bound applies if the CFT operators S ⊂ P. The set P +depends on f(z). +By construction the functional ω(∆, ℓ) is non-negative for ∆ ≥ 2∆φ (for all ℓ). The sign +of the functional ω(∆, ℓ) for ∆ < 2∆φ depends only on the sign of Re (f (zb)) and hence +can be positive or negative depending on the choice of kernel f(z). Ideally one would want +Re (f (zb)) ≥ 0 for all operators (except identity) satisfying unitarity bounds with ∆ < 2∆φ. +This would show that the central charge is exponentially large in D in any large dimensional +conformal field theory except for the free theory. We will now show that this condition, +Re (f (zb)) ≥ 0 +1 +2 ≤ δ < 2δφ, +(3.6) +is impossible to achieve with the kind of functionals we are using i.e. (2.13). To see this, +notice that the kernel f(z) is necessarily anti-symmetric around z = 1/2 (see equation (2.18)). +This means f(1/2) = 0, and the Re(f(1/2 ± a)) are of opposite sign for any a. The functional +– 10 – + +δ +ω +δϕ<3/4 +δ* +1/2 +1 +2δϕ +Double Trace +δ +ω +δϕ>3/4 +δ* +1/2 +1 +2δϕ +Double Trace +Figure 3: Schematic plots of functional ω(δ, ℓ) with kernel f(z) = fopt(z), for δφ < 3/4 (left) +and δφ > 3/4 (right). Functional at stress tensor i.e. with δ = 1, is positive and functional at +identity (not shown) is negative, in both cases. The ℓ dependence is not shown because the +signature of ω(δ, ℓ) is independent of ℓ. +ω(∆, ℓ) for ∆ < 2∆φ is proportional to Re (f (zb)) (times a positive factor), and of opposite +sign for zb = 1/2±a. The functional vanishes for zb = 1/2. Solving the condition z∗(δb) = 1/2, +we get a critical value of δb where the functional must vanish,4 +δcrit = 1 +2 +� +1 + +� +2δφ − 1 +� +4δφ − 1 +� +. +(3.7) +This means that the functional can never be of the same signature in the whole range 1/2 ≤ +δ < 2δφ. Its sign has to change at δcrit. This means we can not satisfy (3.6). With this +analysis we also conclude that we do not get any bound on the central charge for δφ = 3/4. +This is because for this value of δφ, δcrit = 1. As the functional vanishes for δ = 1, the central +charge is unbounded from below. For other values of δφ we can certainly obtain exponentially +large lower bound on the central charge given the set of CFT operators S ⊂ P. We will +characterize the set P below. As discussed above, the operators with δ ≥ 2δφ belong to P so +we will only be concerned with characterizing P of operators with δ < 2δφ. +Case I: δφ < 3/4 +In this case, the ω(∆) is positive for ∆ > δcritD. The schematic plot of the functional is +shown in figure 3 on the left. Hence P consists of all the operators with ∆ > δcritD. We +can further optimize over the kernel f(z) to produce the lowest upper bound on the OPE +coefficient squared c2 +φφOb. This optimization problem is the same one as is [11]. Borrowing +their result, +fopt(z) = +(1 − 2z) +[z(z − 1)]1/2 �� +z(1 − z) + +� +zb(1 − zb) +�2 , +zb = z∗(δb). +(3.8) +For the case of stress tensor, we use δb = 1 and the equation (3.3) that relates c2 +φφT µν. +4The other solution 1 +2 +� +1 − √2δφ − 1√4δφ − 1 +� +lies outside the unitary region so we ignore it. +– 11 – + +Case II: δφ > 3/4 +In this case, the kernel fopt(z) leads to a functional that is negative for the stress tensor. +However, we need the functional to be positive on the stress tensor to obtain lower bound +on the central charge. This is achieved simply by using f(z) = −fopt(z) as the kernal. With +this choice, the functional is positive for δ < δcrit and also for δ > 2δφ but it is negative for +δcrit < δ < 2δφ. The schematic plot of this functional is shown in figure 3 on the right. The +region (δcrit, 2δφ) is excluded from the set P. In other words, for the case δφ > 3/4, the set P +consists of operators with δ < δcrit along with operators satisfying δ > 2δφ. +3.1 +Modifying the set P +We can modify the set P by changing the functional in both of these cases. This is important +for widening the applicability of the bound i.e. to make it so that all the CFT operators +S ⊂ P. Let us call the function that satisfies all the properties listed below equation (2.35) +a kernel function. Note that if we multiply any kernel function by the so called CDD factor +α(z, zi) we get a new kernel function. The CDD factor is given by +α(z, zi) = ˜α(x(z), x(zi)), +˜α(x, y) = x − y +xy − 1, +x(z) = +� +zb(1 − zb) − +� +z(1 − z) +� +zb(1 − zb) + +� +z(1 − z) +(3.9) +where zi ∈ (0, 1). It has the property that |α(z, zi)| = 1 for z ∈ (−∞, 0). The α(z, zi) factor +endows the kernel with a pair of additional single zeros in the unitarity region. These two new +zeros δi are at the two solutions of the equation zi = z∗(δi) and 1 − zi = z∗(¯δi), respectively. +Each of these equations have two solutions but we are interested in the ones that lie in the +unitarity domain. The pair of zeros (δi, ¯δi) are related to each other as +¯δi = 1 +2 +� +1 + +� +1 − 4(δi − 1)δi + 4δφ(4δφ − 3) +� +. +(3.10) +This follows easily from equation (2.27). It is easy to see that when both the roots are real, +they are on opposite sides of δcrit. +We can think of any one of the zeroes as the mirror +image of the other across the point δcrit. This is because a point that is closer to δcrit on +the right side is mirrored to a point that is closer to δcrit on the right. One can also see +(by definition) that the mirror image of δcrit is δcrit. Also, the point δ = 1/2 is mirrored +to the point δ = ¯δ1/2 ≡ 1 +2 +� +1 + +� +2 + 4δφ(4δφ − 3) +� +. Hence, this mirror maps the interval +D1 ≡ (0, δcrit) to the interval D2 ≡ (δcrit, ¯δ1/2). Also for δ ∈ D3 ≡ (¯δ1/2, 2δφ) the solution ¯δ +is complex and does not correspond to any zero of the functional for real δ. In figure 4 we +have displayed a schematic plot of the functional obtained after multiplying fopt(z) by three +CDD factors. We can see that the set P now consists of points on either sides of δcrit. As +we change the number of CDD factors and their parameters zi the set P changes. This gives +us a lot of freedom to tune the P so that the set of CFT operators S ⊂ P. However, as we +will see below, one can not do so in all the cases. To this end, let us characterize the CFT +spectrum S for which it is not possible, in our framework, to obtain a functional with P such +that S ⊂ P. +– 12 – + +δ +ω +δ* +1/2 +1 +2δϕ +Double Trace +δ1 δ2 +δ2 +δ1 +δ3 +Figure 4: Schematic plot of functional ω(δ, ℓ) where the kernel function f(z) is taken to be +fopt(z) times three CDD factors. There are two single zeros in D1 and two “mirror image” +single zero in D2. The function as double zeros for δ > 2δφ. The unpaired zero δ3 is in region +D3. The CDD factors are chosen in a way that the functional is positive for stress tensor i.e. +δ = 1. +3.2 +Applicability of the bound +First notice that even if we use the CDD factors to multiply the kernel (3.8) the functional +takes opposite values on the mirror pairs. This is simply a consequence of anti-symmetry +f(z) = −f(1 − z). +For the rest of the discussion, it is useful to think graphically. Let us denote all the CFT +operators on the δ axis with red dots. Let us reflect all the red dots in the D1 region onto +the D2 region. Let us color these reflections blue. Now focus only on the region D2 ∪ D3. +It has some distribution of red dots and blue dots, with only red dots in region D3. We +will be successful in finding the desired P if the functional is positive on the red dots and +negative on the blue dots. To achieve this we multiply the kernel (3.8) by CDD factors that +give a single zero whenever we transit from red dots to blue dots and vice versa so that the +functional is positive on the red dots and negative on the blue ones. This construction is +illustrated in figure 5. In the unlikely case there are pairs of blue dot and red dot that are +coincident we need to have a single zero precisely at that point. This functional achieves our +objective of S ⊂ P with the minimum number of CDD factors. When is this construction +not admissible? If the total number of CDD factors used are O(D) then the saddle point +computation performed in equation (2.27) is invalid as it is no longer determined only by the +conformal block. So we conclude that the exponential lower bound (3.4) on the central charge +is applicable if the following conditions are met: +• Condition 1: The exponential lower bound (3.4) on the central charge is applicable if +there are less than O(D) transitions between red dots and blue dots. +– 13 – + +풟1 +풟2 +δcrit +¯δ1/2 +2δϕ +1 +2 +풟3 +Figure 5: The CFT operators are denoted on the δ axis with red dots. The blue dots are the +“mirror reflection” of the red dots in the region D1 to the region D2. In order to construct +a functional with S ∈ P, we pick five CDD factors with zeros in the red-to-blue transition +regions. These are as shown in the figure schematically with crosses. +• Condition 2: There is no CFT operator that is in the O(1/D) neighborhood in the δ +space of the mirror image of the stress tensor (δ = ¯δT ≡ 1 +2 +� +1 + +� +1 + 4δφ(4δφ − 3) +� +). +The second condition, in particular implies that for δφ = 3/4, ¯δT = δT = 1 and hence the +bound is not obtained. Is the first condition reasonable? We can answer this question with the +same level of (im)precision with which it is asked. The following discussion of this condition +is somewhat impressionistic and is only meant to offer some intuition. +Let φ be the CFT operator with smallest scaling dimension δφD. For δφ < 3/4, we have +δcrit < 1. In this case, the operators in the region D1 i.e. between (1/2, δcrit) are only scalars +because of the unitarity bound on operators with spin. Because they are only scalars, it is +reasonable to assume that these are less than O(D) in number. On the other side of δcrit, +beyond δ = 1, we have operators that have arbitrary spin. It is perhaps natural that there are +O(D) of them in the regions D2 and D3 i.e. between (δcrit, ¯δ1/2) and (¯δ1/2, 2δφ) respectively. +Because the applicability of the bound only depends on the number of red-blue transitions +as described above and that the number of operators are expected to be less than O(D) +operators in D1, we expect the bound on the central charge to be applicable. +For δφ > 3/4, we have δcrit > 1. +In this case, on both side of δcrit i.e. +in all three +regions D1, D2 and D3 there are spinning operators. Operators in both these regions could +perhaps be O(D) in number. In this case, it is not reasonable expect that the number of +red-blue transitions are less than O(D) in number. So in this case we have to assume, perhaps +somewhat unnaturally, that the spectrum operators in the window (1, δcrit) is sparse i.e. of +less than O(D) for the central charge bound (3.4) to be valid. As δφ increases above 3/4, +at δφ ∼ 1.44 we get δcrit = 2. At this point, we definitely have multi stress-tensor operators +appearing in the region D1 in a dense fashion. We don’t expect the central charge bound +to be valid in this case. However, we don’t need to probe the regime δφ > 1 at all. If the +lightest scalar has δφ > 1, we can simply use the stress tensor component, say T 00 as a scalar +operator in D − 1 dimensional subspace. As D is taken to be large, this change in dimension +– 14 – + +is of little significance. For δφ=T 00 = 1, δcrit ∼ 1.37. For the bound to be applicable we have +to require that the operator spectrum should be sparse up to this value of δcrit. +This discussion also makes it clear why the free theory is allowed in large D from this +point of view. If we consider the four point function of the free field φ, we have 1 = 2δφ. At +this value of δ, the functional vanishes and we do not get any bound on the central charge. We +could also consider the four point function of φ2 operators. In this case, δφ2 = 1, this means +δcrit = 1.37. However, it is evident that a dense double-trace spectrum starts from δ = 1 and +hence the sparseness assumption and in turn the central charge bound is not applicable. +In [8] a different approach was taken to constrain the space of conformal theories in large +dimension. The conclusions from that paper agree with the results presented in this paper. +In [8] it was argued that for δφ < 3/4, the four point function of φ’s is exponentially close +to that of the generalized free field theory. For 3/4 < δφ < 1, this conclusion required a +sparseness assumption on the spectrum akin to the one argued here. +4 +Corrections at large but finite D +Now that we have obtained exponential bound in large D at leading order, it is a natural to +ask how the bound gets corrected at finite D. This is essentially a question about correction +to the functional value ω(∆, ℓ). There are two sources of corrections. +• Corrections to the kernel h(z). +• Corrections to the integral (2.13) with the corrected kernel. +Let us discuss the first source of corrections. We would like the kernel to obey the gluing +condition (2.20) and the extremality condition. Note that the condition that the inequality +in euqation (2.30) is saturated is not a true extremal condition in finite dimension since it +does not give rise to double zeros of ω(∆, ℓ) for ∆ ≥ 2∆φ. It does however insure that the +functional is non-negative for ∆ ≥ 2∆φ. It is in principle possible to solve the true extremality +condition in perturbation theory in 1/D but it is quite cumbersome to do so. We will instead +take the equation (2.30) as a necessary condition on the kernel in finite dimension as it at +least guarantees positivity of ω(∆, ℓ) for ∆ ≥ 2∆φ. The gluing condition and “extremality +condition” together imply +Im(f(z)) = z2Dδφ−2|f(1/z)| + (1 − z)2Dδφ−2|f(1/(1 − z))| +z ∈ (0, 1). +(4.1) +In our analysis we could simply set the right hand side to zero in the D → ∞ limit. In large +but finite D, the right hand side does give small non-perturbative corrections. We will ignore +these corrections and continue to use the kernel fopt(z) that satisfies Imf(z) = 0 for z ∈ (0, 1) +even in finite dimension. +The second set of corrections, viz. the corrections to the saddle point integral (2.13) are +both perturbative as well as non-perturbative. Instead of treating them differently, we deal +with them by simply performing the integral numerically. We expect to get close to accurate +– 15 – + +bounds (with errors that are exponentially small in D) in large but finite dimension in this +way. The results for D = 50, 20, 10, 6, 4 are presented in figure 6, 7. We have also given the +bound obtained by setting the appropriate value of D in equation (3.4) with f(z) = fopt(z) +for reference. As we have not made use of the CDD factors in the functional, we have to +assume absence of operators in certain ∆ range. This range is given by gray shaded region +in the figures. In principle, we can tune the functional to the operator spectrum using the +CDD factors as described in section 3.2. +The only reason that these bounds are not completely trustworthy for small D is because +the right hand side of equation (4.1) can not be approximated by zero for small D. Also note +that D controls these errors only because we have take ∆φ to be of the same order as D. +This implies that the numerical bounds are good approximations even in small D as long as +∆φ is taken to be large. In this way we can repurpose these bounds as bounds on the central +charge for theories with large gap. We are currently investigating this direction. In order to +get completely trustworthy bounds for small D and small ∆φ, we need to solve the condition +(4.1) exactly. +– 16 – + + + + + + + + + + + + + + + +30 +35 +40 +45 +50 +55 +60 +0 +10 +20 +30 +40 +50 +60 +Δϕ +log[CT]Min +D=50 +Δϕ +D +30 +35 +40 +45 +50 +55 +60 +40 +60 +80 +100 +120 +Δϕ +Δ +D=50 +Δϕ +D + + + + + + + + + + + + + + + +12 +14 +16 +18 +20 +22 +24 +0 +5 +10 +15 +20 +25 +Δϕ +log[CT]Min +D=20 +Δϕ +D +12 +14 +16 +18 +20 +22 +24 +10 +20 +30 +40 +50 +Δϕ +Δ +D=20 +Δϕ +D + + + + + + + + + + + + + + + +6 +8 +10 +12 +-2 +0 +2 +4 +6 +8 +10 +12 +Δϕ +log[CT]Min +D=10 +Δϕ +D +6 +8 +10 +12 +5 +10 +15 +20 +25 +Δϕ +Δ +D=10 +Δϕ +D +Figure 6: Approximate numerical bounds on the central charge of CFTs in D = 50, 20, 10. +The green curve is obtained by substituting appropriate value of D in equation (3.4). The +bounds are obtained with the assumption that no scalar operators are present in the gray +shaded range for a given ∆φ (similar but less constraining assumptions are also made for +spinning operators). The red line is ∆ = 2∆φ and the solid blue line is ∆ = Dδcrit. +– 17 – + + + + + + + + + + + + + + +4 +5 +6 +7 +8 +-4 +-2 +0 +2 +4 +6 +8 +Δϕ +log[CT]Min +D=6 +Δϕ +D +4 +5 +6 +7 +8 +4 +6 +8 +10 +12 +14 +16 +Δϕ +Δ +D=6 +Δϕ +D + + + + + + + + + + + + + + +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +-6 +-4 +-2 +0 +2 +4 +Δϕ +log[CT]Min +D=4 +Δϕ +D +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +2 +4 +6 +8 +10 +Δϕ +Δ +D=4 +Δϕ +D +Figure 7: Approximate numerical bounds on the central charge of CFTs in D = 6, 4. The +green curve is obtained by substituting appropriate value of D in equation (3.4). The bounds +are obtained with the assumption that no scalar operators are present in the gray shaded +range for a given ∆φ (similar but less constraining assumptions are also made for spinning +operators). The red line is ∆ = 2∆φ and the solid blue line is ∆ = Dδcrit. +Acknowledgement +We would like to thank the TIFR string theory group, in particular Gautam Mandal, Shiraz +Minwalla, Onkar Parrikar, Sandip Trivedi for useful discussions. +We would like to thank +Adwait Gaikwad for collaboration in related projects. This work is supported by the Infosys +Endowment for the study of the Quantum Structure of Spacetime. The work of A.G. is also +supported by the SERB Ramanujan fellowship. We acknowledge support of the Department +of Atomic Energy, Government of India, under Project Identification No. RTI 4002. We +would also like to acknowledge our debt to the people of India for their steady support to the +study of the basic sciences. +A +Simplifying the functional +In this appendix, we will simplify the functional given in equation (2.13) to the expression +in (2.14) using some simple properties of the conformal block specialized to the locus z = ¯z. +– 18 – + +We denote the block specialized to this locus as G∆,ℓ(z). +G∆,ℓ(z ± iϵ) = e±iπ∆ ˆG∆,ℓ(z) , +z ∈ (−∞, 0) +G∆,ℓ(z) = ˆG∆,ℓ +� +z +z − 1 +� +, +z ∈ (0, 1) +(u-symmetry). +(A.1) +Where ˆG∆,ℓ = |G∆,ℓ(z + iϵ)|, for z ∈ (−∞, 0). Using these properties it is easy to see that +the first term in (2.13) simplifies to, +� 0 +−∞ +dz h(1 − z)Disc +�G∆,ℓ(z) +z2∆φ +� += +� +eiπ(∆−2∆φ) − e−iπ(∆−2∆φ)� � 0 +−∞ +dz h(1 − z) +ˆG∆,ℓ(z) +|z|2∆φ . +(A.2) +To simplify the second term we first write the integral over the discontinuity as a contour +integral wrapping the branch cut from (−∞, 0). Then we do a variable change z → 1 − z and +deform the new integration contour that wraps the cut (1, ∞) back to the one that wraps the +cut (−∞, 0). In doing so we use the fall off condition of h(z) given above equation (2.13) to +neglect the contribution of the arc at infinity. In this deformation process, we also collect the +contribution of the branch cut of h(z) in (0, 1). +� 0 +−∞ +dz h(1 − z)Disc +�G∆,ℓ(1 − z) +(1 − z)2∆φ +� +(A.3) += +� 0 +−∞ +dz Disc +� +h(z)G∆,ℓ(z) +z2∆φ +� ++ +� 1 +0 +dz Disc [h(z)] G∆,ℓ(z) +z2∆φ += +� 0 +−∞ +dz +� +h(z + iϵ)eiπ(∆−2∆φ) − c.c. +� ˆG∆,ℓ(z) +|z|2∆φ + +� 1 +0 +dz Disc [h(z)] G∆,ℓ(z) +z2∆φ += +� 0 +−∞ +dz +� +h(z + iϵ)eiπ(∆−2∆φ) − c.c. +� ˆG∆,ℓ(z) +|z|2∆φ ++ +� 0 +−∞ +dz (1 − z)2∆φ−2Disc +� +h +� +z +z − 1 +� � ˆG∆,ℓ(z) +|z|2∆φ +In second term of last line we have done variable transform from z → z/(z − 1) and used +the u-symmetry of block from equation (A.1). +Since h(z) is real for z ∈ (1, ∞) we have +h(z∗) = h(z)∗. This implies Disc [h(z)] is purely imaginary. Combining (A.2) and (A.3), and +using the definitions of f(z) and g(z) in equations (2.18) and (2.19) respectively, gives the +simplified expression in equation (2.14). +Action on operators with ∆ > 2∆φ +The simplified integral (2.14) can be sued to compute the functional for operators with ∆ ≥ +2∆φ. That is because both f and g terms are separately convergent in this range. In the +large D limit, these integrals are performed using the saddle point approximation. As ˜g(z) ≡ +(1 − z)2∆φg(z) is take to be O(1), the saddle point is controlled by ˆG∆(z)/|z|2∆φ. This is +computed in appendix B in equation (B.7). The saddle point z∗ ≤ 0 for all ∆ ≥ 2∆φ. +– 19 – + +Figure 8: The top two contours are the original contours of integration in equation (A.4). +The bottom two contours are their respective deformations to saddle points. +Action on operators with ∆ < 2∆φ +In this regime, the two integrals of equation (2.14) are not individually convergent. We need +to resort to the original expression (2.13) to compute the integral. +ω(∆) = +1 +2πi +� 0 +−∞ +dz h(1 − z) Disc +�G∆(z) +z2∆Φ − G∆(1 − z) +(1 − z)2∆Φ +� +(A.4) +If we look at the saddle point (B.7), it is easy to see that it lies in the range (0, 1) for ∆ < 2∆φ. +This motivates the contour deformation shown in figure 8. The functional is written as, +ω(∆) = − +1 +2πi +� z0 +0 +dz Disc(h(1 − z)) G∆(z) +z2∆Φ ++ +1 +2πi +� 1 +z0 +dz Disc(h(z)) G∆(z) +z2∆Φ ++ +1 +2πi +� i∞ +z0 +dz (h(z) − h(1 − z)) G∆(z) +z2∆Φ + +1 +2πi +� z0 +−i∞ +dz (h(z) − h(1 − z)) G∆(z) +z2∆Φ += − +� z0 +0 +dz g(1 − z) G∆(z) +z2∆Φ +− +� 1 +z0 +dz g(z) G∆(z) +z2∆Φ ++ 1 +2i +� i∞ +z0 +dz f(z) G∆(z) +z2∆Φ ++ +1 +2i +� z0 +−i∞ +dz f(z) G∆(z) +z2∆Φ +(A.5) +Here, in the second equality we have used the definitions of kernels f(z) (2.18) and g(z) (2.19). +Using the scaling (2.23), the first two terms in second equality (i.e. the g terms) vanishes in +the limit ∆φ → ∞. We can perform the saddle point integral in the direction perpendicular +to real line at z = z∗ giving our integral, +ω(∆) = (f(z∗ + iϵ) + f(z∗ − iϵ)) +2 +G∆(z∗) +z2∆Φ +∗ +� +2π +∆φq′′ +δ(z∗). +(A.6) +In appendix B we show that the steepest descent at z∗ is indeed perpendicular to the real +axis. Here the function qδ is defined in equation (B.6). Using the gluing condition (2.20) +– 20 – + +0 +1 +Zo +1 +00 +1 +Zo +0 +7Im(f(z)) = 0, for z ∈ (0, 1). Thus, the functional in (A.6) becomes +ω(∆) = Re (f(z∗)) G∆(z∗) +z2∆Φ +∗ +� +2π +∆φp′′(z∗). +(A.7) +Action on Identity +Here we will look at the action of functional (2.13) on identity block given by, +ω(1) = +1 +2πi +� ∞ +1 +dz h(z)Disc +� +1 +z2∆φ − +1 +(1 − z)2∆φ +� +(A.8) +The first term is zero since z−2∆φ does not have any discontinuity in the region (1, ∞). In +the second term, we deform the contour from (1, ∞) to (−∞, 1), where h has a discontinuity +but (1 − z)−2∆φ does not. +ω(1) = +1 +2πi +� 1 +−∞ +dz Disc [h(z)] +(1 − z)2∆φ += +1 +2πi +� 0 +−∞ +dz Disc [h(z)] +(1 − z)2∆φ + +1 +2πi +� 1 +0 +dz Disc [h(z)] +(1 − z)2∆φ += +1 +2πi +� 0 +−∞ +dz Disc [h(z) − h(1 − z)] +(1 − z)2∆φ ++ +1 +2πi +� 1 +0 +dz Disc [h(z)] +(1 − z)2∆φ += − +� 0 +−∞ +dz Im(f(z)) +(1 − z)2∆φ − +� 1 +0 +dz +g(z) +(1 − z)2∆φ +(A.9) +where, in the third equality we have added h(1 − z) inside the discontinuity because h(1 − z) +does not have any discontinuity on z ∈ (−∞, 0). In the fourth equality we have used the +definitions of kernels f(z) (2.18) and g(z) (2.19). Using the scaling (2.23), the first term +vanishes in the limit ∆φ → ∞, and we get +ω(1) = − +� 1 +0 +dz +g(z) +(1 − z)2∆φ = − +� 1 +0 +dz ˜g(z). +(A.10) +Now using extremality condition where the inequality (2.30) is saturated the above functional +becomes, +ω(1) = − +� 1 +0 +dz (1 − z)−2 +����f +� +z +z − 1 +����� = − +� 0 +−∞ +dz |f(z)|. +(A.11) +B +Conformal blocks in large D and saddle points +The conformal block in large dimension is computed in [16]. On the z = ¯z locus it is given by +G∆,ℓ(z) = +2∆ +� +1 − +� +z +2−z +�2 A1−ℓ(1) +� +z +2 − z +�∆ +2F 1 +� +∆ +2 , ∆ − 1 +2 +, ∆ − D +2 + 1, +� +z +2 − z +�2� +(B.1) +– 21 – + +where, the function Aβ(x) is given in equation (2.25). Using the integral representation of +the hypergeometric function +B(b, c − b) 2F 1 (a, b, c, z) = +� 1 +0 +dt tb−1(1 − t)c−b−1(1 − zt)−a +(B.2) +and the scaling ∆ = δD, the conformal block (B.1) becomes, +GδD,ℓ(z) = +2δD +� +1 − +� +z +2−z +�2 A1−ℓ(1) +� +z +2−z +�δD +B +� δD−1 +2 +, δD−D+3 +2 +� +� 1 +0 +dt +� +1 − t +t3 +� +� +� +� +� +� +t +δ +2 (1 − t) +δ−1 +2 +� +1 − +� +z +2−z +�2 +t +� δ +2 +� +� +� +� +� +� +D +(B.3) +This integral can be done by saddle point approximation in large D limit. This has been +done in [8], but we will reproduce it here for convenience. The saddle point equation and its +solution is, +d +dt log +� +� +� +� +� +� +t +δ +2 (1 − t) +δ−1 +2 +� +1 − +� +z +2−z +�2 +t +� δ +2 +� +� +� +� +� +� += 0 +⇒ +t± +∗ = +2δ − 1 ± +� +4δ(δ − 1) +� +1 − +� +z +2−z +�2� ++ 1 +2(δ − 1) +� +z +2−z +�2 +(B.4) +For z ∈ (−∞, 1], t− +∗ lies in the integration range (0, 1) while t+ +∗ lies outside. Picking up the +saddle t− +∗ the block at leading order in large D is, +GδD,ℓ(z) = +1 +� +1 − +� +z +2−z +�2 A1−ℓ(1) vδ(z) eDqδ(z) +(B.5) +where, +qδ(z) = log +�� +2(δ − 1)2ˆy+ +(2δ − 1) (B − 2δ + 2(δ − 1)y+ + 1) +�2(2δ − 1)2(B − 2δ + 1)(B − 2δ + 2(δ − 1)y+ + 1) +(1 − B)δ(δ − 1)2y+ +�δ/2 � +vδ(z) = +� +(16δ)−1(δ − 1)−3(B − 1)3δ(2δ − 1)(B − 2δ + 2(δ − 1)y+ + 1)2 +2(δ − 1)2(B − 4δ)y2 ++ + 2(δ − 1) (B (1 − 3δ) + (8δ2 − 6δ + 1)) y+ + (1 − 2δ)2(1 + B − 2δ) +(B.6) +with, B = +� +1 + 4(1 − y+)δ(δ − 1) and y+ = +� +z +2−z +�2 +. +Given the large dimensional block (B.5) in leading order in large D limit, we will now compute +the saddle point approximation of +ˆG∆,ℓ(z) +z2∆φ +with respect to z in large D limit. The saddle point +equation and the solution for this is +d +dz (qδ(z) − 2δφ log(z)) = 0 +⇒ +z∗ = (2δφ − δ)(2δφ + δ − 1) +δφ(4δφ − 1) +. +(B.7) +– 22 – + +It is easy to verify that, +d2 +dz2 (qδ(z) − 2δφ log(z)) +�� +z=z∗ ≶ 0 +for δ ≷ 2δφ +(B.8) +This means that the path of steepest descent from the saddle point z∗ is along the real axis +for δ > 2δφ while it is perpendicular to the real axis for δ < 2δφ. +References +[1] R. Rattazzi, S. Rychkov and A. Vichi, Central Charge Bounds in 4D Conformal Field Theory, +Phys. Rev. D 83 (2011) 046011 [1009.2725]. +[2] D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field +Theories, JHEP 05 (2011) 017 [1009.2087]. +[3] D. Simmons-Duffin, The Conformal Bootstrap, in Theoretical Advanced Study Institute in +Elementary Particle Physics: New Frontiers in Fields and Strings, pp. 1–74, 2017, DOI +[1602.07982]. +[4] S. 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Poland, Conformal Blocks in the Large D Limit, JHEP 08 +(2013) 107 [1305.0004]. +– 23 – + diff --git a/ddE4T4oBgHgl3EQfQQxz/content/tmp_files/load_file.txt b/ddE4T4oBgHgl3EQfQQxz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a967c4636b447ca988cc555d1a73ac18f7f3ece --- /dev/null +++ b/ddE4T4oBgHgl3EQfQQxz/content/tmp_files/load_file.txt @@ -0,0 +1,880 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf,len=879 +page_content='Prepared for submission to JHEP Bound on the central charge of CFTs in large dimension Abhijit Gaddea, Mrunmay Jagadaleb, Shraiyance Jaina and Trakshu Sharmaa a Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400005, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' b Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' E-mail: abhijit@theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='tifr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='in, mjagadal@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='edu, shraiyance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='jain@tifr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='in, trakshu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='sharma@tifr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='in Abstract: In this paper, we use crossing symmetry and unitarity constraints to put a lower bound on the central charge of conformal field theories in large space-time dimensions D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Specifically, we work with the four-point function of identical scalars φ with scaling dimension ∆φ, and use a certain class of analytic functionals to show that the OPE coefficient squared c2 φφT µν must be exponentially small in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For this to hold, we need to make a mild assumption about the nature of the spectrum below 2∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Our argument is robust and can be applied to any OPE coefficient squared c2 φφO with ∆O < 2∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This suggests that conformal field theories in large dimensions (if they exist) must be exponentially close to generalized free field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='04980v1 [hep-th] 12 Jan 2023 Contents 1 Introduction 1 2 Review of analytic functional bootstrap 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1 Functionals for OPE coefficient maximization 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='2 Analytic functionals for z = ¯z 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='3 Computing the functional in large D 7 3 Bounds on Central Charge in Large D CFTs 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1 Modifying the set P 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='2 Applicability of the bound 13 4 Corrections at large but finite D 15 A Simplifying the functional 18 B Conformal blocks in large D and saddle points 21 1 Introduction The central charge of a conformal field theory is a measure of its degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For a unit-normalized stress tensor T µν, the central charge is inversely proportional to the three- point function coefficient squared c2 φφT µν for any operator φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Hence, it governs the strength of the gravitational coupling in the dual theory in anti-de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Therefore, bounding the central charge is important from the point of view of charting not only the space of CFTs but also the landscape of quantum gravity in AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Lower bounds on the central charge of CFTs in two and four dimensions have been computed using numerical bootstrap [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1 It is believed that no non-trivial conformal field theory exists in dimensions greater than six2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Here, a non-trivial CFT means that it is (a) unitary, (b) not free, and (c) contains the stress tensor in its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' One can alternatively formulate this conjecture as: In D > 6, a unitary CFT that is not free must have c2 φφTµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The advantage of this formulation is that it opens a way of addressing this problem in a quantitative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In this paper, we will show: 1See [3–6] and references therein for an introduction and review of the vast literature on the conformal bootstrap program and [7] and references therein for a focus on analytic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 2See [8] for discussion regarding this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 1 – In large D, a unitary CFT - with a reasonable condition on the low lying spectrum (this includes not being free) - must have c2 φφTµν < Aφe−αφD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The exponent αφ is an O(1) number given in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The problem of constraining CFTs in large dimensions was considered in [8] by two of the present authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It was concluded that the unitary CFTs in large dimensions must be expo- nentially close to generalized free field theories in a certain sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Happily, our results in this paper concur with the results of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We will comment more on the connection towards the end of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The exponential lower bound on the central charge may lead one to naively conclude that there is an exponential hierarchy between the cosmological constant scale and the Planck scale in large dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is not so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For a D-dimensional CFT, MD−1 Λ GN ∼ c2 φφTµν = Aφe−αφD ⇒ MΛ MP ∼ e−αφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1) We will use the crossing symmetry and unitarity of the four-point function of identical scalars φ in a large D CFT to put upper bounds on the OPE coefficient c2 φφTµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is accomplished using analytic functional bootstrap [9–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In particular, we will use the analytic functionals in [11] that were used to bound certain OPE coefficients in one-dimensional CFTs in the large ∆φ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We will review these tools in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Their application to CFTs in large dimensions is made in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The lower bound on the central charge for CFTs in large dimensions is obtained in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In section 4, we use the same functionals to obtain approximate numerical bounds for CFTs in large but finite dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The paper contains two appendices that supplement the discussion in the bulk of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 2 Review of analytic functional bootstrap Consider the four-point function of identical scalar primary operators φ of dimension ∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This four-point function is fixed by conformal symmetry up to a function of cross-ratios (u, v) as, ⟨φ(x1)φ(x2)φ(x3)φ(x4)⟩ = 1 |x12x34|2∆φ G(u, v) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1) where u ≡ z¯z = x2 12x2 34 x2 13x2 24 , v ≡ (1 − z)(1 − ¯z) = x2 14x2 23 x2 13x2 24 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The s-channel OPE expansion i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' corresponding to x1 → x2, x3 → x4 is convergent in z, ¯z ∈ C\\[1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The t-channel OPE expansion x1 → x4, x2 → x3 is convergent in z, ¯z ∈ C\\(−∞, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We will consider the correlator in the overlapping region of convergence (z, ¯z) ∈ R×R where R = C \\ {(−∞, 0] ∪ [1, ∞)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The OPE expansions take the form, G(u, v) =s � O∈φ×φ c2 φφOG∆O,ℓO(z, ¯z) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='2) G(u, v) =t (z¯z)∆φ ((1 − z)(1 − ¯z))∆φ � O∈φ×φ c2 φφOG∆O,ℓO(1 − z, 1 − ¯z) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='3) – 2 – Due to conformal symmetry and permutation symmetry only operators with even spin ℓ appear in these expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The equality of expansions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='3) is called the crossing equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It is convenient to express the crossing equation in terms of an elegant sum rule, � O∈φ×φ c2 φφOF ∆φ ∆O,ℓO(z, ¯z) = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4) where F ∆φ ∆O,ℓO(z, ¯z) ≡ G∆O,ℓO(z, ¯z) (z¯z)∆φ − G∆O,ℓO(1 − z, 1 − ¯z) ((1 − z)(1 − ¯z))∆φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The functions F ∆φ ∆O,ℓO(z, ¯z) are holomorphic in R × R and obey F(z, ¯z) = F(¯z, z) = −F(1 − z, 1 − ¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5) Let us call the vector space of such functions V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Unitarity implies cφφO ∈ R and hence c2 φφO ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Therefore, the sum rule sets a positive linear combination of the vectors F ∆φ ∆O,ℓO(z, ¯z) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Consider a linear functional ω, that is an element of the dual of the vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' One can act this functional ω on the sum rule (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4) to get, � O∈φ×φ c2 φφO ω � F ∆φ ∆O,ℓO(z, ¯z) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='6) Some simple example of such functionals include evaluation and taking derivatives at a point in R×R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The numerical bootstrap typically uses ω to be derivatives at the crossing symmetric point z = ¯z = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Note that to get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='6) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4) we have swapped the action of functional ω with an infinite sum over operators appearing in the OPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Not all functionals satisfy this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Following [10] we call this property of ω, the swapping condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Further we want the functionals to be finite on F ∆φ ∆,ℓ(z, ¯z) with ∆, ℓ satisfying the unitarity bound i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' ∆ ≥ d−2 2 for ℓ = 0 and ∆ ≥ ℓ + d − 2 for ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We will only consider functionals which satisfy the swapping and finiteness conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1 Functionals for OPE coefficient maximization In this paper, we will be concerned with obtaining an upper bound on the OPE coefficient squared c2 φφOb of a primary operator Ob ∈ φ × φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Let P be the set of all (∆, ℓ) values where the functional is non-negative and S be the set of all CFT operators except for identity 1 and Ob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The sum rule constraining CFT data is, F ∆φ 1 (z, ¯z) + c2 φφObF ∆φ ∆b,ℓb(z, ¯z) + � (∆,ℓ)∈S c2 φφOF ∆φ ∆,ℓ(z, ¯z) = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='7) The action of a functional ω on the sum rule is, ω(1) + c2 φφObω(∆b, ℓb) + � S c2 φφOω(∆, ℓ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='8) – 3 – 0 1 2 3 4 5 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 Δ ω 1 2 S Figure 1: Extremal Functional example Therefore, the OPE coefficient c2 φφOb can be expressed as c2 φφOb = − ω(1) ω(∆b, ℓb) − � S c2 φφOω(∆, ℓ) ω(∆b, ℓb) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='9) At this point, it is easy to see that we can obtain an upper bound on c2 φφOb by constructing a functional that satisfies, ω(1) < 0, ω(∆b, ℓb) > 0, S ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='10) Existence of such functional would give us the bound, c2 φφOb ≤ −ω(1) ω(∆b, ℓb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='11) Moreover this inequality is saturated when ω(∆, ℓ) = 0, ∀(∆, ℓ) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Such functionals are called extremal functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' A cartoon of an extremal functional for one dimensional CFT is given in figure 1, where ∆b is taken to lie between 1 and 2 and S is {∆ : ∆ > 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The set S is precisely the set of double zeros of the functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='2 Analytic functionals for z = ¯z In this section we will consider analytical functionals that act on the functions restricted to the locus z = ¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' These functionals were first constructed in [11] for one dimensional CFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It is straightforward to repurpose them as functionals for CFTs in general dimensions but acting only on the specialization z = ¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Let ˜V be the vector space of functions F(z) that is holomorphic in R and obey F(z) = −F(1 − z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' After specializing a function in V to z = ¯z, we precisely get a function in ˜V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 4 – The authors of [11] consider a class of functionals acting on ˜V given by the integral of the discontinuity along the branch cut [1, ∞) weighted by a kernel h(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' ω(F) = 1 2πi � ∞ 1 dz h(z)Disc[F(z)] = 1 2πi � 0 −∞ dz h(1 − z)Disc[F(z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='12) Here Disc[F(z)] = limϵ→0+ F(z + iϵ) − F(z − iϵ) is the discontinuity along the branch cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In the second equality we have done the change of variables from z → 1 − z and used F(1 − z) = −F(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The kernel h(z) is analytic on C \\ (−∞, 1) with possible branch cuts at (−∞, 0] and [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Without loss of generality, we can assume h(z) ∈ R for z ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This implies h∗(z) = h(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The kernel h(z) satisfies the properties, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' h(z) is analytic away from possible poles or branch points at z = 0, 1 and ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' h(z) is bounded by A1|z|−1−ϵ1 for some A1, ϵ1 > 0 as z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The discontinuity of h(z) around z = 1 is bounded by A2|z − 1|2∆φ−1+ϵ2 for some A2, ϵ2 > 0 as z → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The second and the third properties follow from finiteness and swapping conditions respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' See [10, 11] for details regarding this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The action of the functional in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='12) on the function F ∆φ ∆,ℓ(z, z) is given by ω(∆, ℓ) = 1 2πi � 0 −∞ dz h(1 − z)Disc �G∆,ℓ(z) z2∆φ − G∆,ℓ(1 − z) (1 − z)2∆φ � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='13) Here, we use G∆,ℓ(z) = G∆,ℓ(z, z) for the conformal block specialized to z = ¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' After certain contour manipulations detailed in appendix A, the functional reduces to, ω(∆, ℓ) = g(∆, ℓ) − Im � eiπ(∆−2∆φ)f(∆, ℓ) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='14) where, f and g are given by, f(∆, ℓ) = � 0 −∞ dz f(z) ˆG∆,ℓ(z) |z|2∆φ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='15) g(∆, ℓ) = � 0 −∞ dz (1 − z)2∆φ−2g � z z − 1 � ˆG∆,ℓ(z) |z|2∆φ = � 1 0 dz g(z)G∆,ℓ(z) z2∆φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='16) Here we have introduced3 ˆG∆,ℓ(z) = |G∆,ℓ(z + iϵ)| for z ∈ (−∞, 0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='17) f(z) = h(z) − h(1 − z) π for Im(z) > 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='18) g(z) = −Disc [h(z)] 2πi for z ∈ (0, 1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='19) 3Our definition of f(z) differs from the one used in [11] by a factor of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' That is the reason Im[f(z)] appears in our gluing condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='20) rather than Re[f(z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 5 – The function g is analytically continued in the z variable and the function f is continued to the region with Im(z) < 0 via f(z) ≡ −f(1 − z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' With this analytic continuation, we have f∗(z) = f(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The definitions of the kernels f and g in terms of the kernel h imply that they obey the following relation called the gluing condition, Im[f(z)] + g(z) + g(1 − z) = 0 for z ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='20) This manipulation is valid for ∆ > ∆c, where ∆c is some positive scaling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is because although the functional ω(∆) is always finite by definition, the individual integrals in equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='16) can diverge near z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Since the conformal block goes like |z|∆ as z → 0, the integrals (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='16) are convergent for ∆ greater than certain ∆c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The precise value of ∆c depends on the behavior of the kernel h(z) near z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Outside this range i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' for ∆ < ∆c, we need to resort to the manifestly finite expression of the functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='13) for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Since Disc[h(z)] is purely imaginary and G∆,ℓ(z) is real for z ∈ (0, 1), this means g(∆, ℓ) is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' However h(z + iϵ) generically has both real and imaginary parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' So f(∆, ℓ) can be written as f(∆, ℓ) = ir(∆, ℓ)eiπγ(∆,ℓ) with r(∆, ℓ) ∈ R+ and γ(∆, ℓ) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' With this, the functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='14) becomes, ω(∆, ℓ) = g(∆, ℓ) − r(∆, ℓ) cos(π(∆ − 2∆φ + γ(∆, ℓ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='21) If the choice of h is such that g, r, γ are slowly varying compared to the oscillations of the cosine in above equation and further if r(∆, ℓ) = g(∆, ℓ), then one gets an extremal functional which has double zeros at ∆n = 2∆φ + 2n − γ(∆n, ℓ) n ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='22) The functional, in particular the integrals f(∆, ℓ) and g(∆, ℓ) are difficult to compute analytically in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' If the conformal dimensions ∆φ and ∆ both are taken to be large then these integrals can be performed via saddle point approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' So at this stage, we will take the limit of large D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This will also set all the conformal dimensions to be large, thanks to the unitarity bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' If we want the saddle points of f(∆, ℓ) and g(∆, ℓ) to be universal and not depend on the kernel f and g, and if we further want f(∆, ℓ) and g(∆, ℓ) to be of the same order (which is necessary to get the double zeroes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='22)) then we need to take f(z) ∼ O(1) for Im(z) > 0, g(z) = (1 − z)2∆φ˜g(z), with ˜g(z) ∼ O(1) for z ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='23) In the limit of large ∆, ∆φ and with the scaling of kernels f(z) and g(z) given above, it is easy to see that the integrals for f and g are convergent for ∆ below some ∆c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' If f(z) = O(z−1+ϵ) near z = 0 then ∆c = 2∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We will assume that f(z) obeys this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Therefore, the computation of the functional ω(∆) can be divided into three regions, depending on the method of computation, namely (I) ω(∆ > 2∆φ), (II) ω(0 < ∆ < 2∆φ) and (III) ω(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 6 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='3 Computing the functional in large D In this section we will compute the functional ω(∆, ℓ) in the large D limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For this we will need the conformal block in the large D limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' These blocks were first computed in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' They gave an explicit expression in terms of the hypergeometric function, G∆,ℓ(y+, y−) = 2∆ √y− − y+ A∆(y+)A1−ℓ(y−) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='24) with, Aβ(x) = x β 2 2F 1 �β 2 , β − 1 2 , β − D 2 + 1, x � , y± = z¯z (1 ± � (1 − z)(1 − ¯z))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='25) One immediate observation is that on the z = ¯z slice, y− = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Hence, the spin dependance of the block trivializes as we get A1−ℓ(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We will drop the spin label from now on as the functional ω(∆, ℓ) does not depend on ℓ in the large D limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We are interested in scaling all ∆ = δD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In this limit the large D block simplifies even further on z = ¯z locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We get GδD(z) = � 1 − � z 2 − z �2 �− 1 2 vδ(z) eDqδ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='26) The functions vδ(z) and qδ(z) are given in equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The saddle point in integrals (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='16) come from extremizing GδD(z)/z2δφD with respect to z in the large D limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is computed in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We get, z∗(δ) = (2δφ − δ)(2δφ + δ − 1) δφ(4δφ − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='27) No we will evaluate the functional in the three regions mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (I) δ > 2δφ: For this range of δ the saddle point z∗ ∈ (−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The functional is ω(∆ > 2∆φ) = µ(δφ, z∗) � (1 − z∗)−2˜g � z∗ z∗ − 1 � − cos(π(∆ − 2∆φ + γ(z∗))) |f(z∗)| � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='28) where, µ(δφ, z∗) is a positive pre-factor given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It is independent of f(z) and ˜g(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' µ(δφ, z∗) = ˆGδD(z∗) |z∗|2δφD � 2π −δφDq′′ δ(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='29) The phase γ(z) is defined by f(z + i0+) = i|f(z)|eiπγ(z) and is of O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It is clear from the expression that the functional ω(∆) for ∆ ≥ 2∆φ is oscillating because of the cosine factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Since we want positive functional for ∆ > 2∆φ, this means the first term in the square brackets should be greater than or equal to the second, (1 − z)−2˜g � z z − 1 � ≥ |f(z)| for z ∈ (−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='30) – 7 – To get an extremal functional we will require this inequality be saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Extremality along with the gluing condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='20) yields the constraint on the kernel f(z), Im(f(z)) = z2∆φ−2|f(1/z)| + (1 − z)2∆φ−2|f(1/(1 − z))| = 0 z ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='31) In the second equality we use the fact that ∆φ is large, z ∈ (0, 1) and f(z) ∼ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The functional takes the form ω(∆ > 2∆φ) = 2µ(δφ, z∗)|f(z∗)| sin2 �π 2 (∆ − 2∆φ + γ(z∗)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='32) It has double zeroes at ∆ = 2∆φ + 2n − γ(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (II) 0 < δ < 2δφ: For this range of δ the integrals (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='16) are divergent so we have to use the original definition of the functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='13) to evaluate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is done in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The result is, ω(0 < ∆ < 2∆φ) = Re (f(z∗)) GδD(z∗) z2δφD ∗ � 2π δφDq′′ δ(z∗) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='33) Here z∗ is the one defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Notice that the value of the functional scale exponentially with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (III) Identity operator 1: This functional is also evaluated in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The result is, ω(1) = − � 0 −∞ dz|f(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='34) Recall the conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='10) on ω to get a bound on the OPE coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It is clear from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='34) that ω(1) < 0 is already obeyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' To impose the condition ω(∆b, ℓ) > 0, we need to impose Re(f(z)) > 0 for z ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' As the functional is non-negative for δ > 2δφ, the set of operators with quantum numbers {(∆, ℓ) : ∆ > 2∆φ} is definitely contained in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' If the rest of the CFT operators also lie in the set P then the bound on the OPE coefficient squared for ∆b ≤ 2∆φ is c2 φφOb ≤ − ω(1) ω(∆b) = � 0 −∞ dz|f(z)| Re(f(zb)) �� δφD 2π z2δφD b � q′′ δ(zb) GδbD(zb) , � zb = z∗(δb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='35) The expression inside the big parenthesis is independent of f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It is evaluated explicitly in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Let us collect all the properties of f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It is analytic in C \\ (−∞, ∞) with possible singularities only at 0, 1 and ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' f(z) = −f(1 − z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This follows from the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='18) of f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Im[f(z)] = 0 for z ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is derived in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 8 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' f(z) = O � z−1+ϵ� near z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is needed for ∆c = 2∆φ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' f(z) = O � z−1−ϵ� near z = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This follows from decay property of h(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' If we show the existence of f(z) satisfying above properties then the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='35) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' As shown in appendix B, the bound takes the form c2 φφOb < A(δb, δφ)e−α(δb,δφ)D with α(δb, δφ) > 0 for δb < 2δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The explicit expressions for A(δb, δφ) and α(δb, δφ) are cumbersome and are give in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Note that exponent α is robust and does not depend on f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' However, the set P of the quantum numbers (∆, ℓ) where the functional ω is non-negative depends crucially on f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 3 Bounds on Central Charge in Large D CFTs In this section, we will apply the technology of section 2 to obtain lower bound on the central charge CT of a CFT in large D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' When the stress tensor of the CFT is canonically normalized i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' when it obeys the Ward identity ∂µ⟨T µνφ(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' φ(xn)⟩ = − � i δ(x − xi)⟨φ(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' ∂µφ(xi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' φ(xn)⟩, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1) the central charge governs its two point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' ⟨T µν(x)T λσ(0)⟩ = CT S2 d 1 x2d �1 2(IµλIνσ + IµσIνλ) − 1 Dδµνδλσ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='2) Here Iµν(x) ≡ δµν −2xµxν/x2 and Sd = 2πD/2/Γ(D/2) is the volume of the D−1 dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' With this definition of the central charge, its value for a single free degree of freedom is Cscalar T = D/(D − 1) Cfermion T = D/2 C(D−2)/2 form T = D2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' There is 1 degree of freedom for a free scalar field and this value for a free fermion field and a free (D − 2)/2-form are 2D/2 and Γ(D − 2)/Γ2(D/2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In order to apply the analytic bounds for the stress tensor we must first normalize its two point function to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' With the new normalization, CT appears in the OPE coefficient squared c2 φφT µν as c2 φφT µν = 1 CT � ∆φD (D − 1) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='3) – 9 – 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 δϕ α(δϕ,1) Figure 2: The function α(δφ, 1) is plotted against δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It is always greater than zero for δφ > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' If we now apply the bound computed in section 2, we get CT > � ∆φD (D − 1) �2 1 A(δφ, 1)eα(δφ,1)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4) Here we have set δb = 1 because the role of Ob is played by T µν and because its conformal dimension is D, δb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For δb = 1, the expression for α simplifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' α(δφ, 1) = log � 2(2δφ − 1) �4δφ − 1 � 4δφ − 1 2(2δφ − 1) �2δφ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5) It is easy to see that α > 0 for all values of δφ in the unitary range i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' for δφ > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The plot of α(δφ, 1) against δφ is given in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The function A(δφ, 1) depends on the details of the function f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Recall that this bound applies if the CFT operators S ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The set P depends on f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' By construction the functional ω(∆, ℓ) is non-negative for ∆ ≥ 2∆φ (for all ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The sign of the functional ω(∆, ℓ) for ∆ < 2∆φ depends only on the sign of Re (f (zb)) and hence can be positive or negative depending on the choice of kernel f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Ideally one would want Re (f (zb)) ≥ 0 for all operators (except identity) satisfying unitarity bounds with ∆ < 2∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This would show that the central charge is exponentially large in D in any large dimensional conformal field theory except for the free theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We will now show that this condition, Re (f (zb)) ≥ 0 1 2 ≤ δ < 2δφ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='6) is impossible to achieve with the kind of functionals we are using i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' To see this, notice that the kernel f(z) is necessarily anti-symmetric around z = 1/2 (see equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This means f(1/2) = 0, and the Re(f(1/2 ± a)) are of opposite sign for any a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The functional – 10 – δ ω δϕ<3/4 δ* 1/2 1 2δϕ Double Trace δ ω δϕ>3/4 δ* 1/2 1 2δϕ Double Trace Figure 3: Schematic plots of functional ω(δ, ℓ) with kernel f(z) = fopt(z), for δφ < 3/4 (left) and δφ > 3/4 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Functional at stress tensor i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' with δ = 1, is positive and functional at identity (not shown) is negative, in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The ℓ dependence is not shown because the signature of ω(δ, ℓ) is independent of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' ω(∆, ℓ) for ∆ < 2∆φ is proportional to Re (f (zb)) (times a positive factor), and of opposite sign for zb = 1/2±a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The functional vanishes for zb = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Solving the condition z∗(δb) = 1/2, we get a critical value of δb where the functional must vanish,4 δcrit = 1 2 � 1 + � 2δφ − 1 � 4δφ − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='7) This means that the functional can never be of the same signature in the whole range 1/2 ≤ δ < 2δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Its sign has to change at δcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This means we can not satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' With this analysis we also conclude that we do not get any bound on the central charge for δφ = 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is because for this value of δφ, δcrit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' As the functional vanishes for δ = 1, the central charge is unbounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For other values of δφ we can certainly obtain exponentially large lower bound on the central charge given the set of CFT operators S ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We will characterize the set P below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' As discussed above, the operators with δ ≥ 2δφ belong to P so we will only be concerned with characterizing P of operators with δ < 2δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Case I: δφ < 3/4 In this case, the ω(∆) is positive for ∆ > δcritD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The schematic plot of the functional is shown in figure 3 on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Hence P consists of all the operators with ∆ > δcritD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We can further optimize over the kernel f(z) to produce the lowest upper bound on the OPE coefficient squared c2 φφOb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This optimization problem is the same one as is [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Borrowing their result, fopt(z) = (1 − 2z) [z(z − 1)]1/2 �� z(1 − z) + � zb(1 − zb) �2 , zb = z∗(δb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='8) For the case of stress tensor, we use δb = 1 and the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='3) that relates c2 φφT µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 4The other solution 1 2 � 1 − √2δφ − 1√4δφ − 1 � lies outside the unitary region so we ignore it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 11 – Case II: δφ > 3/4 In this case, the kernel fopt(z) leads to a functional that is negative for the stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' However, we need the functional to be positive on the stress tensor to obtain lower bound on the central charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is achieved simply by using f(z) = −fopt(z) as the kernal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' With this choice, the functional is positive for δ < δcrit and also for δ > 2δφ but it is negative for δcrit < δ < 2δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The schematic plot of this functional is shown in figure 3 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The region (δcrit, 2δφ) is excluded from the set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In other words, for the case δφ > 3/4, the set P consists of operators with δ < δcrit along with operators satisfying δ > 2δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1 Modifying the set P We can modify the set P by changing the functional in both of these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is important for widening the applicability of the bound i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' to make it so that all the CFT operators S ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Let us call the function that satisfies all the properties listed below equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='35) a kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Note that if we multiply any kernel function by the so called CDD factor α(z, zi) we get a new kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The CDD factor is given by α(z, zi) = ˜α(x(z), x(zi)), ˜α(x, y) = x − y xy − 1, x(z) = � zb(1 − zb) − � z(1 − z) � zb(1 − zb) + � z(1 − z) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='9) where zi ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It has the property that |α(z, zi)| = 1 for z ∈ (−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The α(z, zi) factor endows the kernel with a pair of additional single zeros in the unitarity region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' These two new zeros δi are at the two solutions of the equation zi = z∗(δi) and 1 − zi = z∗(¯δi), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Each of these equations have two solutions but we are interested in the ones that lie in the unitarity domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The pair of zeros (δi, ¯δi) are related to each other as ¯δi = 1 2 � 1 + � 1 − 4(δi − 1)δi + 4δφ(4δφ − 3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='10) This follows easily from equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It is easy to see that when both the roots are real, they are on opposite sides of δcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We can think of any one of the zeroes as the mirror image of the other across the point δcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is because a point that is closer to δcrit on the right side is mirrored to a point that is closer to δcrit on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' One can also see (by definition) that the mirror image of δcrit is δcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Also, the point δ = 1/2 is mirrored to the point δ = ¯δ1/2 ≡ 1 2 � 1 + � 2 + 4δφ(4δφ − 3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Hence, this mirror maps the interval D1 ≡ (0, δcrit) to the interval D2 ≡ (δcrit, ¯δ1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Also for δ ∈ D3 ≡ (¯δ1/2, 2δφ) the solution ¯δ is complex and does not correspond to any zero of the functional for real δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In figure 4 we have displayed a schematic plot of the functional obtained after multiplying fopt(z) by three CDD factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We can see that the set P now consists of points on either sides of δcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' As we change the number of CDD factors and their parameters zi the set P changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This gives us a lot of freedom to tune the P so that the set of CFT operators S ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' However, as we will see below, one can not do so in all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' To this end, let us characterize the CFT spectrum S for which it is not possible, in our framework, to obtain a functional with P such that S ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 12 – δ ω δ* 1/2 1 2δϕ Double Trace δ1 δ2 δ2 δ1 δ3 Figure 4: Schematic plot of functional ω(δ, ℓ) where the kernel function f(z) is taken to be fopt(z) times three CDD factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' There are two single zeros in D1 and two “mirror image” single zero in D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The function as double zeros for δ > 2δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The unpaired zero δ3 is in region D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The CDD factors are chosen in a way that the functional is positive for stress tensor i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='2 Applicability of the bound First notice that even if we use the CDD factors to multiply the kernel (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='8) the functional takes opposite values on the mirror pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is simply a consequence of anti-symmetry f(z) = −f(1 − z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For the rest of the discussion, it is useful to think graphically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Let us denote all the CFT operators on the δ axis with red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Let us reflect all the red dots in the D1 region onto the D2 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Let us color these reflections blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Now focus only on the region D2 ∪ D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It has some distribution of red dots and blue dots, with only red dots in region D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We will be successful in finding the desired P if the functional is positive on the red dots and negative on the blue dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' To achieve this we multiply the kernel (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='8) by CDD factors that give a single zero whenever we transit from red dots to blue dots and vice versa so that the functional is positive on the red dots and negative on the blue ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This construction is illustrated in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In the unlikely case there are pairs of blue dot and red dot that are coincident we need to have a single zero precisely at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This functional achieves our objective of S ⊂ P with the minimum number of CDD factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' When is this construction not admissible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' If the total number of CDD factors used are O(D) then the saddle point computation performed in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='27) is invalid as it is no longer determined only by the conformal block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' So we conclude that the exponential lower bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4) on the central charge is applicable if the following conditions are met: Condition 1: The exponential lower bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4) on the central charge is applicable if there are less than O(D) transitions between red dots and blue dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 13 – 풟1 풟2 δcrit ¯δ1/2 2δϕ 1 2 풟3 Figure 5: The CFT operators are denoted on the δ axis with red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The blue dots are the “mirror reflection” of the red dots in the region D1 to the region D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In order to construct a functional with S ∈ P, we pick five CDD factors with zeros in the red-to-blue transition regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' These are as shown in the figure schematically with crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Condition 2: There is no CFT operator that is in the O(1/D) neighborhood in the δ space of the mirror image of the stress tensor (δ = ¯δT ≡ 1 2 � 1 + � 1 + 4δφ(4δφ − 3) � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The second condition, in particular implies that for δφ = 3/4, ¯δT = δT = 1 and hence the bound is not obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Is the first condition reasonable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We can answer this question with the same level of (im)precision with which it is asked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The following discussion of this condition is somewhat impressionistic and is only meant to offer some intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Let φ be the CFT operator with smallest scaling dimension δφD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For δφ < 3/4, we have δcrit < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In this case, the operators in the region D1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' between (1/2, δcrit) are only scalars because of the unitarity bound on operators with spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Because they are only scalars, it is reasonable to assume that these are less than O(D) in number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' On the other side of δcrit, beyond δ = 1, we have operators that have arbitrary spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It is perhaps natural that there are O(D) of them in the regions D2 and D3 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' between (δcrit, ¯δ1/2) and (¯δ1/2, 2δφ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Because the applicability of the bound only depends on the number of red-blue transitions as described above and that the number of operators are expected to be less than O(D) operators in D1, we expect the bound on the central charge to be applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For δφ > 3/4, we have δcrit > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In this case, on both side of δcrit i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' in all three regions D1, D2 and D3 there are spinning operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Operators in both these regions could perhaps be O(D) in number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In this case, it is not reasonable expect that the number of red-blue transitions are less than O(D) in number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' So in this case we have to assume, perhaps somewhat unnaturally, that the spectrum operators in the window (1, δcrit) is sparse i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' of less than O(D) for the central charge bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4) to be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' As δφ increases above 3/4, at δφ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='44 we get δcrit = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' At this point, we definitely have multi stress-tensor operators appearing in the region D1 in a dense fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We don’t expect the central charge bound to be valid in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' However, we don’t need to probe the regime δφ > 1 at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' If the lightest scalar has δφ > 1, we can simply use the stress tensor component, say T 00 as a scalar operator in D − 1 dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' As D is taken to be large, this change in dimension – 14 – is of little significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For δφ=T 00 = 1, δcrit ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For the bound to be applicable we have to require that the operator spectrum should be sparse up to this value of δcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This discussion also makes it clear why the free theory is allowed in large D from this point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' If we consider the four point function of the free field φ, we have 1 = 2δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' At this value of δ, the functional vanishes and we do not get any bound on the central charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We could also consider the four point function of φ2 operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In this case, δφ2 = 1, this means δcrit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' However, it is evident that a dense double-trace spectrum starts from δ = 1 and hence the sparseness assumption and in turn the central charge bound is not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In [8] a different approach was taken to constrain the space of conformal theories in large dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The conclusions from that paper agree with the results presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In [8] it was argued that for δφ < 3/4, the four point function of φ’s is exponentially close to that of the generalized free field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' For 3/4 < δφ < 1, this conclusion required a sparseness assumption on the spectrum akin to the one argued here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 4 Corrections at large but finite D Now that we have obtained exponential bound in large D at leading order, it is a natural to ask how the bound gets corrected at finite D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is essentially a question about correction to the functional value ω(∆, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' There are two sources of corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Corrections to the kernel h(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Corrections to the integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='13) with the corrected kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Let us discuss the first source of corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We would like the kernel to obey the gluing condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='20) and the extremality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Note that the condition that the inequality in euqation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='30) is saturated is not a true extremal condition in finite dimension since it does not give rise to double zeros of ω(∆, ℓ) for ∆ ≥ 2∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It does however insure that the functional is non-negative for ∆ ≥ 2∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' It is in principle possible to solve the true extremality condition in perturbation theory in 1/D but it is quite cumbersome to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We will instead take the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='30) as a necessary condition on the kernel in finite dimension as it at least guarantees positivity of ω(∆, ℓ) for ∆ ≥ 2∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The gluing condition and “extremality condition” together imply Im(f(z)) = z2Dδφ−2|f(1/z)| + (1 − z)2Dδφ−2|f(1/(1 − z))| z ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1) In our analysis we could simply set the right hand side to zero in the D → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In large but finite D, the right hand side does give small non-perturbative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We will ignore these corrections and continue to use the kernel fopt(z) that satisfies Imf(z) = 0 for z ∈ (0, 1) even in finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The second set of corrections, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' the corrections to the saddle point integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='13) are both perturbative as well as non-perturbative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Instead of treating them differently, we deal with them by simply performing the integral numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We expect to get close to accurate – 15 – bounds (with errors that are exponentially small in D) in large but finite dimension in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The results for D = 50, 20, 10, 6, 4 are presented in figure 6, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We have also given the bound obtained by setting the appropriate value of D in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4) with f(z) = fopt(z) for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' As we have not made use of the CDD factors in the functional, we have to assume absence of operators in certain ∆ range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This range is given by gray shaded region in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In principle, we can tune the functional to the operator spectrum using the CDD factors as described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The only reason that these bounds are not completely trustworthy for small D is because the right hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1) can not be approximated by zero for small D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Also note that D controls these errors only because we have take ∆φ to be of the same order as D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This implies that the numerical bounds are good approximations even in small D as long as ∆φ is taken to be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In this way we can repurpose these bounds as bounds on the central charge for theories with large gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We are currently investigating this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In order to get completely trustworthy bounds for small D and small ∆φ, we need to solve the condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1) exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='– 16 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='\uf750 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='\uf750 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='\uf750 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='\uf750 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='\uf750 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='\uf750 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='Δϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='log[CT]Min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='D=50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='Δϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='45 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='Δϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='Δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='D=10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='Δϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='Figure 6: Approximate numerical bounds on the central charge of CFTs in D = 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The green curve is obtained by substituting appropriate value of D in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The bounds are obtained with the assumption that no scalar operators are present in the gray shaded range for a given ∆φ (similar but less constraining assumptions are also made for spinning operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The red line is ∆ = 2∆φ and the solid blue line is ∆ = Dδcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 17 – \uf750 \uf750 \uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750 \uf750 \uf750 \uf750\uf750 \uf750 \uf750 \uf750 \uf750 \uf750 \uf750 \uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750 4 5 6 7 8 4 2 0 2 4 6 8 Δϕ log[CT]Min D=6 Δϕ D 4 5 6 7 8 4 6 8 10 12 14 16 Δϕ Δ D=6 Δϕ D \uf750 \uf750 \uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750 \uf750 \uf750 \uf750 \uf750 \uf750 \uf750 \uf750 \uf750 \uf750 \uf750 \uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 6 4 2 0 2 4 Δϕ log[CT]Min D=4 Δϕ D 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5 2 4 6 8 10 Δϕ Δ D=4 Δϕ D Figure 7: Approximate numerical bounds on the central charge of CFTs in D = 6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The green curve is obtained by substituting appropriate value of D in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The bounds are obtained with the assumption that no scalar operators are present in the gray shaded range for a given ∆φ (similar but less constraining assumptions are also made for spinning operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The red line is ∆ = 2∆φ and the solid blue line is ∆ = Dδcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Acknowledgement We would like to thank the TIFR string theory group, in particular Gautam Mandal, Shiraz Minwalla, Onkar Parrikar, Sandip Trivedi for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We would like to thank Adwait Gaikwad for collaboration in related projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This work is supported by the Infosys Endowment for the study of the Quantum Structure of Spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The work of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' is also supported by the SERB Ramanujan fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We acknowledge support of the Department of Atomic Energy, Government of India, under Project Identification No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' RTI 4002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We would also like to acknowledge our debt to the people of India for their steady support to the study of the basic sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' A Simplifying the functional In this appendix, we will simplify the functional given in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='13) to the expression in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='14) using some simple properties of the conformal block specialized to the locus z = ¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 18 – We denote the block specialized to this locus as G∆,ℓ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' G∆,ℓ(z ± iϵ) = e±iπ∆ ˆG∆,ℓ(z) , z ∈ (−∞, 0) G∆,ℓ(z) = ˆG∆,ℓ � z z − 1 � , z ∈ (0, 1) (u-symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1) Where ˆG∆,ℓ = |G∆,ℓ(z + iϵ)|, for z ∈ (−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Using these properties it is easy to see that the first term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='13) simplifies to, � 0 −∞ dz h(1 − z)Disc �G∆,ℓ(z) z2∆φ � = � eiπ(∆−2∆φ) − e−iπ(∆−2∆φ)� � 0 −∞ dz h(1 − z) ˆG∆,ℓ(z) |z|2∆φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='2) To simplify the second term we first write the integral over the discontinuity as a contour integral wrapping the branch cut from (−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Then we do a variable change z → 1 − z and deform the new integration contour that wraps the cut (1, ∞) back to the one that wraps the cut (−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In doing so we use the fall off condition of h(z) given above equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='13) to neglect the contribution of the arc at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In this deformation process, we also collect the contribution of the branch cut of h(z) in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' � 0 −∞ dz h(1 − z)Disc �G∆,ℓ(1 − z) (1 − z)2∆φ � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='3) = � 0 −∞ dz Disc � h(z)G∆,ℓ(z) z2∆φ � + � 1 0 dz Disc [h(z)] G∆,ℓ(z) z2∆φ = � 0 −∞ dz � h(z + iϵ)eiπ(∆−2∆φ) − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' � ˆG∆,ℓ(z) |z|2∆φ + � 1 0 dz Disc [h(z)] G∆,ℓ(z) z2∆φ = � 0 −∞ dz � h(z + iϵ)eiπ(∆−2∆φ) − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' � ˆG∆,ℓ(z) |z|2∆φ + � 0 −∞ dz (1 − z)2∆φ−2Disc � h � z z − 1 � � ˆG∆,ℓ(z) |z|2∆φ In second term of last line we have done variable transform from z → z/(z − 1) and used the u-symmetry of block from equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Since h(z) is real for z ∈ (1, ∞) we have h(z∗) = h(z)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This implies Disc [h(z)] is purely imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='2) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='3), and using the definitions of f(z) and g(z) in equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='19) respectively, gives the simplified expression in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Action on operators with ∆ > 2∆φ The simplified integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='14) can be sued to compute the functional for operators with ∆ ≥ 2∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' That is because both f and g terms are separately convergent in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In the large D limit, these integrals are performed using the saddle point approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' As ˜g(z) ≡ (1 − z)2∆φg(z) is take to be O(1), the saddle point is controlled by ˆG∆(z)/|z|2∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This is computed in appendix B in equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The saddle point z∗ ≤ 0 for all ∆ ≥ 2∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 19 – Figure 8: The top two contours are the original contours of integration in equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The bottom two contours are their respective deformations to saddle points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Action on operators with ∆ < 2∆φ In this regime, the two integrals of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='14) are not individually convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We need to resort to the original expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='13) to compute the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' ω(∆) = 1 2πi � 0 −∞ dz h(1 − z) Disc �G∆(z) z2∆Φ − G∆(1 − z) (1 − z)2∆Φ � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4) If we look at the saddle point (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='7), it is easy to see that it lies in the range (0, 1) for ∆ < 2∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This motivates the contour deformation shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The functional is written as, ω(∆) = − 1 2πi � z0 0 dz Disc(h(1 − z)) G∆(z) z2∆Φ + 1 2πi � 1 z0 dz Disc(h(z)) G∆(z) z2∆Φ + 1 2πi � i∞ z0 dz (h(z) − h(1 − z)) G∆(z) z2∆Φ + 1 2πi � z0 −i∞ dz (h(z) − h(1 − z)) G∆(z) z2∆Φ = − � z0 0 dz g(1 − z) G∆(z) z2∆Φ − � 1 z0 dz g(z) G∆(z) z2∆Φ + 1 2i � i∞ z0 dz f(z) G∆(z) z2∆Φ + 1 2i � z0 −i∞ dz f(z) G∆(z) z2∆Φ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5) Here, in the second equality we have used the definitions of kernels f(z) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='18) and g(z) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Using the scaling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='23), the first two terms in second equality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' the g terms) vanishes in the limit ∆φ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' We can perform the saddle point integral in the direction perpendicular to real line at z = z∗ giving our integral, ω(∆) = (f(z∗ + iϵ) + f(z∗ − iϵ)) 2 G∆(z∗) z2∆Φ ∗ � 2π ∆φq′′ δ(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='6) In appendix B we show that the steepest descent at z∗ is indeed perpendicular to the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Here the function qδ is defined in equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Using the gluing condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='20) – 20 – 0 1 Zo 1 00 1 Zo 0 7Im(f(z)) = 0, for z ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Thus, the functional in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='6) becomes ω(∆) = Re (f(z∗)) G∆(z∗) z2∆Φ ∗ � 2π ∆φp′′(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='7) Action on Identity Here we will look at the action of functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='13) on identity block given by, ω(1) = 1 2πi � ∞ 1 dz h(z)Disc � 1 z2∆φ − 1 (1 − z)2∆φ � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='8) The first term is zero since z−2∆φ does not have any discontinuity in the region (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In the second term, we deform the contour from (1, ∞) to (−∞, 1), where h has a discontinuity but (1 − z)−2∆φ does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' ω(1) = 1 2πi � 1 −∞ dz Disc [h(z)] (1 − z)2∆φ = 1 2πi � 0 −∞ dz Disc [h(z)] (1 − z)2∆φ + 1 2πi � 1 0 dz Disc [h(z)] (1 − z)2∆φ = 1 2πi � 0 −∞ dz Disc [h(z) − h(1 − z)] (1 − z)2∆φ + 1 2πi � 1 0 dz Disc [h(z)] (1 − z)2∆φ = − � 0 −∞ dz Im(f(z)) (1 − z)2∆φ − � 1 0 dz g(z) (1 − z)2∆φ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='9) where, in the third equality we have added h(1 − z) inside the discontinuity because h(1 − z) does not have any discontinuity on z ∈ (−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' In the fourth equality we have used the definitions of kernels f(z) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='18) and g(z) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Using the scaling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='23), the first term vanishes in the limit ∆φ → ∞, and we get ω(1) = − � 1 0 dz g(z) (1 − z)2∆φ = − � 1 0 dz ˜g(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='10) Now using extremality condition where the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='30) is saturated the above functional becomes, ω(1) = − � 1 0 dz (1 − z)−2 ����f � z z − 1 ����� = − � 0 −∞ dz |f(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='11) B Conformal blocks in large D and saddle points The conformal block in large dimension is computed in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' On the z = ¯z locus it is given by G∆,ℓ(z) = 2∆ � 1 − � z 2−z �2 A1−ℓ(1) � z 2 − z �∆ 2F 1 � ∆ 2 , ∆ − 1 2 , ∆ − D 2 + 1, � z 2 − z �2� (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1) – 21 – where, the function Aβ(x) is given in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Using the integral representation of the hypergeometric function B(b, c − b) 2F 1 (a, b, c, z) = � 1 0 dt tb−1(1 − t)c−b−1(1 − zt)−a (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='2) and the scaling ∆ = δD, the conformal block (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='1) becomes, GδD,ℓ(z) = 2δD � 1 − � z 2−z �2 A1−ℓ(1) � z 2−z �δD B � δD−1 2 , δD−D+3 2 � � 1 0 dt � 1 − t t3 � � � � � � t δ 2 (1 − t) δ−1 2 � 1 − � z 2−z �2 t � δ 2 � � � � � � D (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='3) This integral can be done by saddle point approximation in large D limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' This has been done in [8], but we will reproduce it here for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The saddle point equation and its solution is, d dt log � � � � � � t δ 2 (1 − t) δ−1 2 � 1 − � z 2−z �2 t � δ 2 � � � � � � = 0 ⇒ t± ∗ = 2δ − 1 ± � 4δ(δ − 1) � 1 − � z 2−z �2� + 1 2(δ − 1) � z 2−z �2 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='4) For z ∈ (−∞, 1], t− ∗ lies in the integration range (0, 1) while t+ ∗ lies outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Picking up the saddle t− ∗ the block at leading order in large D is, GδD,ℓ(z) = 1 � 1 − � z 2−z �2 A1−ℓ(1) vδ(z) eDqδ(z) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5) where, qδ(z) = log �� 2(δ − 1)2ˆy+ (2δ − 1) (B − 2δ + 2(δ − 1)y+ + 1) �2(2δ − 1)2(B − 2δ + 1)(B − 2δ + 2(δ − 1)y+ + 1) (1 − B)δ(δ − 1)2y+ �δ/2 � vδ(z) = � (16δ)−1(δ − 1)−3(B − 1)3δ(2δ − 1)(B − 2δ + 2(δ − 1)y+ + 1)2 2(δ − 1)2(B − 4δ)y2 + + 2(δ − 1) (B (1 − 3δ) + (8δ2 − 6δ + 1)) y+ + (1 − 2δ)2(1 + B − 2δ) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='6) with, B = � 1 + 4(1 − y+)δ(δ − 1) and y+ = � z 2−z �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Given the large dimensional block (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='5) in leading order in large D limit, we will now compute the saddle point approximation of ˆG∆,ℓ(z) z2∆φ with respect to z in large D limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' The saddle point equation and the solution for this is d dz (qδ(z) − 2δφ log(z)) = 0 ⇒ z∗ = (2δφ − δ)(2δφ + δ − 1) δφ(4δφ − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='7) – 22 – It is easy to verify that, d2 dz2 (qδ(z) − 2δφ log(z)) �� z=z∗ ≶ 0 for δ ≷ 2δφ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='8) This means that the path of steepest descent from the saddle point z∗ is along the real axis for δ > 2δφ while it is perpendicular to the real axis for δ < 2δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Rattazzi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Rychkov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' Vichi, Central Charge Bounds in 4D Conformal Field Theory, Phys.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content='0004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} +page_content=' – 23 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf'} diff --git a/dtFPT4oBgHgl3EQfyzWo/content/tmp_files/2301.13173v1.pdf.txt b/dtFPT4oBgHgl3EQfyzWo/content/tmp_files/2301.13173v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6733151487287caf78bdbd97dd3f27ac8662b190 --- /dev/null +++ b/dtFPT4oBgHgl3EQfyzWo/content/tmp_files/2301.13173v1.pdf.txt @@ -0,0 +1,981 @@ +Shape-aware Text-driven Layered Video Editing +Yao-Chih Lee +Ji-Ze Genevieve Jang +Yi-Ting Chen +Elizabeth Qiu +Jia-Bin Huang +University of Maryland, College Park +https://text-video-edit.github.io +Figure 1. Shape-aware consistent video editing. Our method enables consistent text-guided video editing with both appearance and +shape changes. The top row shows the input frames. The second and third rows present editing results from two text prompts: “running +sports car” and “running minivan”, respectively. Note that text-driven editing involves both texture and structure editing on the +foreground object. Our method performs consistent edits on sequential frames while preserving the object motion in the input video. +Abstract +Temporal consistency is essential for video editing appli- +cations. Existing work on layered representation of videos +allows propagating edits consistently to each frame. These +methods, however, can only edit object appearance rather +than object shape changes due to the limitation of using +a fixed UV mapping field for texture atlas. +We present +a shape-aware, text-driven video editing method to tackle +this challenge. To handle shape changes in video editing, +we first propagate the deformation field between the in- +put and edited keyframe to all frames. We then leverage +a pre-trained text-conditioned diffusion model as guidance +for refining shape distortion and completing unseen regions. +The experimental results demonstrate that our method can +achieve shape-aware consistent video editing and compare +favorably with the state-of-the-art. +1. Introduction +Image editing. +Recently, image editing [19, 20, 24, 34, +40, 44] has made tremendous progress, especially those +using diffusion models [19, 20, 40, 44]. +With free-form +text prompts, users can obtain photo-realistic edited images +without artistic skills or labor-intensive editing. However, +unlike image editing, video editing is more challenging due +to the requirement of temporal consistency. Independently +editing individual frames leads to undesired inconsistent +frames, as shown in Fig. 2a. A na¨ıve way to deal with tem- +poral consistency in video editing is to edit a single frame +and then propagate the change to all the other frames. Nev- +ertheless, artifacts are presented when there are unseen pix- +els from the edited frame in the other frames, as shown in +Fig. 2b. +Video editing and their limitations. For consistent video +editing, Neural Layered Atlas (NLA) [18] decomposes a +video into unified appearance layers atlas. The layered de- +composition helps consistently propagate the user edit to +arXiv:2301.13173v1 [cs.CV] 30 Jan 2023 + +(a) Multi-frame editing with frame interpolation [42] +(b) Single-frame editing with frame propagation [17] +(c) Text2LIVE [2] with prompt “sports car” +Figure 2. Limitation of existing work. Compare these results from baseline methods with our “sports car” result in Fig. 1. (a) +Multiple frames are edited independently and interpolated by frame interpolation method [42]. Such an approach shows realistic +per-frame results but suffers from temporal flickering. (b) Extracting a single keyframe for image editing, the edits are propagated to each +frame via [17]. The propagated edits are temporally stable. However, it yields visible distortions due to the unseen pixels from the +keyframe. (c) The SOTA Text2LIVE [2] results demonstrate temporally-consistent appearance editing but remain the source shape +“Jeep” instead of the target prompt “sports car” by using the fixed UV mapping of NLA. +individual frames with per-frame UV sampling association. +Based on NLA, Text2LIVE [2] performs text-driven editing +on atlases with the guidance of the Vision-Language model, +CLIP [39]. Although Text2LIVE [2] makes video editing +easier with a text prompt, it can only achieve appearance +manipulation due to the use of fixed-shape associated UV +sampling. Since per-frame UV sampling gathers informa- +tion on motion and shape transformation in each frame to +learn the pixel mapping from the atlas, shape editing is not +feasible, as shown in Fig. 2c. +Our work. In this paper, we propose a shape-aware text- +guided video editing approach. The core idea in our work +lies in a novel UV map deformation formulation. With a se- +lected keyframe and target text prompt, we first generate an +edited frame by image-based editing tool (e.g., Stable Diffu- +sion [44]). We then perform pixel-wise alignment between +the input and edited keyframe pair through a semantic cor- +respondence method [51]. The correspondence specifies the +deformation between the input-edited pair at the keyframe. +According to the correspondence, the shape and appearance +change can then be mapped back to the atlas space. We can +thus obtain per-frame deformation by sampling the defor- +mation from the atlas to the original UV maps. While this +method helps with shape-aware editing, it is insufficient due +to unseen pixels in the edited keyframe. We tackle this by +further optimizing the atlas texture and the deformation us- +ing a pretrained diffusion model by adopting the gradient +update procedure described in DreamFusion [38]. Through +the atlas optimization, we achieve consistent shape and ap- +pearance editing, even in challenging cases where the mov- +ing object undergoes 3D transformation (Fig. 1). +Our contributions. +• We extend the capability of existing video editing +methods to enable shape-aware editing. +• We present a deformation formulation for frame- +dependent shape deformation to handle target shape +edits. +• We demonstrate the use of a pre-trained diffusion +model for guiding atlas completion in layered video +representation. +2. Related Work +Text-driven image synthesis and editing. +Recent years +have witnessed impressive progress in text-guided image +synthesis and manipulation using GANs [24, 25, 27, 41, 43, +55,56,61]. On text-to-image generation, DALL-E [41] first +demonstrates the benefits of training text-to-image models +2 + +using a massive image-text dataset. Most recent text-to- +image generators [6,30] use a pre-trained CLIP [39] as the +guidance. +On text-to-image manipulation/editing, recent +methods also take advantage of the pretrained CLIP embed- +ding for text-driven editing [9,36,58]. These methods either +pretrain the model with CLIP embedding as inputs or use a +test-time optimization approach [2,8,21]. +Recently, diffusion models [7, 14, 50] have shown re- +markable success in both text-guided image generation [1, +35, 44–46] and editing [12, 35, 44] tasks. +Stable Diffu- +sion [44] performs a denoising diffusion process in a latent +space and achieves high-resolution text-to-image genera- +tion and image-to-image translation results. In particular, +the release of the model trained on large-scale text-image +pair dataset [47] facilitates various creative applications +from artists and practitioners in the community. Our work +leverages the state-of-the-art text-to-image model, Stable +Diffusion [44], and extends its semantic image editing ca- +pability to consistent video editing. +Video generation. Building upon the success of photoreal- +istic (text-driven) image generation, recent work has shown +impressive results on video generation, with a focus on gen- +erating long video [5,11,49,60] and videos from free-form +text prompts [13, 48, 52]. Unlike video generation meth- +ods, our work differs in that we perform text-driven video +editing for real videos. +Video editing. +In contrast to the breakthrough of image +editing, video editing methods are faced with two core chal- +lenges: 1) temporal consistency and 2) computational com- +plexity of the additional dimension. To attain temporally +consistent editing effects, EbSynth [17] utilizes keyframes +and propagates the edits to the entire video with opti- +cal flows computed from consecutive frames. Such flow- +based techniques have been applied in other tasks such as +video synthesis [3], video completion [10,15,26], and blind +video consistency [22, 23]. Several studies address tem- +poral inconsistency in the latent space via GAN inversion +[29,54,57]. However, current GAN-based models can only +model datasets with limited diversity (e.g., portrait or ani- +mal faces). Another line of approaches [18, 28, 32, 33, 59] +decomposes a video into unified layer representation for +consistent editing. Neural Layered Atlas (NLA) [18] per- +forms test-time optimization on a given input video to learn +the canonical appearance layer and per-frame UV mapping +using video reconstruction loss. +With layer decomposi- +tion, one can use text-driven image editing techniques to +the unified layers to consistently broadcast the edits to each +frame. The work most relevant to ours is Text2LIVE [2] +and Loeschcke et al. [31]. Both methods build upon NLA +to perform text-driven editing on the learned atlases. A pre- +trained CLIP is used for each input video to guide the atlas +editing via a test-time optimization framework. Yet, limited +by the formulation of NLA, they only allow appearance ed- +its due to the fixed UV mapping from the atlas to frames. +The mapping fields store the original shape information in +each frame so that the fixed UV mapping restricts the free- +dom of shape editing in [2, 18, 31]. Our work also builds +upon NLA for achieving temporally consistent video edit- +ing. In contrast to existing methods [2, 31], we extend the +capability of text-driven editing to enable shape editing. +3. Method +Given an input video Is +1..N and a text prompt, our pro- +posed shape-aware video editing method produces a video +It +1..N with appearance and shape changes while preserving +the motion in the input video. For maintaining temporal +consistency, our method uses the pre-trained video decom- +position method, NLA [18], to acquire the canonical atlas +layer Is +A and the associated per-frame UV map Ws +A→1..N per +motion group. For simplicity, we assume a single moving +object in an input video so that there are two atlases Is,FG +A +and Is,BG +A +for foreground and background contents, respec- +tively. The edits in Is,FG +A +can be consistently transferred to +each frame with UV mapping. To render the image Is +j back, +we use the Ws +A→t and an alpha map αs +t to sample and blend: +Is +j = Is,FG +j +∗αs +j +Is,BG +j +∗(1−αs +j), +Is,g +j += Ws,g +A→j ⊗Is,g +A ,g ∈ {FG,BG}, +(1) +where ⊗ denotes the warping operation. +Following our +shape deformation introduction, we focus on the foreground +atlas and will omit FG from Is,FG for simplicity. +We first select a single source keyframe Is +k to pass into a +text-driven image editing tool (e.g., Stable Diffusion [44]). +The edits in target It +k will then be propagated to It +1..N +through the atlas space with the mapping of Ws +A→1..N. Yet, +the UV mapping cannot work when the edits involve shape +changes since Ws +A→1..N are specifically for reconstructing +the original shapes in the input video. Hence, to associate +the target shape correctly, we propose a UV deformation +formulation (Sec. 3.2) to transform each Ws +A→j into Wt +A→j +according to the deformation between (Is +k,It +k). +In other +words, the keyframe deformation Ds→t +k +between (Is +k,It +k) +serves as the bridge between input and output videos for +changing into the edited target shape while preserving the +source motion in the input. Note that the edits and keyframe +deformation Ds→t +k +alone are insufficient due to some unob- +served areas from the viewpoint of image Is +k. Therefore, to +acquire a complete and consistent editing result, we lever- +age a pre-trained diffusion model to optimize the editing +appearance and deformation parameters in the atlas space +in Sec. 3.3. The process produces the final edited video +It +1..N with desired object shape and appearance changes. +3 + +BG atlas +FG UV +alpha +BG UV +FG atlas +dense semantic +correspondence +deformed FG UV +deformed alpha +input keyframe +edited keyframe +Input frames +NLA pre-processing +keyframe editing (Sec. 3.1) +Optimization (Sec. 3.3) +backpropagate +diffusion-guided gradient +Text prompt "a running sports car" +Deformation +initialization +(Sec. 3.2) +FG appearance & +deformation atlases +image +editing +edited frames +... +... +select +single +keyframe +... +... +pretrained +diffusion model +Eq. 6 +background frames +Figure 3. Method overview. Given an input video and a target edit text prompt, our method first bases on a pre-trained NLA [18] to +decompose the video into unified atlases with the associated per-frame UV mapping. Aside from video decomposition, we use the +text-to-image diffusion model to manipulate a single keyframe in the video (Sec. 3.1). Subsequently, we estimate the dense semantic +correspondence between the input and edited keyframes for shape deformation. The shape deformation of the keyframe serves as the +bridge between input and output videos for per-frame deformation through the UV mapping and atlas. Our deformation module (Sec. 3.2) +transforms the UV map with the semantic correspondence to associate with the edits for each frame. To address the issues of unseen +pixels from the single keyframe, we optimize the edited atlas and the deformation parameters guided by a pre-trained diffusion model +with the input prompt (Sec. 3.3). +3.1. Keyframe editing +With the given text prompt, we edit a representative +keyframe Is +k (e.g., the middle frame of the video) by a +pre-trained Stable Diffusion [44] to obtain target edited +keyframe It +k. Afterward, we leverage a pre-trained semantic +correspondence model [51] to associate the correspondence +between two different objects. The pixel-level semantic cor- +respondence is the deformation that transforms the target +shape in It +k to the source shape in Is +k. +3.2. Deformation formulation +With the estimated semantic correspondence, we can +obtain the pixel-level shape deformation vectors, Dt→s +k +∈ +RH×W×2. The target shape in It +k are then deformed into the +source shapes in Is +k via Dt→s +k +: +It→s +k += Dt→s +k +⊗It +k. +(2) +With the aid of Dt→s +k +, the edited object can be back- +projected to the atlas to form an edited atlas, It→s +A +, by +Ws +k→A. Since it maintains the original shape, we cannot +directly map the edited It +k to the atlas with Ws +k→A. +Given the edited atlas It→s +A +, the appearance edits can al- +ready be propagated to each frame with Ws +A→1..N in source +shapes. However, this needs improvement since our goal is +to generate a new video with the target shape. In addition +to propagating the edited appearance via the atlas space, we +spread the displacement vectors to each frame to obtain per- +frame deformation by back projecting keyframe deforma- +tion Dt→s +k +into atlas space A with Ws +k→A to get Dt→s +A +. Yet, +simply warping into the new image space is insufficient as +the coordinate system also got transformed by the warping +operation. Therefore, we formulate a shape deformation +vector transformation matrix, MW, to handle the deforma- +tion vectors w.r.t. the original coordinate system by a warp +field W: +D′(x′,y′)T = MWD(x,y)T, +(3) +where (x,y) and (x′,y′) represent the corresponding pixels +in the source and target images, respectively, by the warping +field, W (i.e., (x′,y′) = W(x,y)). For pixel-level deforma- +tion, we compute a per-pixel deformation vector MW for +4 + +LL 1.J/CICTEsemantic correspondence +(a) Original UV sampling +(b) Shape-aware UV sampling +Warping field +(c) shape deformation vector transform +warping operation +shape deformation vector transform +atlas deformation map +atlas appearance map +Figure 4. Deformation formulation. Given the semantic correspondence between the input and edited keyframes, we map the edits back +to the atlas via the original UV map (in the shape of the original atlas). Meanwhile, we transform the per-pixel deformation vectors into +the atlas space with the same UV mapping field by (c). Consequently, the UV map samples the color and the deformation vectors onto +each frame to deform the original UV map respecting the edited shape. +each pixel (x,y) by: +MW = +� +W(x+∆x,y)−W(x,y) +W(x,y+∆y)−W(x,y) +�T � +1/∆x +1/∆y +� +, +(4) +where ∆x and ∆y denote small scalar shifts to form the local +coordinate system in the source space. In practice, to avoid +discrete sampling of warping, we use thin-plate spline [4] to +approximate the warping field smoothly. We illustrate the +transformation of the shape deformation vector in Fig. 4c. +With the transformation for the vector, we can obtain the +corresponding deformation in the target warped space with +the warp function W, which is the UV map in the atlas +framework. Thus, the deformation map Dt→s +k +is propagated +to each It +j by: +Dt→s +A += MWs +k→A ⋆(Ws +k→A ⊗Dt→s +k +) +Dt→s +j += MWs +A→j ⋆(Ws +A→j ⊗Dt→s +A +), +(5) +where ⋆ denotes the per-pixel matrix multiplication for the +deformation map. +Hence, we can deform the UV map +Ws +A→ j into Wt +A→j by Wt +A→j = Ds→t +j +⊗ Ws +A→ j. Note that +the alpha map for blending the target-shape object is also +deformed in the same manner by αt +j = Ds→t +j +⊗αs +j. Finally, +the edited It +j with initial deformation on the foreground ob- +ject can be obtained by: +It +j = Wt +A→j ⊗It→s +A +∗αt +j +IBG +A ∗(1−αt +j). +(6) +3.3. Atlas optimization +Through the deformation formulation in Sec. 3.2, we can +already obtain an edited video with the corresponding shape +changes if the semantic correspondence, i.e., Dt→s +k +, is reli- +able. However, the estimated semantic correspondence is +often inaccurate for shape deformation. As a result, it would +yield distortions in some frames. Moreover, the edited atlas +could be incomplete since it only acquires the editing pixels +from the single edited keyframe so the unseen pixels from +the keyframe are missing. Hence, these incomplete pixels +produce visible artifacts in other frames. +To address these issues, we utilize an additional atlas +network FθA and semantic correspondence network FθSC to +fill the unseen pixels and refine the noisy semantic corre- +spondence via an optimization. Here, the atlas network FθA +takes the initial appearance and deformation of the fore- +ground atlas (It→s +A +,Dt→s +A +) as input and outputs the refined +( ˜It→s +A +, ˜Dt→s +A +). Similarly, the semantic correspondence Dt→s +k +is approximated by a thin-plate spline. We feed the con- +trol points into the semantic correspondence network FθSC +to obtain the refined ˜Dt→s +k +. +We select several frames that capture different view- +points for optimization. +Our training of synthesizing +the edited frames, It, is guided by a pre-trained Vision- +Language model with the target prompt. +Inspired by +DreamFusion [38], we leverage a pre-trained diffusion +model [44] to provide pixel-level guidance by backpropa- +gating the gradient of noise residual to the generated im- +ages (without backpropagating through the U-Net model). +Adding a noise ε on It as the input, the pretrained diffusion +UNet outputs a predicted noise ˆε. The gradient of the noise +5 + +residual ˆε −ε is backpropagated to update θ: +∇θLdif f (It) ≜ Ei,ε[w(i)(ˆε −ε)∂It +∂θ ], +(7) +where i stands for the time step for the diffusion model and +the parameter set θ = {θA,θSC}. We update the unified in- +formation in the atlas space to maintain the temporal consis- +tency of the editing appearance and deformation with only +training on a few generated frames It. +In addition to the guidance of the diffusion model on +multiple frames, we also apply several constraints to the +learning of the refinement networks, FθA and FθSC, to pre- +serve the editing effects as in the target edited keyframe It +k. +To ensure that the deformation through the atlas can suc- +cessfully reconstruct the original edited It +k, the keyframe +loss, Lk, measures the error between the original It +k and the +reconstructed ˜It +k by L1 loss: +Lk = | ˜It +k −It +k|. +(8) +Besides, we also apply a total variation loss to encourage +the spatial smoothness of the refined appearance in the atlas. +The atlas loss is as follows: +LA = Ltv( ˜It→s +A +). +(9) +During the optimization, we also refine the semantic cor- +respondence ˜Dt→s +k +of the keyframe pair. An ideal seman- +tic correspondence matches semantically-similar pixels and +perfectly transforms the target shape into the source shape. +Therefore, we compute the errors of the deformed target and +the source object masks, Mt +k and Ms +k: +LSC = |( ˜Dt→s +k +⊗Mt +k)−Ms +k| +(10) +The total loss function L = Ldi f f + λkLk + λALA + +λSCLSC, λk,λA,λSC = 106,103,103. The optimized parame- +ters θ ∗ are then used to generate the final edited video It∗ +1..N. +Implementation details. +We implement our method in PyTorch. We follow the +video configuration in NLA with the resolution of 768 × +432. We use a thin-plate spline to inverse a warping field +to prevent introducing holes by forward warping. The re- +finement networks, FθA and FθSC exploits the architecture of +Text2LIVE [2] and TPS-STN [16], respectively. The opti- +mization performs on 3 to 5 selected frames, including It +1, It +k, +and It +N, for 600 to 1000 iterations. The optimization process +takes 20 mins on a 24GB A5000 GPU. We further utilize +an off-the-shelf super-resolution model [53] to obtain sharp +details in the final edited atlases. +4. Experimental Results +Here we show sample editing results in the paper. We +include additional video results in the supplementary ma- +terial. We will make our source code and editing results +publicly available to foster reproducibility. +4.1. Experimental Setup +Dataset. We select several videos from DAVIS [37]. Each +video contains a moving object in 50 to 70 frames. We edit +each video with a prompt that describes a target object with +a different shape from the original one. +Compared methods. We compare our results with SOTA +and several baseline methods. For fair comparisons, all the +baseline methods use the same image editing method, Sta- +ble Diffusion [44]. +• Multi-frame baseline: Multiple keyframes in a video are +edited individually. The nearby edited keyframes tempo- +rally interpolate the remaining frames with FILM [42]. +• Single-frame baseline: We extract a single keyframe +from a video to be edited. The edited information is then +propagated to each frame with EbSynth [17]. +• Text2LIVE [2]: The SOTA text-driven editing method +with NLA. Note that it utilizes a structure loss to preserve +the original shape. We compare the official Text2LIVE in +this section and show the comparison of removing structure +loss in our supplementary material. +4.2. Visual Comparison +We show a visual comparison with the baseline meth- +ods and Text2LIVE in Fig. 5. In the first example with +“blackswan→duck”, the multi-frame baseline shows +inconsistent editing in different frames. The single-frame +baseline suffers from inaccurate frame motion and thus +yields distortion during propagation. +Text2LIVE shows +a promising target appearance with temporal consistency +but cannot change the shape that matches the target ob- +ject. In contrast, our method provides the desired appear- +ance and consistent shape editing. In the second example +with “boat→yacht”, the single-frame baseline shows an +inconsistent shape since the frame propagation relies on +the frame motion of the source shape. +Consequently, it +cannot propagate the edited pixels correctly in a different +shape. +In the third example with “dog→cat”, the in- +put video contains a non-rigid motion moving object. It +poses further challenges for multi- and single-frame base- +lines. Again, Text2LIVE demonstrates plausible cat ap- +pearance while remaining in the source dog shape. Our +shape-aware method maintains the object motion and ma- +nipulates the texture and shape corresponding to the desired +editing. +6 + +Input +Ours +Multi-frame baseline +Single-frame baseline +Text2LIVE [2] +Figure 5. Visual comparison with baselines and SOTA. We show three examples with edits of “blackswan →duck”, “boat +→yacht”, and “dog →cat”. The multi-frame baseline shows inconsistency in the edited objects. The single-frame method suffers +from the incomplete flow motion of the source object shape and thus could not propagate the edits properly. Text2LIVE demonstrates +consistent appearance editing corresponding to the target edits. Nevertheless, the shape remains the same as the original object. In +contrast, our proposed method outperforms the compared methods with consistent and plausible appearance and shape editing. +Input +(a) fixed NLA +(b) w/ semantic corres. +(c) w/ UV deformation +(d) w/ optimization (full) +Figure 6. Ablation study. We study the effects of removing the deformation and optimization components. (a) Editing with fixed NLA +UV mapping. (b) Using a semantic correspondence with fixed UV, the edits are mapped to the atlas properly but still remains the original +shapes in results. (c) With deformation initialization (Sec. 3.2), the NLA UV maps are deformed to restore the target shape. (d) With +further atlas optimization (Sec. 3.3), the incomplete pixels in edited atlas and distortion (in car’s roof and back wheel) are refined. +7 + +Figure 7. Shape-aware interpolation. Our methods allow interpolation between two shapes by simply interpolating the atlas deformation +maps. The examples demonstrate the gradual changes from source objects to edited objects over the time. +4.3. Ablation Study +We conduct an ablation study in Fig. 6 to validate the +effectiveness of the UV deformation and atlas optimiza- +tion. +With fixed NLA UV mapping, the shape edits in +the keyframe cannot be adequately transformed through +the atlas to each frame (Fig. 6a). +Therefore, by adding +a keyframe semantic correspondence to deform the target +into the source shape, the fixed UV maps the edits correctly +into the atlas but remains source shapes in the edited frames +(Fig. 6b). To restore the target shape, our deformation mod- +ule deforms the UV maps by the semantic correspondence +(Fig. 6c). However, the unseen pixels and inaccurate cor- +respondence yield artifacts in different views (e.g., in the +car’s roof and back wheel). We refine the edited atlas and +deformation with the atlas optimization (Fig. 6d). +4.4. Application +We present an application of shape-aware interpolation +in Fig. 7. Through interpolating the deformation maps, the +object shape can be easily interpolated without additional +frame interpolation methods. Similarly, we can interpolate +atlas textures. Note that we directly apply image editing +on the background atlas since it can be treated as a natu- +ral panorama image (shown in Fig. 3). However, the fore- +ground atlas is an unwrapped object texture, which is unnat- +ural for general pre-trained editing models. Therefore, we +edit the video frame and map it back to the atlas. This ap- +proach is more general and allows users to use their chosen +images for video editing. +4.5. Limitations +Our method strictly relies on the many-to-one mapping +from individual frames to a unified atlas. However, NLA +may fail to get the ideal mapping in challenging scenarios +with complex motions. Therefore, we observe artifacts in +the erroneous mapping regions (e.g., the motion of hind +legs shown in Fig. 8). In addition, it remains difficult to +build semantic correspondence between two different ob- +Input +Edit +Figure 8. Limitations. We visualize a failure example (bear +→lion). The inaccurate NLA mapping in the motion of +crossing hind legs yields distortion in the edited result. +Figure 9. User-guided correspondence. Associating two +different objects remains challenging even for the SOTA semantic +correspondence methods. For a pair of source (a) and target (b), +the severe false matching can be corrected by users’ manual +warping for better results. +jects. While the atlas optimization can improve noisy cor- +respondences, poor semantic correspondence initialization +would hinder the optimization. We show that user manual +correction (in Fig. 9) can lead to better video editing results. +5. Conclusions +We have presented a shape-aware text-driven video edit- +ing method. We tackle the limitation of appearance-only +8 + +ORUGSIOURUGUNabCdmanipulation in existing methods. We propose a deforma- +tion formulation using layered video representation to trans- +form the mapping field corresponding to the target shape +edits. We further refine the unseen regions by utilizing the +guidance from a pre-trained text-to-image diffusion model. +Our method facilitates a variety of shape and texture editing +applications. +Societal impacts. +Our work proposes a tool for enabling +creative video editing applications. 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In ICCV, 2017. 2 +11 + diff --git a/dtFPT4oBgHgl3EQfyzWo/content/tmp_files/load_file.txt b/dtFPT4oBgHgl3EQfyzWo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..91d13243c157f26772537402d3ab3e84e977b3f7 --- /dev/null +++ b/dtFPT4oBgHgl3EQfyzWo/content/tmp_files/load_file.txt @@ -0,0 +1,553 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf,len=552 +page_content='Shape-aware Text-driven Layered Video Editing Yao-Chih Lee Ji-Ze Genevieve Jang Yi-Ting Chen Elizabeth Qiu Jia-Bin Huang University of Maryland, College Park https://text-video-edit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='io Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Shape-aware consistent video editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our method enables consistent text-guided video editing with both appearance and shape changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The top row shows the input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The second and third rows present editing results from two text prompts: “running sports car” and “running minivan”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Note that text-driven editing involves both texture and structure editing on the foreground object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our method performs consistent edits on sequential frames while preserving the object motion in the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Abstract Temporal consistency is essential for video editing appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Existing work on layered representation of videos allows propagating edits consistently to each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' These methods, however, can only edit object appearance rather than object shape changes due to the limitation of using a fixed UV mapping field for texture atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We present a shape-aware, text-driven video editing method to tackle this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' To handle shape changes in video editing, we first propagate the deformation field between the in- put and edited keyframe to all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We then leverage a pre-trained text-conditioned diffusion model as guidance for refining shape distortion and completing unseen regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The experimental results demonstrate that our method can achieve shape-aware consistent video editing and compare favorably with the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Introduction Image editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Recently, image editing [19, 20, 24, 34, 40, 44] has made tremendous progress, especially those using diffusion models [19, 20, 40, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' With free-form text prompts, users can obtain photo-realistic edited images without artistic skills or labor-intensive editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' However, unlike image editing, video editing is more challenging due to the requirement of temporal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Independently editing individual frames leads to undesired inconsistent frames, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' A na¨ıve way to deal with tem- poral consistency in video editing is to edit a single frame and then propagate the change to all the other frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Nev- ertheless, artifacts are presented when there are unseen pix- els from the edited frame in the other frames, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Video editing and their limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' For consistent video editing, Neural Layered Atlas (NLA) [18] decomposes a video into unified appearance layers atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The layered de- composition helps consistently propagate the user edit to arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='13173v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='CV] 30 Jan 2023 (a) Multi-frame editing with frame interpolation [42] (b) Single-frame editing with frame propagation [17] (c) Text2LIVE [2] with prompt “sports car” Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Limitation of existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Compare these results from baseline methods with our “sports car” result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (a) Multiple frames are edited independently and interpolated by frame interpolation method [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Such an approach shows realistic per-frame results but suffers from temporal flickering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (b) Extracting a single keyframe for image editing, the edits are propagated to each frame via [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The propagated edits are temporally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' However, it yields visible distortions due to the unseen pixels from the keyframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (c) The SOTA Text2LIVE [2] results demonstrate temporally-consistent appearance editing but remain the source shape “Jeep” instead of the target prompt “sports car” by using the fixed UV mapping of NLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' individual frames with per-frame UV sampling association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Based on NLA, Text2LIVE [2] performs text-driven editing on atlases with the guidance of the Vision-Language model, CLIP [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Although Text2LIVE [2] makes video editing easier with a text prompt, it can only achieve appearance manipulation due to the use of fixed-shape associated UV sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Since per-frame UV sampling gathers informa- tion on motion and shape transformation in each frame to learn the pixel mapping from the atlas, shape editing is not feasible, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In this paper, we propose a shape-aware text- guided video editing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The core idea in our work lies in a novel UV map deformation formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' With a se- lected keyframe and target text prompt, we first generate an edited frame by image-based editing tool (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=', Stable Diffu- sion [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We then perform pixel-wise alignment between the input and edited keyframe pair through a semantic cor- respondence method [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The correspondence specifies the deformation between the input-edited pair at the keyframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' According to the correspondence, the shape and appearance change can then be mapped back to the atlas space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We can thus obtain per-frame deformation by sampling the defor- mation from the atlas to the original UV maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' While this method helps with shape-aware editing, it is insufficient due to unseen pixels in the edited keyframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We tackle this by further optimizing the atlas texture and the deformation us- ing a pretrained diffusion model by adopting the gradient update procedure described in DreamFusion [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Through the atlas optimization, we achieve consistent shape and ap- pearance editing, even in challenging cases where the mov- ing object undergoes 3D transformation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We extend the capability of existing video editing methods to enable shape-aware editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We present a deformation formulation for frame- dependent shape deformation to handle target shape edits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We demonstrate the use of a pre-trained diffusion model for guiding atlas completion in layered video representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Related Work Text-driven image synthesis and editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Recent years have witnessed impressive progress in text-guided image synthesis and manipulation using GANs [24, 25, 27, 41, 43, 55,56,61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' On text-to-image generation, DALL-E [41] first demonstrates the benefits of training text-to-image models 2 using a massive image-text dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Most recent text-to- image generators [6,30] use a pre-trained CLIP [39] as the guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' On text-to-image manipulation/editing, recent methods also take advantage of the pretrained CLIP embed- ding for text-driven editing [9,36,58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' These methods either pretrain the model with CLIP embedding as inputs or use a test-time optimization approach [2,8,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Recently, diffusion models [7, 14, 50] have shown re- markable success in both text-guided image generation [1, 35, 44–46] and editing [12, 35, 44] tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Stable Diffu- sion [44] performs a denoising diffusion process in a latent space and achieves high-resolution text-to-image genera- tion and image-to-image translation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In particular, the release of the model trained on large-scale text-image pair dataset [47] facilitates various creative applications from artists and practitioners in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our work leverages the state-of-the-art text-to-image model, Stable Diffusion [44], and extends its semantic image editing ca- pability to consistent video editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Video generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Building upon the success of photoreal- istic (text-driven) image generation, recent work has shown impressive results on video generation, with a focus on gen- erating long video [5,11,49,60] and videos from free-form text prompts [13, 48, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Unlike video generation meth- ods, our work differs in that we perform text-driven video editing for real videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Video editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In contrast to the breakthrough of image editing, video editing methods are faced with two core chal- lenges: 1) temporal consistency and 2) computational com- plexity of the additional dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' To attain temporally consistent editing effects, EbSynth [17] utilizes keyframes and propagates the edits to the entire video with opti- cal flows computed from consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Such flow- based techniques have been applied in other tasks such as video synthesis [3], video completion [10,15,26], and blind video consistency [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Several studies address tem- poral inconsistency in the latent space via GAN inversion [29,54,57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' However, current GAN-based models can only model datasets with limited diversity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=', portrait or ani- mal faces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Another line of approaches [18, 28, 32, 33, 59] decomposes a video into unified layer representation for consistent editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Neural Layered Atlas (NLA) [18] per- forms test-time optimization on a given input video to learn the canonical appearance layer and per-frame UV mapping using video reconstruction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' With layer decomposi- tion, one can use text-driven image editing techniques to the unified layers to consistently broadcast the edits to each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The work most relevant to ours is Text2LIVE [2] and Loeschcke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Both methods build upon NLA to perform text-driven editing on the learned atlases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' A pre- trained CLIP is used for each input video to guide the atlas editing via a test-time optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Yet, limited by the formulation of NLA, they only allow appearance ed- its due to the fixed UV mapping from the atlas to frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The mapping fields store the original shape information in each frame so that the fixed UV mapping restricts the free- dom of shape editing in [2, 18, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our work also builds upon NLA for achieving temporally consistent video edit- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In contrast to existing methods [2, 31], we extend the capability of text-driven editing to enable shape editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Method Given an input video Is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='.N and a text prompt, our pro- posed shape-aware video editing method produces a video It 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='.N with appearance and shape changes while preserving the motion in the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' For maintaining temporal consistency, our method uses the pre-trained video decom- position method, NLA [18], to acquire the canonical atlas layer Is A and the associated per-frame UV map Ws A→1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='.N per motion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' For simplicity, we assume a single moving object in an input video so that there are two atlases Is,FG A and Is,BG A for foreground and background contents, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The edits in Is,FG A can be consistently transferred to each frame with UV mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' To render the image Is j back, we use the Ws A→t and an alpha map αs t to sample and blend: Is j = Is,FG j ∗αs j +Is,BG j ∗(1−αs j), Is,g j = Ws,g A→j ⊗Is,g A ,g ∈ {FG,BG}, (1) where ⊗ denotes the warping operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Following our shape deformation introduction, we focus on the foreground atlas and will omit FG from Is,FG for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We first select a single source keyframe Is k to pass into a text-driven image editing tool (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=', Stable Diffusion [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The edits in target It k will then be propagated to It 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='.N through the atlas space with the mapping of Ws A→1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='.N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Yet, the UV mapping cannot work when the edits involve shape changes since Ws A→1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='.N are specifically for reconstructing the original shapes in the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Hence, to associate the target shape correctly, we propose a UV deformation formulation (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='2) to transform each Ws A→j into Wt A→j according to the deformation between (Is k,It k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In other words, the keyframe deformation Ds→t k between (Is k,It k) serves as the bridge between input and output videos for changing into the edited target shape while preserving the source motion in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Note that the edits and keyframe deformation Ds→t k alone are insufficient due to some unob- served areas from the viewpoint of image Is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Therefore, to acquire a complete and consistent editing result, we lever- age a pre-trained diffusion model to optimize the editing appearance and deformation parameters in the atlas space in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The process produces the final edited video It 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='.N with desired object shape and appearance changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3 BG atlas FG UV alpha BG UV FG atlas dense semantic correspondence deformed FG UV deformed alpha input keyframe edited keyframe Input frames NLA pre-processing keyframe editing (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='1) Optimization (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='3) backpropagate diffusion-guided gradient Text prompt "a running sports car" Deformation initialization (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='2) FG appearance & deformation atlases image editing edited frames .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' select single keyframe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' pretrained diffusion model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 6 background frames Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Method overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Given an input video and a target edit text prompt, our method first bases on a pre-trained NLA [18] to decompose the video into unified atlases with the associated per-frame UV mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Aside from video decomposition, we use the text-to-image diffusion model to manipulate a single keyframe in the video (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Subsequently, we estimate the dense semantic correspondence between the input and edited keyframes for shape deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The shape deformation of the keyframe serves as the bridge between input and output videos for per-frame deformation through the UV mapping and atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our deformation module (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='2) transforms the UV map with the semantic correspondence to associate with the edits for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' To address the issues of unseen pixels from the single keyframe, we optimize the edited atlas and the deformation parameters guided by a pre-trained diffusion model with the input prompt (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Keyframe editing With the given text prompt, we edit a representative keyframe Is k (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=', the middle frame of the video) by a pre-trained Stable Diffusion [44] to obtain target edited keyframe It k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Afterward, we leverage a pre-trained semantic correspondence model [51] to associate the correspondence between two different objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The pixel-level semantic cor- respondence is the deformation that transforms the target shape in It k to the source shape in Is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Deformation formulation With the estimated semantic correspondence, we can obtain the pixel-level shape deformation vectors, Dt→s k ∈ RH×W×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The target shape in It k are then deformed into the source shapes in Is k via Dt→s k : It→s k = Dt→s k ⊗It k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (2) With the aid of Dt→s k , the edited object can be back- projected to the atlas to form an edited atlas, It→s A , by Ws k→A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Since it maintains the original shape, we cannot directly map the edited It k to the atlas with Ws k→A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Given the edited atlas It→s A , the appearance edits can al- ready be propagated to each frame with Ws A→1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='.N in source shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' However, this needs improvement since our goal is to generate a new video with the target shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In addition to propagating the edited appearance via the atlas space, we spread the displacement vectors to each frame to obtain per- frame deformation by back projecting keyframe deforma- tion Dt→s k into atlas space A with Ws k→A to get Dt→s A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Yet, simply warping into the new image space is insufficient as the coordinate system also got transformed by the warping operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Therefore, we formulate a shape deformation vector transformation matrix, MW, to handle the deforma- tion vectors w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' the original coordinate system by a warp field W: D′(x′,y′)T = MWD(x,y)T, (3) where (x,y) and (x′,y′) represent the corresponding pixels in the source and target images, respectively, by the warping field, W (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=', (x′,y′) = W(x,y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' For pixel-level deforma- tion, we compute a per-pixel deformation vector MW for 4 LL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='J/CICTEsemantic correspondence (a) Original UV sampling (b) Shape-aware UV sampling Warping field (c) shape deformation vector transform warping operation shape deformation vector transform atlas deformation map atlas appearance map Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Deformation formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Given the semantic correspondence between the input and edited keyframes, we map the edits back to the atlas via the original UV map (in the shape of the original atlas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Meanwhile, we transform the per-pixel deformation vectors into the atlas space with the same UV mapping field by (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Consequently, the UV map samples the color and the deformation vectors onto each frame to deform the original UV map respecting the edited shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' each pixel (x,y) by: MW = � W(x+∆x,y)−W(x,y) W(x,y+∆y)−W(x,y) �T � 1/∆x 1/∆y � , (4) where ∆x and ∆y denote small scalar shifts to form the local coordinate system in the source space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In practice, to avoid discrete sampling of warping, we use thin-plate spline [4] to approximate the warping field smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We illustrate the transformation of the shape deformation vector in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' With the transformation for the vector, we can obtain the corresponding deformation in the target warped space with the warp function W, which is the UV map in the atlas framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Thus, the deformation map Dt→s k is propagated to each It j by: Dt→s A = MWs k→A ⋆(Ws k→A ⊗Dt→s k ) Dt→s j = MWs A→j ⋆(Ws A→j ⊗Dt→s A ), (5) where ⋆ denotes the per-pixel matrix multiplication for the deformation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Hence, we can deform the UV map Ws A→ j into Wt A→j by Wt A→j = Ds→t j ⊗ Ws A→ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Note that the alpha map for blending the target-shape object is also deformed in the same manner by αt j = Ds→t j ⊗αs j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Finally, the edited It j with initial deformation on the foreground ob- ject can be obtained by: It j = Wt A→j ⊗It→s A ∗αt j +IBG A ∗(1−αt j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (6) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Atlas optimization Through the deformation formulation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='2, we can already obtain an edited video with the corresponding shape changes if the semantic correspondence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=', Dt→s k , is reli- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' However, the estimated semantic correspondence is often inaccurate for shape deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' As a result, it would yield distortions in some frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Moreover, the edited atlas could be incomplete since it only acquires the editing pixels from the single edited keyframe so the unseen pixels from the keyframe are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Hence, these incomplete pixels produce visible artifacts in other frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' To address these issues, we utilize an additional atlas network FθA and semantic correspondence network FθSC to fill the unseen pixels and refine the noisy semantic corre- spondence via an optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Here, the atlas network FθA takes the initial appearance and deformation of the fore- ground atlas (It→s A ,Dt→s A ) as input and outputs the refined ( ˜It→s A , ˜Dt→s A ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Similarly, the semantic correspondence Dt→s k is approximated by a thin-plate spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We feed the con- trol points into the semantic correspondence network FθSC to obtain the refined ˜Dt→s k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We select several frames that capture different view- points for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our training of synthesizing the edited frames, It, is guided by a pre-trained Vision- Language model with the target prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Inspired by DreamFusion [38], we leverage a pre-trained diffusion model [44] to provide pixel-level guidance by backpropa- gating the gradient of noise residual to the generated im- ages (without backpropagating through the U-Net model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Adding a noise ε on It as the input, the pretrained diffusion UNet outputs a predicted noise ˆε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The gradient of the noise 5 residual ˆε −ε is backpropagated to update θ: ∇θLdif f (It) ≜ Ei,ε[w(i)(ˆε −ε)∂It ∂θ ], (7) where i stands for the time step for the diffusion model and the parameter set θ = {θA,θSC}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We update the unified in- formation in the atlas space to maintain the temporal consis- tency of the editing appearance and deformation with only training on a few generated frames It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In addition to the guidance of the diffusion model on multiple frames, we also apply several constraints to the learning of the refinement networks, FθA and FθSC, to pre- serve the editing effects as in the target edited keyframe It k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' To ensure that the deformation through the atlas can suc- cessfully reconstruct the original edited It k, the keyframe loss, Lk, measures the error between the original It k and the reconstructed ˜It k by L1 loss: Lk = | ˜It k −It k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (8) Besides, we also apply a total variation loss to encourage the spatial smoothness of the refined appearance in the atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The atlas loss is as follows: LA = Ltv( ˜It→s A ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (9) During the optimization, we also refine the semantic cor- respondence ˜Dt→s k of the keyframe pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' An ideal seman- tic correspondence matches semantically-similar pixels and perfectly transforms the target shape into the source shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Therefore, we compute the errors of the deformed target and the source object masks, Mt k and Ms k: LSC = |( ˜Dt→s k ⊗Mt k)−Ms k| (10) The total loss function L = Ldi f f + λkLk + λALA + λSCLSC, λk,λA,λSC = 106,103,103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The optimized parame- ters θ ∗ are then used to generate the final edited video It∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='.N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We implement our method in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We follow the video configuration in NLA with the resolution of 768 × 432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We use a thin-plate spline to inverse a warping field to prevent introducing holes by forward warping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The re- finement networks, FθA and FθSC exploits the architecture of Text2LIVE [2] and TPS-STN [16], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The opti- mization performs on 3 to 5 selected frames, including It 1, It k, and It N, for 600 to 1000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The optimization process takes 20 mins on a 24GB A5000 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We further utilize an off-the-shelf super-resolution model [53] to obtain sharp details in the final edited atlases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Experimental Results Here we show sample editing results in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We include additional video results in the supplementary ma- terial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We will make our source code and editing results publicly available to foster reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Experimental Setup Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We select several videos from DAVIS [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Each video contains a moving object in 50 to 70 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We edit each video with a prompt that describes a target object with a different shape from the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We compare our results with SOTA and several baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' For fair comparisons, all the baseline methods use the same image editing method, Sta- ble Diffusion [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Multi-frame baseline: Multiple keyframes in a video are edited individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The nearby edited keyframes tempo- rally interpolate the remaining frames with FILM [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Single-frame baseline: We extract a single keyframe from a video to be edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The edited information is then propagated to each frame with EbSynth [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Text2LIVE [2]: The SOTA text-driven editing method with NLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Note that it utilizes a structure loss to preserve the original shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We compare the official Text2LIVE in this section and show the comparison of removing structure loss in our supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Visual Comparison We show a visual comparison with the baseline meth- ods and Text2LIVE in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In the first example with “blackswan→duck”, the multi-frame baseline shows inconsistent editing in different frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The single-frame baseline suffers from inaccurate frame motion and thus yields distortion during propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Text2LIVE shows a promising target appearance with temporal consistency but cannot change the shape that matches the target ob- ject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In contrast, our method provides the desired appear- ance and consistent shape editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In the second example with “boat→yacht”, the single-frame baseline shows an inconsistent shape since the frame propagation relies on the frame motion of the source shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Consequently, it cannot propagate the edited pixels correctly in a different shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In the third example with “dog→cat”, the in- put video contains a non-rigid motion moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' It poses further challenges for multi- and single-frame base- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Again, Text2LIVE demonstrates plausible cat ap- pearance while remaining in the source dog shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our shape-aware method maintains the object motion and ma- nipulates the texture and shape corresponding to the desired editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 6 Input Ours Multi-frame baseline Single-frame baseline Text2LIVE [2] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Visual comparison with baselines and SOTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We show three examples with edits of “blackswan →duck”, “boat →yacht”, and “dog →cat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The multi-frame baseline shows inconsistency in the edited objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The single-frame method suffers from the incomplete flow motion of the source object shape and thus could not propagate the edits properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Text2LIVE demonstrates consistent appearance editing corresponding to the target edits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Nevertheless, the shape remains the same as the original object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In contrast, our proposed method outperforms the compared methods with consistent and plausible appearance and shape editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Input (a) fixed NLA (b) w/ semantic corres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (c) w/ UV deformation (d) w/ optimization (full) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We study the effects of removing the deformation and optimization components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (a) Editing with fixed NLA UV mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (b) Using a semantic correspondence with fixed UV, the edits are mapped to the atlas properly but still remains the original shapes in results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (c) With deformation initialization (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='2), the NLA UV maps are deformed to restore the target shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' (d) With further atlas optimization (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='3), the incomplete pixels in edited atlas and distortion (in car’s roof and back wheel) are refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 7 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Shape-aware interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our methods allow interpolation between two shapes by simply interpolating the atlas deformation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The examples demonstrate the gradual changes from source objects to edited objects over the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Ablation Study We conduct an ablation study in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 6 to validate the effectiveness of the UV deformation and atlas optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' With fixed NLA UV mapping, the shape edits in the keyframe cannot be adequately transformed through the atlas to each frame (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Therefore, by adding a keyframe semantic correspondence to deform the target into the source shape, the fixed UV maps the edits correctly into the atlas but remains source shapes in the edited frames (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' To restore the target shape, our deformation mod- ule deforms the UV maps by the semantic correspondence (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' However, the unseen pixels and inaccurate cor- respondence yield artifacts in different views (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=', in the car’s roof and back wheel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We refine the edited atlas and deformation with the atlas optimization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 6d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Application We present an application of shape-aware interpolation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Through interpolating the deformation maps, the object shape can be easily interpolated without additional frame interpolation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Similarly, we can interpolate atlas textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Note that we directly apply image editing on the background atlas since it can be treated as a natu- ral panorama image (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' However, the fore- ground atlas is an unwrapped object texture, which is unnat- ural for general pre-trained editing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Therefore, we edit the video frame and map it back to the atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' This ap- proach is more general and allows users to use their chosen images for video editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Limitations Our method strictly relies on the many-to-one mapping from individual frames to a unified atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' However, NLA may fail to get the ideal mapping in challenging scenarios with complex motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Therefore, we observe artifacts in the erroneous mapping regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=', the motion of hind legs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' In addition, it remains difficult to build semantic correspondence between two different ob- Input Edit Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We visualize a failure example (bear →lion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' The inaccurate NLA mapping in the motion of crossing hind legs yields distortion in the edited result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' User-guided correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Associating two different objects remains challenging even for the SOTA semantic correspondence methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' For a pair of source (a) and target (b), the severe false matching can be corrected by users’ manual warping for better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' While the atlas optimization can improve noisy cor- respondences, poor semantic correspondence initialization would hinder the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We show that user manual correction (in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 9) can lead to better video editing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Conclusions We have presented a shape-aware text-driven video edit- ing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We tackle the limitation of appearance-only 8 ORUGSIOURUGUNabCdmanipulation in existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We propose a deforma- tion formulation using layered video representation to trans- form the mapping field corresponding to the target shape edits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We further refine the unseen regions by utilizing the guidance from a pre-trained text-to-image diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our method facilitates a variety of shape and texture editing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Societal impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Our work proposes a tool for enabling creative video editing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' Nevertheless, similar to many image/video synthesis applications, care should be taken to prevent misuse or malicious use of such techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' We will release our code under a similar license as Stable Diffusion that focuses on ethical and legal use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content='1 References [1] Yogesh Balaji, Seungjun Nah, Xun Huang, Arash Vahdat, Jiaming Song, Karsten Kreis, Miika Aittala, Timo Aila, Samuli Laine, Bryan Catanzaro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' ediffi: Text-to-image diffusion models with an ensemble of expert denoisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFPT4oBgHgl3EQfyzWo/content/2301.13173v1.pdf'} +page_content=' 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new file mode 100644 index 0000000000000000000000000000000000000000..6365ccdae63d5ed93bf68c7e0d769b6fe562809e --- /dev/null +++ b/gdE4T4oBgHgl3EQfrA0Y/content/tmp_files/2301.05204v1.pdf.txt @@ -0,0 +1,612 @@ +Super-chirality of paraxial higher order Poincar´e modes +M. Babiker1,∗, J. Yuan1, K. Koksal2, V. E. Lembessis3 +1School of Physics, Engineering and Technology, University of York, YO10 5DD, UK +2Physics Department, Bitlis Eren University, Bitlis, Turkey +3Quantum Technology Group, Department of Physics and Astronomy, +College of Science, King Saud University, Riyadh 11451, Saudi Arabia and +∗ Corresponding author: m.babiker@york.ac.uk +(Dated: January 13, 2023) +We demonstrate that higher order Poincar´e modes of order m are super-chiral, displaying enhance- +ment factors proportional to m and m2 in their helicity/chirality. With m having arbitrarily large +integer values, such modes, in principle, possess unlimited super-chirality. These findings pave the +way to applications, including the strong enhancements of optical interactions with chiral matter. +The work indicates considerable flexibility in controlling the helicity of any higher order paraxial +twisted light mode and it incorporates a very wide range of physical scenarios. +Recent research has highlighted the fundamental sig- +nificance and the potential for applications of higher- +order optical vector modes, also called Poincar´e modes +[1–8]. +The overall polarisation state of such modes is +formally identified as characteristically non-separable su- +perpositions of solutions involving circular polarisation +(ˆx±iˆy)/ +√ +2 and spatial phase functions e±imφ with inte- +ger m the higher order and φ the azimuthal angular vari- +able. The polarisation is represented by a point (ΘP , ΦP ) +on the surface of a unit Poincar´e sphere, as explained in +the caption to Fig.1. +Although the higher order vector modes have already +been realised experimentally [4, 5, 9], their properties +have not yet been explored for arbitrary m. In particu- +lar, the question arises as to whether and how the higher +order modes can offer enhanced beam properties such as +higher order encoding schemes for enhanced bandwidth +optical communications [10] and whether they could lead +to enhanced optical angular momentum, spin and chiral- +ity which could influence optical interaction with chiral +matter [11]. Increased optical chirality is highly desirable +in order to engage effectively with chiropical processes. +Could it be the case that higher order modes would pro- +vide sufficiently strong chirality to engage effectively with +chiral molecules and be able to achieve a high degree of +enantioselectivity? +In this Letter we focus on the prospect of the existence +of super-chirality, which, we envisage, maybe one of the +major properties of the higher order modes. To this end, +we have aimed to evaluate the helicity density and its +spatial integral for the most general paraxial mode of ar- +bitrary order m ≥ 0, which covers all possible scenarios. +In cylindrical coordinates the electric and magnetic +fields of a general paraxial twisted light mode with the +most general polarisation are derivable from a vector po- +tential in the form +A = ˆϵ ˜F{ℓ}(ρ)eiℓφeikzz +(1) +Here kz is the axial wavevector with the light travelling +along the +z axis and ˜F{ℓ} is the paraxial mode func- +tion which depends only on the radial coordinate ρ. The +mode is labelled by the group of indices generically de- +noted by {ℓ}, which includes integer ℓ, the winding num- +ber, and integer p the radial number, as in the case of +Laguerre-Gaussian (LG) optical vortex modes. However, +the treatment is not restricted to LG modes and is ap- +plicable, in general, to other vortex modes. The higher +order polarisation state vector can be written as +ˆϵ = eimφ(ˆx − iˆy)UP + e−imφ(ˆx + iˆy)VP +(2) +where m is a positive integer, unlike ℓ which spans all +real integers; UP and VP are Poincar´e functions given by +UP = +1 +√ +2 cos +�ΘP +2 +� +e−iΦP /2; +VP = +1 +√ +2 sin +�ΘP +2 +� +eiΦP /2 +(3) +The above polarisation state ˆϵ is the most general po- +larisation vector, similarly defined by Milione et al [1] in +terms of the higher order Poincar´e sphere. The validity +of the higher order polarisation states, has already been +confirmed experimentally [4, 5, 12]. +At a general point on the surface of the higher order +Poincar´e sphere, we have for the vector potential +A = +� +(ˆx − iˆy)ei(ℓ+m))φUP + (ˆx + iˆy)ei(ℓ−m)φVP +� +˜F{ℓ}(ρ)eikzz +(4) +An important requirement of free-space paraxial optical +fields is that the electric field must be derivable from the +arXiv:2301.05204v1 [physics.optics] 12 Jan 2023 + +2 +S01 +S02 +S03 +𝛷P +𝜣P +|R0> +|L0> +|H0> +|V0> +|A0> +|D0> +S11 +S12 +S13 +𝛷P +𝜣P +|R1> +|L1> +|H1> +|V1> +|D1> +|A1> +(a) 0-order PS +(b) 1st-order PS +FIG. 1: 0th order, (a), and 1st-order, (b), Pioncar´e Sphere (PS) representation of the polarisation state in which +optical polarisation is coupled with vortex phase, characterized by a unit sphere with respect to the corresponding +Stokes-parameter(-like) Cartesian coordinates (S0 +1, S0 +2, S0 +3) and (S1 +1, S1 +2, S1 +3) respectively. It is seen that the 0th +order PS is equivalent to the conventional PS where |H > and |V > are commonly used to denote the vertically and +horizontally linearly polarized light, |A > and |D > for ±45o tilted linearly polarized light, |R > and |L > for +right-hand and left-hand circularly-polarized light, respectively. The 1st order PS figure is related to the +corresponding figure by Milione et al [1] with slightly different conventions for S1 +1 and S1 +2. Six sets of special vector +modes are drawn in different colours next to each sphere for illustration. Their positions on the Poincar´e sphere are +indicated by dots of the same colour. +magnetic field using the first Maxwell curl equation and +that the electric field must produce the same magnetic +field via the second Maxwell curl equation. We write the +vector potential as the sum of two terms +A = A1 + A2 +(5) +A1 = (ˆx − iˆy)F(1) +{ℓ}(ρ, φ)eikzz +(6) +A2 = (ˆx + iˆy)F(2) +{ℓ}(ρ, φ)eikzz +(7) +where we have introduced F(i) +{ℓ} with i = 1, 2 as the func- +tions of ρ and φ as follows +F(1) +{ℓ}(ρ, φ) = UP ei(ℓ+m)φ ˜F{ℓ}(ρ); +F(2) +{ℓ}(ρ, φ) = VP ei(ℓ−m)φ ˜F{ℓ}(ρ) +(8) +The electric and magnetic fields of this generally- +polarised mode are similarly written as the sums B = +B1 +B2 and E = E1 +E2 where Bi = ∇×Ai; +i = 1, 2. +The sequence of steps involve dealing first with the two +parts of the magnetic field and from those use Maxwell’s +curl B equation to derive the corresponding electric field +parts. We have for B1 and E1 +B1 = {ikz(ˆy + iˆx) − ˆz (i∂x + ∂y)} F(1)eikzz +E1 = c {ikz(ˆx − iˆy) − ˆz (∂x − i∂y)} F(1)eikzz +(9) +It is easy to see that B2 and E2 follow, respectively, +from B1 and E1 by the following substitution +B2 = B1(i → −i; F(1)eikzz → F(2)eikzz) +E2 = E1(i → −i; F(1)eikzz → F(2)eikzz) +(10) +where we have dropped the subscript label {ℓ} in F(1),(2) +and in ˜F for ease of notation and the notation can be +restored when the need arises. It is easy to see that the +procedure we have followed amounts to ensuring that the +fields satisfy the wave equation ∇×∇×E−ω2E/c2 = 0 +to the leading derivative order. The fields we now have +form the basis for the derivation of the optical properties +of the higher order modes. + +3 +The cycle-averaged optical densities of the helicity ¯η +and chirality ¯χ are defined by +¯η(r) = −ϵ0c +2ω ℑ +2 +� +i=1 +2 +� +j=1 +(E∗ +i · Bj) = c +ω2 ¯χ +(11) +where ω is the frequency of the light, Ei and Bi, with +i = 1, 2 are as given in Eqs.(9) and (10).The symbol ℑ[...] +in Eq.(11) stands for the imaginary part of [...] and the +superscript * in E∗ stands for the complex conjugate of +E. In what follows, we focus on the helicity from which +the chirality can be determined using Eq.(11). +We seek to evaluate both the helicity density and its +space integral specifically in relation to the most gen- +eral higher order optical vortex modes. The four terms +arising from the summation in Eq.(11) require separate +evaluations. +The evaluations are straightforward and +require, as a first step, expressions for the x- and y- +derivatives of F(1) and F(2) in polar coordinates. Note +from Eqs.(8) that F(1) is distinguished by the phase +factor exp [i(ℓ + m)φ] and F(2) is distinguished by the +phase factor exp [i(ℓ − m)φ]. It turns out that the sum +E∗ +1 · B2 + E∗ +2 · B1 does not contribute an imaginary part +and only the two direct terms contribute. We find after +some algebra +¯η(r) = ϵ0c2 +4ω +� +cos (ΘP ) +�� +2k2 +z| ˜F|2 + | ˜F′|2� ++ ℓ2 +ρ2 | ˜F|2 +� +− 2ℓ +˜F′ ˜F +ρ +� ++ϵ0c2 +4ω +� +| ˜F|2 +ρ2 [m2 cos (ΘP ) − 2mℓ] + +˜F′ ˜F +ρ +m cos (ΘP ) +� +(12) +where we have set ˜F′ = d ˜F/dρ and chosen to sepa- +rate the m-dependent terms from the other terms. The +first set of terms in Eq.(12) (enclosed between the curly +brackets) coincides with the zero order (m = 0) helic- +ity density in the case of elliptical polarisation. The rest +are the m-dependent higher-order terms and are capable +for sufficiently large m of dominating the zero-order he- +licity, leading to super-chirality. The Poincar´e function +cos(ΘP ) takes real values from +1.0 (ΘP = 0; right-hand +circular polarisation at the north pole of the Poincar´e +sphere) to -1.0 (ΘP = π; left-hand circular polarisation +at the south pole) with intermediate points 0 < ΘP < π +representing elliptical polarisation and the special points +where ΘP = π/2 representing radial and azimuthal po- +larisation. Thus we can immediately infer that we have +a very general result which is applicable to any parax- +ial optical vortex of a general polarisation defined by a +point on the surface of a higher order Poincare sphere. +To obtain the helicity density for any specific case all we +need to do is simply specify the order m, the Poincar´e +polarisation angles (ΘP , ΦP ) and the amplitude function +˜F, with its winding number ℓ and its radial number p, +if applicable. Note, however, that the helicity does not +contain any dependence on the Poincar´e angle ΦP , so +that, for example, all points on the equatorial circle have +the same helicity. +Setting m = 0 and ΘP = 0, π in Eq.(12) we imme- +diately identify the exact expression between the curly +brackets as the helicity density of the basic circularly- +polarised general optical vortex mode [13], interpreting +cos(ΘP ) = ±1 as σ = ±1 for circular polarisation. There +is also an additional term involving −ℓ +˜ +F′ ˜ +F +ρ , which is ap- +propriate for uniform linear polarisation and has been +shown to lead to zero chirality on spatial integration [13– +16]. The basic circularly-polarised helicity defined by the +terms in the curly brackets has been fully evaluated for +Laguerre-Gaussian light [14]. +However, for m ̸= 0 there are now additional m- +dependent terms in the helicity density for all values of +cos (ΘP ) = (+1.0 to − 1.0), which means that ellipti- +cally polarised modes (including circular, linear, as well +as radial and azimuthal) have additional m-dependent +density contributions. In particular, For m ≥ 1, as ΘP +increases the Poincar´e function cos(ΘP ) passes through +zero at ΘP = π/2 for all points ΦP on the equatorial cir- +cle. Only for m = 1, this helicity density coincides with +the case of radially-polarised optical vortex modes [17]. +We have for m > 1 +¯ηmℓ(r) = −ℓϵ0c2 +2ω +� +m| ˜F{ℓ}|2 +ρ2 ++ +˜F′ +{ℓ} ˜F{ℓ} +ρ +� +(13) +Clearly, since m can in principle take any integer value +greater than 1, the first term in Eq.(13) increases with +increasing m. +This means that the magnitude of the +term is m times larger than for the case m = 1, which +corresponds to lowest order radially-polarised paraxial +modes. The general case for which m > 1 and for any +point ΘP , ΦP , the helicity density is given by Eq.(12) +and it constitutes the most general result for the helicity +density of a paraxial vortex mode of any order m. +We may now evaluate the super-chirality properties +of higher order modes for the special case of a parax- +ial Laguerre-Gaussian mode of winding number ℓ, radial +number p and waist w0, which has an amplitude function +given by +˜Fℓ,p(ρ) = E0 +� +p! +(p + |ℓ|)!e +− ρ2 +w2 +0 +�√ +2ρ +w0 +�|ℓ| +L|ℓ| +p +�2ρ2 +w2 +0 +� +(14) + +4 +where L|ℓ| +p +is the associated Laguerre polynomial of in- +dices |ℓ| and p. The overall factor E0 is a normalisation +constant which is determined in terms of the applied +power P, evaluated as the integral of the z-component +of the Poynting vector over the beam cross-section. We +have +P = +1 +2µ0 +� 2π +0 +dφ +� ∞ +0 +|(E∗ × B)z|ρdρ = +�πω2ϵ0cw2 +0 +4 +� +E2 +0 +(15) +from which we can obtain E0 in terms of the power P. +We first illustrate how the higher order affects radially- +polarised Laguerre-Gaussian modes identified from the +general formalism by setting cos (ΘP ) = 0, so the helicity +density is given by Eq.(13). Figure 2 displays the helicity +density for the case m = 4 along with that of the first +order (m = 1) for ℓ = +1 and ℓ = +2. It is clear that +for ℓ = +1, m = 4 the helicity density is super-chiral. +The chirality density here is always negative for ℓ = +|ℓ| +and is concentrated on the core at ρ = 0. +The case +ℓ = +2, m = 4 is also super-chiral, but concentrated off- +axis ρ > 0. +Consider now the full higher order helicity desnity +Eq.(12) for the particular case ℓ = 1 and the order is +m = 10, for illustration, and we choose cos (ΘP ) = +1, +corresponding to right-circular polarisation. The varia- +tions of the helicity density with ρ/w0 for the case where +ℓ = +1, +2, are shown in Fig. 3. We find, as in Fig. 2, +that for ℓ = +1 the helicity density does not vanish at +ρ = 0, in contrast to the case ℓ ≥ 2 where it always does. +The behaviour in the case ℓ = 1 can be explained +by inspecting the general form of the helicity density +terms which appear in Eq.(13) and also on the m- +dependent terms in Eq.(12). +When applied to the +Laguerre-Gaussian F for ℓ = 1, we have from Eq.(14) +| ˜Fℓ=1|2 ∝ ρ2 and also we have [ ˜F′ ˜F]ℓ=1 ∝ ρ. Once sub- +stituted in the relevant terms in the helicity density we +see that the factor 1/ρ2 in the first term cancels the fac- +tor ρ2 in the numerator and the 1/ρ in the second term +cancels with the factor ρ in the numerator. The overall +variation amounts to a non-zero value of the helicity at +ρ = 0 only in the case ℓ = 1. This variation contrasts +with the case ℓ ≥ 2 in which the numerators in the two +terms have higher powers of ρ, guaranteeing that the he- +licity density vanishes at ρ = 0. +Since the higher order helicity for m > 2|ℓ| is domi- +nated by the m-dependent terms we can evaluate the to- +tal integral of the helicity density due to the m-dependent +terms in Eq.(12) over the x − y plane. +First we note +that the radial integral of all terms in the form +˜ +F′ ˜ +F +ρ +are identically zero for all mode functions which satisfy +˜F{ℓ}(0) = 0 = ˜F{ℓ}(∞). We then have, for any ˜F{ℓ}, the +helicity per unit length is +¯Cm = ϵ0c2 +4ω [m2 cos (ΘP ) − 2mℓ] +� ∞ +0 +ρdρ +� 1 +ρ2 | ˜F{ℓ}|2 +� +(16) +FIG. 2: Variations with ρ/w0 of the helicity density, +Eq.(13), due to modes of orders m = 0, 1, 4, 10. The +plots concern radially-polarised Laguerre-Gaussians for +which cos (ΘP ) = 0 and the winding numbers are +ℓ = +1 and ℓ = +2. When compared with the m = 0 +and m = 1 plots we see that for ℓ = +1, m = 4 and +ℓ = +1, m = 10 the helicity density in each case is +super-chiral. It is negative and concentrated on the core +at ρ = 0. Also by comparison, the ℓ = +2, m = 4 and +ℓ = +2, m = 10 higher order modes are also +super-chiral, but concentrated off-axis ρ > 0. +FIG. 3: Variations with ρ/w0 of the full helicity density, +Eq.(12), due to modes of order m = 0, 1, 4, 10. The +plots concern circularly-polarised Laguerre-Gaussians +for which cos (ΘP ) = 1 and the winding numbers are +ℓ = +1 and ℓ = +2. When compared with the +zero-order m = 0 plot we see that for ℓ = +1, m = 4 and +ℓ = +1, m = 10 the higher order helicity density is +strongly super-chiral and is concentrated on the core at +ρ = 0. Also by comparison, the ℓ = +2, m = 4, 10 are +also super-chiral, but concentrated off-axis ρ > 0. +Substituting from Eq.(14) and using the integration vari- +able x = 2ρ2/w2 +0 we have for the radial integral in Eq.(16) +I = +p! +2(p + |ℓ|)! +� ∞ +0 +x|ℓ|−1e−x[L|ℓ| +p (x)]2dx = +1 +2|ℓ| + +p/wo +5 +2.0 +/=1;m=10 +./=1:m=1 +-6 +/=2;m=10 +/=2;m=1 +-8 +--/=1:m=4 +---/=1;m=0 +-- /=2;m=4 +--- /=2;m=0 +-1040 +/=1;m=10 +......-1.m=1 +/=2;m=10 +.../=2;m=1 +30 +--/=1:m=4 +.-.-.- /=1;m=0 +20 +/=2:m=4 +/=2;m=0 +10 +p/wo +0.5 +1.0 +1.5 +2.05 +We finally obtain +¯Cm = L0 +�m2 cos (ΘP ) − 2mℓ +|ℓ| +� � +1 +k2zw2 +0 +� +(17) +where L0 = P/(kzc2) is a constant for a fixed power P +and we have substituted for E0 using Eq.(15). It is easy to +check that ¯Cm has the dimensions of angular momentum +per unit length. Note that although the factor 1/k2 +zw2 +0 +in Eq.(17) is typically small for w2 +0 >> 1/k2 +z, the higher +order helicity for which m ≫ 2|ℓ| would ensure super- +chirality for relatively large w0 ≫ λ/2π. Also we note +from Fig. +3 in which ΘP = 0 that two of the curves +are identical, namely the one for the m = 4, ℓ = +2 +case which is seen to have the same helicity curve as +for m = 0, ℓ = 2. +This can be verified from Eq.(12) +but directly from Eq.(17), both indicating that the m- +dependent terms of the helicity vanish for m = 2ℓ with +ℓ > 0 and we are left with the m-independent chiral- +ity. Also it can be seen that for m < 2ℓ with ℓ > 0 we +have negative contributions to the helicity from the m- +dependent terms. Super-chirality arises when m ≫ 2|ℓ|. +We have verified by direct analysis that the energy den- +sity behaves in the same manner as the helicity density +as its expression contains similar terms to those entering +the helicity density. +In conclusion, we have evaluated the chirality/helicity +densities for general paraxial light modes in which the +state of polarisation is specified by a general point +(ΘP , ΦP ) on the surface of the order m Poincar´e unit +sphere, where m is a positive integer. The general results +obtained encompass a wide range of scenarios governed +by their dependence on the Poincare sphere angles, the +winding number ℓ, the mode amplitude function and the +higher order m. In particular, for points (π/2, ΦP ) on +the equatorial circle, the helicity/chirality is found to be +proportional to m, which means that the higher order +modes exhibit super-chirality since it is enhanced m-fold +relative to the helicity of an ordinary (order m = 1) ra- +dially polarised mode. For all other points on the sur- +face of the Poincar´e sphere the helicity is enhanced fur- +ther by terms proportional to m2. We have also shown +that a higher order m Laguerre-Gaussian mode for which +ℓ = +1 is a strongly super-chiral vortex beam which is +dominated by the vortex core at ρ = 0 and the helic- +ity at the core increases with increasing m. +We have +found that other higher order Laguerre-Gaussian modes +for which ℓ > +1 have off-axis maximum helicity which +is also super-chiral. These results strongly indicate the +existence of a highly desirable super-chirality property of +the higher order modes which, we suggest, is now ripe for +direct experimental investigation. There are diverse ap- +plications that can be envisaged, including improved in- +teractions with chiral matter and stronger trapping and +manipulation using optical spanners and tweezers, for ex- +ample in micro-fluidics and improved encoding schemes +for higher bandwidth optical communications. +[1] G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, +Phys. Rev. Lett. 107, 053601 (2011). +[2] E. J. Galvez, B. Khajavi, and B. M. Holmes, in Struc- +tured Light for Optical Communication, Nanophotonics, +edited by M. D. Al-Amri, D. L. Andrews, and M. Babiker +(Elsevier, 2021) pp. 95–106. +[3] C. Maurer, A. Jesacher, S. F¨urhapter, S. Bernet, +and +M. Ritsch-Marte, New Journal of Physics 9, 78 (2007). +[4] Y. Liu, +X. Ling, +X. Yi, +X. Zhou, +H. Luo, +and +S. Wen, Applied Physics Letters 104, 191110 (2014), +https://doi.org/10.1063/1.4878409. +[5] D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, +L. Marrucci, and A. Forbes, Nature Photonics 10, 327 +(2016). +[6] Q. Zhan, Advances in Optics and Photonics 1, 1 (2009). +[7] G. Volpe and D. Petrov, Optics Communications 237, 89 +(2004). +[8] B. M. Holmes and E. J. Galvez, Journal of Optics 21, +104001 (2019). +[9] C. Chen, Y. Zhang, L. Ma, Y. Zhang, Z. Li, R. Zhang, +X. Zeng, Z. Zhan, C. He, X. Ren, C. Cheng, and C. Liu, +Opt. Express 28, 10618 (2020). +[10] M. D. Al-Amri, D. L. Andrews, and M. Babiker, Struc- +tured Light for Optical Communication (Elsevier, 2021). +[11] Y. Tang and A. E. Cohen, Phys. Rev. Lett. 104, 163901 +(2010). +[12] Y. Shen, Z. Wang, X. Fu, D. Naidoo, +and A. Forbes, +Physical Review A 102, 31501 (2020). +[13] M. Babiker, J. Yuan, V. Lembessis, and K. Koksal, Op- +tics Communications 525, 128846 (2022). +[14] K. Koksal, M. Babiker, V. E. Lembessis, +and J. Yuan, +J. Opt. Soc. Am. B 39, 459 (2022). +[15] K. A. Forbes and G. A. Jones, Journal of Optics 23, +115401 (2021). +[16] K. A. Forbes, Phys. Rev. A 105, 023524 (2022). +[17] K. Koksal, M. Babiker, V. E. Lembessis, +and J. Yuan, +Manuscript submitted for publication (2022). + diff --git a/gdE4T4oBgHgl3EQfrA0Y/content/tmp_files/load_file.txt b/gdE4T4oBgHgl3EQfrA0Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ddeadbf7134e7ef76acc6e30a639bc95ebd4e8dc --- /dev/null +++ b/gdE4T4oBgHgl3EQfrA0Y/content/tmp_files/load_file.txt @@ -0,0 +1,305 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf,len=304 +page_content='Super-chirality of paraxial higher order Poincar´e modes M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Babiker1,∗, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Yuan1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Koksal2, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Lembessis3 1School of Physics, Engineering and Technology, University of York, YO10 5DD, UK 2Physics Department, Bitlis Eren University, Bitlis, Turkey 3Quantum Technology Group, Department of Physics and Astronomy, College of Science, King Saud University, Riyadh 11451, Saudi Arabia and ∗ Corresponding author: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='babiker@york.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='uk (Dated: January 13, 2023) We demonstrate that higher order Poincar´e modes of order m are super-chiral, displaying enhance- ment factors proportional to m and m2 in their helicity/chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' With m having arbitrarily large integer values, such modes, in principle, possess unlimited super-chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' These findings pave the way to applications, including the strong enhancements of optical interactions with chiral matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The work indicates considerable flexibility in controlling the helicity of any higher order paraxial twisted light mode and it incorporates a very wide range of physical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Recent research has highlighted the fundamental sig- nificance and the potential for applications of higher- order optical vector modes, also called Poincar´e modes [1–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The overall polarisation state of such modes is formally identified as characteristically non-separable su- perpositions of solutions involving circular polarisation (ˆx±iˆy)/ √ 2 and spatial phase functions e±imφ with inte- ger m the higher order and φ the azimuthal angular vari- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The polarisation is represented by a point (ΘP , ΦP ) on the surface of a unit Poincar´e sphere, as explained in the caption to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Although the higher order vector modes have already been realised experimentally [4, 5, 9], their properties have not yet been explored for arbitrary m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' In particu- lar, the question arises as to whether and how the higher order modes can offer enhanced beam properties such as higher order encoding schemes for enhanced bandwidth optical communications [10] and whether they could lead to enhanced optical angular momentum, spin and chiral- ity which could influence optical interaction with chiral matter [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Increased optical chirality is highly desirable in order to engage effectively with chiropical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Could it be the case that higher order modes would pro- vide sufficiently strong chirality to engage effectively with chiral molecules and be able to achieve a high degree of enantioselectivity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' In this Letter we focus on the prospect of the existence of super-chirality, which, we envisage, maybe one of the major properties of the higher order modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' To this end, we have aimed to evaluate the helicity density and its spatial integral for the most general paraxial mode of ar- bitrary order m ≥ 0, which covers all possible scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' In cylindrical coordinates the electric and magnetic fields of a general paraxial twisted light mode with the most general polarisation are derivable from a vector po- tential in the form A = ˆϵ ˜F{ℓ}(ρ)eiℓφeikzz (1) Here kz is the axial wavevector with the light travelling along the +z axis and ˜F{ℓ} is the paraxial mode func- tion which depends only on the radial coordinate ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The mode is labelled by the group of indices generically de- noted by {ℓ}, which includes integer ℓ, the winding num- ber, and integer p the radial number, as in the case of Laguerre-Gaussian (LG) optical vortex modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' However, the treatment is not restricted to LG modes and is ap- plicable, in general, to other vortex modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The higher order polarisation state vector can be written as ˆϵ = eimφ(ˆx − iˆy)UP + e−imφ(ˆx + iˆy)VP (2) where m is a positive integer, unlike ℓ which spans all real integers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' UP and VP are Poincar´e functions given by UP = 1 √ 2 cos �ΘP 2 � e−iΦP /2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' VP = 1 √ 2 sin �ΘP 2 � eiΦP /2 (3) The above polarisation state ˆϵ is the most general po- larisation vector, similarly defined by Milione et al [1] in terms of the higher order Poincar´e sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The validity of the higher order polarisation states, has already been confirmed experimentally [4, 5, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' At a general point on the surface of the higher order Poincar´e sphere, we have for the vector potential A = � (ˆx − iˆy)ei(ℓ+m))φUP + (ˆx + iˆy)ei(ℓ−m)φVP � ˜F{ℓ}(ρ)eikzz (4) An important requirement of free-space paraxial optical fields is that the electric field must be derivable from the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='05204v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='optics] 12 Jan 2023 2 S01 S02 S03 𝛷P 𝜣P |R0> |L0> |H0> |V0> |A0> |D0> S11 S12 S13 𝛷P 𝜣P |R1> |L1> |H1> |V1> |D1> |A1> (a) 0-order PS (b) 1st-order PS FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' 1: 0th order, (a), and 1st-order, (b), Pioncar´e Sphere (PS) representation of the polarisation state in which optical polarisation is coupled with vortex phase, characterized by a unit sphere with respect to the corresponding Stokes-parameter(-like) Cartesian coordinates (S0 1, S0 2, S0 3) and (S1 1, S1 2, S1 3) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' It is seen that the 0th order PS is equivalent to the conventional PS where |H > and |V > are commonly used to denote the vertically and horizontally linearly polarized light, |A > and |D > for ±45o tilted linearly polarized light, |R > and |L > for right-hand and left-hand circularly-polarized light, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The 1st order PS figure is related to the corresponding figure by Milione et al [1] with slightly different conventions for S1 1 and S1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Six sets of special vector modes are drawn in different colours next to each sphere for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Their positions on the Poincar´e sphere are indicated by dots of the same colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' magnetic field using the first Maxwell curl equation and that the electric field must produce the same magnetic field via the second Maxwell curl equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We write the vector potential as the sum of two terms A = A1 + A2 (5) A1 = (ˆx − iˆy)F(1) {ℓ}(ρ, φ)eikzz (6) A2 = (ˆx + iˆy)F(2) {ℓ}(ρ, φ)eikzz (7) where we have introduced F(i) {ℓ} with i = 1, 2 as the func- tions of ρ and φ as follows F(1) {ℓ}(ρ, φ) = UP ei(ℓ+m)φ ˜F{ℓ}(ρ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' F(2) {ℓ}(ρ, φ) = VP ei(ℓ−m)φ ˜F{ℓ}(ρ) (8) The electric and magnetic fields of this generally- polarised mode are similarly written as the sums B = B1 +B2 and E = E1 +E2 where Bi = ∇×Ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The sequence of steps involve dealing first with the two parts of the magnetic field and from those use Maxwell’s curl B equation to derive the corresponding electric field parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We have for B1 and E1 B1 = {ikz(ˆy + iˆx) − ˆz (i∂x + ∂y)} F(1)eikzz E1 = c {ikz(ˆx − iˆy) − ˆz (∂x − i∂y)} F(1)eikzz (9) It is easy to see that B2 and E2 follow, respectively, from B1 and E1 by the following substitution B2 = B1(i → −i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' F(1)eikzz → F(2)eikzz) E2 = E1(i → −i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' F(1)eikzz → F(2)eikzz) (10) where we have dropped the subscript label {ℓ} in F(1),(2) and in ˜F for ease of notation and the notation can be restored when the need arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' It is easy to see that the procedure we have followed amounts to ensuring that the fields satisfy the wave equation ∇×∇×E−ω2E/c2 = 0 to the leading derivative order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The fields we now have form the basis for the derivation of the optical properties of the higher order modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' 3 The cycle-averaged optical densities of the helicity ¯η and chirality ¯χ are defined by ¯η(r) = −ϵ0c 2ω ℑ 2 � i=1 2 � j=1 (E∗ i · Bj) = c ω2 ¯χ (11) where ω is the frequency of the light, Ei and Bi, with i = 1, 2 are as given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='The symbol ℑ[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (11) stands for the imaginary part of [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='] and the superscript * in E∗ stands for the complex conjugate of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' In what follows, we focus on the helicity from which the chirality can be determined using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We seek to evaluate both the helicity density and its space integral specifically in relation to the most gen- eral higher order optical vortex modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The four terms arising from the summation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (11) require separate evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The evaluations are straightforward and require, as a first step, expressions for the x- and y- derivatives of F(1) and F(2) in polar coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Note from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (8) that F(1) is distinguished by the phase factor exp [i(ℓ + m)φ] and F(2) is distinguished by the phase factor exp [i(ℓ − m)φ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' It turns out that the sum E∗ 1 · B2 + E∗ 2 · B1 does not contribute an imaginary part and only the two direct terms contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We find after some algebra ¯η(r) = ϵ0c2 4ω � cos (ΘP ) �� 2k2 z| ˜F|2 + | ˜F′|2� + ℓ2 ρ2 | ˜F|2 � − 2ℓ ˜F′ ˜F ρ � +ϵ0c2 4ω � | ˜F|2 ρ2 [m2 cos (ΘP ) − 2mℓ] + ˜F′ ˜F ρ m cos (ΘP ) � (12) where we have set ˜F′ = d ˜F/dρ and chosen to sepa- rate the m-dependent terms from the other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The first set of terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (12) (enclosed between the curly brackets) coincides with the zero order (m = 0) helic- ity density in the case of elliptical polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The rest are the m-dependent higher-order terms and are capable for sufficiently large m of dominating the zero-order he- licity, leading to super-chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The Poincar´e function cos(ΘP ) takes real values from +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='0 (ΘP = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' right-hand circular polarisation at the north pole of the Poincar´e sphere) to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='0 (ΘP = π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' left-hand circular polarisation at the south pole) with intermediate points 0 < ΘP < π representing elliptical polarisation and the special points where ΘP = π/2 representing radial and azimuthal po- larisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Thus we can immediately infer that we have a very general result which is applicable to any parax- ial optical vortex of a general polarisation defined by a point on the surface of a higher order Poincare sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' To obtain the helicity density for any specific case all we need to do is simply specify the order m, the Poincar´e polarisation angles (ΘP , ΦP ) and the amplitude function ˜F, with its winding number ℓ and its radial number p, if applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Note, however, that the helicity does not contain any dependence on the Poincar´e angle ΦP , so that, for example, all points on the equatorial circle have the same helicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Setting m = 0 and ΘP = 0, π in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (12) we imme- diately identify the exact expression between the curly brackets as the helicity density of the basic circularly- polarised general optical vortex mode [13], interpreting cos(ΘP ) = ±1 as σ = ±1 for circular polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' There is also an additional term involving −ℓ ˜ F′ ˜ F ρ , which is ap- propriate for uniform linear polarisation and has been shown to lead to zero chirality on spatial integration [13– 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The basic circularly-polarised helicity defined by the terms in the curly brackets has been fully evaluated for Laguerre-Gaussian light [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' However, for m ̸= 0 there are now additional m- dependent terms in the helicity density for all values of cos (ΘP ) = (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='0 to − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='0), which means that ellipti- cally polarised modes (including circular, linear, as well as radial and azimuthal) have additional m-dependent density contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' In particular, For m ≥ 1, as ΘP increases the Poincar´e function cos(ΘP ) passes through zero at ΘP = π/2 for all points ΦP on the equatorial cir- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Only for m = 1, this helicity density coincides with the case of radially-polarised optical vortex modes [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We have for m > 1 ¯ηmℓ(r) = −ℓϵ0c2 2ω � m| ˜F{ℓ}|2 ρ2 + ˜F′ {ℓ} ˜F{ℓ} ρ � (13) Clearly, since m can in principle take any integer value greater than 1, the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (13) increases with increasing m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' This means that the magnitude of the term is m times larger than for the case m = 1, which corresponds to lowest order radially-polarised paraxial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The general case for which m > 1 and for any point ΘP , ΦP , the helicity density is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (12) and it constitutes the most general result for the helicity density of a paraxial vortex mode of any order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We may now evaluate the super-chirality properties of higher order modes for the special case of a parax- ial Laguerre-Gaussian mode of winding number ℓ, radial number p and waist w0, which has an amplitude function given by ˜Fℓ,p(ρ) = E0 � p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (p + |ℓ|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='e − ρ2 w2 0 �√ 2ρ w0 �|ℓ| L|ℓ| p �2ρ2 w2 0 � (14) 4 where L|ℓ| p is the associated Laguerre polynomial of in- dices |ℓ| and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The overall factor E0 is a normalisation constant which is determined in terms of the applied power P, evaluated as the integral of the z-component of the Poynting vector over the beam cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We have P = 1 2µ0 � 2π 0 dφ � ∞ 0 |(E∗ × B)z|ρdρ = �πω2ϵ0cw2 0 4 � E2 0 (15) from which we can obtain E0 in terms of the power P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We first illustrate how the higher order affects radially- polarised Laguerre-Gaussian modes identified from the general formalism by setting cos (ΘP ) = 0, so the helicity density is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='(13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Figure 2 displays the helicity density for the case m = 4 along with that of the first order (m = 1) for ℓ = +1 and ℓ = +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' It is clear that for ℓ = +1, m = 4 the helicity density is super-chiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The chirality density here is always negative for ℓ = +|ℓ| and is concentrated on the core at ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The case ℓ = +2, m = 4 is also super-chiral, but concentrated off- axis ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Consider now the full higher order helicity desnity Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (12) for the particular case ℓ = 1 and the order is m = 10, for illustration, and we choose cos (ΘP ) = +1, corresponding to right-circular polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The varia- tions of the helicity density with ρ/w0 for the case where ℓ = +1, +2, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We find, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' 2, that for ℓ = +1 the helicity density does not vanish at ρ = 0, in contrast to the case ℓ ≥ 2 where it always does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The behaviour in the case ℓ = 1 can be explained by inspecting the general form of the helicity density terms which appear in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (13) and also on the m- dependent terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' When applied to the Laguerre-Gaussian F for ℓ = 1, we have from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (14) | ˜Fℓ=1|2 ∝ ρ2 and also we have [ ˜F′ ˜F]ℓ=1 ∝ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Once sub- stituted in the relevant terms in the helicity density we see that the factor 1/ρ2 in the first term cancels the fac- tor ρ2 in the numerator and the 1/ρ in the second term cancels with the factor ρ in the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The overall variation amounts to a non-zero value of the helicity at ρ = 0 only in the case ℓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' This variation contrasts with the case ℓ ≥ 2 in which the numerators in the two terms have higher powers of ρ, guaranteeing that the he- licity density vanishes at ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Since the higher order helicity for m > 2|ℓ| is domi- nated by the m-dependent terms we can evaluate the to- tal integral of the helicity density due to the m-dependent terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (12) over the x − y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' First we note that the radial integral of all terms in the form ˜ F′ ˜ F ρ are identically zero for all mode functions which satisfy ˜F{ℓ}(0) = 0 = ˜F{ℓ}(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We then have, for any ˜F{ℓ}, the helicity per unit length is ¯Cm = ϵ0c2 4ω [m2 cos (ΘP ) − 2mℓ] � ∞ 0 ρdρ � 1 ρ2 | ˜F{ℓ}|2 � (16) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' 2: Variations with ρ/w0 of the helicity density, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (13), due to modes of orders m = 0, 1, 4, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The plots concern radially-polarised Laguerre-Gaussians for which cos (ΘP ) = 0 and the winding numbers are ℓ = +1 and ℓ = +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' When compared with the m = 0 and m = 1 plots we see that for ℓ = +1, m = 4 and ℓ = +1, m = 10 the helicity density in each case is super-chiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' It is negative and concentrated on the core at ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Also by comparison, the ℓ = +2, m = 4 and ℓ = +2, m = 10 higher order modes are also super-chiral, but concentrated off-axis ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' 3: Variations with ρ/w0 of the full helicity density, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (12), due to modes of order m = 0, 1, 4, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The plots concern circularly-polarised Laguerre-Gaussians for which cos (ΘP ) = 1 and the winding numbers are ℓ = +1 and ℓ = +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' When compared with the zero-order m = 0 plot we see that for ℓ = +1, m = 4 and ℓ = +1, m = 10 the higher order helicity density is strongly super-chiral and is concentrated on the core at ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Also by comparison, the ℓ = +2, m = 4, 10 are also super-chiral, but concentrated off-axis ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Substituting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (14) and using the integration vari- able x = 2ρ2/w2 0 we have for the radial integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (16) I = p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' 2(p + |ℓ|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' � ∞ 0 x|ℓ|−1e−x[L|ℓ| p (x)]2dx = 1 2|ℓ| p/wo 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='0 /=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='/=1:m=1 6 /=2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=10 /=2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=1 8 --/=1:m=4 ---/=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=0 -- /=2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=4 --- /=2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=0 1040 /=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='.-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=1 /=2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='/=2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=1 30 --/=1:m=4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='- /=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=0 20 /=2:m=4 /=2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='m=0 10 p/wo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='05 We finally obtain ¯Cm = L0 �m2 cos (ΘP ) − 2mℓ |ℓ| � � 1 k2zw2 0 � (17) where L0 = P/(kzc2) is a constant for a fixed power P and we have substituted for E0 using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content='(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' It is easy to check that ¯Cm has the dimensions of angular momentum per unit length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Note that although the factor 1/k2 zw2 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (17) is typically small for w2 0 >> 1/k2 z, the higher order helicity for which m ≫ 2|ℓ| would ensure super- chirality for relatively large w0 ≫ λ/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Also we note from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' 3 in which ΘP = 0 that two of the curves are identical, namely the one for the m = 4, ℓ = +2 case which is seen to have the same helicity curve as for m = 0, ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' This can be verified from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (12) but directly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' (17), both indicating that the m- dependent terms of the helicity vanish for m = 2ℓ with ℓ > 0 and we are left with the m-independent chiral- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Also it can be seen that for m < 2ℓ with ℓ > 0 we have negative contributions to the helicity from the m- dependent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Super-chirality arises when m ≫ 2|ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We have verified by direct analysis that the energy den- sity behaves in the same manner as the helicity density as its expression contains similar terms to those entering the helicity density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' In conclusion, we have evaluated the chirality/helicity densities for general paraxial light modes in which the state of polarisation is specified by a general point (ΘP , ΦP ) on the surface of the order m Poincar´e unit sphere, where m is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' The general results obtained encompass a wide range of scenarios governed by their dependence on the Poincare sphere angles, the winding number ℓ, the mode amplitude function and the higher order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' In particular, for points (π/2, ΦP ) on the equatorial circle, the helicity/chirality is found to be proportional to m, which means that the higher order modes exhibit super-chirality since it is enhanced m-fold relative to the helicity of an ordinary (order m = 1) ra- dially polarised mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' For all other points on the sur- face of the Poincar´e sphere the helicity is enhanced fur- ther by terms proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We have also shown that a higher order m Laguerre-Gaussian mode for which ℓ = +1 is a strongly super-chiral vortex beam which is dominated by the vortex core at ρ = 0 and the helic- ity at the core increases with increasing m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' We have found that other higher order Laguerre-Gaussian modes for which ℓ > +1 have off-axis maximum helicity which is also super-chiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' These results strongly indicate the existence of a highly desirable super-chirality property of the higher order modes which, we suggest, is now ripe for direct experimental investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' There are diverse ap- plications that can be envisaged, including improved in- teractions with chiral matter and stronger trapping and manipulation using optical spanners and tweezers, for ex- ample in micro-fluidics and improved encoding schemes for higher bandwidth optical communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Milione, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' Sztul, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE4T4oBgHgl3EQfrA0Y/content/2301.05204v1.pdf'} diff --git a/gtE_T4oBgHgl3EQf3ByR/content/tmp_files/2301.08344v1.pdf.txt b/gtE_T4oBgHgl3EQf3ByR/content/tmp_files/2301.08344v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1350260a31323da474fb8a7dedc0dea79efa975 --- /dev/null +++ b/gtE_T4oBgHgl3EQf3ByR/content/tmp_files/2301.08344v1.pdf.txt @@ -0,0 +1,1438 @@ +MNRAS 000, 1–10 (2022) +Preprint 23 January 2023 +Compiled using MNRAS LATEX style file v3.0 +On the cusp of cusps: a universal model for extreme scattering +events in the ISM +Dylan L. Jow,2,3,6★ Ue-Li Pen,1,2,3,4,5,6 and Daniel Baker2,3 +1Institute of Astronomy and Astrophysics, Academia Sinica, Astronomy-Mathematics Building, No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan +2Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada +3Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON M5S 1A7, Canada +4Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON, Canada N2L 2Y5 +5Canadian Institute for Advanced Research, CIFAR program in Gravitation and Cosmology +6Dunlap Institute for Astronomy & Astrophysics, University of Toronto, AB 120-50 St. George Street, Toronto, ON M5S 3H4, Canada +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +The scattering structures in the ISM responsible for so-called “extreme scattering events" +(ESEs), observed in quasars and pulsars, remain enigmatic. Current models struggle to explain +the high-frequency light curves of ESEs, and a recent analysis of a double lensing event in +PSR B0834+06 reveals features of ESEs that may also be challenging to accommodate via +existing models. We propose that these features arise naturally when the lens has a cusp-like +profile, described by the elementary 𝐴3 cusp catastrophe. This is an extension of previous +work describing pulsar scintillation as arising from 𝐴2 fold catastrophes in thin, corrugated +plasma sheets along the line of sight. We call this framework of describing the lens potentials +via elementary catastrophes “doubly catastrophic lensing", as catastrophes (e.g. folds and +cusps) have long been used to describe universal features in the light curves of lensing events +that generically manifest, regardless of the precise details of the lens. Here, we argue that +the lenses themselves may be described by these same elementary structures. If correct, the +doubly catastrophic lensing framework would provide a unified description of scintillation and +ESEs, where the lenses responsible for these scattering phenomena are universal and can be +fully described by a small number of unfolding parameters. This could enable their application +as giant cosmic lenses for precision measurements of coherent sources, including FRBs and +pulsars. +Key words: waves – radio continuum: ISM – pulsars:general – fast radio bursts +1 +INTRODUCTION +The enigmatic extreme scattering events (ESEs) that were first dis- +covered in quasars in the late 80s (Fiedler et al. 1987) and in pulsars a +few years later (Cognard et al. 1993) have presented a long-standing +mystery in observations of radio sources. While they are known to +be caused by scattering in the interstellar medium (ISM), the pre- +cise form of the plasma structures that cause these events and the +physical origin of these structures remains unknown. Interest in ob- +serving and understanding ESEs has increased, with recent work +highlighting their relative ubiquity and setting the stage for future +surveys of these mysterious events (Bannister et al. 2016). More- +over, the excess time delays induced by ESEs have implications for +precision gravitational wave detection through pulsar timing arrays. +Precise modelling of these excess delays will be necessary to move +beyond detection of a stochastic gravitational wave background to +individual detections (Burke-Spolaor et al. 2019). Recently, novel +★ E-mail: djow@physics.utoronto.ca +phase retrieval techniques have been used for precision localization +of the refractive images formed by the ESE lens (Zhu et al. 2022). +In conjunction with such techniques, new observations from current +and next-generation radio telescopes built for pulsar timing arrays +and fast radio burst (FRB) detections, among other purposes, will +allow us to test the variety of models that have been proposed to +explain ESEs. +While future observations will hopefully shed light on the +plasma structures causing ESEs, current observations pose sev- +eral theoretical challenges. Assuming ESEs are caused by three- +dimensional plasma inhomogeneities in the ISM (i.e. an over- or +under-dense cloud of ionized plasma) leads to inferences for the +pressure of such clouds that are several orders of magnitude in ex- +cess of typical pressures in the diffuse ISM (Clegg et al. 1998). If +such highly pressurized clouds existed, then they would be unsta- +ble on the time-scales needed to explain ESE observations; this is +known as the over-pressure problem. Thin, two-dimensional current +sheets that are aligned with the line of sight have been proposed as +a potential resolution to the over-pressure problem (Romani et al. +© 2022 The Authors +arXiv:2301.08344v1 [astro-ph.HE] 19 Jan 2023 + +2 +Jow et al. +1987; Pen & King 2012), but, thus far, such models struggle to ex- +plain certain features of the ESE light curves, in particular their rich +frequency structure (Walker & Wardle 1998). For example, while +a two-dimensional Gaussian profile may fit the low-frequency light +curves observed in ESEs, they fail to match the high-frequency light +curves (Clegg et al. 1998). Typically, such models invoke substruc- +ture in the ISM that becomes resolved at high frequencies to explain +the complex morphologies of the high-frequency light curves; how- +ever, it remains desirable to be able to explain both the time and +frequency structure of ESEs with a single lens model. Cold, self- +gravitating clouds of neutral gas with an ionized skin have been +proposed to explain the frequency structure of ESEs (Henriksen & +Widrow 1995; Walker & Wardle 1998), but if correct would imply +that a substantial fraction of the galaxy’s mass is contained within +these clouds. +ESEs are not the only scattering phenomenon associated with +the ISM that radio sources are observed to undergo. Pulsars are ob- +served to scintillate due to multi-path scattering in the ISM. It has +generally been assumed that pulsar scintillation and extreme scat- +tering events are distinct phenomena, caused by different plasma +structures in the ISM. However, just as thin plasma sheets have +been proposed as an explanation for ESEs, in recent decades, there +has been growing observational evidence that a substantial fraction +of scintillation observations (if not all) can be explained by refrac- +tive plasma sheets along the line of sight (Stinebring et al. 2001; +Walker et al. 2004; Goldreich & Sridhar 2006; Brisken et al. 2010; +Pen & Levin 2014). This is in contrast to traditional models of an +extended Kolmogorov turbulent medium. A similar story has been +playing out in the study of the turbulent ISM through magnetohy- +drodynamic (MHD) simulations. Recent MHD simulations suggest +that the turbulent cascade is driven by intermittent sheet-like struc- +tures in the ISM (Dong et al. 2022). +If thin plasma sheets explain scintillation observations and are +consistent with current understandings of the physics of turbulence +in the ISM, might they not also explain ESEs? Here we propose +a model for ESEs that arises naturally from the thin sheet picture +which qualitatively explains several features of current ESE obser- +vations, including the complex frequency structure. The model we +propose is a simple application of catastrophe theory at the density +level of the lens description. That is, lensing by thin sheets can +be effectively described by the projected density of the sheet onto +a plane perpendicular to the line of sight. Mathematically, singu- +larities in the projection map can be classified and described by a +small set of elementary catastrophes (Thom et al. 1975). Fold (𝐴2) +catastrophes in corrugated plasma sheets have been proposed as an +explanation for pulsar scintillation observations (Goldreich & Srid- +har 2006; Pen & Levin 2014; Simard & Pen 2018). Here we propose +the next higher-order catastrophe, the 𝐴3 cusp, as an explanation for +ESEs. We call this framework “doubly catastrophic" lensing, since +catastrophe theory has long been applied to the theory of lensing +to describe the magnification of sources near singularities in the +lens map (Nye 1999). In addition to describing the magnification +as a network of catastrophes, here we describe the lens itself as +a catastrophe. One of the advantages of such a framework, is that +the elementary catastrophes are universal and described by a small +number of unfolding parameters. Therefore, if correct, the effec- +tive lenses describing ESEs may be exceptionally simple in form, +even if the physical plasma sheets are formed by complex physical +processes. +This paper is structured as follows. In Section 2 we introduce +our model for ESEs and discuss some of its qualitative features. In +Section 4 we discuss the possibility of using the doubly catastrophic +Lens +Projected 
 +Density +Source +Observer +̂xs +̂x +̂xo +ds +dl +dsl +Figure 1. Diagram of a corrugated sheet lens. When the distances between +the observer, source, and lens are large compared to the extent of the sheet +along the line of sight (as in most astrophysical scenarios), the lensing is +effectively described by the projected density (shown in gray) of the sheet +onto the lens plane perpendicular to the line of sight. As the sheet is rotated +to be more aligned with the line of sight, the peaks of the projected density +become larger. The refraction of light due to the lens causes multi-path +propagation from source to observer, shown by the grey, dashed lines. +lensing framework to explain both scintillation and ESEs. In Sec- +tion 3 we analyse in detail observations of an extreme scattering +event in the pulsar PSR B0834+06 with our model, and in Section 5 +we discuss potential applications of this framework. +2 +THE 𝐴3 LENS +In geometric optics, the effect of a plasma lens localized to a single +plane along the line of sight is determined by the lens equation +ˆ𝒚 = ˆ𝒙 + +𝑑𝑐𝑒2 +𝑚𝑒𝜖0𝜔2 ∇ ˆ𝒙Σ𝑒( ˆ𝒙), +(1) +where 𝜔 is the angular frequency of the light, ˆ𝒚 = ( ˆ𝒙𝑠𝑑𝑙+ ˆ𝒙𝑜𝑑𝑠𝑙)/𝑑𝑠 +is a weighted average of the transverse displacement between the +source and observer, 𝑑 = 𝑑𝑠𝑙𝑑𝑙/𝑑𝑠 is an effective distance, and +Σ𝑒( ˆ𝒙) is the excess surface electron in the lens plane. The coordi- +nates and distances involved are shown in Fig. 1. The lens equation +determines a mapping between the source plane and the lens plane, +determining the set of rays that connect the source and observer. +In astrophysical lensing, due to the vast distances between the +source and the observer, it is often sufficient to treat lensing in this +“thin lens" approximation, where the lens is taken to be localized +to a single plane perpendicular to the line of sight (the lens plane). +The physical plasma that produces the lens effect is, of course, not a +fully two-dimensional screen, but has some extent along the line of +sight. As such, the surface density, Σ𝑒, is a projection of the actual +density onto the lens plane: +Σ𝑒( ˆ𝒙) = +∫ +𝛿𝑛𝑒( ˆ𝒙, 𝑧)𝑑𝑧, +(2) +where 𝛿𝑛𝑒 is the excess electron density. +Fig. 1 shows an example of this projection process. Consider +MNRAS 000, 1–10 (2022) + +Cusp of cusps +3 +a thin sheet with a periodic profile that is inclined by some angle +with respect to the line of sight (the blue curve in Fig. 1). The +surface density along the lens plane (the grey curve) is obtained +by projecting the sheet onto a plane perpendicular to the line of +sight. In particular, for an infinitely thin sheet with a shape given by +𝑥 = 𝑓 (𝑧), the projected density is proportional to | 𝑑𝑥 +𝑑𝑧 |−1. Thus, the +density is formally infinite at singularities of the projection. +Catastrophe theory describes the mathematics of such singu- +larities. Powerfully, catastrophe theory shows that the topological +structure of singularities must conform to a few fundamental forms +(the “elementary catastrophes") regardless of the precise details of +the map in which the singularities arise. Catastrophe theory has +already been used to great effect in lensing theory to predict the +magnification of a source near a lens’ caustics without needing to +know the precise details of the lens potential. Our proposal here is to +extend this use of catastrophe theory to the density level. That is, we +wish to describe not only the magnification via elementary catastro- +phes, but the lens potential itself. We expect these elementary forms +are likely to appear in the projected plasma density, as catastrophes +generically arise when projecting thin, sheet-like structures in the +ISM along the line of sight. We will argue that by modelling the +plasma structures responsible for ESEs by these catastrophes, it is +possible to explain aspects of observations that have thus far been +challenging to model. We call this framework of describing the pro- +jected plasma density by a network of caustics “doubly catastrophic +lensing", as catastrophes arise both in the magnification produced +by the lens and in the lens potential, itself. +2.1 +Modelling the 𝐴3 lens +In this paper we will focus on the 𝐴3 catastrophe, a.k.a. the cusp +catastrophe. The fold (𝐴2) and cusp catastrophes are the simplest of +the elementary catastrophes. Lensing by a fold has been discussed +elsewhere, and has been proposed as an explanation for pulsar scin- +tillation (Goldreich & Sridhar 2006; Pen & Levin 2014; Simard & +Pen 2018). Here we propose lensing by an 𝐴3 catastrophe as an ex- +planation for ESEs in pulsars and quasars. The basic idea is shown +in Fig. 2; when the folds of a thin, folded sheet come to an end, they +meet in a cusp. The sheet, when viewed under projection, forms +an 𝐴3 cusp density profile. The cusp is described by two unfolding +parameters: 𝑥1 and 𝑥2. The bottom panel of Fig. 2 shows the cusp +density as a function of 𝑥1 for fixed 𝑥2. +The idea is to use the canonical intensity of the 𝐴3 catastrophe, +which we will call 𝜇𝐴3 (𝑥1, 𝑥2), as the lens potential. That is, our +lens equation is: +𝒚 = 𝒙 + 𝛼∇𝜇𝐴3 (𝒙). +(3) +The intensity of the 𝐴3 catastrophe is described by the canonical +phase +𝜙𝐴3 (𝑡; 𝑥1, 𝑥2) = 𝑡4 +4 − 𝑥2𝑡2 +2 ++ 𝑥1𝑡, +(4) +where 𝑥1 and 𝑥2 are the unfolding parameters, and 𝑡 is the coordinate +in the phase screen. It follows that the cusp intensity is given by +𝜇𝐴3 (𝑥1, 𝑥2) = +∑︁ +𝑡𝑖 +|3𝑡2 +𝑖 − 𝑥2|−1 +(5) +where 𝑡𝑖 are the solutions to the stationary phase equation 𝑡3 −𝑥2𝑡 − +𝑥1 = 0. We will also introduce an additional parameter 𝜎 and define +˜𝜇𝐴3 (𝑥1, 𝑥2; 𝜎) ≡ 𝜇𝐴3 (𝑥1, 𝑥2) ★ 𝑊(𝑥1; 𝜎), , +(6) +x2 > 0 +x2 = 0 +x2 < 0 +x2 +Folded sheet +Projected density +Folded sheet +Projected density +x1 +x1 +x1 +x1 +x2 > 0 +x2 = 0 +x2 < 0 +Figure 2. Diagram showing how the 𝐴3 cusp catastrophe arises when a +folded sheet is projected onto the lens plane. The grey mesh shows the +physical sheet and the colour map below shows the density of the sheet +when projected onto the lens plane. The cusp profile arises generically when +two folds in the sheet come to an end. The variables 𝑥1 and 𝑥2 are the +two unfolding parameters of the 𝐴3 catastrophe and can be thought of as +the physical coordinates in the lens plane up to some arbitrary scaling. The +red lines show cross sections through the sheet for fixed 𝑥2. In the bottom +panel, we show the cross section through the physical sheet in red, and the +projected density for an infinitely thin sheet below that in grey. The blue +curves show the projected density for a sheet of finite thickness; the effect +of which is to smooth out the sharply peaked grey curves. For 𝑥2 < 0, the +sheet is made up of two folds that converge at 𝑥2 = 0 at the cusp point. For +𝑥2 > 0, the two folds disappear as the sheet flattens out. +where𝑊(𝑥1; 𝜎) is taken to be a simple Gaussian smoothing function +with standard deviation 𝜎. This smoothing is performed to remove +the infinite densities that arise in the cusp catastrophe and represents +the fact that the physical sheet that gives rise to the cusp has a finite +thickness. +Together, Eqs. 3 and 6 provide a full description for the 𝐴3 lens +in geometric optics. However, to further simplify our analysis we +will treat the lens as quasi-one-dimensional. That is, we will assume +that the direction of the rays is only modified in the 𝑥1 direction. +In other words, we treat 𝑥2 as a fixed parameter of the lens, and we +only need to solve the one-dimensional lens equation: +𝑦1 = 𝑥 + 𝛼𝜕𝑥𝜙(𝑥; 𝑥2). +(7) +The second lens equation is simply 𝑦2 = 𝑥2. This simplification +is possible because the derivative of ˜𝜇𝐴3 is typically much larger +in the 𝑥1 direction than it is in the 𝑥2 direction. While not strictly +necessary here, this simplification will come in handy when we +consider a multi-plane lens in Section 3. +In this limit, the magnification due to the 𝐴3 lens is given by +𝜇(𝑦1; 𝑥2) = +∑︁ +𝑥 +|1 + 𝛼𝜕2 +𝑥𝜙(𝑥; 𝑥2)|−1, +(8) +where the sum is taken over solutions to the lens equation, Eq. 7. +MNRAS 000, 1–10 (2022) + +4 +Jow et al. +10 +5 +0 +5 +10 +x1 (AU) +4 +2 +0 +2 +4 +6 +8 +10 +x2 (AU) +0.10 +0.05 +0.00 +0.05 +0.10 +t (year) +100 +101 +102 +103 +104 +f (GHz) +Cusp of cusp +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +Figure 3. The right panel shows the dynamic spectrum (intensity as a +function of time and frequency) one would observe for the source trajectory +shown by the black arrows on the left. The blue curve on the left shows the +caustics of the 𝐴3 cusp profile, i.e., the points of maximum projected density +shown in Fig. 2. We have chosen the trajectory to just graze the cusp point. +Here we are describing the lens potential as an cusp catastrophe, but cusps +also generically arise in the dynamic spectrum. This is clearly seen in the +right panel, where the bright peaks of the magnification converge towards a +cusp at high frequencies: this is the titular "cusp of the cusp". +It turns out that for the 𝐴3 potential, changes along the 𝑥2 +direction can be described by changes in the amplitude of the lens, +and a re-scaling of the 𝑥 coordinate. This follows from the identity +𝛼𝜇𝐴3 (𝑥, 𝑥2 = ±1) = 𝜇𝐴3 +�𝛼−3/2𝑥, 𝑥2 = ± 1 +𝛼 +�, +(9) +or, equivalently, +𝜇𝐴3 (𝑥, 𝑥2) = 1 +𝑥2 +𝜇�𝑥−3/2 +2 +𝑥, sign(𝑥2)�. +(10) +In other words, once a sign is chosen for 𝑥2, one is free to re-scale +the amplitude 𝛼 and the 𝑥 coordinate so that |𝑥2| = 1. This means +that the effect of changing the amplitude 𝛼 is simply to re-scale the +coordinates. +Fig. 4 shows the lens map described by Eq. 7 for fixed 𝑥2 = 1 +and different values of 𝛼. One may be interested in the location of +the caustics that are formed by the lens (i.e. the turning points in the +lens map). The location of these turning points depends on 𝛼 and the +smoothing scale 𝜎, as the un-smoothed potential is formally infinite +at certain points. Thus, the maximum amplitude of the smoothed +potential depends strongly on 𝜎. Fig. 5 shows the level curves of +the lens map (Eq. 7), as a function of the unfolding parameters, +𝑥1 and 𝑥2, for fixed 𝛼 = ±1, where the sign of alpha determines +whether the lens is convergent or divergent (note that because of +the scaling relations shown in Eqs. 9 and 10 varying either 𝛼 or 𝑥2, +while holding the other fixed, covers the entirety of the parameter +space of the lens). Fig. 5 tells us where a given source position 𝑦1 +gets mapped to in the lens plane. That is, consider 𝑥2 to be some +fixed value. Then we can read off the image positions in 𝑥1 for a +given value of 𝑦1 by looking at the 𝑥1 value the contour associated +with 𝑦1 reaches for that value of 𝑥2. A peculiar feature of the 𝐴3 lens +is that for a fixed source position, for a small region about 𝑥2 = 0, +the distance between the image positions in 𝑥1 increases. This is +in contrast to large values of 𝑥2, for which a decrease in 𝑥2 leads +to the images moving closer together. This feature will lead to the +characteristic hockey-stick shape shown later in Figs. 9 and 10. +Now, it is straightforward to compute how the location of the +outermost caustics scale with 𝜎 and 𝛼. Roughly, the location of +the outermost caustic is given by 𝑦∗ +1 ∼ max{𝛼𝜕𝑥𝜙}. For the un- +smoothed lens, the maximum derivative of the lens potential is +infinite. For the smoothed lens, the maximum value of the derivative +is nevertheless attained close to where the unsmoothed lens diverges. +Since the lens potential near the divergence is described by a fold +20 +15 +10 +5 +0 +5 +10 +15 +20 +y +4 +2 +0 +2 +4 +x += 1 += 0.1 += 0.01 +Figure 4. The lens map of the 𝐴3 lens for fixed 𝑥2 = 1 and varying 𝛼. The +effect of changing 𝛼 is to effectively re-scale the coordinates as shown in +Eq. 9. +1.0 +0.5 +0.0 +0.5 +1.0 +x1 +0.0 +0.5 +1.0 +1.5 +2.0 +x2 +Convergent Lens +1.0 +0.5 +0.0 +0.5 +1.0 +x1 +0.0 +0.5 +1.0 +1.5 +2.0 +x2 +Divergent Lens +70 +60 +50 +40 +25 +10 +1 +1 +10 +25 +40 +50 +60 +70 +70 +60 +50 +40 +25 +10 +1 +1 +10 +25 +40 +50 +60 +70 +Figure 5. The level curves of the lens map, Eq. 7, as a function of the +unfolding parameters 𝑥1 and 𝑥2. +catastrophe, the lens potential is given by 𝜙(𝑥; 𝑥2) ∝ 𝑥−1/2 on +one side of the divergence and zero on the other side. Thus, near +the divergence, the smoothed derivative is given by 𝜕𝑥𝜙(𝑥; 𝑥2) ∝ +𝜙(𝑥; 𝑥2) ★ 𝜕𝑥𝑊(𝑥; 𝜎). It follows from this that the location of the +outermost caustic scales as +𝑦∗ +1 ∼ 𝛼𝜎−3/2. +(11) +Knowing the location of the outermost caustic will be useful later +when we are trying to infer the lens parameters from observed time +delays in Section 3. +Fig. 6 shows the critical curve and caustic structures that arise +due to magnification by the 𝐴3 lens. The top row shows the results for +the divergent lens (𝛼 < 0), corresponding to an over-dense lens, and +the bottom row shows the convergent lens (𝛼 > 0), corresponding +to an under-dense lens. The light curve that arises as one changes +impact parameter, 𝑦1, (i.e. as the source moves relative to the lens), +are effectively entirely determined by the number and location of +the caustics. When 𝜎 ≪ 1 and 𝛼 ≳ 1, the caustics tend to be located +far from the axis; for example, the outermost caustic is located at +𝑦∗ +1 ≫ 1. This means that in between the caustics the slope of the +inverse lens map is large, meaning that the total magnification is +close to one (see, for example, the blue curve in Fig. 4, where the +inverse lens map, 𝑥(𝑦), is effectively flat for most of the region +between the caustics). The result is that the light curve as the source +moves relative to the lens is close to unity except at the caustics +where the magnification suddenly diverges. Fig. 7 shows an example +MNRAS 000, 1–10 (2022) + +Cusp of cusps +5 +2 +1 +0 +1 +2 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +x2 +Critical Curves +60 +40 +20 +0 +20 +40 +60 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Caustics +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +x1 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +x2 +60 +40 +20 +0 +20 +40 +60 +y1 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Figure 6. The critical curves and caustics of the 𝐴3 lens for fixed |𝛼| = 1. +The top row shows the results for the divergent lens, 𝛼 < 0, and the bottom +row shows the results for the convergent lens, 𝛼 > 0. +light curve for 𝛼 = 0.3, 𝜎 = 0.03. The lens is taken to be of size +ℓ = 10 AU and moving with velocity 𝑣 = 200 kms−1 relative to +the lens. The parameters are chosen to match the 𝑓 = 2.7 GHz +light curve of the original ESE observation presented in Fiedler +et al. (1987). Since for plasma lensing 𝛼 ∝ 𝑓 −2, we can compute +the light curve for multiple frequency bands. Computing the light +curve for 𝑓 = 8.1 GHz (shown in the bottom panel of Fig. 7), we +see that, at high frequencies, multiple caustics in the light curve +become apparent. It is not the case that these magnification caustics +appear because new features in the lens become resolved as the +frequency increases. Rather, as can be seen from the scaling relation +in Eq. 9, an increase in frequency leads to an effective rescaling of +the 𝑥1 coordinate. Thus, the additional magnification caustics seen +at 8.1 GHz are still present at 2.7 GHz, but are further out, and are +not seen at the impact parameters spanned by the observation. In +other words, the low-frequency light curve is effectively a zoomed +in version of the high-frequency light curve. +The high-frequency light curve shown in Fig. 7 shares quali- +tative similarities to the high-frequency observation of the original +ESE in Fiedler et al. (1987). While we have not undertaken a quan- +titative best-fit analysis of the observations with our model, we +argue that the 𝐴3 lens naturally explains the appearance of multi- +ple magnification caustics at high frequencies without the need to +invoke unknown substructure, as is often done in many attempts to +model ESEs. This also leads to a concrete prediction of our model: +large-bandwidth observations of ESEs across multiple frequency +bands should reveal that the high-frequency magnification caustics +do not appear spontaneously as one crosses some focal frequency, +but rather they should gradually move inwards from infinity. +2.2 +A physical picture +We will now briefly discuss a possible physical origin for 𝐴3 lenses. +First, however, we will note that such lenses will generically arise +for any lens that can be described by a thin-lens approximation. +This is because the mathematics of catastrophe theory requires sin- +gularities of projection maps to take on a small number of generic +forms (the elementary catastrophes). Given the vast distances in- +volved in astrophysical lensing, all but the most extended lenses +will be adequately described by the thin-lens approximation. Thus, +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +2.7 GHz +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +t (years) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +8.1 GHz +Figure 7. An example light curve for two frequency bands, 𝑓 = 2.7 GHz and +𝑓 = 8.1 GHz, for the 𝐴3 lens. The light curve is computed for 𝑑𝑙 = 1 kpc, +𝑑𝑠 = 1 Gpc, and fixed 𝑥2 = 10 AU. We choose 𝛼 = 0.3 at 𝑓 = 2.7 GHz, and +a smoothing scale 𝜎𝑥 = 0.03, which for the chosen parameters corresponds +to a peak electron surface density of Σ𝑒 ≈ 0.02 pc cm−2. The source position +as a function of time is given by 𝑦 = 𝑣𝑡 where 𝑣 = 200 km s−1. +the presence of 𝐴3 lenses does not strongly depend on a particular +physical model; 𝐴3 lenses should arise generically in most models +of the ISM. The question is not whether or not there are 𝐴3 lenses, +but whether or not the 𝐴3 lenses predicted by a given physical model +can explain the observed properties of ESEs. +In order for 𝐴3 lenses to be a reasonable candidate to explain +observed ESEs, we require that the transverse physical size of the +lenses (projected onto the plane of the sky) be roughly on the order +of 1 AU, as this is the typical physical scale that can be inferred +from ESE observations. We also require that the thickness of the +sheet that is projected to produce these lenses be much less than +1 AU. In other words, we require 𝜎 ≪ 1. While, in principle, there +is nothing stopping us from modelling lensing events with a highly +smoothed 𝐴3 lens, when the smoothing scale is large (𝜎 ≳ 1), +the unique features of the cusp become washed out, lessening the +explanatory power of such a model. A third requirement is that +whatever physical process causes the 𝐴3 lenses, the lenses must +persist on a timescale of months to years in order to match the +timescale of ESE observations. +A physical picture for corrugated sheets that satisfies these +properties has already been proposed (Goldreich & Sridhar 2006) +and has been suggested as a potential explanation for pulsar scintilla- +tion (Pen & Levin 2014; Simard & Pen 2018). The basic idea, which +we will summarize here, is that magnetic re-connection sheets in +the ISM (boundaries between oppositely oriented magnetic field +lines) sustain plasma current sheets. Ducted waves, driven by the +tension produced by the magnetic fields, propagate through the cur- +rent sheets, forming a corrugated pattern in the plasma density along +the sheet. When the current sheet is close to aligned with the line of +sight, these corrugated patterns produce the fold (𝐴2) and cusp (𝐴3) +lenses when projected onto the lens plane (see e.g. Fig. 1). While +the ducted waves propagate at the speed of sound in the plasma +(𝑐𝑠 ∼ 10𝑇1/2 +4 +km s−1, 𝑇4 ≡ 𝑇/10 K), when the sheet is aligned with +the line of sight, the transverse speed of the waves projected onto +the lens plane can be made arbitrarily small, depending on the de- +gree of alignment. This leads to the long timescales over which +the ESE lens structures are observed to persist. These magnetic re- +connection sheets are also predicted to arise on the spatial scales +MNRAS 000, 1–10 (2022) + +6 +Jow et al. +required to explain ESE observations. While the energetic processes +that stir the ISM (e.g. supernovae, ionization fronts, spiral density +waves, etc.) are typically short lived and occur on parsec scales or +larger, magneto-hydrodynamic simulations of turbulent dynamos +demonstrate that stable magnetic re-connection sheets may occur +well below the stirring scale, and, in particular, on the several AU +scale required by ESEs. Moreover, these sheets are indeed predicted +to be “thin" relative to the transverse AU scale. +It remains to be seen how realistic such a model of the small- +scale ISM is. Realistic, high-resolution simulations are needed. Re- +cent magnetohydrodynamic simulations of the turbulent ISM have +revealed the ubiquity of thin, filamentary-like structures on small +scales intermittently permeating the diffuse medium (Dong et al. +2022; Fielding et al. 2022). However, the resolution of these sim- +ulations are typically at much larger scales than the ∼AU scales +we require to explain ESEs. Nevertheless, these recent simulations +give some confidence that these thin, intermittent structures may +plausibly exist. However, we stress again that one of the strengths of +the doubly catastrophic lensing framework is that it does not depend +crucially on the details of the underlying physical model. We have +summarized this particular model to give an outline of a plausible, +but not necessary scenario that could give rise to the kinds of lenses +we are considering. +3 +A DOUBLE LENSING EVENT +Now that we have outlined the details of the 𝐴3 lens, we will turn to a +particular lensing event of interest in the pulsar PSR B0834+06.This +event has been a particularly fruitful object of study since its ob- +servation in 2005 (Brisken et al. 2010) with the William E.Gordon +Telescope at the Arecibo Observatory, whose data we use here. The +data was taken in a 32 MHz band centred at 316.5 MHz over the +course of ∼ 2 hours. The dynamic spectrum was created using 5s +integrations with ∼ 0.25 kHz channels. In order to collapse the +inverted arclets into single points to more easily identify images, +we use the conjugate wavefield produced by Baker et al. (2022) +using phase retrieval techniques to recover the electric field from +the dynamic spectrum. The conjugate wave-field (the top-left panel +of Fig. 9) shows the main parabolic arc that is ubiquitous in scintil- +lation observations, in addition to a peculiar island of power located +at a delay of roughly 1 ms and Doppler shift of −40 mHz. We will +refer to this feature as the “millisecond feature". Zhu et al. (2022) +use observations over four epochs in roughly fifteen-day intervals +to demonstrate that this event is best explained by a double lens +system. That is, they argue the pulsar is lensed by a main scattering +screen, producing the primary scintillation arc, and a second lens +producing the millisecond features (a schematic of this is shown in +Fig, 8). Zhu et al. (2022) use novel phase retrieval techniques to +infer the distances to the two screens, as well as the angular position +of the many images produced by this lensing system. From this they +argue that the secondary lens associated with the millisecond fea- +ture has similar properties to the plasma structures responsible for +ESEs. In particular, its persistence over the more-than-month-long +observation and large bending angles (𝜃 ≈ 83 mas) are consistent +with other ESE observations. +Here we will consider the possibility that this millisecond fea- +ture is actually produced by an 𝐴3 lens. In order to do this, we +first need to introduce the double lensing formalism. For a multi- +plane lens, the induced phase along a particular path from source to + Lens +A3 +Main Scattering
 +Screen +Source +Observer +Figure 8. Diagram showing the geometry of the extreme scattering event +observed in PSR B0834+06. The pulsar is first scattered by the ESE lens, +which we propose is an 𝐴3 lens, and is then scattered by the primary +scattering screen which results in the parabolic arc that is ubiquitous in +scintillation observations. +observer is given by +𝑆 = 𝜔 +𝑁 +∑︁ +𝑖=1 +𝑑0𝑖𝑑0𝑖+1 +𝑐𝑑𝑖𝑖+1 +� 1 +2 (𝜽𝑖+1 − 𝜽𝑖)2 + 𝑑𝑖𝑖+1𝑑0𝑛+1 +𝑑0𝑖+1𝑑𝑖𝑛+1 +ˆ𝜙𝑖(𝜽𝑖) +� +, +(12) +where 𝑑𝑖 𝑗 is the distance from the 𝑖th lens plane to the 𝑗th lens +plane, and 𝑖 = 0, 𝑁 refer to the observer and source, respectively. +The angular coordinates associated with the 𝑖th lens plane is given +by 𝜽𝑖 and ˆ𝜙𝑖 is the 𝑖th lens potential. +For a two-plane system, we can re-write this phase in terms of +dimensionless parameters as follows: +𝑆 = 𝜈 +� 𝑑2 +𝑑1 +� 1 +2 (𝒙 − 𝒛)2 + 𝜌𝜙1(𝒛) +� ++ 1 +2 (𝒙 − 𝒚)2 + 𝛼𝜙2(𝒙) +� +, +(13) +where we have defined the co-ordinates 𝒛 ≡ 𝜽2𝑑01/ℓ, 𝒙 ≡ 𝜽2𝑑02/ℓ, +and 𝒚 ≡ 𝜽3𝑑02/ℓ to be the physical distance in the respective +lens/source plane, re-scaled by some physical scale ℓ. For our pur- +poses, we will take the first lens plane to be the main scattering +screen, and the second lens plane to be the 𝐴3 lens. It is, there- +fore, convenient to choose ℓ to be a physical scale associated with +the 𝐴3 lens. The barred distances are combined distances given by +𝑑1 = 𝑑12𝑑02/𝑑01 and 𝑑2 = 𝑑23𝑑02/𝑑03. The phase is multiplied +by an overall factor of 𝜈 = 𝜔ℓ2/𝑑2𝑐 and the amplitudes of the lens +potentials are given by +𝛼 = +𝑑2𝑒2Σ∗ +2 +2𝑚𝑒𝜖0𝜔2ℓ2 , +(14) +𝜌 = 𝑑12𝑑03 +𝑑02𝑑13 +𝑑1𝑒2Σ∗ +1 +2𝑚𝑒𝜖0𝜔2ℓ2 , +(15) +where Σ∗ +1 and Σ∗ +2 are the projected electron density of the lenses. +MNRAS 000, 1–10 (2022) + +Cusp of cusps +7 +We could just have easily defined 𝜈 = 𝜔ℓ2/𝑑1𝑐, factoring out an +overall factor of 𝑑1 as opposed to 𝑑2; however, for our purposes it +is convenient to treat the 𝐴3 lens as the primary lens, absorbing the +geometric factors that appear in Eq. 15 into 𝜌 rather than 𝛼. This, +however, is purely a choice of convention, as the image locations +and magnifications in geometric optics do not depend on the overall +factor 𝜈. +The locations of the geometric images are given by the lens +equations, ∇𝒙/𝒛𝑆 = 0, which are: +𝐷(𝑥1 − 𝑧1) + (𝑥1 − 𝑦1) + 𝑑𝜙2 +𝑑𝑥1 += 0, +(16) +𝐷(𝑥2 − 𝑧2) + (𝑥2 − 𝑦2) + 𝑑𝜙2 +𝑑𝑥2 += 0, +(17) +𝑥1 − 𝑧1 + 𝑑𝜙1 +𝑑𝑧1 += 0, +(18) +𝑥2 − 𝑧2 + 𝑑𝜙1 +𝑑𝑧2 += 0, +(19) +where we have defined the ratio 𝐷 ≡ 𝑑2/𝑑1. +Now, for the sake of simplicity, we will assume that the lenses +are both highly anisotropic (i.e. one-dimensional) and that they are +perpendicular to each other. That is, we will assume 𝜙2(𝑥1, 𝑥2) = +𝜙2(𝑥2) and 𝜙1(𝑥1, 𝑥2) = 𝜙1(𝑥1). In this way, the lens equations +simplify to +𝑥2 − 𝑦2 + 𝑑𝜙2 +𝑑𝑥2 += 0, +(20) +𝑥1 − 𝑧1 + 𝑑𝜙1 +𝑑𝑧1 += 0, +(21) +𝑥1 − 𝑦1 + 𝐷𝑧1 +𝐷 + 1 += 0, +(22) +𝑥2 − 𝑧2 = 0. +(23) +It is convenient to do this because the result is that the two lenses +act independently from each other; that is, we can solve the lens +equations for 𝑥2 and 𝑧1 co-ordinates of the images independently +using Eqs. 20 and 21, respectively, which then directly give us the +𝑥1 and 𝑧2 co-ordinates through Eqs. 22 and 23. In general, this +simple separation of the lens equations into independent equations +is not possible since the two lenses will not generically be perfectly +perpendicular to each other. However, for our purposes in this work, +we are primarily interested in the qualitative aspects of the 𝐴3 lens, +as opposed to a precise quantitative comparison, and Zhu et al. +(2022) show that the two lenses are, indeed, roughly perpendicular +to each other. +In order to simulate the lensing event of PSR B0834+06, we +will take 𝜙2(𝑥2) = 𝜇𝐴3 (𝑥2; 𝑥1): the 𝐴3 lens. Again, we stress that +we are treating the 𝐴3 lens as a quasi-one-dimensional lens, where +the second co-ordinate 𝑥1 is treated as a lens parameter. For the +main scattering screen, instead of specifying a lens potential, we +will simply specify a set of co-ordinates, 𝑧1, fixing the location (in +the 𝑧1 direction) on the main scattering screen the rays must pass +through. The goal of this is to re-produce the main scintillation +arc seen in the conjugate wave-field without having to over-commit +ourselves, as it were, to a particular scintillation model for the main +screen. +Once we have the location of the images by solving the lens +equations, it is straightforward to compute where the images should +appear in Doppler-delay space. The group delay of the images is +given by +𝜏 = 𝜕𝑆 +𝜕𝜔 = 𝜈 +𝜔 +� +𝐷 +� 1 +2 (𝒙 − 𝒛)2 − 𝜌𝜙1(𝒛) +� ++ 1 +2 (𝒙 − 𝒚)2 − 𝛼𝜙2(𝒙) +� +, +≈ 𝜈 +𝜔 +� 𝐷 +2 (𝒙 − 𝒛)2 + 1 +2 (𝒙 − 𝒚)2 − 𝛼𝜙2(𝒙) +� +. +(24) +Note that the dispersive terms appear with a relative minus sign +compared to Eq. 13 since for plasma lensing the amplitudes of the +lens potential have a frequency dependence 𝛼, 𝜌 ∼ 𝜔−2. We drop +the dispersive term related to the main scattering screen as we have +not specified the lens potential 𝜙1. We assume that the delay from +the main scattering screen is primarily geometric. +The Doppler shift of the images is given by 𝑓𝐷 = 𝑑𝒚 +𝑑𝑡 · ∇𝒚𝑆, +which can be computed from the following: +𝜕𝑆 +𝜕𝑦1 +≈ − 𝜈𝐷 +𝐷 + 1 (𝑧1 − 𝑦1), +𝜕𝑆 +𝜕𝑦2 +≈ −𝜈(𝑥2 − 𝑦2). +(25) +Note that Eq. 25 follows from the assumption that 𝜕𝑥2 +𝜕𝑦2 = 𝜕𝑧1 +𝜕𝑦1 = 0. +This assumption is equivalent to stating that the lensed images are +stationary transverse to the lens. Alternatively, this is equivalent to +stating that the lensed images are only weakly magnified, which for +the 𝐴3 lens follows from the fact that the lens map is essentially flat +away from the folds (see Fig. 4). This assumption fails only for a +small region near the folds. We are also assuming that the two lens +screens are stationary relative to each other. Namely, we assume that +the velocity 𝜕𝒚 +𝑑𝑡 is dominated by the velocity of the source relative +to the combined position of the two screens. +We now have the tools to simulate the conjugate wave-spectra +of our 𝐴3 lens plus scattering screen system. The top and middle +panel of the right column of Fig. 9 shows the location of the lensed +images in Doppler-delay space. The plotted points correspond to +where the power would be localized in the conjugate wave-field. +The conjugate wave-field for the actual observation is shown in the +left column as comparison. For our simulation, we have chosen the +dimensionless parameters 𝛼 = 0.7, 𝜎 = 0.05, 𝜈 = 31, 000, 𝐷 = 5, +𝑦1 = 0, and 𝑦2 = 11. The locations of the images on the scattering +screen are chosen to be distributed uniformly random over a range +𝑧1 ∈ [−16, 16]. These values are chosen to be consistent with the +observing frequency 𝑓 = 311 MHz and the distances, 𝑑01 = 389 pc, +𝑑02 = 415 pc, and 𝑑03 = 620 pc measured by Zhu et al. (2022). +We also choose the velocity to be in the direction 𝜕𝒚 +𝜕𝑡 ∥ [1, 0.1] to +be consistent with the velocity measured by Zhu et al. (2022). In +order to convert the dimensionless time delay and Doppler shift to +dimensionful quantities, we take the physical scale of the lens to be +ℓ = 1 AU, and the magnitude of the relative velocity of the source +to the lens to be 𝑣 = 23 km s−1 (again, consistent with the velocity +measured by Zhu et al. (2022)). From the parameters, we can also +infer the value for the amplitude of the plasma under-density, Σ∗ +2 ∼ +0.0001 pc cm−3. It is important to note that these chosen parameters +are not the best-fit parameters for this model, but rather a reasonable +estimate of the parameters in order to qualitatively compare the +features of the model with the data. +There are two features of the data that we wish to point out that +our model naturally reproduces. Firstly, we note that the millisecond +feature is highly asymmetric. That is, if the two lenses responsible +for the main scattering screen and the millisecond feature were truly +one-dimensional (i.e. translationally invariant along one axis), then +the millisecond feature would simply be a lensed copy of the main +MNRAS 000, 1–10 (2022) + +8 +Jow et al. +parabolic arc, centred at a different delay and Doppler shift. This, +however, is not the case; the millisecond feature is trunctaed as it +moves towards larger Doppler shifts. This truncation is naturally +accommodated by our model as the 𝐴3 lens, by design, ends in a +cusp. The folds that produce the lensed images meet at the cusp +rather than continuing on forever. Secondly, as the lensed images +approach this truncation point (the cusp) in Doppler-delay space, +the separation between pairs of images in delay first becomes larger +before converging at the cusp. This is the characteristic hockey +stick shape that can be seen both in the data and simulation in the +middle row of Fig. 9. This increase in the delay between images as +one approaches the cusp is a generic feature of our model as the +contours of the lens map effectively form a loop around the cusp, +as seen in Fig. 5. That is the images have a tendency to spread out +before converging at the cusp. +We may also wish to examine the behaviour of the magnifi- +cation of the images. The bottom left panel of Fig. 9 shows the +total flux of the millisecond feature as a function of observing fre- +quency, computed by summing the total power in the millisecond +feature, normalized to a value of unity at some reference frequency. +Essentially what happens is that as the frequency increases, pairs of +localized islands of power in the conjugate wave-field merge succes- +sively. When each pair merges they rapidly increase in brightness +before disappearing. This happens multiple times as one varies the +frequency, creating the structure in seen in the bottom left panel +of Fig. 9: each spike in magnification corresponds to one of these +mergers. A sequence of successive peaks in magnification as a func- +tion of frequency is the generic expectation of our 𝐴3 model. As the +frequency decreases, different pairs of images encounter the fold +catastrophe of the lens, thereby merging and attaining a formally +infinite magnification briefly before disappearing. This happens re- +peatedly until all the images associated with the millisecond feature +disappear. +Although we have not undertaken the more involved process +of quantitatively determining the best-fit parameters of our model +to the data, we hope to have demonstrated that qualitatively the +𝐴3 lens can naturally accommodate features of the PSR B0834+06 +event that have previously been challenging to accommodate. +4 +ESES AND SCINTILLATION +One of the more powerful aspects of the doubly catastrophic lensing +framework is that it has the potential to explain both ESEs and +scintillation with a single, unified framework. While in this work +we have focused on cusp (𝐴3) lenses as a natural explanation for +ESEs, previous works have explored the possibility of explaining +scintillation observations with ensembles of fold (𝐴2) lenses (Pen +& Levin 2014; Simard & Pen 2018). The basic picture we propose +is that corrugated plasma sheets are responsible for both scattering +phenomena. When corrugated sheets are closely aligned with the +line of sight, they form folds under projection. These folds result +in the multi-path propagation that is seen in pulsar scintillation. +While these folds are required to be highly elongated (i.e. effectively +one-dimensional) to explain scintillation observations, they cannot +continue forever. When folds end, they are mathematically required +to merge in cusp (𝐴3) catastrophes. It is these 𝐴3 catastrophes that +we propose as the origin of ESEs. +One immediate question that arises is why, if both scintillation +and ESEs are caused by the same ISM structures, is one phenomenon +so much more common than the other. Within the PSR B0834+06 +observation we have been discussing, there is only one feature asso- +40 +20 +0 +20 +40 +0 +200 +400 +600 +800 +1000 +1200 +1400 +Delay (ms) +60 +40 +20 +0 +20 +40 +60 +0 +200 +400 +600 +800 +1000 +1200 +1400 +46 +44 +42 +40 +38 +36 +34 +Doppler (mHz) +950 +1000 +1050 +1100 +1150 +Delay (ms) +50 +45 +40 +35 +30 +Doppler (mHz) +900 +1000 +1100 +1200 +1300 +1400 +310 +315 +320 +325 +330 +335 +340 +freq (GHz) +0 +2 +4 +6 +8 +280 +290 +300 +310 +320 +330 +340 +freq (GHz) +80.0 +80.2 +80.4 +80.6 +80.8 +81.0 +81.2 +81.4 +Figure 9. The left column shows data for an observation of PSR B0834+06. +The top panel shows the conjugate wave-field of the observation in Doppler- +delay space. The middle panel is the same as the top panel, zoomed in on the +millisecond-delay feature associated with the ESE lens. The bottom panel +shows the sum of the power in the conjugate wave-field of the millisecond +wave-field as a function of frequency, normalized to unity when the signal +falls below the noise threshold. This is taken as a proxy for the magnification +induced by the lens. The top and middle panel of the right column shows +the location of the images in Doppler-delay space for our double lensing +model with an 𝐴3 lens plus a primary scattering screen. We choose the +dimensionless parameters 𝛼 = 0.7, 𝜎 = 0.05, 𝜈 = 31, 000, 𝐷 = 5, 𝑦1 = 0, +𝑦2 = 11, and the direction of the velocity to be 𝜕𝒗 +𝜕𝑡 ∥ [1, 0.1]. The locations +of the images on the scattering screen are chosen to be distributed uniformly +over 𝑧1 ∈ [−16, 16]. To convert to dimensionful parameters, we choose +𝑓 += 311 MHz, 𝑑01 = 389 pc, 𝑑02 = 415 pc, and 𝑑03 = 620 pc and a +physical scale of the lens ℓ = 1 AU. The magnitude of the velocity is chosen +to be 23 km s−1. These parameters correspond to an amplitude of the plasma +under-density of Σ∗ +2 ∼ 0.0001 pc cm−3 (note that the maximum density at +the fold is about an order of magnitude higher than this). Since we choose the +two lenses to be perpendicular to each other, it is well-defined to identify each +image with one of the three images produced by the 𝐴3 lens: distinguished +in the figure by the green, orange, and blue colours. The bottom right panel +shows the total flux of the green and orange images (the images associated +with the millisecond feature) as a function of frequency. +ciated with an 𝐴3 lens (namely, the millisecond feature), whereas the +each of the scattered images along the main scintillation arc, in our +picture, would be associated with a fold. Since there are hundreds +of scattered images, this suggests that fold lenses are much more +common than cusp lenses. Moreover, most pulsars are observed to +scintillate, but are only observed to undergo ESE-like scattering +about one percent of the time. +While this relative rarity of ESEs compared to scintillation +might initially seem to pose an issue for any attempt to explain +MNRAS 000, 1–10 (2022) + +Cusp of cusps +9 +these two phenomena with a unified model, the doubly catastrophic +framework actually provides a natural explanation. It is a well- +known result of catastrophe theory that the cross-section for folds +is much larger than the cross-section for cusps. That is, consider +a projected plasma surface density profile given by Σ𝑒. We can +define the area on the sky such that the density is greater than +some threshold, Δ, to be 𝜎(Σ𝑒 > Δ). The scaling as a function of +threshold for this cross-section can be computed for the fold and +cusp catastrophes as (Narayan & Wallington 1993): +𝜎𝐴2 (Σ𝑒 > Δ) ∼ Δ−2, +(26) +𝜎𝐴3 (Σ𝑒 > Δ) ∼ Δ−5/2. +(27) +That is, the cross-section for the cusp decreases faster as a function +of threshold than the fold cross-section. Therefore, it is a generic +expectation of catastrophe theory that folds will contribute more +to the observed density than cusps. This is also, notably, a precise +and testable prediction of our model; the number of ESEs with an +inferred maximum column density above some threshold should +scale according to this power law. +5 +APPLICATIONS +We have argued that doubly catastrophic lensing is a potentially +powerful framework for analyzing scattering phenomena in pul- +sars and other radio sources, as it provides a unified explanation +for both scintillation and ESE observations, and also naturally ac- +commodates qualitative features of the data that have thus far been +challenging to explain. Another powerful aspect of lenses as catas- +trophes is that the mathematics of catastrophe theory constrains the +form of the lenses to a small set of elementary catastrophes. These +elementary catastrophes are universal and are described by a small +number of unfolding parameters. +If the plasma structures in the ISM responsible for these scat- +tering phenomena are indeed catastrophes, then this would represent +a significant advancement in our ability to unambiguously infer the +physical properties of the lenses and also to use them as astrometric +tools. Contrast this with the present situation. Since we lack any +prior information on the form of the lenses, in principle one has an +infinite number of degrees-of-freedom when attempting to build a +model to match ESE or scintillation data. As a result, inferences +of, say, the electron density, Σ𝑒, may vary by orders-of-magnitude +between different lens models. Moreover, there has been little obser- +vational evidence, so far, that has allowed us to distinguish between +the many proposed models. At the very least, the doubly catastrophic +lensing formalism makes precise predictions which we will be able +to test soon. If confirmed, then the space of potential lens shapes +collapses from infinite, to a small number of catastrophes. +This would have particularly important implications for pulsar +astrometry. One of the primary limitations of our ability to use lens- +ing to obtain precise astrometric data is the fact that we typically +do not know the dispersive contribution of the lens to the observed +time delays. That is, we are typically forced to ascribe the observed +time delays entirely to the geometric part of the time delay, e.g. +the quadratic terms in Eq. 24. If the lenses are catastrophes, de- +scribed by a small number of parameters, then it becomes possible +to unambiguously infer the dispersive contributions to the delay. +Especially for a system such as PSR B0834+06 which is highly +over-determined, it would potentially be possible to infer the full +lens potential. +One concrete example of an application of the doubly catas- +trophic lensing framework to infer the physics of scattering struc- +50 +45 +40 +35 +30 +freq (GHz) +1000 +1050 +1100 +1150 +1200 +1250 +1300 +1350 +Delay (ms) +60 +55 +50 +45 +40 +35 +30 +freq (GHz) +1000 +1050 +1100 +1150 +1200 +Delay (ms) +Figure 10. The simulated millisecond feature using the same parameters +for our double-lens model as Fig. 9, except the left panel is for 𝛼 > 0 +(the convergent lens) and the right panel is for 𝛼 < 0 (the divergent lens). +The crosses show the value of the geometric delay, whereas the solid dots +show the total delay (geometric plus group delay). The red and blue colours +indicate the parity of the images, which is either positive or negative, given +by the sign of the determinant of the Jacobian of the lens map. +tures in the ISM is its potential ability to unambiguously distin- +guish between convergent (under-dense) and divergent (over-dense) +lenses. Fig. 10 shows the millisecond feature of the PSR B0834+06 +lensing event, modelled with the 𝐴3 lens for using 𝛼 > 0 (left +panel) and 𝛼 < 0 (right panel). The crosses show the value of the +geometric delay, whereas the solid dots show the total delay. Note +that since the group delay is negative for positive 𝛼, and positive +for negative 𝛼, for a convergent (under-dense) lens, the total delay +is less than the geometric delay, and for a divergent (over-dense) +lens, the total delay is greater than the geometric delay. The red and +blue colours indicate the parity of the images, which refers to the +direction of motion of the images with respect to the source. If the +image moves in the same direction as the source then it is said to +have positive parity, and if it moves in the opposite direction it is +said to have negative parity. Note that in either the over- or under- +dense case, the positive parity images have much smaller group +delay than the negative parity images. As a result, the orientation +of the characteristic hockey-stick shape of the millisecond feature +is flipped between the convergent and divergent lenses. Inspecting +the data shown in Fig. 9 visually, the millisecond feature appears +to be closer to the convergent (under-dense) case. However, since +we have not attempted to precisely fit the data, it is not possible to +make any strong inferences. +An additional possibility that the doubly catastrophic frame- +work opens up, is that if pulsar scintillation is caused by 𝐴2 folds +and ESEs are caused by 𝐴3 cusps by the same sheet structures in the +ISM as viewed under projection, then many observations of these +phenomena will allow us to take advantage of the rich mathemat- +ics of catastrophe theory to probe the turbulent ISM on scales that +are inaccessible to simulations. That is, with many scintillation and +ESE observations, we can effectively generate a sky-map of caus- +tic networks in the ISM. Such a map may be a powerful tool for +studying the physics of the ISM. +6 +CONCLUSION +In this paper, we have presented a model based on a simple applica- +tion of catastrophe theory to thin plasma sheets to explain extreme +scattering events. That is, we propose that several aspects of ESE +observations can be explained using lens potential with an 𝐴3 cusp +profile. This is an extension of previous work (Pen & Levin 2014; +Simard & Pen 2018) suggesting that 𝐴2 folds arising from corru- +gated plasma sheets may explain pulsar scintillation. We call this +application of catastrophe theory to the lens potential “doubly catas- +MNRAS 000, 1–10 (2022) + +10 +Jow et al. +trophic" lensing, as catastrophes also generically appear in the light +curves of lensed sources. +The doubly catastrophic framework is well-motivated for sev- +eral reasons. Firstly, the past decade of pulsar scintillation observa- +tions suggest the ubiquity of thin plasma sheets in the ISM. Since +lensing is well-described by an effective projected density perpen- +dicular to the line of sight, and since catastrophes generically arise +when thin sheets are viewed under projection, 𝐴2 and 𝐴3 lenses +(in addition to higher order catastrophes which have not considered +here) should naturally arise. We argue that these catastrophic lens +potentials should exist in the ISM, whether or not they are abundant +enough to explain all scintillation or ESE observations. Secondly, +recent work on the physics of the turbulent ISM through MHD sim- +ulations suggest that corrugated plasma sheets of the kind we con- +sider are physically well-motivated. Lastly, the doubly catastrophic +framework has several desirable theoretical features: it provides a +universal framework that describes both scintillation and ESEs as +aspects of the same phenomenon, and the application of catastrophe +theory means that the lens potentials are generic and well-described +by a small number of parameters. +In this work, we have described the features of the simplest +𝐴3 lens and have argued that it can explain many of the qualitative +features of ESE observations, including the frequency structure of +ESEs. The inability to account for features of ESE light curves +at high frequencies has been a roadblock for thin sheet models of +these events. We argue that the 𝐴3 lens overcomes this issue. We +also argue that the 𝐴3 lens provides a natural explanation for features +seen in the lensing of PSR B0834+06. +DATA AVAILABILITY +No new data were generated or analysed in support of this research. +ACKNOWLEDGEMENTS +We receive support from Ontario Research Fund—research Ex- +cellence Program (ORF-RE), Natural Sciences and Engineering +Research Council of Canada (NSERC) [funding reference number +RGPIN-2019-067, CRD 523638-18, 555585-20], Canadian Insti- +tute for Advanced Research (CIFAR), Canadian Foundation for In- +novation (CFI), the National Science Foundation of China (Grants +No. 11929301), Thoth Technology Inc, Alexander von Humboldt +Foundation, and the Ministry of Science and Technology(MOST) +of Taiwan(110-2112-M-001-071-MY3). Computations were per- +formed on the SOSCIP Consortium’s [Blue Gene/Q, Cloud Data +Analytics, Agile and/or Large Memory System] computing plat- +form(s). SOSCIP is funded by the Federal Economic Development +Agency of Southern Ontario, the Province of Ontario, IBM Canada +Ltd., Ontario Centres of Excellence, Mitacs and 15 Ontario aca- +demic member institutions. Cette recherche a été financée par le +Conseil de recherches en sciences naturelles et en génie du Canada +(CRSNG), [numéro de référence 523638-18,555585-20 RGPIN- +2019-067]. +REFERENCES +Baker D., Brisken W., van Kerkwijk M. H., Main R., Pen U.-L., Sprenger +T., Wucknitz O., 2022, MNRAS, 510, 4573, arXiv:2101.04646 +Bannister K. W., Stevens J., Tuntsov A. V., Walker M. A., Johnston S., +Reynolds C., Bignall H., 2016, Science, 351, 354, arXiv:1601.05876 +Brisken W. F., Macquart J. P., Gao J. J., Rickett B. J., Coles W. A., Deller +A. T., Tingay S. J., West C. J., 2010, ApJ, 708, 232, arXiv:0910.5654 +Burke-Spolaor S., et al., 2019, AAPR, 27, 5, arXiv:1811.08826 +Clegg A. W., Fey A. L., Lazio T. J. 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H., 2022, +arXiv e-prints, p. arXiv:2208.06884, arXiv:2208.06884 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–10 (2022) + diff --git a/gtE_T4oBgHgl3EQf3ByR/content/tmp_files/load_file.txt b/gtE_T4oBgHgl3EQf3ByR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88e3431a9cebf4f941f85e2813d75d815a8fcf06 --- /dev/null +++ b/gtE_T4oBgHgl3EQf3ByR/content/tmp_files/load_file.txt @@ -0,0 +1,815 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf,len=814 +page_content='MNRAS 000, 1–10 (2022) Preprint 23 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 On the cusp of cusps: a universal model for extreme scattering events in the ISM Dylan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Jow,2,3,6★ Ue-Li Pen,1,2,3,4,5,6 and Daniel Baker2,3 1Institute of Astronomy and Astrophysics, Academia Sinica, Astronomy-Mathematics Building, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan 2Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' George Street, Toronto, ON M5S 3H8, Canada 3Department of Physics, University of Toronto, 60 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' George Street, Toronto, ON M5S 1A7, Canada 4Perimeter Institute for Theoretical Physics, 31 Caroline St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' North, Waterloo, ON, Canada N2L 2Y5 5Canadian Institute for Advanced Research, CIFAR program in Gravitation and Cosmology 6Dunlap Institute for Astronomy & Astrophysics, University of Toronto, AB 120-50 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' George Street, Toronto, ON M5S 3H4, Canada Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The scattering structures in the ISM responsible for so-called “extreme scattering events" (ESEs), observed in quasars and pulsars, remain enigmatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Current models struggle to explain the high-frequency light curves of ESEs, and a recent analysis of a double lensing event in PSR B0834+06 reveals features of ESEs that may also be challenging to accommodate via existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We propose that these features arise naturally when the lens has a cusp-like profile, described by the elementary 𝐴3 cusp catastrophe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This is an extension of previous work describing pulsar scintillation as arising from 𝐴2 fold catastrophes in thin, corrugated plasma sheets along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We call this framework of describing the lens potentials via elementary catastrophes “doubly catastrophic lensing", as catastrophes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' folds and cusps) have long been used to describe universal features in the light curves of lensing events that generically manifest, regardless of the precise details of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Here, we argue that the lenses themselves may be described by these same elementary structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' If correct, the doubly catastrophic lensing framework would provide a unified description of scintillation and ESEs, where the lenses responsible for these scattering phenomena are universal and can be fully described by a small number of unfolding parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This could enable their application as giant cosmic lenses for precision measurements of coherent sources, including FRBs and pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Key words: waves – radio continuum: ISM – pulsars:general – fast radio bursts 1 INTRODUCTION The enigmatic extreme scattering events (ESEs) that were first dis- covered in quasars in the late 80s (Fiedler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1987) and in pulsars a few years later (Cognard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1993) have presented a long-standing mystery in observations of radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' While they are known to be caused by scattering in the interstellar medium (ISM), the pre- cise form of the plasma structures that cause these events and the physical origin of these structures remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Interest in ob- serving and understanding ESEs has increased, with recent work highlighting their relative ubiquity and setting the stage for future surveys of these mysterious events (Bannister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' More- over, the excess time delays induced by ESEs have implications for precision gravitational wave detection through pulsar timing arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Precise modelling of these excess delays will be necessary to move beyond detection of a stochastic gravitational wave background to individual detections (Burke-Spolaor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Recently, novel ★ E-mail: djow@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='ca phase retrieval techniques have been used for precision localization of the refractive images formed by the ESE lens (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In conjunction with such techniques, new observations from current and next-generation radio telescopes built for pulsar timing arrays and fast radio burst (FRB) detections, among other purposes, will allow us to test the variety of models that have been proposed to explain ESEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' While future observations will hopefully shed light on the plasma structures causing ESEs, current observations pose sev- eral theoretical challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Assuming ESEs are caused by three- dimensional plasma inhomogeneities in the ISM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' an over- or under-dense cloud of ionized plasma) leads to inferences for the pressure of such clouds that are several orders of magnitude in ex- cess of typical pressures in the diffuse ISM (Clegg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' If such highly pressurized clouds existed, then they would be unsta- ble on the time-scales needed to explain ESE observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' this is known as the over-pressure problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Thin, two-dimensional current sheets that are aligned with the line of sight have been proposed as a potential resolution to the over-pressure problem (Romani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='08344v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='HE] 19 Jan 2023 2 Jow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Pen & King 2012), but, thus far, such models struggle to ex- plain certain features of the ESE light curves, in particular their rich frequency structure (Walker & Wardle 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' For example, while a two-dimensional Gaussian profile may fit the low-frequency light curves observed in ESEs, they fail to match the high-frequency light curves (Clegg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Typically, such models invoke substruc- ture in the ISM that becomes resolved at high frequencies to explain the complex morphologies of the high-frequency light curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' how- ever, it remains desirable to be able to explain both the time and frequency structure of ESEs with a single lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Cold, self- gravitating clouds of neutral gas with an ionized skin have been proposed to explain the frequency structure of ESEs (Henriksen & Widrow 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Walker & Wardle 1998), but if correct would imply that a substantial fraction of the galaxy’s mass is contained within these clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' ESEs are not the only scattering phenomenon associated with the ISM that radio sources are observed to undergo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Pulsars are ob- served to scintillate due to multi-path scattering in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' It has generally been assumed that pulsar scintillation and extreme scat- tering events are distinct phenomena, caused by different plasma structures in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' However, just as thin plasma sheets have been proposed as an explanation for ESEs, in recent decades, there has been growing observational evidence that a substantial fraction of scintillation observations (if not all) can be explained by refrac- tive plasma sheets along the line of sight (Stinebring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Goldreich & Sridhar 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Brisken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Pen & Levin 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This is in contrast to traditional models of an extended Kolmogorov turbulent medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' A similar story has been playing out in the study of the turbulent ISM through magnetohy- drodynamic (MHD) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Recent MHD simulations suggest that the turbulent cascade is driven by intermittent sheet-like struc- tures in the ISM (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' If thin plasma sheets explain scintillation observations and are consistent with current understandings of the physics of turbulence in the ISM, might they not also explain ESEs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Here we propose a model for ESEs that arises naturally from the thin sheet picture which qualitatively explains several features of current ESE obser- vations, including the complex frequency structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The model we propose is a simple application of catastrophe theory at the density level of the lens description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, lensing by thin sheets can be effectively described by the projected density of the sheet onto a plane perpendicular to the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Mathematically, singu- larities in the projection map can be classified and described by a small set of elementary catastrophes (Thom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Fold (𝐴2) catastrophes in corrugated plasma sheets have been proposed as an explanation for pulsar scintillation observations (Goldreich & Srid- har 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Pen & Levin 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Simard & Pen 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Here we propose the next higher-order catastrophe, the 𝐴3 cusp, as an explanation for ESEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We call this framework “doubly catastrophic" lensing, since catastrophe theory has long been applied to the theory of lensing to describe the magnification of sources near singularities in the lens map (Nye 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In addition to describing the magnification as a network of catastrophes, here we describe the lens itself as a catastrophe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' One of the advantages of such a framework, is that the elementary catastrophes are universal and described by a small number of unfolding parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Therefore, if correct, the effec- tive lenses describing ESEs may be exceptionally simple in form, even if the physical plasma sheets are formed by complex physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In Section 2 we introduce our model for ESEs and discuss some of its qualitative features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In Section 4 we discuss the possibility of using the doubly catastrophic Lens Projected Density Source Observer ̂xs ̂x ̂xo ds dl dsl Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Diagram of a corrugated sheet lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' When the distances between the observer, source, and lens are large compared to the extent of the sheet along the line of sight (as in most astrophysical scenarios), the lensing is effectively described by the projected density (shown in gray) of the sheet onto the lens plane perpendicular to the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' As the sheet is rotated to be more aligned with the line of sight, the peaks of the projected density become larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The refraction of light due to the lens causes multi-path propagation from source to observer, shown by the grey, dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' lensing framework to explain both scintillation and ESEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In Sec- tion 3 we analyse in detail observations of an extreme scattering event in the pulsar PSR B0834+06 with our model, and in Section 5 we discuss potential applications of this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2 THE 𝐴3 LENS In geometric optics, the effect of a plasma lens localized to a single plane along the line of sight is determined by the lens equation ˆ𝒚 = ˆ𝒙 + 𝑑𝑐𝑒2 𝑚𝑒𝜖0𝜔2 ∇ ˆ𝒙Σ𝑒( ˆ𝒙), (1) where 𝜔 is the angular frequency of the light, ˆ𝒚 = ( ˆ𝒙𝑠𝑑𝑙+ ˆ𝒙𝑜𝑑𝑠𝑙)/𝑑𝑠 is a weighted average of the transverse displacement between the source and observer, 𝑑 = 𝑑𝑠𝑙𝑑𝑙/𝑑𝑠 is an effective distance, and Σ𝑒( ˆ𝒙) is the excess surface electron in the lens plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The coordi- nates and distances involved are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The lens equation determines a mapping between the source plane and the lens plane, determining the set of rays that connect the source and observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In astrophysical lensing, due to the vast distances between the source and the observer, it is often sufficient to treat lensing in this “thin lens" approximation, where the lens is taken to be localized to a single plane perpendicular to the line of sight (the lens plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The physical plasma that produces the lens effect is, of course, not a fully two-dimensional screen, but has some extent along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' As such, the surface density, Σ𝑒, is a projection of the actual density onto the lens plane: Σ𝑒( ˆ𝒙) = ∫ 𝛿𝑛𝑒( ˆ𝒙, 𝑧)𝑑𝑧, (2) where 𝛿𝑛𝑒 is the excess electron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1 shows an example of this projection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Consider MNRAS 000, 1–10 (2022) Cusp of cusps 3 a thin sheet with a periodic profile that is inclined by some angle with respect to the line of sight (the blue curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The surface density along the lens plane (the grey curve) is obtained by projecting the sheet onto a plane perpendicular to the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In particular, for an infinitely thin sheet with a shape given by 𝑥 = 𝑓 (𝑧), the projected density is proportional to | 𝑑𝑥 𝑑𝑧 |−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Thus, the density is formally infinite at singularities of the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Catastrophe theory describes the mathematics of such singu- larities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Powerfully, catastrophe theory shows that the topological structure of singularities must conform to a few fundamental forms (the “elementary catastrophes") regardless of the precise details of the map in which the singularities arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Catastrophe theory has already been used to great effect in lensing theory to predict the magnification of a source near a lens’ caustics without needing to know the precise details of the lens potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Our proposal here is to extend this use of catastrophe theory to the density level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, we wish to describe not only the magnification via elementary catastro- phes, but the lens potential itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We expect these elementary forms are likely to appear in the projected plasma density, as catastrophes generically arise when projecting thin, sheet-like structures in the ISM along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We will argue that by modelling the plasma structures responsible for ESEs by these catastrophes, it is possible to explain aspects of observations that have thus far been challenging to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We call this framework of describing the pro- jected plasma density by a network of caustics “doubly catastrophic lensing", as catastrophes arise both in the magnification produced by the lens and in the lens potential, itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='1 Modelling the 𝐴3 lens In this paper we will focus on the 𝐴3 catastrophe, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' the cusp catastrophe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The fold (𝐴2) and cusp catastrophes are the simplest of the elementary catastrophes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Lensing by a fold has been discussed elsewhere, and has been proposed as an explanation for pulsar scin- tillation (Goldreich & Sridhar 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Pen & Levin 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Simard & Pen 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Here we propose lensing by an 𝐴3 catastrophe as an ex- planation for ESEs in pulsars and quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The basic idea is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' when the folds of a thin, folded sheet come to an end, they meet in a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The sheet, when viewed under projection, forms an 𝐴3 cusp density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The cusp is described by two unfolding parameters: 𝑥1 and 𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2 shows the cusp density as a function of 𝑥1 for fixed 𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The idea is to use the canonical intensity of the 𝐴3 catastrophe, which we will call 𝜇𝐴3 (𝑥1, 𝑥2), as the lens potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, our lens equation is: 𝒚 = 𝒙 + 𝛼∇𝜇𝐴3 (𝒙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (3) The intensity of the 𝐴3 catastrophe is described by the canonical phase 𝜙𝐴3 (𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝑥1, 𝑥2) = 𝑡4 4 − 𝑥2𝑡2 2 + 𝑥1𝑡, (4) where 𝑥1 and 𝑥2 are the unfolding parameters, and 𝑡 is the coordinate in the phase screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' It follows that the cusp intensity is given by 𝜇𝐴3 (𝑥1, 𝑥2) = ∑︁ 𝑡𝑖 |3𝑡2 𝑖 − 𝑥2|−1 (5) where 𝑡𝑖 are the solutions to the stationary phase equation 𝑡3 −𝑥2𝑡 − 𝑥1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We will also introduce an additional parameter 𝜎 and define ˜𝜇𝐴3 (𝑥1, 𝑥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝜎) ≡ 𝜇𝐴3 (𝑥1, 𝑥2) ★ 𝑊(𝑥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝜎), , (6) x2 > 0 x2 = 0 x2 < 0 x2 Folded sheet Projected density Folded sheet Projected density x1 x1 x1 x1 x2 > 0 x2 = 0 x2 < 0 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Diagram showing how the 𝐴3 cusp catastrophe arises when a folded sheet is projected onto the lens plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The grey mesh shows the physical sheet and the colour map below shows the density of the sheet when projected onto the lens plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The cusp profile arises generically when two folds in the sheet come to an end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The variables 𝑥1 and 𝑥2 are the two unfolding parameters of the 𝐴3 catastrophe and can be thought of as the physical coordinates in the lens plane up to some arbitrary scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The red lines show cross sections through the sheet for fixed 𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In the bottom panel, we show the cross section through the physical sheet in red, and the projected density for an infinitely thin sheet below that in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The blue curves show the projected density for a sheet of finite thickness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' the effect of which is to smooth out the sharply peaked grey curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' For 𝑥2 < 0, the sheet is made up of two folds that converge at 𝑥2 = 0 at the cusp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' For 𝑥2 > 0, the two folds disappear as the sheet flattens out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' where𝑊(𝑥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝜎) is taken to be a simple Gaussian smoothing function with standard deviation 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This smoothing is performed to remove the infinite densities that arise in the cusp catastrophe and represents the fact that the physical sheet that gives rise to the cusp has a finite thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Together, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 3 and 6 provide a full description for the 𝐴3 lens in geometric optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' However, to further simplify our analysis we will treat the lens as quasi-one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, we will assume that the direction of the rays is only modified in the 𝑥1 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In other words, we treat 𝑥2 as a fixed parameter of the lens, and we only need to solve the one-dimensional lens equation: 𝑦1 = 𝑥 + 𝛼𝜕𝑥𝜙(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝑥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (7) The second lens equation is simply 𝑦2 = 𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This simplification is possible because the derivative of ˜𝜇𝐴3 is typically much larger in the 𝑥1 direction than it is in the 𝑥2 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' While not strictly necessary here, this simplification will come in handy when we consider a multi-plane lens in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In this limit, the magnification due to the 𝐴3 lens is given by 𝜇(𝑦1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝑥2) = ∑︁ 𝑥 |1 + 𝛼𝜕2 𝑥𝜙(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝑥2)|−1, (8) where the sum is taken over solutions to the lens equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022) 4 Jow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 10 5 0 5 10 x1 (AU) 4 2 0 2 4 6 8 10 x2 (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='10 t (year) 100 101 102 103 104 f (GHz) Cusp of cusp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The right panel shows the dynamic spectrum (intensity as a function of time and frequency) one would observe for the source trajectory shown by the black arrows on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The blue curve on the left shows the caustics of the 𝐴3 cusp profile, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=', the points of maximum projected density shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We have chosen the trajectory to just graze the cusp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Here we are describing the lens potential as an cusp catastrophe, but cusps also generically arise in the dynamic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This is clearly seen in the right panel, where the bright peaks of the magnification converge towards a cusp at high frequencies: this is the titular "cusp of the cusp".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' It turns out that for the 𝐴3 potential, changes along the 𝑥2 direction can be described by changes in the amplitude of the lens, and a re-scaling of the 𝑥 coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This follows from the identity 𝛼𝜇𝐴3 (𝑥, 𝑥2 = ±1) = 𝜇𝐴3 �𝛼−3/2𝑥, 𝑥2 = ± 1 𝛼 �, (9) or, equivalently, 𝜇𝐴3 (𝑥, 𝑥2) = 1 𝑥2 𝜇�𝑥−3/2 2 𝑥, sign(𝑥2)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (10) In other words, once a sign is chosen for 𝑥2, one is free to re-scale the amplitude 𝛼 and the 𝑥 coordinate so that |𝑥2| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This means that the effect of changing the amplitude 𝛼 is simply to re-scale the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 4 shows the lens map described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 7 for fixed 𝑥2 = 1 and different values of 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' One may be interested in the location of the caustics that are formed by the lens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' the turning points in the lens map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The location of these turning points depends on 𝛼 and the smoothing scale 𝜎, as the un-smoothed potential is formally infinite at certain points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Thus, the maximum amplitude of the smoothed potential depends strongly on 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 5 shows the level curves of the lens map (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 7), as a function of the unfolding parameters, 𝑥1 and 𝑥2, for fixed 𝛼 = ±1, where the sign of alpha determines whether the lens is convergent or divergent (note that because of the scaling relations shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 9 and 10 varying either 𝛼 or 𝑥2, while holding the other fixed, covers the entirety of the parameter space of the lens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 5 tells us where a given source position 𝑦1 gets mapped to in the lens plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, consider 𝑥2 to be some fixed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Then we can read off the image positions in 𝑥1 for a given value of 𝑦1 by looking at the 𝑥1 value the contour associated with 𝑦1 reaches for that value of 𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' A peculiar feature of the 𝐴3 lens is that for a fixed source position, for a small region about 𝑥2 = 0, the distance between the image positions in 𝑥1 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This is in contrast to large values of 𝑥2, for which a decrease in 𝑥2 leads to the images moving closer together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This feature will lead to the characteristic hockey-stick shape shown later in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Now, it is straightforward to compute how the location of the outermost caustics scale with 𝜎 and 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Roughly, the location of the outermost caustic is given by 𝑦∗ 1 ∼ max{𝛼𝜕𝑥𝜙}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' For the un- smoothed lens, the maximum derivative of the lens potential is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' For the smoothed lens, the maximum value of the derivative is nevertheless attained close to where the unsmoothed lens diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Since the lens potential near the divergence is described by a fold 20 15 10 5 0 5 10 15 20 y 4 2 0 2 4 x = 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='01 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The lens map of the 𝐴3 lens for fixed 𝑥2 = 1 and varying 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The effect of changing 𝛼 is to effectively re-scale the coordinates as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 x1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 x2 Convergent Lens 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 x1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 x2 Divergent Lens 70 60 50 40 25 10 1 1 10 25 40 50 60 70 70 60 50 40 25 10 1 1 10 25 40 50 60 70 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The level curves of the lens map, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 7, as a function of the unfolding parameters 𝑥1 and 𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' catastrophe, the lens potential is given by 𝜙(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝑥2) ∝ 𝑥−1/2 on one side of the divergence and zero on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Thus, near the divergence, the smoothed derivative is given by 𝜕𝑥𝜙(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝑥2) ∝ 𝜙(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝑥2) ★ 𝜕𝑥𝑊(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝜎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' It follows from this that the location of the outermost caustic scales as 𝑦∗ 1 ∼ 𝛼𝜎−3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (11) Knowing the location of the outermost caustic will be useful later when we are trying to infer the lens parameters from observed time delays in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 6 shows the critical curve and caustic structures that arise due to magnification by the 𝐴3 lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The top row shows the results for the divergent lens (𝛼 < 0), corresponding to an over-dense lens, and the bottom row shows the convergent lens (𝛼 > 0), corresponding to an under-dense lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The light curve that arises as one changes impact parameter, 𝑦1, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' as the source moves relative to the lens), are effectively entirely determined by the number and location of the caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' When 𝜎 ≪ 1 and 𝛼 ≳ 1, the caustics tend to be located far from the axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' for example, the outermost caustic is located at 𝑦∗ 1 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This means that in between the caustics the slope of the inverse lens map is large, meaning that the total magnification is close to one (see, for example, the blue curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 4, where the inverse lens map, 𝑥(𝑦), is effectively flat for most of the region between the caustics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The result is that the light curve as the source moves relative to the lens is close to unity except at the caustics where the magnification suddenly diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 7 shows an example MNRAS 000, 1–10 (2022) Cusp of cusps 5 2 1 0 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 x2 Critical Curves 60 40 20 0 20 40 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 Caustics 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 x1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 x2 60 40 20 0 20 40 60 y1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The critical curves and caustics of the 𝐴3 lens for fixed |𝛼| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The top row shows the results for the divergent lens, 𝛼 < 0, and the bottom row shows the results for the convergent lens, 𝛼 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' light curve for 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='3, 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The lens is taken to be of size ℓ = 10 AU and moving with velocity 𝑣 = 200 kms−1 relative to the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The parameters are chosen to match the 𝑓 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='7 GHz light curve of the original ESE observation presented in Fiedler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Since for plasma lensing 𝛼 ∝ 𝑓 −2, we can compute the light curve for multiple frequency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Computing the light curve for 𝑓 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='1 GHz (shown in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 7), we see that, at high frequencies, multiple caustics in the light curve become apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' It is not the case that these magnification caustics appear because new features in the lens become resolved as the frequency increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Rather, as can be seen from the scaling relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 9, an increase in frequency leads to an effective rescaling of the 𝑥1 coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Thus, the additional magnification caustics seen at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='1 GHz are still present at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='7 GHz, but are further out, and are not seen at the impact parameters spanned by the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In other words, the low-frequency light curve is effectively a zoomed in version of the high-frequency light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The high-frequency light curve shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 7 shares quali- tative similarities to the high-frequency observation of the original ESE in Fiedler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' While we have not undertaken a quan- titative best-fit analysis of the observations with our model, we argue that the 𝐴3 lens naturally explains the appearance of multi- ple magnification caustics at high frequencies without the need to invoke unknown substructure, as is often done in many attempts to model ESEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This also leads to a concrete prediction of our model: large-bandwidth observations of ESEs across multiple frequency bands should reveal that the high-frequency magnification caustics do not appear spontaneously as one crosses some focal frequency, but rather they should gradually move inwards from infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='2 A physical picture We will now briefly discuss a possible physical origin for 𝐴3 lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' First, however, we will note that such lenses will generically arise for any lens that can be described by a thin-lens approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This is because the mathematics of catastrophe theory requires sin- gularities of projection maps to take on a small number of generic forms (the elementary catastrophes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Given the vast distances in- volved in astrophysical lensing, all but the most extended lenses will be adequately described by the thin-lens approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Thus, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='7 GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='00 t (years) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='1 GHz Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' An example light curve for two frequency bands, 𝑓 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='7 GHz and 𝑓 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='1 GHz, for the 𝐴3 lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The light curve is computed for 𝑑𝑙 = 1 kpc, 𝑑𝑠 = 1 Gpc, and fixed 𝑥2 = 10 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We choose 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='3 at 𝑓 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='7 GHz, and a smoothing scale 𝜎𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='03, which for the chosen parameters corresponds to a peak electron surface density of Σ𝑒 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='02 pc cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The source position as a function of time is given by 𝑦 = 𝑣𝑡 where 𝑣 = 200 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' the presence of 𝐴3 lenses does not strongly depend on a particular physical model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝐴3 lenses should arise generically in most models of the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The question is not whether or not there are 𝐴3 lenses, but whether or not the 𝐴3 lenses predicted by a given physical model can explain the observed properties of ESEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In order for 𝐴3 lenses to be a reasonable candidate to explain observed ESEs, we require that the transverse physical size of the lenses (projected onto the plane of the sky) be roughly on the order of 1 AU, as this is the typical physical scale that can be inferred from ESE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We also require that the thickness of the sheet that is projected to produce these lenses be much less than 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In other words, we require 𝜎 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' While, in principle, there is nothing stopping us from modelling lensing events with a highly smoothed 𝐴3 lens, when the smoothing scale is large (𝜎 ≳ 1), the unique features of the cusp become washed out, lessening the explanatory power of such a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' A third requirement is that whatever physical process causes the 𝐴3 lenses, the lenses must persist on a timescale of months to years in order to match the timescale of ESE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' A physical picture for corrugated sheets that satisfies these properties has already been proposed (Goldreich & Sridhar 2006) and has been suggested as a potential explanation for pulsar scintilla- tion (Pen & Levin 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Simard & Pen 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The basic idea, which we will summarize here, is that magnetic re-connection sheets in the ISM (boundaries between oppositely oriented magnetic field lines) sustain plasma current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Ducted waves, driven by the tension produced by the magnetic fields, propagate through the cur- rent sheets, forming a corrugated pattern in the plasma density along the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' When the current sheet is close to aligned with the line of sight, these corrugated patterns produce the fold (𝐴2) and cusp (𝐴3) lenses when projected onto the lens plane (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' While the ducted waves propagate at the speed of sound in the plasma (𝑐𝑠 ∼ 10𝑇1/2 4 km s−1, 𝑇4 ≡ 𝑇/10 K), when the sheet is aligned with the line of sight, the transverse speed of the waves projected onto the lens plane can be made arbitrarily small, depending on the de- gree of alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This leads to the long timescales over which the ESE lens structures are observed to persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' These magnetic re- connection sheets are also predicted to arise on the spatial scales MNRAS 000, 1–10 (2022) 6 Jow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' required to explain ESE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' While the energetic processes that stir the ISM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' supernovae, ionization fronts, spiral density waves, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=') are typically short lived and occur on parsec scales or larger, magneto-hydrodynamic simulations of turbulent dynamos demonstrate that stable magnetic re-connection sheets may occur well below the stirring scale, and, in particular, on the several AU scale required by ESEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Moreover, these sheets are indeed predicted to be “thin" relative to the transverse AU scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' It remains to be seen how realistic such a model of the small- scale ISM is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Realistic, high-resolution simulations are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Re- cent magnetohydrodynamic simulations of the turbulent ISM have revealed the ubiquity of thin, filamentary-like structures on small scales intermittently permeating the diffuse medium (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Fielding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' However, the resolution of these sim- ulations are typically at much larger scales than the ∼AU scales we require to explain ESEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Nevertheless, these recent simulations give some confidence that these thin, intermittent structures may plausibly exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' However, we stress again that one of the strengths of the doubly catastrophic lensing framework is that it does not depend crucially on the details of the underlying physical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We have summarized this particular model to give an outline of a plausible, but not necessary scenario that could give rise to the kinds of lenses we are considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 3 A DOUBLE LENSING EVENT Now that we have outlined the details of the 𝐴3 lens, we will turn to a particular lensing event of interest in the pulsar PSR B0834+06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='This event has been a particularly fruitful object of study since its ob- servation in 2005 (Brisken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 2010) with the William E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='Gordon Telescope at the Arecibo Observatory, whose data we use here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The data was taken in a 32 MHz band centred at 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='5 MHz over the course of ∼ 2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The dynamic spectrum was created using 5s integrations with ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='25 kHz channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In order to collapse the inverted arclets into single points to more easily identify images, we use the conjugate wavefield produced by Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (2022) using phase retrieval techniques to recover the electric field from the dynamic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The conjugate wave-field (the top-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 9) shows the main parabolic arc that is ubiquitous in scintil- lation observations, in addition to a peculiar island of power located at a delay of roughly 1 ms and Doppler shift of −40 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We will refer to this feature as the “millisecond feature".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (2022) use observations over four epochs in roughly fifteen-day intervals to demonstrate that this event is best explained by a double lens system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, they argue the pulsar is lensed by a main scattering screen, producing the primary scintillation arc, and a second lens producing the millisecond features (a schematic of this is shown in Fig, 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (2022) use novel phase retrieval techniques to infer the distances to the two screens, as well as the angular position of the many images produced by this lensing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' From this they argue that the secondary lens associated with the millisecond fea- ture has similar properties to the plasma structures responsible for ESEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In particular, its persistence over the more-than-month-long observation and large bending angles (𝜃 ≈ 83 mas) are consistent with other ESE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Here we will consider the possibility that this millisecond fea- ture is actually produced by an 𝐴3 lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In order to do this, we first need to introduce the double lensing formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' For a multi- plane lens, the induced phase along a particular path from source to Lens A3 Main Scattering Screen Source Observer Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Diagram showing the geometry of the extreme scattering event observed in PSR B0834+06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The pulsar is first scattered by the ESE lens, which we propose is an 𝐴3 lens, and is then scattered by the primary scattering screen which results in the parabolic arc that is ubiquitous in scintillation observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' observer is given by 𝑆 = 𝜔 𝑁 ∑︁ 𝑖=1 𝑑0𝑖𝑑0𝑖+1 𝑐𝑑𝑖𝑖+1 � 1 2 (𝜽𝑖+1 − 𝜽𝑖)2 + 𝑑𝑖𝑖+1𝑑0𝑛+1 𝑑0𝑖+1𝑑𝑖𝑛+1 ˆ𝜙𝑖(𝜽𝑖) � , (12) where 𝑑𝑖 𝑗 is the distance from the 𝑖th lens plane to the 𝑗th lens plane, and 𝑖 = 0, 𝑁 refer to the observer and source, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The angular coordinates associated with the 𝑖th lens plane is given by 𝜽𝑖 and ˆ𝜙𝑖 is the 𝑖th lens potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' For a two-plane system, we can re-write this phase in terms of dimensionless parameters as follows: 𝑆 = 𝜈 � 𝑑2 𝑑1 � 1 2 (𝒙 − 𝒛)2 + 𝜌𝜙1(𝒛) � + 1 2 (𝒙 − 𝒚)2 + 𝛼𝜙2(𝒙) � , (13) where we have defined the co-ordinates 𝒛 ≡ 𝜽2𝑑01/ℓ, 𝒙 ≡ 𝜽2𝑑02/ℓ, and 𝒚 ≡ 𝜽3𝑑02/ℓ to be the physical distance in the respective lens/source plane, re-scaled by some physical scale ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' For our pur- poses, we will take the first lens plane to be the main scattering screen, and the second lens plane to be the 𝐴3 lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' It is, there- fore, convenient to choose ℓ to be a physical scale associated with the 𝐴3 lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The barred distances are combined distances given by 𝑑1 = 𝑑12𝑑02/𝑑01 and 𝑑2 = 𝑑23𝑑02/𝑑03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The phase is multiplied by an overall factor of 𝜈 = 𝜔ℓ2/𝑑2𝑐 and the amplitudes of the lens potentials are given by 𝛼 = 𝑑2𝑒2Σ∗ 2 2𝑚𝑒𝜖0𝜔2ℓ2 , (14) 𝜌 = 𝑑12𝑑03 𝑑02𝑑13 𝑑1𝑒2Σ∗ 1 2𝑚𝑒𝜖0𝜔2ℓ2 , (15) where Σ∗ 1 and Σ∗ 2 are the projected electron density of the lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022) Cusp of cusps 7 We could just have easily defined 𝜈 = 𝜔ℓ2/𝑑1𝑐, factoring out an overall factor of 𝑑1 as opposed to 𝑑2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' however, for our purposes it is convenient to treat the 𝐴3 lens as the primary lens, absorbing the geometric factors that appear in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 15 into 𝜌 rather than 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This, however, is purely a choice of convention, as the image locations and magnifications in geometric optics do not depend on the overall factor 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The locations of the geometric images are given by the lens equations, ∇𝒙/𝒛𝑆 = 0, which are: 𝐷(𝑥1 − 𝑧1) + (𝑥1 − 𝑦1) + 𝑑𝜙2 𝑑𝑥1 = 0, (16) 𝐷(𝑥2 − 𝑧2) + (𝑥2 − 𝑦2) + 𝑑𝜙2 𝑑𝑥2 = 0, (17) 𝑥1 − 𝑧1 + 𝑑𝜙1 𝑑𝑧1 = 0, (18) 𝑥2 − 𝑧2 + 𝑑𝜙1 𝑑𝑧2 = 0, (19) where we have defined the ratio 𝐷 ≡ 𝑑2/𝑑1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Now, for the sake of simplicity, we will assume that the lenses are both highly anisotropic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' one-dimensional) and that they are perpendicular to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, we will assume 𝜙2(𝑥1, 𝑥2) = 𝜙2(𝑥2) and 𝜙1(𝑥1, 𝑥2) = 𝜙1(𝑥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In this way, the lens equations simplify to 𝑥2 − 𝑦2 + 𝑑𝜙2 𝑑𝑥2 = 0, (20) 𝑥1 − 𝑧1 + 𝑑𝜙1 𝑑𝑧1 = 0, (21) 𝑥1 − 𝑦1 + 𝐷𝑧1 𝐷 + 1 = 0, (22) 𝑥2 − 𝑧2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (23) It is convenient to do this because the result is that the two lenses act independently from each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' that is, we can solve the lens equations for 𝑥2 and 𝑧1 co-ordinates of the images independently using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 20 and 21, respectively, which then directly give us the 𝑥1 and 𝑧2 co-ordinates through Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 22 and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In general, this simple separation of the lens equations into independent equations is not possible since the two lenses will not generically be perfectly perpendicular to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' However, for our purposes in this work, we are primarily interested in the qualitative aspects of the 𝐴3 lens, as opposed to a precise quantitative comparison, and Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (2022) show that the two lenses are, indeed, roughly perpendicular to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In order to simulate the lensing event of PSR B0834+06, we will take 𝜙2(𝑥2) = 𝜇𝐴3 (𝑥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 𝑥1): the 𝐴3 lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Again, we stress that we are treating the 𝐴3 lens as a quasi-one-dimensional lens, where the second co-ordinate 𝑥1 is treated as a lens parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' For the main scattering screen, instead of specifying a lens potential, we will simply specify a set of co-ordinates, 𝑧1, fixing the location (in the 𝑧1 direction) on the main scattering screen the rays must pass through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The goal of this is to re-produce the main scintillation arc seen in the conjugate wave-field without having to over-commit ourselves, as it were, to a particular scintillation model for the main screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Once we have the location of the images by solving the lens equations, it is straightforward to compute where the images should appear in Doppler-delay space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The group delay of the images is given by 𝜏 = 𝜕𝑆 𝜕𝜔 = 𝜈 𝜔 � 𝐷 � 1 2 (𝒙 − 𝒛)2 − 𝜌𝜙1(𝒛) � + 1 2 (𝒙 − 𝒚)2 − 𝛼𝜙2(𝒙) � , ≈ 𝜈 𝜔 � 𝐷 2 (𝒙 − 𝒛)2 + 1 2 (𝒙 − 𝒚)2 − 𝛼𝜙2(𝒙) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (24) Note that the dispersive terms appear with a relative minus sign compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 13 since for plasma lensing the amplitudes of the lens potential have a frequency dependence 𝛼, 𝜌 ∼ 𝜔−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We drop the dispersive term related to the main scattering screen as we have not specified the lens potential 𝜙1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We assume that the delay from the main scattering screen is primarily geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The Doppler shift of the images is given by 𝑓𝐷 = 𝑑𝒚 𝑑𝑡 · ∇𝒚𝑆, which can be computed from the following: 𝜕𝑆 𝜕𝑦1 ≈ − 𝜈𝐷 𝐷 + 1 (𝑧1 − 𝑦1), 𝜕𝑆 𝜕𝑦2 ≈ −𝜈(𝑥2 − 𝑦2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (25) Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 25 follows from the assumption that 𝜕𝑥2 𝜕𝑦2 = 𝜕𝑧1 𝜕𝑦1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This assumption is equivalent to stating that the lensed images are stationary transverse to the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Alternatively, this is equivalent to stating that the lensed images are only weakly magnified, which for the 𝐴3 lens follows from the fact that the lens map is essentially flat away from the folds (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This assumption fails only for a small region near the folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We are also assuming that the two lens screens are stationary relative to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Namely, we assume that the velocity 𝜕𝒚 𝑑𝑡 is dominated by the velocity of the source relative to the combined position of the two screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We now have the tools to simulate the conjugate wave-spectra of our 𝐴3 lens plus scattering screen system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The top and middle panel of the right column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 9 shows the location of the lensed images in Doppler-delay space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The plotted points correspond to where the power would be localized in the conjugate wave-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The conjugate wave-field for the actual observation is shown in the left column as comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' For our simulation, we have chosen the dimensionless parameters 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='7, 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='05, 𝜈 = 31, 000, 𝐷 = 5, 𝑦1 = 0, and 𝑦2 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The locations of the images on the scattering screen are chosen to be distributed uniformly random over a range 𝑧1 ∈ [−16, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' These values are chosen to be consistent with the observing frequency 𝑓 = 311 MHz and the distances, 𝑑01 = 389 pc, 𝑑02 = 415 pc, and 𝑑03 = 620 pc measured by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We also choose the velocity to be in the direction 𝜕𝒚 𝜕𝑡 ∥ [1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='1] to be consistent with the velocity measured by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In order to convert the dimensionless time delay and Doppler shift to dimensionful quantities, we take the physical scale of the lens to be ℓ = 1 AU, and the magnitude of the relative velocity of the source to the lens to be 𝑣 = 23 km s−1 (again, consistent with the velocity measured by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' From the parameters, we can also infer the value for the amplitude of the plasma under-density, Σ∗ 2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0001 pc cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' It is important to note that these chosen parameters are not the best-fit parameters for this model, but rather a reasonable estimate of the parameters in order to qualitatively compare the features of the model with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' There are two features of the data that we wish to point out that our model naturally reproduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Firstly, we note that the millisecond feature is highly asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, if the two lenses responsible for the main scattering screen and the millisecond feature were truly one-dimensional (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' translationally invariant along one axis), then the millisecond feature would simply be a lensed copy of the main MNRAS 000, 1–10 (2022) 8 Jow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' parabolic arc, centred at a different delay and Doppler shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This, however, is not the case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' the millisecond feature is trunctaed as it moves towards larger Doppler shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This truncation is naturally accommodated by our model as the 𝐴3 lens, by design, ends in a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The folds that produce the lensed images meet at the cusp rather than continuing on forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Secondly, as the lensed images approach this truncation point (the cusp) in Doppler-delay space, the separation between pairs of images in delay first becomes larger before converging at the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This is the characteristic hockey stick shape that can be seen both in the data and simulation in the middle row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This increase in the delay between images as one approaches the cusp is a generic feature of our model as the contours of the lens map effectively form a loop around the cusp, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is the images have a tendency to spread out before converging at the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We may also wish to examine the behaviour of the magnifi- cation of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The bottom left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 9 shows the total flux of the millisecond feature as a function of observing fre- quency, computed by summing the total power in the millisecond feature, normalized to a value of unity at some reference frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Essentially what happens is that as the frequency increases, pairs of localized islands of power in the conjugate wave-field merge succes- sively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' When each pair merges they rapidly increase in brightness before disappearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This happens multiple times as one varies the frequency, creating the structure in seen in the bottom left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 9: each spike in magnification corresponds to one of these mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' A sequence of successive peaks in magnification as a func- tion of frequency is the generic expectation of our 𝐴3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' As the frequency decreases, different pairs of images encounter the fold catastrophe of the lens, thereby merging and attaining a formally infinite magnification briefly before disappearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This happens re- peatedly until all the images associated with the millisecond feature disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Although we have not undertaken the more involved process of quantitatively determining the best-fit parameters of our model to the data, we hope to have demonstrated that qualitatively the 𝐴3 lens can naturally accommodate features of the PSR B0834+06 event that have previously been challenging to accommodate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 4 ESES AND SCINTILLATION One of the more powerful aspects of the doubly catastrophic lensing framework is that it has the potential to explain both ESEs and scintillation with a single, unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' While in this work we have focused on cusp (𝐴3) lenses as a natural explanation for ESEs, previous works have explored the possibility of explaining scintillation observations with ensembles of fold (𝐴2) lenses (Pen & Levin 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Simard & Pen 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The basic picture we propose is that corrugated plasma sheets are responsible for both scattering phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' When corrugated sheets are closely aligned with the line of sight, they form folds under projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' These folds result in the multi-path propagation that is seen in pulsar scintillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' While these folds are required to be highly elongated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' effectively one-dimensional) to explain scintillation observations, they cannot continue forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' When folds end, they are mathematically required to merge in cusp (𝐴3) catastrophes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' It is these 𝐴3 catastrophes that we propose as the origin of ESEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' One immediate question that arises is why, if both scintillation and ESEs are caused by the same ISM structures, is one phenomenon so much more common than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Within the PSR B0834+06 observation we have been discussing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' there is only one feature asso- 40 20 0 20 40 0 200 400 600 800 1000 1200 1400 Delay (ms) 60 40 20 0 20 40 60 0 200 400 600 800 1000 1200 1400 46 44 42 40 38 36 34 Doppler (mHz) 950 1000 1050 1100 1150 Delay (ms) 50 45 40 35 30 Doppler (mHz) 900 1000 1100 1200 1300 1400 310 315 320 325 330 335 340 freq (GHz) 0 2 4 6 8 280 290 300 310 320 330 340 freq (GHz) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='8 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='4 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The left column shows data for an observation of PSR B0834+06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The top panel shows the conjugate wave-field of the observation in Doppler- delay space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The middle panel is the same as the top panel, zoomed in on the millisecond-delay feature associated with the ESE lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The bottom panel shows the sum of the power in the conjugate wave-field of the millisecond wave-field as a function of frequency, normalized to unity when the signal falls below the noise threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This is taken as a proxy for the magnification induced by the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The top and middle panel of the right column shows the location of the images in Doppler-delay space for our double lensing model with an 𝐴3 lens plus a primary scattering screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We choose the dimensionless parameters 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='7, 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='05, 𝜈 = 31, 000, 𝐷 = 5, 𝑦1 = 0, 𝑦2 = 11, and the direction of the velocity to be 𝜕𝒗 𝜕𝑡 ∥ [1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The locations of the images on the scattering screen are chosen to be distributed uniformly over 𝑧1 ∈ [−16, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' To convert to dimensionful parameters, we choose 𝑓 = 311 MHz, 𝑑01 = 389 pc, 𝑑02 = 415 pc, and 𝑑03 = 620 pc and a physical scale of the lens ℓ = 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The magnitude of the velocity is chosen to be 23 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' These parameters correspond to an amplitude of the plasma under-density of Σ∗ 2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='0001 pc cm−3 (note that the maximum density at the fold is about an order of magnitude higher than this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Since we choose the two lenses to be perpendicular to each other, it is well-defined to identify each image with one of the three images produced by the 𝐴3 lens: distinguished in the figure by the green, orange, and blue colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The bottom right panel shows the total flux of the green and orange images (the images associated with the millisecond feature) as a function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' ciated with an 𝐴3 lens (namely, the millisecond feature), whereas the each of the scattered images along the main scintillation arc, in our picture, would be associated with a fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Since there are hundreds of scattered images, this suggests that fold lenses are much more common than cusp lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Moreover, most pulsars are observed to scintillate, but are only observed to undergo ESE-like scattering about one percent of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' While this relative rarity of ESEs compared to scintillation might initially seem to pose an issue for any attempt to explain MNRAS 000, 1–10 (2022) Cusp of cusps 9 these two phenomena with a unified model, the doubly catastrophic framework actually provides a natural explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' It is a well- known result of catastrophe theory that the cross-section for folds is much larger than the cross-section for cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, consider a projected plasma surface density profile given by Σ𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We can define the area on the sky such that the density is greater than some threshold, Δ, to be 𝜎(Σ𝑒 > Δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The scaling as a function of threshold for this cross-section can be computed for the fold and cusp catastrophes as (Narayan & Wallington 1993): 𝜎𝐴2 (Σ𝑒 > Δ) ∼ Δ−2, (26) 𝜎𝐴3 (Σ𝑒 > Δ) ∼ Δ−5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' (27) That is, the cross-section for the cusp decreases faster as a function of threshold than the fold cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Therefore, it is a generic expectation of catastrophe theory that folds will contribute more to the observed density than cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This is also, notably, a precise and testable prediction of our model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' the number of ESEs with an inferred maximum column density above some threshold should scale according to this power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 5 APPLICATIONS We have argued that doubly catastrophic lensing is a potentially powerful framework for analyzing scattering phenomena in pul- sars and other radio sources, as it provides a unified explanation for both scintillation and ESE observations, and also naturally ac- commodates qualitative features of the data that have thus far been challenging to explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Another powerful aspect of lenses as catas- trophes is that the mathematics of catastrophe theory constrains the form of the lenses to a small set of elementary catastrophes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' These elementary catastrophes are universal and are described by a small number of unfolding parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' If the plasma structures in the ISM responsible for these scat- tering phenomena are indeed catastrophes, then this would represent a significant advancement in our ability to unambiguously infer the physical properties of the lenses and also to use them as astrometric tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Contrast this with the present situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Since we lack any prior information on the form of the lenses, in principle one has an infinite number of degrees-of-freedom when attempting to build a model to match ESE or scintillation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' As a result, inferences of, say, the electron density, Σ𝑒, may vary by orders-of-magnitude between different lens models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Moreover, there has been little obser- vational evidence, so far, that has allowed us to distinguish between the many proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' At the very least, the doubly catastrophic lensing formalism makes precise predictions which we will be able to test soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' If confirmed, then the space of potential lens shapes collapses from infinite, to a small number of catastrophes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This would have particularly important implications for pulsar astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' One of the primary limitations of our ability to use lens- ing to obtain precise astrometric data is the fact that we typically do not know the dispersive contribution of the lens to the observed time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, we are typically forced to ascribe the observed time delays entirely to the geometric part of the time delay, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' the quadratic terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' If the lenses are catastrophes, de- scribed by a small number of parameters, then it becomes possible to unambiguously infer the dispersive contributions to the delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Especially for a system such as PSR B0834+06 which is highly over-determined, it would potentially be possible to infer the full lens potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' One concrete example of an application of the doubly catas- trophic lensing framework to infer the physics of scattering struc- 50 45 40 35 30 freq (GHz) 1000 1050 1100 1150 1200 1250 1300 1350 Delay (ms) 60 55 50 45 40 35 30 freq (GHz) 1000 1050 1100 1150 1200 Delay (ms) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The simulated millisecond feature using the same parameters for our double-lens model as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 9, except the left panel is for 𝛼 > 0 (the convergent lens) and the right panel is for 𝛼 < 0 (the divergent lens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The crosses show the value of the geometric delay, whereas the solid dots show the total delay (geometric plus group delay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The red and blue colours indicate the parity of the images, which is either positive or negative, given by the sign of the determinant of the Jacobian of the lens map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' tures in the ISM is its potential ability to unambiguously distin- guish between convergent (under-dense) and divergent (over-dense) lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 10 shows the millisecond feature of the PSR B0834+06 lensing event, modelled with the 𝐴3 lens for using 𝛼 > 0 (left panel) and 𝛼 < 0 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The crosses show the value of the geometric delay, whereas the solid dots show the total delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Note that since the group delay is negative for positive 𝛼, and positive for negative 𝛼, for a convergent (under-dense) lens, the total delay is less than the geometric delay, and for a divergent (over-dense) lens, the total delay is greater than the geometric delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The red and blue colours indicate the parity of the images, which refers to the direction of motion of the images with respect to the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' If the image moves in the same direction as the source then it is said to have positive parity, and if it moves in the opposite direction it is said to have negative parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Note that in either the over- or under- dense case, the positive parity images have much smaller group delay than the negative parity images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' As a result, the orientation of the characteristic hockey-stick shape of the millisecond feature is flipped between the convergent and divergent lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Inspecting the data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 9 visually, the millisecond feature appears to be closer to the convergent (under-dense) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' However, since we have not attempted to precisely fit the data, it is not possible to make any strong inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' An additional possibility that the doubly catastrophic frame- work opens up, is that if pulsar scintillation is caused by 𝐴2 folds and ESEs are caused by 𝐴3 cusps by the same sheet structures in the ISM as viewed under projection, then many observations of these phenomena will allow us to take advantage of the rich mathemat- ics of catastrophe theory to probe the turbulent ISM on scales that are inaccessible to simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, with many scintillation and ESE observations, we can effectively generate a sky-map of caus- tic networks in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Such a map may be a powerful tool for studying the physics of the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 6 CONCLUSION In this paper, we have presented a model based on a simple applica- tion of catastrophe theory to thin plasma sheets to explain extreme scattering events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' That is, we propose that several aspects of ESE observations can be explained using lens potential with an 𝐴3 cusp profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' This is an extension of previous work (Pen & Levin 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Simard & Pen 2018) suggesting that 𝐴2 folds arising from corru- gated plasma sheets may explain pulsar scintillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We call this application of catastrophe theory to the lens potential “doubly catas- MNRAS 000, 1–10 (2022) 10 Jow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' trophic" lensing, as catastrophes also generically appear in the light curves of lensed sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The doubly catastrophic framework is well-motivated for sev- eral reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Firstly, the past decade of pulsar scintillation observa- tions suggest the ubiquity of thin plasma sheets in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Since lensing is well-described by an effective projected density perpen- dicular to the line of sight, and since catastrophes generically arise when thin sheets are viewed under projection, 𝐴2 and 𝐴3 lenses (in addition to higher order catastrophes which have not considered here) should naturally arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We argue that these catastrophic lens potentials should exist in the ISM, whether or not they are abundant enough to explain all scintillation or ESE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Secondly, recent work on the physics of the turbulent ISM through MHD sim- ulations suggest that corrugated plasma sheets of the kind we con- sider are physically well-motivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Lastly, the doubly catastrophic framework has several desirable theoretical features: it provides a universal framework that describes both scintillation and ESEs as aspects of the same phenomenon, and the application of catastrophe theory means that the lens potentials are generic and well-described by a small number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' In this work, we have described the features of the simplest 𝐴3 lens and have argued that it can explain many of the qualitative features of ESE observations, including the frequency structure of ESEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' The inability to account for features of ESE light curves at high frequencies has been a roadblock for thin sheet models of these events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We argue that the 𝐴3 lens overcomes this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' We also argue that the 𝐴3 lens provides a natural explanation for features seen in the lensing of PSR B0834+06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' DATA AVAILABILITY No new data were generated or analysed in support of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We receive support from Ontario Research Fund—research Ex- cellence Program (ORF-RE), Natural Sciences and Engineering Research Council of Canada (NSERC) [funding reference number RGPIN-2019-067, CRD 523638-18, 555585-20], Canadian Insti- tute for Advanced Research (CIFAR), Canadian Foundation for In- novation (CFI), the National Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' 11929301), Thoth Technology Inc, Alexander von Humboldt Foundation, and the Ministry of Science and Technology(MOST) of Taiwan(110-2112-M-001-071-MY3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' Computations were per- formed on the SOSCIP Consortium’s [Blue Gene/Q, Cloud Data Analytics, Agile and/or Large Memory System] computing plat- form(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=' SOSCIP is funded by the Federal Economic Development Agency of Southern Ontario, the Province of Ontario, IBM Canada Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} +page_content=', Ontario Centres of Excellence, Mitacs and 15 Ontario aca- demic member institutions.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE_T4oBgHgl3EQf3ByR/content/2301.08344v1.pdf'} diff --git a/h9AzT4oBgHgl3EQfbPwA/vector_store/index.pkl b/h9AzT4oBgHgl3EQfbPwA/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..5504160369c18e43316ad8c387c12307aebef66b --- /dev/null +++ b/h9AzT4oBgHgl3EQfbPwA/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:78184ef6d1697971f24a030db5de4eec8c53658c82b2345d250d206fc52340b1 +size 151721 diff --git a/htE1T4oBgHgl3EQfMwMt/content/tmp_files/2301.02992v1.pdf.txt b/htE1T4oBgHgl3EQfMwMt/content/tmp_files/2301.02992v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..57ed6b1cd17edb6c05339ee9aa33f04752216e3b --- /dev/null +++ b/htE1T4oBgHgl3EQfMwMt/content/tmp_files/2301.02992v1.pdf.txt @@ -0,0 +1,2543 @@ +MATHEMATICS OF COMPUTATION +Volume 00, Number 0, Pages 000–000 +S 0025-5718(XX)0000-0 +ERROR ESTIMATES OF THE TIME-SPLITTING METHODS +FOR THE NONLINEAR SCHR¨ODINGER EQUATION WITH +SEMI-SMOOTH NONLINEARITY +WEIZHU BAO AND CHUSHAN WANG +Abstract. We establish error bounds of the Lie-Trotter time-splitting sine +pseudospectral method for the nonlinear Schr¨odinger equation (NLSE) with +semi-smooth nonlinearity f(ρ) = ρσ, where ρ = |ψ|2 is the density with ψ +the wave function and σ > 0 is the exponent of the semi-smooth nonlinearity. +Under the assumption of H2-solution of the NLSE, we prove error bounds +at O(τ +1 +2 +σ + h1+2σ) and O(τ + h2) in L2-norm for 0 < σ ≤ +1 +2 and σ ≥ +1 +2 , respectively, and an error bound at O(τ +1 +2 + h) in H1-norm for σ ≥ +1 +2 , +where h and τ are the mesh size and time step size, respectively. In addition, +when 1 +2 < σ < 1 and under the assumption of H3-solution of the NLSE, we +show an error bound at O(τ σ + h2σ) in H1-norm. Two key ingredients are +adopted in our proof: one is to adopt an unconditional L2-stability of the +numerical flow in order to avoid an a priori estimate of the numerical solution +for the case of 0 < σ ≤ +1 +2 , and to establish an l∞-conditional H1-stability +to obtain the l∞-bound of the numerical solution by using the mathematical +induction and the error estimates for the case of σ ≥ 1 +2 ; and the other one is +to introduce a regularization technique to avoid the singularity of the semi- +smooth nonlinearity in obtaining improved local truncation errors. +Finally, +numerical results are reported to demonstrate our error bounds. +1. Introduction +In this paper, we consider the following nonlinear Schr¨odinger equation (NLSE) +(1.1) i∂tψ(x, t) = −∆ψ(x, t)+V (x)ψ(x, t)+f(|ψ(x, t)|2)ψ(x, t), +x ∈ Ω, +t > 0, +with the initial data +(1.2) +ψ(x, 0) = ψ0(x), +x ∈ Ω, +and the homogeneous Dirichlet boundary condition +(1.3) +ψ(x, t) = 0, +x ∈ ∂Ω, +t ≥ 0; +2020 Mathematics Subject Classification. Primary 35Q55, 65M15, 65M70, 81Q05. +Key words and phrases. nonlinear Schr¨odinger equation, semi-smooth nonlinearity, time- +splitting pseudospectral method, error estimate, local regularization. +This work was partially supported by the Ministry of Education of Singapore grant MOE- +000357-00 (W. Bao). +©XXXX American Mathematical Society +1 +arXiv:2301.02992v1 [math.NA] 8 Jan 2023 + +2 +W. BAO AND C. WANG +where t is time, x ∈ Rd (d = 1, 2, 3) is the spatial coordinate, ψ := ψ(x, t) is a +complex-valued wave function, V := V (x) : Ω → R is a time-independent real- +valued potential. Here Ω = Πd +i=1(ai, bi) ⊂ Rd is a bounded domain and the nonlin- +earity is given as +(1.4) +f(ρ) = βρσ, +ρ := |ψ|2 ≥ 0, +where β ∈ R is a given constant and σ > 0 is the exponent of the nonlinearity. The +NLSE (1.1) conserves the mass +(1.5) +M(ψ(·, t)) = +� +Ω +|ψ(x, t)|2dx ≡ M(ψ0), +t ≥ 0, +and the energy +(1.6) +E(ψ(·, t)) = +� +Ω +� +|∇ψ(x, t)|2 + V (x)|ψ(x, t)|2 + F(|ψ(x, t)|2) +� +dx +≡ E(ψ0), +t ≥ 0, +where the interaction energy density F(ρ) is given as +(1.7) +F(ρ) = +� ρ +0 +f(s)ds = +β +σ + 1ρσ+1, +ρ ≥ 0. +When σ = 1 in (1.4), i.e. f(ρ) = βρ and F(ρ) = β +2 ρ2, (1.1) collapses to the well- +known nonlinear Schr¨odinger equation with cubic nonlinearity (or smooth nonlin- +earity) or the Gross-Pitaevskii equation (GPE), which has been widely adopted for +modeling and simulation in quantum mechanics, nonlinear optics and Bose-Einstein +condensation [6, 22, 43]. Arising from different physics applications, semi-smooth +nonlinearity is introduced in the NLSE (1.1), i.e. +σ is taken as a non-integer +in (1.4). +Typical examples include, in the Schr¨odinger-Poisson-Xα model with +f(ρ) = −αρ1/d(α > 0) [13, 15], i.e. σ = 1 +3 and σ = 1 +2 in three dimensions (3D) and +two dimensions (2D), respectively; in the LHY correction (a next-order correction +of the ground state energy proposed by Lee, Huang and Yang in 1957 [31]) for a +beyond-mean-field term which is widely adopted in modeling and simulation for +quantum droplets [28, 16, 4, 38, 26] with f(ρ) = ρ3/2 in 3D, i.e. σ = 3 +2, f(ρ) = √ρ +in one dimension (1D), i.e. σ = 1 +2, and f(ρ) = ρ ln ρ in 2D; and in the mean field +model for Bose-Fermi mixture [23, 17], f(ρ) = ρ2/3, i.e. σ = 2 +3. For all the afore- +mentioned nonlinearity (actually for σ > 0), the NLSE (1.1) is well-posed in H2 +[29, 18]. However, to our best knowledge, there is no guarantee of higher regularity +to be propagated due to the low regularity of the semi-smooth nonlinearity, which +is similar to the case of the logarithmic Schr¨odinger equation (LogSE) [9, 10, 11]. +In fact, similar to the LogSE, the low regularity of the solution of the NLSE with +semi-smooth nonlinearity is mainly due to the low regularity of the nonlinearity. +For the cubic NLSE, i.e. σ = 1, many accurate and efficient numerical meth- +ods have been proposed and analyzed in the last two decades, including the finite +difference method [1, 7, 6, 3], the exponential wave integrator [8, 25, 19], the time- +splitting method [12, 14, 32, 21, 6, 33, 3], the finite element method [2, 41, 44, 45, 24], +etc. Recently, new low regularity integrators or non-resonance Fourier integrators +are designed and analyzed for the cubic NLSE with low regularity initial data since +the important work by Ostermann and Schratz [35], followed by [30, 34, 40, 37, 36] +and references therein for different dispersive partial differential equations. For all + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +3 +these numerical methods, optimal error bounds were rigorously established under +different regularity assumptions of the cubic NLSE. +Most numerical methods for the cubic NLSE can be extended straightforwardly +to solve the NLSE (1.1) with non-integer σ > 0, e.g. semi-smooth nonlinearity +with 0 < σ < 1. +However, due to the low regularity of solution of the NLSE +(1.1) with semi-smooth nonlinearity and the low regularity of the semi-smooth +nonlinearity (1.4) in the NLSE (1.1) which causes order reduction in local truncation +errors, error analysis for different numerical methods for (1.1) with non-integer +σ > 0 is a very subtle and challenging question! For example, first order temporal +convergence of the finite difference method requires boundedness of the second- +order time derivative, which roughly requires the exact solution to be in H4, which +is beyond the regularity property of the NLSE (1.1) with semi-smooth nonlinearity. +In fact, based on our numerical experiments with a smooth initial datum ψ0(x) = +xe−x2/2, it indicates that ψ(·, t) ̸∈ H4 for t > 0 and σ small! Since the time-splitting +methods usually need lower regularity requirements on the exact solution than the +finite difference methods, in this work, we consider the time-splitting method and +in particular the first-order Lie-Trotter splitting method due to the low regularity +of the semi-smooth nonlinearity and the low regularity of the exact solution of the +(1.1). +Error estimates of the time-splitting methods with different orders for the cubic +NLSE i.e. σ = 1 have been well understood and we refer the readers to [32, 21, 33, 3] +and therein. However, for the NLSE with non-integer σ, only limited results are +established for the filtered Lie-Trotter splitting scheme which requires a coupling +constraint between τ and h at τ = O(h2). In [27], first order convergence in L2- +norm is established for H2-solution and σ ≥ 1/2. Then generalized in [20], half +order convergence in L2-norm is established for H1-solution and σ > 0. These +convergence rates are optimal with respect to the regularity assumptions on the +exact solution. However, there are still some questions related to error estimates +to be addressed: (i) it is unclear whether higher convergence order can be obtained +for H2-solution when 0 < σ < 1/2; (ii) their results are established for the filtered +Lie-Trotter scheme, which is a semi-discretization scheme with a specific coupling +constraint between τ and h and it loses mass conservation and time symmetric +property in the discretizaed level; and (iii) there is no optimal error estimate in +H1-norm, which is the natural norm of the NLSE. +The main aim of this paper is to establish error estimates for the time-splitting +sine pseudospectral (TSSP) method (2.13) for the NLSE (1.1) with semi-smooth +nonlinearity. We remark here that the TSSP is a full-discretization scheme and +it preserves many good properties of the original NLSE in the discretized level, +including mass conservation and time symmetric as well as dispersion relation. +When 0 < σ ≤ 1 +2, under the assumption of H2-solution of the NLSE, we prove error +bounds at O(τ +1 +2 +σ + h1+2σ) in L2-norm without any coupling condition between +the time step size τ and the mesh size h, which fills the gap between the results in +[27, 20]. When σ ≥ 1/2, under the assumption of H2-solution again, we prove error +bounds at O(τ + h2) and O(τ +1 +2 + h) in L2-norm and H1-norm, respectively, with +a very mild coupling condition between τ and h, which generalize the result in [27] +to mass-conservative full discretization scheme. In addition, when 1 +2 < σ < 1 and +under the assumption of H3-solution, we show a new error bound at O(τ σ + h2σ) +in H1-norm. + +4 +W. BAO AND C. WANG +The rest of the paper is organized as follows. In Section 2, we present the time- +splitting sine pseudospectral (TSSP) method, introduce a local regularization for +the semi-smooth nonlinearity to be used for obtaining improved local truncation +errors and state our main results. Section 3 is devoted to error estimates of the +TSSP method for 0 < σ ≤ 1/2 and Section 4 is devoted to error estimates for +σ ≥ 1/2. Numerical results are reported in Section 5 to confirm the error estimates. +Finally some conclusions are drawn in Section 6. Throughout the paper, we adopt +the standard Sobolev spaces as well as the corresponding norms, and denote by C +a generic positive constant independent of the mesh size h, time step τ, and by +C(c) a generic positive constant depending on c. The notation A ≲ B is used to +represent that there exists a generic constant C > 0, such that |A| ≤ CB. +2. Numerical methods and main results +2.1. The TSSP method. We shall use the Lie-Trotter splitting method for the +temporal discretization and use the sine pseudospectral method for the spatial +discretization. The operator splitting technique is based on the decomposition of +the flow of (1.1) +(2.1) +∂tψ = A(ψ) + B(ψ), +where +(2.2) +A(ψ) = i∆ψ, +B(ψ) = B1(ψ) + B2(ψ) := −iV ψ − if(|ψ|2)ψ, +into two sub-problems. The first one is +(2.3) +�∂tψ(x, t) = A(ψ) = i∆ψ(x, t), +x ∈ Ω, +t > 0, +ψ(x, 0) = ψ0(x), +x ∈ Ω, +which can be formally integrated exactly in time as +(2.4) +ψ(·, t) = eit∆ψ0(·), +t ≥ 0. +The second one is to solve +(2.5) +� +∂tψ(x, t) = B(ψ) = −iV (x)ψ(x, t) − if(|ψ(x, t)|2)ψ(x, t), +t > 0, +ψ(x, 0) = ψ0(x), +x ∈ Ω, +which, by using the fact |ψ(x, t)| = |ψ0(x)| for t ≥ 0, can be integrated exactly in +time as +(2.6) +ψ(x, t) = Φt +B(ψ0) := e−itV (x)Φt +B2(ψ0(x)), +x ∈ Ω, +t ≥ 0, +where +(2.7) +Φt +B2(z) = ze−itf(|z|2), +z ∈ C, +t ≥ 0. +Choose a time step size τ > 0, denote time steps as tk = kτ for k = 0, 1, ..., and +let ψ[k] := ψ[k](x) be the approximation of ψ(x, tk) for k ≥ 0. Then a first order +semi-discretization of the NLSE (1.1) via the Lie-Trotter splitting is given as: +(2.8) +ψ[k+1] = eiτ∆Φτ +B(ψ[k]), +with ψ[0](x) = ψ0(x) for x ∈ Ω. Then we discretize (2.8) in space by the sine +pseudospectral method to obtain a full discretization for the NLSE (1.1). +For +simplicity of notations, here we only present the spatial discretization in 1D (taking +Ω = (a, b)), and the generalization to higher dimensions is straightforward. Choose + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +5 +a mesh size h = (b − a)/N with N being a positive integer and denote grid points +as +xj = a + jh, +j = 0, 1, · · · , N. +Define the index sets +TN = {1, 2, · · · , N − 1}, +T 0 +N = {0, 1, · · · , N}, +and denote +XN = span {sin(µl(x − a)) : l ∈ TN} , +µl = +πl +b − a, +(2.9) +YN = +� +v = (v0, v1, · · · , vN)T ∈ CN+1 : v0 = vN = 0 +� +. +(2.10) +We define the lp(1 ≤ p ≤ ∞) norm on YN as +∥v∥lp = +� +�h +N−1 +� +j=0 +|vj|p +� +� +1 +p +, +1 ≤ p < ∞, +∥v∥l∞ = +max +0≤j≤N−1 |vj|, +v ∈ YN. +We shall sometimes identify a function φ(·) ∈ C0(Ω) with a vector φ = (φ0, φ1, · · · , +φN)T ∈ YN with φj = φ(xj) and then the discrete norm ∥ · ∥lp can also be defined +on XN. For v ∈ YN, we define the forward finite difference operator as +(δ+ +x v)j = δ+ +x vj = vj+1 − vj +h +, +0 ≤ j ≤ N − 1. +Let PN : L2(Ω) → XN be the standard L2 projection onto XN and IN : YN → XN +be the standard sine interpolation operator as +(2.11) +(PNv)(x) = +� +l∈TN +�vl sin(µl(x − a)), +(INw)(x) = +� +l∈TN +�wl sin(µl(x − a)), +x ∈ Ω = [a, b], +where v ∈ L2(Ω), w ∈ YN, and +(2.12) +�vl = +2 +b − a +� b +a +v(x) sin(µl(x − a))dx, +�wl = 2 +N +� +j∈TN +wj sin(jπl/N), +l ∈ TN. +Let ψk +j be the numerical approximations of ψ(xj, tk) for j ∈ T 0 +N and k ≥ 0, and +denote ψk := (ψk +0, ψk +1, . . . , ψk +N)T ∈ YN. Then the time-splitting sine pseudospectral +(TSSP) method for discretizing the NLSE (1.1) can be given for k ≥ 0 as +(2.13) +ψ(1) +j += e−iτ(V (xj)+f(|ψn +j |2))ψk +j , +ψk+1 +j += +� +l∈TN +e−iτµ2 +l � +(ψ(1))l sin(µl(xj − a)), +j ∈ T 0 +N +where ψ0 +j = ψ0(xj) for j ∈ T 0 +N. +Let Φτ : XN → XN be the numerical integrator defined as +(2.14) +Φτ(φ) = eiτ∆INΦτ +B(φ), +φ ∈ XN, + +6 +W. BAO AND C. WANG +where Φτ +B is defined in (2.6). Then one has +(2.15) +INψk+1 = Φτ(INψk), +k ≥ 0, +INψ0 = INψ0. +Remark 2.1. In applications, the NLSE (1.1) can also be discretized by the Lie- +Trotter splitting via a different order as: +(2.16) +ψ[k+1] = Φτ +B(eiτ∆ψ[k]), +k ≥ 0. +Then a full discretization can be obtained straightforward by using the sine pseu- +dospectral method in space. +2.2. A local regularization for f(ρ) = βρσ. When 0 < σ < 1 in (1.4), f(ρ) +is a semi-smooth function and it is not differentiable at ρ = 0. +Following the +regularization methods used in [11] for the logarithmic Schr¨odinger equation, we +regularize the semi-smooth nonlinearity f(ρ) only locally in a small region near +ρ = 0. Take 0 < ε ≪ 1 as a regularization parameter, we approximate f(ρ) locally +in the region {ρ < ε2} by a polynomial and leave it unchanged in {ρ ≥ ε2}, i.e. +(2.17) +fε(ρ) = +� +f(ρ), +ρ ≥ ε2 +ρQε(ρ), +0 ≤ ρ < ε2, +where Qε(ρ) is a polynomial with degree at most 3 such that +(2.18) +fε ∈ C3([0, ∞)). +Note that fε given by (2.17) is uniquely determined by the interpolation conditions +(2.18) and it satisfies fε(0) = f(0) = 0. Actually, the explicit formula of Qε(ρ) can +be given as +(2.19) +Qε(ρ) = βε2σ−2 +3 +� +k=0 +�k − σ +k +� � +1 − ρ +ε2 +�k +, +0 ≤ ρ < ε2. +In fact, fε ∈ C3([0, ∞)) can be regarded as a local regularization of the semi- +smooth nonlinearity f(ρ) ∈ C0([0, ∞)), which has much better regularity near +ρ = 0. For fε, we have the following estimates +Lemma 2.2. Assume 0 < σ < 1, we have +(2.20) +|fε(ρ)| + |ρf ′ +ε(ρ)| ≤ C1ρσ, +ρ ≥ 0, +(2.21) +|√ρf ′ +ε(ρ)| + |ρ +3 +2 f ′′ +ε (ρ)| ≤ C2 +� +� +� +� +� +1 +ε1−2σ , +0 ≤ σ ≤ 1 +2, +ρσ− 1 +2 , +1 +2 < σ < 1, +, +ρ ≥ 0, +and +(2.22) +|f ′ +ε(ρ)| + |ρf ′′ +ε (ρ)| + |ρ2f ′′′ +ε (ρ)| ≤ +C3 +ε2−2σ , +ρ ≥ 0, +where C1, C2 and C3 depend exclusively on σ and β. +Proof. When ρ ≥ ε2, by (2.17), we have f (k)(ρ) = f (k) +ε +(ρ) for 0 ≤ k ≤ 3, and +(2.20)–(2.22) follows immediately from f(ρ) = βρσ and 0 < ε < 1. +In the following, we assume that 0 ≤ ρ < ε2. From (2.19), we easily obtain that +(2.23) +|Q(k) +ε (ρ)| ≲ ε2σ−2−2k, +0 ≤ ρ < ε2, +0 ≤ k ≤ 3. + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +7 +From (2.17), using (2.23), one gets +(2.24) +|fε(ρ)| ≤ ρ|Qε(ρ)| ≲ ρε2σ−2 = ρσ � ρ +ε2 +�1−σ +≤ ρσ. +Similarly, one has +(2.25) +|ρf ′ +ε(ρ)| ≤ |ρ2Q′ +ε(ρ)| + |ρQε(ρ)| ≲ +� ρ +ε2 + 1 +� +ρε2σ−2 ≤ 2ρσ, +which proves (2.20). +From (2.17), using (2.23), one gets, when 0 < σ ≤ 1/2, +|√ρf ′ +ε(ρ)| ≲ ρ +1 +2 � +|Qε(ρ)| + ρ|Qε +′(ρ)| +� +≤ ε +� +ε2σ−2 + ε2ε2σ−4� +≲ ε2σ−1, +and when 1/2 < σ < 1, +(2.26) |√ρf ′ +ε(ρ)| ≲ ρ +1 +2 � +|Qε(ρ)| + ρ|Qε +′(ρ)| +� +≲ ρ +1 +2 ε2σ−2 = ρσ− 1 +2 +� ρ +ε2 +�1−σ +≤ ρσ− 1 +2 . +The estimate of |ρ +3 +2 f ′′ +ε (ρ)| can be obtained similarly, which completes the proof of +(2.21). +For (2.22), using (2.23) again, one has +(2.27) +|f ′ +ε(ρ)| ≤ |Qε(ρ)| + ρ|Q′ +ε(ρ)| ≲ ε2σ−2 + ε2ε2σ−4 ≲ ε2σ−2. +The estimate of |ρf ′′ +ε (ρ)| and |ρ2f ′′′ +ε (ρ)| can be obtained similarly, which completes +the proof of (2.22). +□ +Corollary 2.3. Assume 0 < σ ≤ 1/2, we have +∥fε(|v|2)v∥L2 ≤ C1(∥v∥L∞)∥v∥L2, +v ∈ L∞(Ω), +(2.28) +∥fε(|v|2)v∥H1 ≤ C2(∥v∥L∞)∥v∥H1, +v ∈ H1(Ω) ∩ L∞(Ω), +(2.29) +∥fε(|v|2)v∥H2 ≤ C3 (∥v∥H2) +ε1−2σ +, +v ∈ H2(Ω). +(2.30) +Assume 0 < σ < 1, we have +(2.31) +∥fε(|v|2)v∥H3 ≤ C4 (∥v∥H3) +ε2−2σ +, +v ∈ H3(Ω). +Proof. By (2.20), one has +(2.32) +∥fε(|v|2)v∥L2 ≤ ∥fε(|v|2)∥L∞∥v∥L2 ≲ ∥v∥2σ +L∞∥v∥L2, +which proves (2.28). +By direct calculation, using (2.20), one gets +��∇ +� +fε(|v|2)v +��� +L2 += +��fε(|v|2)∇v + f ′ +ε(|v|2)v(v∇v + v∇v) +�� +L2 +≤ +� +∥fε(|v|2)∥L∞ + ∥f ′ +ε(|v|2)v2∥L∞ + ∥f ′ +ε(|v|2)|v|2∥L∞� +∥∇v∥L2 +≲ +∥v∥2σ +L∞∥v∥H1, +(2.33) +which shows (2.29). +To show (2.30), we note that +∂jk(fε(|v|2)v) += +∂j +� +(fε(|v|2) + f ′ +ε(|v|2)|v|2)∂kv + f ′ +ε(|v|2)v2∂kv +� += +(2f ′ +ε(|v|2) + f ′′ +ε (|v|2)|v|2)∂j|v|2∂kv + (fε(|v|2) + f ′ +ε(|v|2)|v|2)∂jkv ++f ′′ +ε (|v|2)v2∂j|v|2∂kv + 2f ′ +ε(|v|2)v∂jv∂kv + f ′ +ε(|v|2)v2∂jkv, +(2.34) +where ∂j = ∂xj and ∂jk = ∂xj∂xk for 1 ≤ j, k ≤ d. Here we adopt the notations x = +x1 (or x) when d = 1, x = (x1, x2)T (or (x, y)T ) when d = 2, and x = (x1, x2, x3)T + +8 +W. BAO AND C. WANG +(or (x, y, z)T ) when d = 3. From Lemma 2.2, using (2.20) and (2.21) and noting +that |∂j|v|2| ≤ 2|v| |∂jv|, one gets +(2.35) +��∂jk(fε(|v|2)v) +�� ≲ +� +f ′ +ε(|v|2)|v| + f ′′ +ε (|v|2)|v|3� +|∂jv| |∂kv| ++ +� +fε(|v|2) + f ′ +ε(|v|2)|v|2� +|∂jkv| +≲ |∂jv| |∂kv| +ε1−2σ ++ |v|2σ|∂jkv|, +which, by using H¨older’s inequality and Sobolev embedding, yields +(2.36) +∥∂jk(fε(|v|2)v)∥L2 ≲ ∥∂jv∥L4∥∂kv∥L4 +ε1−2σ ++ ∥v∥2σ +L∞∥∂jkv∥L2 ≤ C(∥v∥H2) +ε1−2σ +, +which implies (2.30). +Following Lemma 2.2, noting (2.35) and (2.36) and using (2.22), we can similarly +obtain (2.31) and the details are omitted here for brevity. +□ +Lemma 2.4. Assume 0 < σ < 1, we have +|f(ρ) − fε(ρ)| ≤ Cε2σ1ρ<ε2, +ρ ≥ 0. +Proof. Recalling (2.17), we have +(2.37) +|f(ρ) − fε(ρ)| = 0, +ρ ≥ ε2, +and, by (1.4) and (2.20), +(2.38) +|f(ρ) − fε(ρ)| ≤ |f(ρ)| + |fε(ρ)| ≲ ρσ ≤ ε2σ, +0 ≤ ρ < ε2, +which completes the proof. +□ +2.3. Main results. Let Tmax be the maximal existing time for the solution of the +NLSE (1.1) with (1.2) and (1.3) and take 0 < T < Tmax be a fixed time. Based on +the known existence and regularity results in [29, 18] for the solution of (1.1), we +make the assumption that the solution ψ satisfies ψ ∈ C([0, T]; H1 +0(Ω) ∩ H2(Ω)) ∩ +C1([0, T]; L2(Ω)) such that +(A) +∥ψ∥L∞([0,T ];H2) + ∥∂tψ∥L∞([0,T ];L2) ≲ 1. +Define +(2.39) +M2 := max +� +∥ψ∥L∞([0,T ];H2), ∥ψ∥L∞([0,T ];L∞), ∥V ∥H2� +, +and assume the following coupling condition between τ and h < 1 +(B) +τ ≲ +� +� +� +� +� +� +� +1, +d = 1, +1 +| ln h|2 , +d = 2, +h, +d = 3. +For the TSSP method (2.13), we can establish the following error estimates. +Theorem 2.5. When 0 < σ ≤ 1/2, assume V ∈ H2(Ω) and under the assumption +(A), for 0 < τ < 1 and 0 < h < 1, we have +(2.40) +∥ψ(·, tk) − INψk∥L2 ≲ τ 1/2+σ + h1+2σ, +0 ≤ k ≤ T +τ . + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +9 +Corollary 2.6. When d = 1 and 0 < σ ≤ 1/2, under the following much weaker +assumption +V ∈ H1(Ω), +ψ ∈ C([0, T]; H1 +0(Ω)), +we have for 0 < τ < 1 and 0 < h < 1, +(2.41) +∥ψ(·, tk) − INψk∥L2 ≲ τ 1/2 + h, +0 ≤ k ≤ T +τ . +Theorem 2.7. When σ ≥ 1/2, assume V ∈ H2(Ω) ∩ W 1,∞(Ω) and under the +assumption (A), there exist τ0 > 0 and h0 > 0 sufficiently small and depending +on M2, ∥V ∥W 1,∞ and T such that for τ ≤ τ0 and h ≤ h0 satisfying the coupling +condition (B), we have +(2.42) +∥ψ(·, tk) − INψk∥L2 ≲ τ + h2, +∥ψk∥l∞ ≤ 1 + M2 +∥ψ(·, tk) − INψk∥H1 ≲ τ +1 +2 + h, +0 ≤ k ≤ T +τ . +Moreover, when 1/2 < σ < 1, under the additional assumptions that V ∈ H3(Ω), +∇V ∈ H1 +0(Ω) and ψ ∈ C([0, T]; H3 +∗(Ω)) ∩ C1([0, T]; H1(Ω)), we have +(2.43) +∥ψ(·, tk) − INψk∥H1 ≲ τ σ + h2σ, +0 ≤ k ≤ T +τ , +where H3 +∗(Ω) := {φ ∈ H3(Ω) | φ(x)|∂Ω = ∆φ(x)|∂Ω = 0}. +Remark 2.8. When σ ≥ 1, under the same assumptions as those for (2.43), one can +obtain the following error bound for the TSSP method (2.13) as +∥ψ(·, tk) − INψk∥H1 ≲ τ + h2, +0 ≤ k ≤ T +τ . +3. Proof of Theorem 2.5 for the case 0 < σ ≤ 1/2 +Throughout this section, we assume that V ∈ H2(Ω), 0 < σ ≤ 1/2 and the +assumption (A). +3.1. Some estimates for the operator B. For the operator B defined in (2.2), +we have +Lemma 3.1. Let v ∈ H1(Ω) such that ∥v∥L∞ ≤ M, then for σ > 0, we have +∥B(v)∥L2 ≤ C1(M, ∥V ∥L∞)∥v∥L2, +(3.1) +∥B(v)∥H1 ≤ ∥v∥H1 +� C2(M, ∥V ∥H1), +d = 1, +C2(M, ∥V ∥W 1,4), +d = 2, 3. +(3.2) +Proof. From the definition of B in (2.2), we have +(3.3) +∥B(v)∥L2 ≤ ∥V ∥L∞∥v∥L2 + C(∥v∥L∞)∥v∥L2, +which implies (3.1). +Introduce a continuous function G : C → C as +(3.4) +G(z) = +� +f ′(|z|2)z2 = βσ|z|2σ−2z2, +z ̸= 0, +0, +z = 0, +z ∈ C, +and identify f ′(|z|2)|z|2 with σf(|z|2). Note that +(3.5) +f(|z|2) + |G(z)| ≲ |z|2σ, +z ∈ C, +σ > 0. + +10 +W. BAO AND C. WANG +Direct calculation yields +(3.6) +∇B(v) = −i +� +V ∇v + v∇V + f(|v|2)∇v + f ′(|v|2)v(v∇v + v∇v) +� += −i +� +V ∇v + v∇V + (1 + σ)f(|v|2)∇v + G(v)∇v +� +, +where G(v)(x) := G(v(x)) for x ∈ Ω. From (3.6), using H¨older’s inequality and +noticing (3.5), we obtain +(3.7) +∥∇B(v)∥L2 ≲ ∥V ∥L∞∥∇v∥L2 + ∥v∥2σ +L∞∥∇v∥L2 + +� ∥v∥L∞∥∇V ∥L2, +d = 1, +∥v∥L4∥∇V ∥L4, +d = 2, 3, , +where different estimates are used for v∇V for d = 1 and d = 2, 3. Thus we have, +by Sobolev embedding, +∥∇B(v)∥L2 ≤ C(∥v∥L∞)∥v∥H1 + ∥v∥H1 +� +C(∥V ∥H1), +d = 1, +C(∥V ∥W 1,4), +d = 2, 3, +which completes the proof. +□ +Lemma 3.2. Let v, w ∈ L∞(Ω) such that ∥v∥L∞ ≤ M and ∥w∥L∞ ≤ M. For any +σ > 0, we have +∥B(v) − B(w)∥L2 ≤ C(M, ∥V ∥L∞)∥v − w∥L2. +Proof. From (2.2), we have +(3.8) +∥B(v) − B(w)∥L2 ≤ ∥V ∥L∞∥v − w∥L2 + ∥f(|v|2)v − f(|w|2)w∥L2. +For any z1, z2 ∈ C, let zθ = (1 − θ)z1 + θz2 and let γ(θ) = f(|zθ|2)zθ for 0 ≤ θ ≤ 1, +we have +(3.9) +f(|z2|2)z2 − f(|z1|2)z1 = γ(1) − γ(0) = +� 1 +0 +γ′(θ)dθ. +Recalling (1.4) and (3.4), we have +(3.10) +γ′(θ) = (1 + σ)f(|zθ|2)(z2 − z1) + G(zθ)(z2 − z1). +Plugging (3.10) into (3.9), noticing (3.5), we have +(3.11) +|f(|z1|2)z1 − f(|z2|2)z2| ≤ sup +0≤θ≤1 +|γ′(θ)| ≲ max{|z1|, |z2|}2σ|z1 − z2|. +Thus we have +(3.12) +∥f(|v|2)v − f(|w|2)w∥L2 ≤ C(max{∥v∥L∞, ∥v∥L∞})∥v − w∥L2, +which combined with (3.8) completes the proof. +□ +Let dB(·)[·] be the Gˆateaux derivative defined as +(3.13) +dB(v)[w] := lim +ε→0 +B(v + εw) − B(v) +ε +, +then we have +Lemma 3.3. Let v ∈ L∞(Ω) such that ∥v∥L∞ ≤ M and w ∈ L2(Ω). When σ > 0, +we have +∥dB(v)[w]∥L2 ≤ C(M, ∥V ∥L∞)∥w∥L2. + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +11 +Proof. Plugging (2.2) into (3.13), we obtain +(3.14) +dB(v)[w] = −iV w + dB2(v)[w] += −i +� +V w + (1 + σ)f(|v|2)w + G(v)w +� +, +where G is defined in (3.4). From (3.14), noting (3.5), we have +∥dB(v)[w]∥L2 ≤ ∥V ∥L∞∥w∥L2 + C(∥v∥L∞)∥w∥L2, +which concludes the proof. +□ +Lemma 3.4. When 0 < σ ≤ 1/2, we have +|Φτ +B2(z1) − Φτ +B2(z2)| ≤ (1 + Cτ) |z1 − z2|, +∀z1, z2 ∈ C, +where C = 2σ|β| min{|z1|, |z2|}2σ. +Proof. Recall that Φτ +B2(z) = ze−iτf(|z|2) in (2.7). Without loss of generality, we +assume that |z2| ≤ |z1|. +If z2 = 0, the conclusion follows immediately. +In the +following, we assume that z2 ̸= 0. Then, by noting that |1 − eiθ| ≤ |θ| for all θ ∈ R, +we have +(3.15) +|Φτ +B2(z1) − Φτ +B2(z2)| = |z1e−iτf(|z1|2) − z2e−iτf(|z2|2)| +≤ |z1 − z2| + |z2| +���1 − e−iτ(f(|z1|2)−f(|z2|2))��� +≤ |z1 − z2| + τ|z2| +��f(|z1|2) − f(|z2|2) +�� . +When 0 < σ ≤ 1/2, since 0 < |z2| ≤ |z1|, by the mean value theorem and the +definition of f in (1.4), we have +(3.16) +��f(|z1|2) − f(|z2|2) +�� = |β| +��|z1|2σ − |z2|2σ�� +≤ +2σ|β||z1 − z2| +min{|z1|, |z2|}1−2σ = 2σ|β||z1 − z2| +|z2|1−2σ . +Plugging (3.16) into (3.15), we get the desired result immediately. +□ +3.2. Local truncation error. +Lemma 3.5. Let φ ∈ XN such that ∥φ∥H2 ≤ M and let 0 < τ < 1 and 0 < h < 1. +Assume that V ∈ H2(Ω). When 0 < σ ≤ 1/2, we have +∥(I − eiτ∆)PNB(φ)∥L2 ≤ C1(M, ∥V ∥H2)τ 1/2+σ, +(3.17) +∥INB(φ) − PNB(φ)∥L2 ≤ C2(M, ∥V ∥H2)h1+2σ. +(3.18) +Proof. Recalling the well-known facts that (see, e.g., [42, 8, 10]) +∥v − PNv∥L2 ≲ h2|v|H2, +∥INv − PNv∥L2 ≲ h2|v|H2, +(3.19) +∥v − eit∆v∥L2 ≲ t∥v∥H2, +v ∈ H1 +0(Ω) ∩ H2(Ω), +(3.20) +noting that H2(Ω) is an algebra when 1 ≤ d ≤ 3, we have +(3.21) +∥(I − eiτ∆)(V φ)∥L2 ≲ τ∥V ∥H2∥φ∥H2, +∥(IN − PN)(V φ)∥L2 ≲ h2∥V ∥H2∥φ∥H2. +According to (2.2), it remains to show (3.17) and (3.18) with f(|φ|2)φ replacing +B(φ). Using the regularized function fε defined in (2.17) with 0 < ε ≪ 1 and the + +12 +W. BAO AND C. WANG +triangle inequality, we have +(3.22) +∥(I − eiτ∆)(f(|φ|2)φ)∥L2 +≤ ∥(I − eiτ∆)(f(|φ|2)φ − fε(|φ|2)φ)∥L2 + ∥(I − eiτ∆)(fε(|φ|2)φ)∥L2. +From (3.22), using ∥(I − eiτ∆)v∥L2 ≤ 2∥v∥L2 for the first term and (3.20) for the +second term, we have +(3.23) +∥(I − eiτ∆)(f(|φ|2)φ)∥L2 ≤ 2∥f(|φ|2)φ − fε(|φ|2)φ∥L2 + τ∥fε(|φ|2)φ∥H2. +By Lemma 2.4 and (2.30), we have +∥f(|φ|2)φ − fε(|φ|2)φ∥L2 ≲ ε2σ∥φ1|φ|<ε∥L2 ≤ |Ω| +1 +2 ε1+2σ, +(3.24) +∥fε(|φ|2)φ∥H2 ≤ C(M) +ε1−2σ . +(3.25) +Plugging (3.24) and (3.25) into (3.23), we have +∥(I − eiτ∆)(f(|φ|2)φ)∥L2 ≤ C(M) inf +0<ε<1 +� +ε1+2σ + +τ +ε1−2σ +� +≤ C(M)τ 1/2+σ, +which combined with (3.21) yields (3.17). +Then we shall prove (3.18). +Similar to (3.22) and (3.23), using the triangle +inequality, the L2-projection property of PN, (3.24), (3.19) and (2.30), we have +(3.26) +∥(IN − PN)(f(|φ|2)φ)∥L2 +≤ ∥(IN − PN)(f(|φ|2)φ − fε(|φ|2)φ)∥L2 + ∥(IN − PN)(fε(|φ|2)φ)∥L2 +≤ ∥IN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 + ∥PN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 ++ h2∥fε(|φ|2)φ∥H2 +≤ ∥IN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 + |Ω| +1 +2 ε1+2σ + h2 C(M) +ε1−2σ . +By Parseval’s identity, +(3.27) +∥INv∥L2 = +� +� +� +�h +N−1 +� +j=1 +|v(xj)|2 ≤ +� +� +� +�h +N−1 +� +j=1 +∥v∥2 +l∞ ≤ |Ω| +1 +2 ∥v∥l∞, +v ∈ C0(Ω), +which implies, by using Lemma 2.4 again, +(3.28) +∥IN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 ≤ |Ω| +1 +2 ∥(f(|φ|2) − fε(|φ|2))φ∥l∞ +≤ |Ω| +1 +2 ε2σ∥φ1|φ|<ε∥l∞ ≤ |Ω| +1 +2 ε1+2σ. +Plugging (3.28) into (3.26), we have +∥(IN − PN)(f(|φ|2)φ)∥L2 ≤ C(M) inf +0<ε<1 +� +ε1+2σ + +h2 +ε1−2σ +� +≤ C(M)h1+2σ, +which completes the proof. +□ +Then we are able to show the local truncation error of the TSSP method. +Proposition 3.6 (local truncation error). Assume that V ∈ H2 and under the +assumption (A), for 0 ≤ k ≤ T/τ − 1, we have +∥PNψ(·, tk+1) − Φτ(PNψ(·, tk))∥L2(Ω) ≤ C(M2)τ +� +τ +1 +2 +σ + h1+2σ� +. + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +13 +Proof. For the simplicity of notations, we define v(t) := ψ(·, tk + t) for 0 ≤ t ≤ τ +and v0 := v(0) = ψ(·, tk). By the Sobolev embedding theorem, noting that eit∆ +preserves the Hk-norm and PN doesn’t increase the Hk-norm for k ≥ 0, we have +∥eis∆v(t)∥L∞ ≲ ∥eis∆v(t)∥H2 = ∥v(t)∥H2 ≤ M2, +(3.29) +∥PNv(t)∥L∞ ≲ ∥PNv(t)∥H2 ≤ ∥v(t)∥H2 ≤ M2, +0 ≤ s, t ≤ τ. +(3.30) +It is well-known that (see [32, 11]) +(3.31) +ψ(tk+1) = eiτ∆v0 + +� τ +0 +ei(τ−s)∆B(eis∆v0)ds ++ +� τ +0 +� s +0 +ei(τ−s)∆dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))]dσds, +where dB(·)[·] is the Gˆateaux derivative defined in (3.13). Applying PN on both +sides of (3.31), one gets +PNψ(tk+1) += +eiτ∆PNv0 + +� τ +0 +ei(τ−s)∆PNB(eis∆v0)ds ++ +� τ +0 +� s +0 +ei(τ−s)∆PN +� +dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] +� +dσds. +(3.32) +From (2.6), recalling that v0 = ψ(tk), we have +(3.33) +Φτ(PNψ(tk)) = eiτ∆INΦτ +B(PNv0). +Applying the first-order Taylor expansion +(3.34) +Φτ +B(w) = w + τB(w) + τ 2 +� 1 +0 +(1 − θ)dB(Φθτ +B (w))[B(Φθτ +B (w))]dθ +for w = PNv0 and plugging it into (3.33), we have +Φτ(PNψ(tk)) += +eiτ∆PNv0 + τeiτ∆INB(PNv0) ++τ 2eiτ∆IN +�� 1 +0 +(1 − θ) +� +dB(Φθτ +B (PNv0))[B(Φθτ +B (PNv0))] +� +dθ +� +. +(3.35) +Subtracting (3.35) from (3.32), we have +(3.36) +PNψ(tk+1) − Φτ(PNψ(tk)) = e1 − e2 + e3, +where +e1 = +� τ +0 +� s +0 +ei(τ−s)∆PN +� +dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] +� +dσds, +(3.37) +e2 = τ 2eiτ∆IN +�� 1 +0 +(1 − θ) +� +dB(Φθτ +B (PNv0))[B(Φθτ +B (PNv0))] +� +dθ +� +, +(3.38) +e3 = +� τ +0 +ei(τ−s)∆PNB(eis∆v0)ds − τeiτ∆INB(PNv0). +(3.39) + +14 +W. BAO AND C. WANG +Next, we shall first estimate e1 and e2. Noticing the property of eit∆ and PN, +using Lemma 3.3 and (3.29), we have +(3.40) +���ei(τ−s)∆PN +� +dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] +���� +L2 +≤ ∥dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))]∥L2 +≤ C(∥V ∥L∞, ∥ei(s−σ)∆v(σ)∥L∞)∥ei(s−σ)∆B(v(σ))∥L2 +≤ C(M2)∥B(v(σ))∥L2. +From (3.37), using (3.40) and (3.1), we get +(3.41) +∥e1∥L2 ≤ +� τ +0 +� s +0 +���ei(τ−s)∆PN +� +dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] +���� +L2 dσds +≤ C(M2) +� τ +0 +� s +0 +∥B(v(σ))∥L2dσds ≤ C(M2)τ 2 max +0≤σ≤τ ∥B(v(σ))∥L2 +≤ C(M2)τ 2C(M2) max +0≤σ≤τ ∥v(σ)∥L2 ≤ C(M2)τ 2. +From (2.6) and (2.2), using (3.30), one gets, +(3.42) +∥Φθτ +B (PNv0)∥L∞ = ∥PNv0∥L∞ ≤ C(M2), +0 ≤ θ ≤ 1, +∥B(Φθτ +B (PNv0))∥L∞ ≤ C(∥V ∥L∞, ∥PNv0∥L∞)∥PNv0∥L∞ ≤ C(M2). +From (3.14), noticing (3.5), one easily gets +(3.43) +∥dB(w1)[w2]∥L∞ ≤ C(∥V ∥L∞, ∥w1∥L∞)∥w2∥L∞, +w1, w2 ∈ L∞(Ω), +which combined with (3.27) and (3.42), yields the estimate for e2 in (3.38) as +(3.44) +∥e2∥L2 ≤ τ 2 +����IN +�� 1 +0 +(1 − θ) +� +dB(Φθτ +B (PNv0))[B(Φθτ +B (PNv0))] +� +dθ +����� +L2 +≤ τ 2|Ω| +1 +2 max +0≤θ≤1 +��dB(Φθτ +B (PNv0))[B(Φθτ +B (PNv0))] +�� +l∞ +≤ τ 2|Ω| +1 +2 C +� +∥V ∥L∞, ∥Φθτ +B (PNv0)∥L∞� +∥B(Φθτ +B (PNv0))∥L∞ +≤ C(M2)τ 2. +Then we shall estimate e3 in (3.39), which can be written as +e3 = +� τ +0 +� +ei(τ−s)∆PNB(eis∆v0) − eiτ∆INB(PNv0) +� +ds, +which yields +(3.45) +∥e3∥L2 ≤ τ max +0≤s≤τ ∥ei(τ−s)∆PNB(eis∆v0) − eiτ∆INB(PNv0)∥L2. + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +15 +Using the standard property of eit∆ and PN, one gets +(3.46) +∥ei(τ−s)∆PNB(eis∆v0) − eiτ∆INB(PNv0)∥L2 += ∥PNB(eis∆v0) − eis∆INB(PNv0)∥L2 +≤ ∥PNB(eis∆v0) − PNB(v0)∥L2 + ∥PNB(v0) − PNB(PNv0)∥L2 ++ ∥PNB(PNv0) − eis∆PNB(PNv0)∥L2 ++ ∥eis∆PNB(PNv0) − eis∆INB(PNv0)∥L2 +≤ ∥B(eis∆v0) − B(v0)∥L2 + ∥B(φ) − B(PNv0)∥L2 ++ ∥(I − eis∆)PNB(PNv0)∥L2 + ∥(PN − IN)B(PNv0)∥L2 +=: ∥e1 +3∥L2 + ∥e2 +3∥L2 + ∥e3 +3∥L2 + ∥e4 +3∥L2. +For e1 +3 and e2 +3 in (3.46), using Lemma 3.2, recalling (3.19), (3.20), (3.29), and (3.30), +we obtain +(3.47) +∥e1 +3∥L2 = ∥B(eis∆v0) − B(v0)∥L2 ≤ C(M2)∥(I − eis∆)v0∥L2 ≤ C(M2)τ, +∥e2 +3∥L2 = ∥B(v0) − B(PNv0)∥L2 ≤ C(M2)∥v0 − PNv0∥L2 ≤ C(M2)h2. +For e3 +3 and e4 +3 in (3.46), using Lemma 3.5, we get +(3.48) +∥e3 +3∥L2 = ∥(I − eis∆)PNB(PNv0)∥L2 ≤ C(M2)τ +1+2σ +2 +, +∥e4 +3∥L2 = ∥(IN − PN)B(PNv0)∥L2 ≤ C(M2)h1+2σ. +Plugging (3.47) and (3.48) into (3.46) and noticing (3.45), we get +(3.49) +∥e3∥L2 ≤ C(M2)τ +� +τ +1+2σ +2 ++ h1+2σ� +. +Combing (3.41), (3.44), and (3.49) and noting (3.36), we get the desired result. +□ +Remark 3.7. The proof of Proposition 3.6 can be generalized to 2D and 3D di- +rectly. +Moreover, in 1D, under much weaker assumption that V ∈ H1(Ω) and +ψ ∈ C([0, T]; H1 +0(Ω)), by using Sobolev embedding H1 �→ L∞ and the estimate +(see, e.g., [10, 11]) +(3.50) +∥v − eit∆v∥L2 ≲ √τ∥v∥H1, +∥v − PNv∥L2 ≲ h|v|H1, +v ∈ H1 +0(Ω), +and following the proof of Proposition 3.6, we can obtain +(3.51) +∥PNψ(tk+1) − Φτ(PNψ(tk))∥L2(Ω) ≤ Cτ +�√τ + h +� +, +where C depends on ∥V ∥H1 and ∥ψ∥L∞([0,T ];H1). +3.3. Unconditional L2-stability and proof of Theorem 2.5. We shall show +the unconditional L2-stability of the numerical flow by using Lemma 3.4. With +the estimate of the local truncation error and the unconditional L2-stability of the +numerical flow, we are able to obtain the error estimates. +Proposition 3.8 (unconditional L2-stability). Let v ∈ XN and w ∈ XN such that +min{∥v∥L∞, ∥w∥L∞} ≤ M. When 0 < σ ≤ 1/2, we have +∥Φτ(v) − Φτ(w)∥L2 ≤ (1 + C(M)τ)∥v − w∥L2, +where C(M) ∼ M 2σ. + +16 +W. BAO AND C. WANG +Proof. Recalling (2.14), noting that eiτ∆ preserves the L2 norm, one gets +(3.52) +∥Φτ(v) − Φτ(w)∥L2 = ∥eiτ∆INΦτ +B(v) − eiτ∆INΦτ +B(w)∥L2 += ∥INΦτ +B(v) − INΦτ +B(w)∥L2. +From (3.52), by (3.27) and Lemma 3.4, noting that IN is an identity on XN and +recalling (2.6), we have +(3.53) +∥INΦτ +B(v) − INΦτ +B(w)∥2 +L2 += h +N−1 +� +j=1 +|Φτ +B(v)(xj) − Φτ +B(w)(xj)|2 += h +N−1 +� +j=1 +���e−iτV (xj)Φτ +B2(v)(xj) − e−iτV (xj)Φτ +B2(w)(xj) +��� +2 += h +N−1 +� +j=1 +��Φτ +B2(v)(xj) − Φτ +B2(w)(xj) +��2 +≤ (1 + C(M)τ)2h +N−1 +� +j=1 +|v(xj) − w(xj)|2 += (1 + C(M)τ)2∥INv − INw∥2 +L2 += (1 + C(M)τ)2∥v − w∥2 +L2. +The proof is completed. +□ +Remark 3.9. In the error estimates, v and w in Proposition 3.8 are related to +the exact solution and the numerical solution, respectively. Hence, to control the +constant C(M) in Proposition 3.8, we can assume bound of the exact solution and +thus get rid of the a priori estimate of the numerical solution, which explains why +Proposition 3.8 is called the unconditional L2-stability. +Proof of Theorem 2.5. Under the assumption (A) and using (3.19), one gets +(3.54) +∥ψ(·, tk) − PNψ(·, tk)∥L2 ≤ C(M2)h2. +Hence it suffices to estimate ek := INψk − PNψ(·, tk) ∈ XN for 0 ≤ k ≤ T/τ. By +(2.15), for 0 ≤ k ≤ T/τ − 1, one has +(3.55) +∥ek+1∥L2 = ∥INψk+1 − PNψ(·, tk+1)∥L2 = ∥Φτ(INψk) − PNψ(·, tk+1)∥L2 +≤ ∥Φτ(INψk) − Φτ(PNψ(·, tk))∥L2 + ∥Φτ(PNψ(·, tk)) − PNψ(·, tk+1)∥L2. +By Propositions 3.6 and 3.8. +noting that ∥PNψ(·, tk)∥L∞ ≲ ∥PNψ(·, tk)∥H2 ≤ +∥ψ(·, tk)∥H2 ≤ M2, one has +∥ek∥L2(Ω) ≤ eC(M2)τ∥ek−1∥L2(Ω) + C(M2)τ +� +τ 1/2+σ + h1+2σ� +, +1 ≤ k ≤ T/τ. +It follows from the discrete Gronwall’s inequality and ∥e0∥L2 = ∥INψ0 − PNψ0∥ ≤ +C(M2)h2 that +∥ek∥L2(Ω) ≤ C(T, M2) +� +τ +1+2σ +2 ++ h1+2σ� +, +0 ≤ k ≤ T/τ, +which completes the proof. +□ + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +17 +The proof of Corollary 2.6 follows the proof of Theorem 2.5 by replacing Propo- +sition 3.6 with (3.51) and we shall omit it for brevity. +4. Proof of Theorem 2.7 for the case σ ≥ 1/2 +In this section, we assume that V ∈ H2(Ω) ∩ W 1,∞(Ω), σ ≥ 1/2 and the as- +sumption (A). The assumption V ∈ W 1,∞(Ω) is only used in Proposition 4.8 and +can be obtained from V ∈ H2(Ω) in 1D or V ∈ H3(Ω) in 2D and 3D. +4.1. Some estimates for the operator B. +Lemma 4.1. Let v ∈ H2(Ω) such that ∥v∥H2 ≤ M. When σ ≥ 1/2, we have +∥B(v)∥H2(Ω) ≤ C(M, ∥V ∥H2). +Proof. Recalling (2.2), noting that H2(Ω) is an algebra when 1 ≤ d ≤ 3, we have +(4.1) +∥B(v)∥H2 ≤ ∥V v∥H2 + ∥f(|v|2)v∥H2 ≤ ∥V ∥H2∥v∥H2 + ∥f(|v|2)v∥H2. +When σ ≥ 1/2, recalling (1.4) and (3.4), by similar calculation as (2.34) and (2.35) +and noting (3.5) as well as +(4.2) +��f ′(|z|2)z +��+ +��f ′(|z|2)z +��+ +��f ′′(|z|2)z3��+ +��f ′′(|z|2)z2z +�� ≲ |z|2σ−1, z ∈ C, σ ≥ 1 +2, +we have +(4.3) +��∂jk(f(|v|2)v) +�� ≲ |v|2σ|∂jkv| + |v|2σ−1|∂jv| |∂kv|, +which yields, by Sobolev embedding, that +(4.4) +∥∂jk(f(|v|2)v)∥L2 ≲ ∥v∥2σ +L∞∥∂jkv∥L2 + ∥v∥2σ−1 +L∞ ∥∇v∥2 +L4 ≤ C(M). +Combing (4.4) and Lemma 3.1, noting (4.1), we obtain the desired result. +□ +Lemma 4.2. Let v, w ∈ H2(Ω) such that ∥v∥H2 ≤ M and ∥w∥H2 ≤ M. When +σ ≥ 1/2, we have +∥B(v) − B(w)∥H1 ≤ C(M, ∥V ∥W 1,4)∥v − w∥H1. +Proof. From (3.6), one gets +(4.5) +∇ (B(v) − B(w)) = −i +� +∇V (v − w) + i(1 + σ)(f(|v|2)∇v − f(|w|2)∇w) ++G(v)∇v − G(w)∇w] . +Using H¨older’s inequality and Sobolev embedding, we have +(4.6) +∥∇(V (v − w))∥L2 ≤ ∥∇V ∥L4∥v − w∥L4 + ∥V ∥L∞∥∇(v − w)∥L2 +≲ ∥V ∥W 1,4∥v − w∥H1. +By (4.5), it remains to show that +∥f(|v|2)∇v − f(|w|2)∇w∥L2 ≤ C(M)∥v − w∥H1, +(4.7) +∥G(v)∇v − G(w)∇w∥L2 ≤ C(M)∥v − w∥H1. +(4.8) +When σ ≥ 1/2, following the proof of (3.11), we have, for z1, z2 ∈ C, +|f(|z1|2) − f(|z2|2)| ≲ max{|z1|, |z2|}2σ−1|z1 − z2|, +(4.9) +|G(z1) − G(z2)| ≲ max{|z1|, |z2|}2σ−1|z1 − z2|. +(4.10) + +18 +W. BAO AND C. WANG +Using (4.9) and Sobolev embedding, we have +∥f(|v|2)∇v − f(|w|2)∇w∥L2 +≤ ∥f(|v|2)∇(v − w)∥L2 + ∥(f(|v|2) − f(|w|2))∇w∥L2 +≤ C(∥v∥L∞)∥v − w∥H1 + C(max{∥v∥L∞, ∥w∥L∞})∥(v − w)∇w∥L2 +≤ C(M)∥v − w∥H1 + C(M)∥v − w∥L4∥∇w∥L4 +≤ C(M)∥v − w∥H1, +which proves (4.7). Similarly, we can prove (4.8), which completes the proof. +□ +Lemma 4.3. Let v, w ∈ H1(Ω) ∩ L∞(Ω) such that ∥v∥L∞ + ∥v∥H1 ≤ M and +∥w∥L∞ + ∥w∥H1 ≤ M. When σ ≥ 1/2, we have +∥dB(v)[w]∥H1 ≤ C(M, ∥V ∥W 1,4). +Proof. From (3.14), using (4.6), we have +(4.11) +∥dB(v)[w]∥H1 ≤ ∥V w∥H1 + (1 + σ)∥f(|v|2)w∥H1 + ∥G(v)w∥H1 +≲ ∥V ∥W 1,4∥w∥H1 + ∥f(|v|2)w∥H1 + ∥G(v)w∥H1. +When σ ≥ 1/2, recalling (4.2), we have +(4.12) +∥f(|v|2)∥H1 = ∥f(|v|2)∥L2 + ∥∇f(|v|2)∥L2 ≲ ∥v∥2σ +L∞ + ∥f ′(|v|2)v∇v∥L2 +≤ ∥v∥2σ +L∞ + ∥v∥2σ−1 +L∞ ∥∇v∥L2 ≤ C(M). +Similarly, one gets ∥G(v)∥H1 ≤ C(M). Then using +(4.13) ∥u1u2∥H1 ≤ ∥u1∥L∞∥u2∥H1 + ∥u2∥L∞∥u1∥H1, +u1, u2 ∈ H1(Ω) ∩ L∞(Ω), +and recalling (3.5), we have +∥f(|v|2)w∥H1 ≤ ∥f(|v|2)∥L∞∥w∥H1 + ∥w∥L∞∥f(|v|2)∥H1 ≤ C(M), +(4.14) +∥G(v)w∥H1 ≤ ∥G(v)∥L∞∥w∥H1 + ∥w∥L∞∥G(v)∥H1 ≤ C(M). +(4.15) +Plugging (4.14) and (4.15) into (4.11) yields the desired result. +□ +Lemma 4.4. Let v, w ∈ H2(Ω) such that ∥v∥H2 ≤ M and ∥w∥H2 ≤ M. +If +|w(x)| ≤ C|v(x)| for all x ∈ Ω, when σ ≥ 1/2, we have +∥dB(v)[w]∥H2 ≤ C (M, ∥V ∥H2) . +Proof. The proof can be obtained similarly as the proof of Lemma 4.1 and we shall +omit it here for brevity. +□ +Lemma 4.5. Let 0 < τ < 1 and v ∈ XN such that ∥v∥L∞ ≤ M and ∥v∥H2 ≤ M1. +When σ > 0, we have +(4.16) +∥Φτ +B(v)∥H1 ≤ (1 + C1(M, ∥V ∥W 1,4)τ) ∥v∥H1(Ω), +and when σ ≥ 1/2, we have +(4.17) +∥Φτ +B(v)∥H2 ≤ C2(M1, ∥V ∥H2). +Proof. Recalling that Φτ +B(v) = ve−iτ(V +f(|v|2)) in (2.6), the proof of (4.16) and +(4.17) follows similarly from the proof of Lemma 3.1 and Lemma 4.1, respectively. +□ + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +19 +Lemma 4.6. Let z1, z2 ∈ C. When σ ≥ 1/2, one has +|Φτ +B2(z1) − Φτ +B2(z2)| ≤ (1 + Cτ)|z1 − z2|, +where C ∼ max{|z1|, |z2|}2σ. +Proof. The proof follows from the proof of Lemma 3.4 by replacing (3.16) with +(4.9). +□ +4.2. Local truncation error. +Proposition 4.7 (local truncation error). Assume that 0 < τ < 1, 0 < h < 1, +V ∈ H2 and σ ≥ 1/2, under the assumption (A), for 0 ≤ k ≤ T/τ − 1, we have +∥PNψ(·, tk+1) − Φτ(PNψ(·, tk))∥L2(Ω) ≤ C1(M2)τ +� +τ + h2� +, +(4.18) +∥PNψ(·, tk+1) − Φτ(PNψ(·, tk))∥H1(Ω) ≤ C2(M2)τ +� +τ +1 +2 + h +� +. +(4.19) +Proof. Following the notation in the proof of Proposition 3.6, we let v(t) = ψ(·, tk + +t) for 0 ≤ t ≤ τ and v0 := v(0) = ψ(·, tk). When σ ≥ 1/2, (3.29) and (3.30) are +also valid and we have the same error decomposition (3.36). When σ ≥ 1/2, the L2 +estimate (4.18) follows from the proof of Proposition 3.6 by replacing (3.48) with +(4.20) +∥e3 +3∥L2 ≲ τ∥PNB(PNv0)∥H2 ≤ τ∥B(PNv0)∥H2 ≤ C(M2)τ, +∥e4 +3∥L2 ≲ h2∥B(PNv0)∥H2 ≤ C(M2)h2, +where (3.20), (3.19) and Lemma 4.1 are used. +In the following, we shall show (4.19). Using Sobolev embedding, the isometry +property of eit∆ and Lemmas 3.1 and 4.1, one gets +(4.21) +∥ei(s−σ)∆B(v(σ))∥H1 = ∥B(v(σ))∥H1 ≤ C(M2), +∥ei(s−σ)∆B(v(σ))∥L∞ ≲ ∥ei(s−σ)∆B(v(σ))∥H2 = ∥B(v(σ))∥H2 ≤ C(M2). +Recalling the property of eit∆ and PN, using Lemma 4.3, noticing (3.29) and (4.21), +we have +(4.22) +���ei(τ−s)∆PN +� +dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] +���� +H1 +≤ ∥dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))]∥H1 +≤ C(∥V ∥W 1,4, ∥ei(s−σ)∆v(σ)∥L∞∩H1, ∥ei(s−σ)∆B(v(σ))∥L∞∩H1) +≤ C(M2), +which yields, for e1 in (3.37), +(4.23) +∥e1∥H1 ≤ +� τ +0 +� s +0 +���ei(τ−s)∆PN +� +dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] +���� +H1 dσds +≤ C(M2)τ 2. +For e2 in (3.38), recalling the standard estimate of the interpolation operator [8, 42] +(4.24) +∥INφ∥H1 ≲ ∥φ∥H1 + h|φ|H2 ≲ ∥φ∥H2, +φ ∈ H1 +0(Ω) ∩ H2(Ω), +one gets +(4.25) +∥e2∥H1 = τ 2 +����IN +�� 1 +0 +(1 − θ) +� +dB(Φθτ +B (PNv0))[B(Φθτ +B (PNv0))] +� +dθ +����� +H1 +≲ τ 2∥dB(Φθτ +B (PNv0))[B(Φθτ +B (PNv0))]∥H2. + +20 +W. BAO AND C. WANG +From (4.25), noting that +(4.26) +��� +B(Φθτ +B (PNv0)) +� +(x) +�� ≲ +��Φθτ +B (PNv0)(x) +�� = (PNv0)(x), +x ∈ Ω, +and using Lemma 4.4, we have +(4.27) +∥e2∥H1 ≤ C(M2)τ 2. +Then we shall estimate e3 in (3.39). Similar to (3.45) and (3.46), it suffices for +us to bound the H1-norm of the four terms ej +3(1 ≤ j ≤ 4) defined in (3.46). Using +the standard estimates, +∥φ − PNφ∥H1 ≲ h|φ|H2, +∥INφ − PNφ∥H1 ≲ h|φ|H2, +(4.28) +∥φ − eit∆φ∥H1 ≲ +√ +t∥φ∥H2, +φ ∈ H1 +0(Ω) ∩ H2(Ω), +(4.29) +and Lemmas 4.1 and 4.2, we have +∥e1 +3∥H1 ≤ C(M2)∥eis∆v0 − v0∥H1 ≤ C(M2)√τ∥v0∥H2 ≤ C(M2)√τ, +∥e2 +3∥H1 ≤ C(M2)∥v0 − PNv0∥H1 ≤ C(M2)h∥v0∥H2 ≤ C(M2)h, +(4.30) +∥e3 +3∥H1 ≲ √τ∥PNB(PNv0)∥H2 ≤ √τ∥B(PNv0)∥H2 ≤ C(M2)√τ, +∥e4 +3∥H1 ≲ h∥B(PNv0)∥H2 ≤ C(M2)h, +(4.31) +which yields immediately +(4.32) +∥e3∥H1 ≤ C(M2)τ +�√τ + h +� +. +Combining (4.23), (4.27), and (4.32), we obtain (4.19), which completes the proof. +□ +4.3. l∞-conditional L2- and H1-stability. +Proposition 4.8 (l∞-conditional stability). Let 0 < τ < 1 and v, w ∈ XN such +that ∥v∥l∞ ≤ M, ∥w∥l∞ ≤ M and ∥v∥H2 ≤ M1. When σ ≥ 1/2, +∥Φτ(v) − Φτ(w)∥L2 ≤ (1 + C1(M)τ)∥v − w∥L2, +(4.33) +∥Φτ(v) − Φτ(w)∥H1 ≤ (1 + C2(M, M1, ∥V ∥W 1,∞)τ)∥v − w∥H1, +(4.34) +Proof. The L2-stability (4.33) can be obtained from (3.53) by using Lemma 4.6 +instead of Lemma 3.4. In the following, we show the H1-stability (4.34). Recalling +(2.6) and that eiτ∆ preserves the H1-norm, (4.34) reduce to +(4.35) +∥INΦτ +B(v) − INΦτ +B(w)∥H1 ≤ (1 + C(M, M1, ∥V ∥W 1,∞)τ)∥v − w∥H1. +The proof is based on the following well-known equivalence relation (see, e.g., +Lemma 3.2 in [8]) +(4.36) +∥δ+ +x φ∥l2 ≤ ∥∇INφ∥L2 ≤ π +2 ∥δ+ +x φ∥l2, +which implies +(4.37) +∥INΦτ +B(v) − INΦτ +B(w)∥H1 ≤ ∥v − w∥H1 + ∥IN(Φτ +B(v) − v) − IN(Φτ +B(w) − w)∥H1 +≤ ∥v − w∥H1 + π +2 ∥δ+ +x (Φτ +B(v) − v) − δ+ +x (Φτ +B(w) − w)∥l2. +We define +vθ +j = (1 − θ)vj + θvj+1, +wθ +j = (1 − θ)wj + θwj+1, +V θ +j = (1 − θ)V (xj) + θV (xj+1), +0 ≤ θ ≤ 1, +j = 0, · · · , N − 1. + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +21 +By some elementary computation, recalling (3.4) and f ′(|z|2)|z|2 = σf(|z|2), one +gets, for 0 ≤ j ≤ N − 1, +(4.38) +δ+ +x (Φτ +B(v) − v)j += δ+ +x +� +v(e−iτ(V +f(|v|2)) − 1) +� +j = 1 +h +� 1 +0 +d +dθ +� +vθ +j (e−iτ(V θ +j +f(|vθ +j |2)) − 1) +� +dθ += +� 1 +0 +δ+ +x vj(e−iτ(V θ +j +f(|vθ +j |2)) − 1)dθ − iτ +� 1 +0 +e−iτV θ +j vθ +j δ+ +x V (xj)e−iτf(|vθ +j |2)dθ +− iτ +� 1 +0 +e−iτV θ +j (σf(|vθ +j |2)δ+ +x vj + G(vθ +j )δ+ +x vj)e−iτf(|vθ +j |2)dθ, +Similarly, for 0 ≤ j ≤ N − 1, +(4.39) +δ+ +x (Φτ +B(w) − w)j += +� 1 +0 +δ+ +x wj(e−iτ(V θ +j +f(|wθ +j |2)) − 1)dθ − iτ +� 1 +0 +e−iτV θ +j wθ +jδ+ +x V (xj)e−iτf(|wθ +j |2)dθ +− iτ +� 1 +0 +e−iτV θ +j (σf(|wθ +j|2)δ+ +x wj + G(wθ +j)δ+ +x wj)e−iτf(|wθ +j |2)dθ. +We define the function e ∈ YN with +ej = vj − wj, +j = 0, · · · , N. +Subtracting (4.39) from (4.38), for 0 ≤ j ≤ N − 1, we have +(4.40) +��δ+ +x (Φτ +B(v) − v)j − δ+ +x (Φτ +B(w) − w)j +�� +≤ +� 1 +0 +����δ+ +x vj(e−iτ(V θ +j +f(|vθ +j |2)) − 1) − δ+ +x wj(e−iτ(V θ +j +f(|wθ +j |2)) − 1) +��� ++ τ +��δ+ +x V (xj) +�� +���vθ +j e−iτf(|vθ +j |2) − wθ +je−iτf(|wθ +j |2)��� ++ στ +���δ+ +x vjf(|vθ +j |2)e−iτf(|vθ +j |2) − δ+ +x wjf(|wθ +j|2)e−iτf(|wθ +j |2)��� ++τ +���δ+ +x vjG(vθ +j )e−iτf(|vθ +j |2) − δ+ +x wjG(wθ +j)e−iτf(|wθ +j |2)��� +� +dθ +=: +� 1 +0 +� +J1 +j + J2 +j + J3 +j + J4 +j +� +dθ. +For J1 +j , by (4.9), one gets +(4.41) +J1 +j ≤ +��δ+ +x vj +�� +���e−iτ(V θ +j +f(|vθ +j |2)) − e−iτ(V θ +j +f(|wθ +j |2))��� ++ +��δ+ +x vj − δ+ +x wj +�� +���e−iτ(V θ +j +f(|wθ +j |2)) − 1 +��� +≤ τ +��δ+ +x vj +�� ��f(|vθ +j |2) − f(|wθ +j|2) +�� + τ +��V θ +j + f(|wθ +j|2) +�� ��δ+ +x vj − δ+ +x wj +�� +≤ C(M)τ +��δ+ +x vj +�� (|ej| + |ej+1|) + C(M, ∥V ∥L∞)τ|δ+ +x ej|. +For J2 +j , recalling (2.7), by Lemma 4.6 and 0 < τ < 1, one gets +(4.42) +J2 +j = τ +��δ+ +x V (xj) +�� ��Φτ +B2(vθ +j ) − Φτ +B2(wθ +j) +�� ≤ τ +��δ+ +x V (xj) +�� (1 + C(M)τ)(|ej| + |ej+1|) +≤ C(M)τ +��δ+ +x V (xj) +�� (|ej| + |ej+1|). + +22 +W. BAO AND C. WANG +For J3 +j , by (4.9) and 0 < τ < 1, one gets +(4.43) +J3 +j ≲ τ +��δ+ +x vj +�� +���f(|vθ +j |2)e−iτf(|vθ +j |2) − f(|wθ +j|2)e−iτf(|wθ +j |2)��� ++ τ +��δ+ +x vj − δ+ +x wj +�� +���f(|wθ +j|2)e−iτf(|wθ +j |2)��� +≤ τ +��δ+ +x vj +�� +���f(|vθ +j |2) − f(|wθ +j|2) +�� + |f(|wθ +j|2)| +���e−iτf(|vθ +j |2) − e−iτf(|wθ +j |2)��� +� ++ τC(M) +��δ+ +x vj − δ+ +x wj +�� +≤ τ +��δ+ +x vj +�� C(M)(1 + τ)|vθ +j − wθ +j| + τC(M) +��δ+ +x vj − δ+ +x wj +�� +≤ C(M)τ +��δ+ +x vj +�� (|ej| + |ej+1|) + C(M)τ +��δ+ +x ej +�� . +Similar to (4.43), using (4.10) instead of (4.9), one gets, for J4 +j , +(4.44) +J4 +j ≤ C(M)τ +��δ+ +x vj +�� (|ej| + |ej+1|) + C(M)τ +��δ+ +x ej +�� . +Plugging (4.41)–(4.44) into (4.40), we have +��δ+ +x (Φτ +B(v) − v)j − δ+ +x (Φτ +B(w) − w)j +�� +≤ C(M)τ +���δ+ +x V (xj) +�� + +��δ+ +x vj +��� +(|ej| + |ej+1|) + C(M, ∥V ∥L∞)τ +��δ+ +x ej +�� , +which yields +(4.45) +∥δ+ +x (Φτ +B(v) − v) − δ+ +x (Φτ +B(w) − w)∥2 +l2 += h +N−1 +� +j=0 +��δ+ +x (Φτ +B(v) − v)j − δ+ +x (Φτ +B(w) − w)j +��2 +≤ C(M)τ 2h +N−1 +� +j=0 +���δ+ +x V (xj) +��2 + +��δ+ +x vj +��2� +(|ej|2 + |ej+1|2) ++ C(M, ∥V ∥L∞)τ 2h +N−1 +� +j=0 +��δ+ +x ej +��2 +≤ C(M)τ 2 +� +�∥∇V ∥2 +L∞∥e∥2 +l2 + h +N−1 +� +j=0 +��δ+ +x vj +��2 (|ej|2 + |ej+1|2) +� +� ++ C(M, ∥V ∥L∞)τ 2∥δ+ +x e∥2 +l2. +When d = 1, one has |δ+ +x vj| ≤ ∥∇v∥L∞ ≤ C(M1), which yields directly that +(4.46) +h +N−1 +� +j=0 +��δ+ +x vj +��2 (|ej|2 + |ej+1|2) ≤ C(M1)∥e∥2 +l2. +However, (4.46) cannot be directly generalized to 2D and 3D without assuming +higher regularity on v. Here, we present an alternative approach that can be gener- +alized to 2D and 3D (see also Remark 4.9). Using the discrete Gargliardo-Nireberg +inequality and the discrete Poincare’s inequality (see Lemma 3.1 of [7]), we have +(4.47) +∥φ∥l4 ≲ ∥φ∥ +3 +4 +l2∥δ+ +x φ∥ +1 +4 +l2 ≲ ∥δ+ +x φ∥l2, +φ ∈ YN, + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +23 +which implies, by first applying H¨older’s inequality in (4.46), +(4.48) +h +N−1 +� +j=0 +��δ+ +x vj +��2 (|ej|2 + |ej+1|2) ≲ ∥δ+ +x v∥2 +l4∥e∥2 +l4 ≲ ∥δ+ +x v∥2 +l4∥δ+ +x e∥2 +l2. +Using the following discrete version of the Sobolev embedding H2 �→ W 1,4 +(4.49) +∥δ+ +x φ∥l4 ≲ ∥φ∥H2, +φ ∈ XN, +we have ∥δ+ +x vj∥l4 ≲ ∥v∥H2 = M1, which yields from (4.45) that +(4.50) +∥δ+ +x (Φτ +B(v) − v) − δ+ +x (Φτ +B(w) − w)∥2 +l2 ≤ C(M, M1, ∥V ∥W 1,∞) +� +∥e∥2 +l2 + ∥δ+ +x e∥2 +l2 +� +. +From (4.50), using the discrete Poincare’s inequality and (3.27) and (4.36), we have +(4.51) +∥δ+ +x (Φτ +B(v) − v) − δ+ +x (Φτ +B(w) − w)∥2 +l2 ≤ C(M, M1, ∥V ∥W 1,∞)∥δ+ +x e∥2 +l2 +≤ C(M, M1, ∥V ∥W 1,∞)∥∇e∥2 +L2, +which plugged into (4.37) yields (4.35), which completes the proof. +□ +Remark 4.9. The 2D case follows exactly (4.47)–(4.49). The proof of (4.49) in 2D +follows from the proof of (3.3) of Lemma 3.1 in [7] with additional attention paid +to the boundary terms. The 3D case follows (4.47)–(4.49) with slight modification: +using H¨older’s inequality with index (3/2, 3) in (4.48). Then the discrete version +of H1 �→ L6 and H2 �→ W 1,3 in 3D are needed. The proof of the first one can be +found in [39] while the proof of the second one will follow the proof of (4.49) in 2D, +which is the reason why we modify the estimates in 3D. +4.4. Proof of (2.42) in Theorem 2.7. With Propositions 4.7 and 4.8, we are +able to obtain (2.42). +Proof of (2.42) in Theorem 2.7. Following the proof of Theorem 2.7, we only need +to estiamte ek = INψk − PNψ(·, tk) for 0 ≤ k ≤ T/τ. We shall first prove the error +estimate in H1 norm by the standard argument of the mathematical induction. +Replacing ∥ · ∥L2 with ∥ · ∥H1 in (3.55), one has for 0 ≤ k ≤ T/τ − 1, +(4.52) +∥ek+1∥H1 ≤ ∥Φτ(INψk) − Φτ(PNψ(·, tk))∥H1 + ∥Φτ(PNψ(·, tk)) − PNψ(·, tk+1)∥H1. +When k = 0, by (4.28), one gets +∥e0∥H1 = ∥INψ0 − PNψ0∥H1 ≲ h∥ψ0∥H2 ≤ C(M2)h, ∥ψ0∥l∞ ≤ ∥ψ0∥L∞ ≤ 1 + M2. +We assume that for 0 ≤ k ≤ m ≤ T/τ − 1, +(4.53) +∥ek∥H1 ≲ τ +1 +2 + h, +∥ψk∥l∞ ≤ 1 + M2. +We shall prove (4.53) for m + 1. From (4.52), using (4.19) and (4.34) and noting +the assumption (4.53), we have +(4.54) +∥em+1∥H1 ≤ (1 + C1τ)∥em∥H1 + C2τ +� +τ +1 +2 + h +� +, +where C1 and C2 are the costants in (4.19) and (4.34) respectively, which depends +exclusively on M2 and ∥V ∥W 1,∞. From (4.54), standard discrete Gronwall’s in- +equality yields +(4.55) +∥em+1∥H1 ≤ 2eC0T C1 +� +τ +1 +2 + h +� +. + +24 +W. BAO AND C. WANG +Recalling that ek = INψk − PNψ(tk) and ∥ψ(·, tk)∥L∞ ≤ M2, using the inverse +inequality ∥φ∥L∞ ≲ h−1/2∥φ∥L2, ∀φ ∈ XN, we have +∥ψm+1∥l∞ = ∥INψm+1∥l∞ ≤ ∥em+1∥l∞ + ∥PNψ(·, tm+1)∥l∞ +≤ ∥em+1∥l∞ + ∥ψ(·, tm+1) − PNψ(·, tm+1)∥l∞ + ∥ψ(·, tm+1)∥l∞ +≤ ∥em+1∥l∞ + h− 1 +2 ∥ψ(·, tm+1) − PNψ(·, tm+1)∥L2 + M2. +Hence, for τ ≤ τ0 and h ≤ h0 with τ0 > 0 and h0 > 0 depending on M2 and T, by +Sobolev embedding and (3.19), we have +(4.56) +∥ψm+1∥l∞ ≤ C∥em+1∥H1 + Ch2−1/2 + M2 ≤ 1 + M2. +Combining (4.55) and (4.56), we proves (4.53) for k = m + 1 and thus for all +0 ≤ k ≤ T/τ by mathematical induction. With the l∞-bound of the numerical +solution, the L2 estimate of ek follows the proof of Theorem 2.5 by using (4.18) and +(4.33), which completes the proof of (2.42). +□ +Remark 4.10. In 2D and 3D, we no longer have H1 �→ L∞. To obtain the l∞-bound +of ψm+1 in (4.56), we use the discrete Sobolev inequalities as in [5, 7, 8] +∥v∥l∞ ≤ C| ln h| ∥INv∥H1, +∥w∥l∞ ≤ Ch−1/2∥INw∥H1, +where v and w are 2D and 3D mesh functions with zero at the boundary, respec- +tively, and the interpolation operator IN can be defined similarly in 2D and 3D as in +1D. Thus by requiring that the time step size τ satisfies the additional assumption +(B), we can control the l∞-norm of the numerical solution. +4.5. Proof of (2.43) in Theorem 2.7. In the following, we assume that 1/2 < +σ < 1, V ∈ H3(Ω), ∇V ∈ H1 +0(Ω), ψ ∈ C([0, T]; H3 +∗(Ω)) ∩ C1([0, T]; H1(Ω)) and let +(4.57) +M3 := max +� +∥ψ∥L∞([0,T ];H3), ∥ψ∥L∞([0,T ];L∞), ∥V ∥H3� +. +We first show an analogous result of Lemma 3.5. +Lemma 4.11. Let φ ∈ XN such that ∥φ∥H3 ≤ M and let 0 < τ < 1 and 0 < h < 1. +Assume that V ∈ H3(Ω) and ∇V ∈ H1 +0(Ω). When 1/2 < σ < 1, we have +∥(I − eiτ∆)PNB(φ)∥H1 ≤ C1(M, ∥V ∥H3)τ σ, +(4.58) +∥INB(φ) − PNB(φ)∥H1 ≤ C2(M, ∥V ∥H3)h2σ. +(4.59) +Proof. Similar to (3.21), noting that V φ ∈ H3 +∗(Ω), we have +(4.60) +∥(I − eiτ∆)(V φ)∥H1 ≲ τ∥V ∥H3∥φ∥H3, +∥(IN − PN)(V φ)∥H1 ≲ h2∥V ∥H3∥φ∥H3. +Following (3.22) and (3.23) with ∥ · ∥H1 replacing ∥ · ∥L2 and using (2.31), we have +(4.61) +∥(I − eiτ∆)(f(|φ|2)φ)∥H1 ≤ 2∥f(|φ|2)φ − fε(|φ|2)φ∥H1 + C(M) +τ +ε2−2σ . +Recalling (3.6), one gets +(4.62) +∇[f(|φ|2)φ − fε(|φ|2)φ] = (1 + σ)(f(|φ|2) − fε(|φ|2))∇φ ++ (G(φ) − f ′ +ε(|φ|2)φ2)∇φ, +from which, using Lemma 2.4 and noting that G(z) = f ′(|z|2)z2 = f ′ +ε(|z|2)z2 when +|z| ≥ ε and |G(z)| + |f ′ +ε(|z|2)z2| ≲ ε2σ when |z| < ε, one gets +(4.63) +∥∇[f(|φ|2)φ − fε(|φ|2)φ]∥L2 ≲ ε2σ∥∇φ1|φ|<ε∥L2 ≤ ε2σ∥φ∥H1. + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +25 +Plugging (4.63) into (4.61), we have +∥(I − eiτ∆)(f(|φ|2)φ)∥H1 ≤ C(M) inf +0<ε<1 +� +ε2σ + +τ +ε2−2σ +� +≤ C(M)τ σ, +which combined with (4.60) yields (4.58). +Then we shall show (4.28). Following (3.26) with ∥ · ∥H1 replacing ∥ · ∥L2, using +the property of PN and (4.63), one gets +(4.64) +∥(IN − PN)(f(|φ|2)φ)∥H1 +≤ ∥IN(f(|φ|2)φ − fε(|φ|2)φ)∥H1 + ε2σ∥φ∥H1 + h2 C(M) +ε2−2σ . +Using (4.36), one gets +(4.65) +∥∇IN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 ≲ ∥δ+ +x (f(|φ|2)φ − fε(|φ|2)φ)∥l2. +Let φθ +j = (1 − θ)φj + θφj+1 for 0 ≤ θ ≤ 1 and 0 ≤ j ≤ N − 1, direct calculation +gives +(4.66) +δ+ +x +� +f(|φj|2)φj − fε(|φj|2)φj +� += 1 +h +� 1 +0 +d +dθ +� +f(|φθ +j|2)φθ +j − fε(|φθ +j|2)φθ +j +� +dθ += +� 1 +0 +��� +f(|φθ +j|2) + f ′(|φθ +j|2)|φθ +j|2� +− +� +fε(|φθ +j|2) + f ′ +ε(|φθ +j|2)|φθ +j|2�� +δ+ +x φj ++(G(φθ +j) − f ′ +ε(|φθ +j|2)(φθ +j)2)δ+ +x φj +� +dθ, +which implies, similar to (4.63), +(4.67) +��δ+ +x (f(|φj|2)φj − fε(|φj|2)φj) +�� ≲ ε2σ|δ+ +x φj|. +From (4.65), using (4.67) and recalling (4.36) and φ ∈ XN, we obatin +(4.68) +∥∇IN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 ≲ ε2σ∥δ+ +x φ∥l2 ≤ ε2σ∥∇φ∥L2, +which plugged into (4.64) yields +∥(IN − PN)(f(|φ|2)φ)∥H1 ≤ C(M) inf +0<ε<1 +� +ε2σ + +h2 +ε2−2σ +� +≤ C(M)h2σ, +which combined with (4.60) yields (4.59) and completes the proof. +□ +Proposition 4.12 (local truncation error). Assume that V +∈ H3(Ω), ∇V +∈ +H1 +0(Ω), ψ ∈ C([0, T]; H3 +∗(Ω)) ∩ C1([0, T]; H1(Ω)) and 1/2 < σ < 1, for 0 ≤ k ≤ +T/τ − 1, we have +∥PNψ(·, tk+1) − Φτ(PNψ(·, tk))∥H1(Ω) ≤ C(M3)τ +� +τ σ + h2σ� +. +Proof. Following the proof of Proposition 4.7, we only need to modify the esti- +mate (4.30) and (4.31), which can be easily done by using the assumption ψ ∈ +C([0, T]; H3) and Lemma 4.11 and the standard estimates of the operators IN −PN, +I − PN and I − eiτ∆. +□ +Proof of (2.43) in Theorem 2.7. Using Proposition 4.12 and (4.34) in (4.52), and +noting the l∞-bound of the numerical solution in (2.42), then (2.43) follows from +the discrete Gronwall’s inequality immediately. +□ + +26 +W. BAO AND C. WANG +5. Numerical results +In this section, we present some numerical examples for the NLSE with 0 < σ < 1 +to confirm our error estimates. In the following, we fix β = −1, V (x) ≡ 0, d = 1 +and T = 1 and consider the following two initial set-ups: +Type I: We consider the smooth initial datum +(5.1) +ψ0(x) = xe− x2 +2 , +x ∈ Ω = (−16, 16). +Type II: We consider the initial datum in H2(Ω) as in [30] +(5.2) +ψ0 = +φ(1) +∥φ(1)∥L2 , +φ(1)(x) = +� +l∈TN +�φ(1) +l +sin(µl(x − a)), +x ∈ Ω = (−1, 1) +�φ(1) +l += +�φl +|µl|2.5 , +�φl = +� +rand(−1, 1) + i rand(−1, 1), +l even, +0, +l odd, +l ∈ TN. +where rand(−1, 1) returns a uniformly distributed random number between +−1 and 1. +Note that both Types I and II initial data are chosen as odd functions to demon- +strate the influence of the semi-smoothness of f at the origin since with an odd +initial datum, the exact solution satisfies ψ(0, t) ≡ 0 for all t ≥ 0. +The NLSE (1.1) is then solved by the TSSP method on the domain Ω with Type +I and Type II initial setups for different σ > 0. The ‘exact’ solution is obtained +numerically by the Strang splitting sine pseudospectral method with a very fine +mesh size he = 2−9 and a small time step size τe = 10−6. +In our numerical +experiments below, when testing the temporal convergence, we always fix the mesh +size h = he. To quantify the error, we introduce the following error functions: +ek +L2 = ∥ψ(·, tk) − INψk∥L2, +ek +H1 = ∥ψ(·, tk) − INψk∥H1, +0 ≤ k ≤ n := T/τ. +Figure 5.1 exhibits the temporal and spatial errors in L2-norm of the TSSP (2.13) +for the NLSE (1.1) with Type I initial datum and different 0 < σ ≤ 1/2. Figure 5.1 +(a) shows that the temporal convergence is first order in L2-norm for all the four σ +and Figure 5.1 (b) shows the spatial convergence is almost third order in L2-norm, +which is also increasing with σ. These results are better than our error estimates +in Theorem 2.5 and suggest that first order temporal convergence in L2-norm may +hold for any σ > 0 and the spatial convergence may be of higher order. However, +we remark that it is impossible to obtain the optimal temporal convergence and +the high order spatial convergence by simply improving the local error estimates +in Proposition 3.6 and there must exist error cancellation between different steps, +which require new techniques and in-depth analysis to handle. +Figure 5.2 plots the temporal and spatial errors in L2- and H1-norm of the +TSSP (2.13) for the NLSE (1.1) with Type II H2 initial datum and fixed σ = 0.5. +Figure 5.2 (a) shows that the temporal convergence is first order in L2-norm and +half order in H1-norm and Figure 5.2 (b) shows the spatial convergence is second +order in L2-norm and first order in H1-norm. These results correspond with our +error estimates (2.42) in Theorem 2.7 very well. +Figure 5.3 displays the temporal and spatial errors in H1-norm of the TSSP +(2.13) for the NLSE (1.1) with Type I smooth initial datum and different 0 < σ < 1. +Figure 5.3 (a) shows that the temporal convergence in H1-norm increases from half +order to first order as σ increase from 0 to 1/2 and remains first order when σ ≥ 1/2. + +ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY +27 +10-4 +10-3 +10-2 +10-1 +10-5 +10-3 +0.1 +0.5 +10-6 +10-4 +Figure 5.1. Temporal errors (a) and spatial errors (b) in L2-norm +for σ = 0.1, 0.2, 0.3, 0.4 with Type I initial datum (5.1) +10-4 +10-3 +10-2 +10-4 +10-2 +10-3 +10-2 +10-5 +10-2 +Figure 5.2. Temporal errors (a) and spatial errors (b) in L2-norm +and H1-norm for σ = 0.5 with Type II initial data (5.2) +Figure 5.3 (b) shows the spatial convergence is almost 2.5 order in H1-norm and is +increasing with σ. Similar to the observation of Figure 5.1, these results are better +than our error estimates (2.43) in Theorem 2.7 and suggest that first order temporal +convergence in H1-norm may hold for any σ ≥ 1/2. We would like to comment +that the order reduction in H1-norm for 0 < σ < 1/2 is indeed resulted from the +semi-smoothness of the nonlinearity instead of the regularity of the exact solution. +Actually, we numerically checked that with the Type I smooth initial datum, the +exact solution is roughly in H3.5+2σ. +6. Conclusion +Error bounds of the Lie-Trotter time-splitting sine pseudospectral method for +the nonlinear Schr¨odinger equation (NLSE) with semi-smooth nonlinearity f(ρ) = +ρσ(σ > 0) were established. For 0 < σ ≤ 1 +2, we prove error bounds at O(τ +1 +2 +σ + +h1+2σ) in L2-norm without any coupling conditions between τ and h. For σ ≥ 1 +2, +error bounds at O(τ +h2) in L2-norm and at O(τ +1 +2 +h) in H1-norm are proved with +mild coupling conditions. 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Comput. 60 (2014), no. 2, 390–407. MR 3225788 +Department of Mathematics, National University of Singapore, Singapore 119076 +Email address: matbaowz@nus.edu.sg +Department of Mathematics, National University of Singapore, Singapore 119076 +Email address: E0546091@u.nus.edu + diff --git a/htE1T4oBgHgl3EQfMwMt/content/tmp_files/load_file.txt b/htE1T4oBgHgl3EQfMwMt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..589240a363c51b45c78f2c92f9751cd69fbbe8ab --- /dev/null +++ b/htE1T4oBgHgl3EQfMwMt/content/tmp_files/load_file.txt @@ -0,0 +1,1403 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf,len=1402 +page_content='MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages 000–000 S 0025-5718(XX)0000-0 ERROR ESTIMATES OF THE TIME-SPLITTING METHODS FOR THE NONLINEAR SCHR¨ODINGER EQUATION WITH SEMI-SMOOTH NONLINEARITY WEIZHU BAO AND CHUSHAN WANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schr¨odinger equation (NLSE) with semi-smooth nonlinearity f(ρ) = ρσ, where ρ = |ψ|2 is the density with ψ the wave function and σ > 0 is the exponent of the semi-smooth nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Under the assumption of H2-solution of the NLSE, we prove error bounds at O(τ 1 2 +σ + h1+2σ) and O(τ + h2) in L2-norm for 0 < σ ≤ 1 2 and σ ≥ 1 2 , respectively, and an error bound at O(τ 1 2 + h) in H1-norm for σ ≥ 1 2 , where h and τ are the mesh size and time step size, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In addition, when 1 2 < σ < 1 and under the assumption of H3-solution of the NLSE, we show an error bound at O(τ σ + h2σ) in H1-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Two key ingredients are adopted in our proof: one is to adopt an unconditional L2-stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of 0 < σ ≤ 1 2 , and to establish an l∞-conditional H1-stability to obtain the l∞-bound of the numerical solution by using the mathematical induction and the error estimates for the case of σ ≥ 1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' and the other one is to introduce a regularization technique to avoid the singularity of the semi- smooth nonlinearity in obtaining improved local truncation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Finally, numerical results are reported to demonstrate our error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Introduction In this paper, we consider the following nonlinear Schr¨odinger equation (NLSE) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) i∂tψ(x, t) = −∆ψ(x, t)+V (x)ψ(x, t)+f(|ψ(x, t)|2)ψ(x, t), x ∈ Ω, t > 0, with the initial data (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2) ψ(x, 0) = ψ0(x), x ∈ Ω, and the homogeneous Dirichlet boundary condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3) ψ(x, t) = 0, x ∈ ∂Ω, t ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Primary 35Q55, 65M15, 65M70, 81Q05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' nonlinear Schr¨odinger equation, semi-smooth nonlinearity, time- splitting pseudospectral method, error estimate, local regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' This work was partially supported by the Ministry of Education of Singapore grant MOE- 000357-00 (W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Bao).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' ©XXXX American Mathematical Society 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='02992v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='NA] 8 Jan 2023 2 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG where t is time, x ∈ Rd (d = 1, 2, 3) is the spatial coordinate, ψ := ψ(x, t) is a complex-valued wave function, V := V (x) : Ω → R is a time-independent real- valued potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Here Ω = Πd i=1(ai, bi) ⊂ Rd is a bounded domain and the nonlin- earity is given as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4) f(ρ) = βρσ, ρ := |ψ|2 ≥ 0, where β ∈ R is a given constant and σ > 0 is the exponent of the nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) conserves the mass (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5) M(ψ(·, t)) = � Ω |ψ(x, t)|2dx ≡ M(ψ0), t ≥ 0, and the energy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6) E(ψ(·, t)) = � Ω � |∇ψ(x, t)|2 + V (x)|ψ(x, t)|2 + F(|ψ(x, t)|2) � dx ≡ E(ψ0), t ≥ 0, where the interaction energy density F(ρ) is given as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7) F(ρ) = � ρ 0 f(s)ds = β σ + 1ρσ+1, ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ = 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' f(ρ) = βρ and F(ρ) = β 2 ρ2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) collapses to the well- known nonlinear Schr¨odinger equation with cubic nonlinearity (or smooth nonlin- earity) or the Gross-Pitaevskii equation (GPE), which has been widely adopted for modeling and simulation in quantum mechanics, nonlinear optics and Bose-Einstein condensation [6, 22, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Arising from different physics applications, semi-smooth nonlinearity is introduced in the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' σ is taken as a non-integer in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Typical examples include, in the Schr¨odinger-Poisson-Xα model with f(ρ) = −αρ1/d(α > 0) [13, 15], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' σ = 1 3 and σ = 1 2 in three dimensions (3D) and two dimensions (2D), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' in the LHY correction (a next-order correction of the ground state energy proposed by Lee, Huang and Yang in 1957 [31]) for a beyond-mean-field term which is widely adopted in modeling and simulation for quantum droplets [28, 16, 4, 38, 26] with f(ρ) = ρ3/2 in 3D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' σ = 3 2, f(ρ) = √ρ in one dimension (1D), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' σ = 1 2, and f(ρ) = ρ ln ρ in 2D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' and in the mean field model for Bose-Fermi mixture [23, 17], f(ρ) = ρ2/3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' σ = 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For all the afore- mentioned nonlinearity (actually for σ > 0), the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) is well-posed in H2 [29, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' However, to our best knowledge, there is no guarantee of higher regularity to be propagated due to the low regularity of the semi-smooth nonlinearity, which is similar to the case of the logarithmic Schr¨odinger equation (LogSE) [9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In fact, similar to the LogSE, the low regularity of the solution of the NLSE with semi-smooth nonlinearity is mainly due to the low regularity of the nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For the cubic NLSE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' σ = 1, many accurate and efficient numerical meth- ods have been proposed and analyzed in the last two decades, including the finite difference method [1, 7, 6, 3], the exponential wave integrator [8, 25, 19], the time- splitting method [12, 14, 32, 21, 6, 33, 3], the finite element method [2, 41, 44, 45, 24], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Recently, new low regularity integrators or non-resonance Fourier integrators are designed and analyzed for the cubic NLSE with low regularity initial data since the important work by Ostermann and Schratz [35], followed by [30, 34, 40, 37, 36] and references therein for different dispersive partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For all ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 3 these numerical methods, optimal error bounds were rigorously established under different regularity assumptions of the cubic NLSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Most numerical methods for the cubic NLSE can be extended straightforwardly to solve the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) with non-integer σ > 0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' semi-smooth nonlinearity with 0 < σ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' However, due to the low regularity of solution of the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) with semi-smooth nonlinearity and the low regularity of the semi-smooth nonlinearity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4) in the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) which causes order reduction in local truncation errors, error analysis for different numerical methods for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) with non-integer σ > 0 is a very subtle and challenging question!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For example, first order temporal convergence of the finite difference method requires boundedness of the second- order time derivative, which roughly requires the exact solution to be in H4, which is beyond the regularity property of the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) with semi-smooth nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In fact, based on our numerical experiments with a smooth initial datum ψ0(x) = xe−x2/2, it indicates that ψ(·, t) ̸∈ H4 for t > 0 and σ small!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Since the time-splitting methods usually need lower regularity requirements on the exact solution than the finite difference methods, in this work, we consider the time-splitting method and in particular the first-order Lie-Trotter splitting method due to the low regularity of the semi-smooth nonlinearity and the low regularity of the exact solution of the (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Error estimates of the time-splitting methods with different orders for the cubic NLSE i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' σ = 1 have been well understood and we refer the readers to [32, 21, 33, 3] and therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' However, for the NLSE with non-integer σ, only limited results are established for the filtered Lie-Trotter splitting scheme which requires a coupling constraint between τ and h at τ = O(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In [27], first order convergence in L2- norm is established for H2-solution and σ ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then generalized in [20], half order convergence in L2-norm is established for H1-solution and σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' These convergence rates are optimal with respect to the regularity assumptions on the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' However, there are still some questions related to error estimates to be addressed: (i) it is unclear whether higher convergence order can be obtained for H2-solution when 0 < σ < 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (ii) their results are established for the filtered Lie-Trotter scheme, which is a semi-discretization scheme with a specific coupling constraint between τ and h and it loses mass conservation and time symmetric property in the discretizaed level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' and (iii) there is no optimal error estimate in H1-norm, which is the natural norm of the NLSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The main aim of this paper is to establish error estimates for the time-splitting sine pseudospectral (TSSP) method (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='13) for the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) with semi-smooth nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We remark here that the TSSP is a full-discretization scheme and it preserves many good properties of the original NLSE in the discretized level, including mass conservation and time symmetric as well as dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When 0 < σ ≤ 1 2, under the assumption of H2-solution of the NLSE, we prove error bounds at O(τ 1 2 +σ + h1+2σ) in L2-norm without any coupling condition between the time step size τ and the mesh size h, which fills the gap between the results in [27, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1/2, under the assumption of H2-solution again, we prove error bounds at O(τ + h2) and O(τ 1 2 + h) in L2-norm and H1-norm, respectively, with a very mild coupling condition between τ and h, which generalize the result in [27] to mass-conservative full discretization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In addition, when 1 2 < σ < 1 and under the assumption of H3-solution, we show a new error bound at O(τ σ + h2σ) in H1-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 4 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In Section 2, we present the time- splitting sine pseudospectral (TSSP) method, introduce a local regularization for the semi-smooth nonlinearity to be used for obtaining improved local truncation errors and state our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Section 3 is devoted to error estimates of the TSSP method for 0 < σ ≤ 1/2 and Section 4 is devoted to error estimates for σ ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Numerical results are reported in Section 5 to confirm the error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Finally some conclusions are drawn in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Throughout the paper, we adopt the standard Sobolev spaces as well as the corresponding norms, and denote by C a generic positive constant independent of the mesh size h, time step τ, and by C(c) a generic positive constant depending on c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The notation A ≲ B is used to represent that there exists a generic constant C > 0, such that |A| ≤ CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Numerical methods and main results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The TSSP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We shall use the Lie-Trotter splitting method for the temporal discretization and use the sine pseudospectral method for the spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The operator splitting technique is based on the decomposition of the flow of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) ∂tψ = A(ψ) + B(ψ), where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2) A(ψ) = i∆ψ, B(ψ) = B1(ψ) + B2(ψ) := −iV ψ − if(|ψ|2)ψ, into two sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The first one is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3) �∂tψ(x, t) = A(ψ) = i∆ψ(x, t), x ∈ Ω, t > 0, ψ(x, 0) = ψ0(x), x ∈ Ω, which can be formally integrated exactly in time as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4) ψ(·, t) = eit∆ψ0(·), t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The second one is to solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5) � ∂tψ(x, t) = B(ψ) = −iV (x)ψ(x, t) − if(|ψ(x, t)|2)ψ(x, t), t > 0, ψ(x, 0) = ψ0(x), x ∈ Ω, which, by using the fact |ψ(x, t)| = |ψ0(x)| for t ≥ 0, can be integrated exactly in time as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6) ψ(x, t) = Φt B(ψ0) := e−itV (x)Φt B2(ψ0(x)), x ∈ Ω, t ≥ 0, where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7) Φt B2(z) = ze−itf(|z|2), z ∈ C, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Choose a time step size τ > 0, denote time steps as tk = kτ for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=', and let ψ[k] := ψ[k](x) be the approximation of ψ(x, tk) for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then a first order semi-discretization of the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) via the Lie-Trotter splitting is given as: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8) ψ[k+1] = eiτ∆Φτ B(ψ[k]), with ψ[0](x) = ψ0(x) for x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then we discretize (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8) in space by the sine pseudospectral method to obtain a full discretization for the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For simplicity of notations, here we only present the spatial discretization in 1D (taking Ω = (a, b)), and the generalization to higher dimensions is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Choose ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 5 a mesh size h = (b − a)/N with N being a positive integer and denote grid points as xj = a + jh, j = 0, 1, · · · , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Define the index sets TN = {1, 2, · · · , N − 1}, T 0 N = {0, 1, · · · , N}, and denote XN = span {sin(µl(x − a)) : l ∈ TN} , µl = πl b − a, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9) YN = � v = (v0, v1, · · · , vN)T ∈ CN+1 : v0 = vN = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='10) We define the lp(1 ≤ p ≤ ∞) norm on YN as ∥v∥lp = � �h N−1 � j=0 |vj|p � � 1 p , 1 ≤ p < ∞, ∥v∥l∞ = max 0≤j≤N−1 |vj|, v ∈ YN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We shall sometimes identify a function φ(·) ∈ C0(Ω) with a vector φ = (φ0, φ1, · · · , φN)T ∈ YN with φj = φ(xj) and then the discrete norm ∥ · ∥lp can also be defined on XN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For v ∈ YN, we define the forward finite difference operator as (δ+ x v)j = δ+ x vj = vj+1 − vj h , 0 ≤ j ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let PN : L2(Ω) → XN be the standard L2 projection onto XN and IN : YN → XN be the standard sine interpolation operator as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='11) (PNv)(x) = � l∈TN �vl sin(µl(x − a)), (INw)(x) = � l∈TN �wl sin(µl(x − a)), x ∈ Ω = [a, b], where v ∈ L2(Ω), w ∈ YN, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='12) �vl = 2 b − a � b a v(x) sin(µl(x − a))dx, �wl = 2 N � j∈TN wj sin(jπl/N), l ∈ TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let ψk j be the numerical approximations of ψ(xj, tk) for j ∈ T 0 N and k ≥ 0, and denote ψk := (ψk 0, ψk 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' , ψk N)T ∈ YN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then the time-splitting sine pseudospectral (TSSP) method for discretizing the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) can be given for k ≥ 0 as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='13) ψ(1) j = e−iτ(V (xj)+f(|ψn j |2))ψk j , ψk+1 j = � l∈TN e−iτµ2 l � (ψ(1))l sin(µl(xj − a)), j ∈ T 0 N where ψ0 j = ψ0(xj) for j ∈ T 0 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let Φτ : XN → XN be the numerical integrator defined as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='14) Φτ(φ) = eiτ∆INΦτ B(φ), φ ∈ XN, 6 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG where Φτ B is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='15) INψk+1 = Φτ(INψk), k ≥ 0, INψ0 = INψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In applications, the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) can also be discretized by the Lie- Trotter splitting via a different order as: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='16) ψ[k+1] = Φτ B(eiτ∆ψ[k]), k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then a full discretization can be obtained straightforward by using the sine pseu- dospectral method in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' A local regularization for f(ρ) = βρσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When 0 < σ < 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4), f(ρ) is a semi-smooth function and it is not differentiable at ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Following the regularization methods used in [11] for the logarithmic Schr¨odinger equation, we regularize the semi-smooth nonlinearity f(ρ) only locally in a small region near ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Take 0 < ε ≪ 1 as a regularization parameter, we approximate f(ρ) locally in the region {ρ < ε2} by a polynomial and leave it unchanged in {ρ ≥ ε2}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17) fε(ρ) = � f(ρ), ρ ≥ ε2 ρQε(ρ), 0 ≤ ρ < ε2, where Qε(ρ) is a polynomial with degree at most 3 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='18) fε ∈ C3([0, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Note that fε given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17) is uniquely determined by the interpolation conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='18) and it satisfies fε(0) = f(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Actually, the explicit formula of Qε(ρ) can be given as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19) Qε(ρ) = βε2σ−2 3 � k=0 �k − σ k � � 1 − ρ ε2 �k , 0 ≤ ρ < ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In fact, fε ∈ C3([0, ∞)) can be regarded as a local regularization of the semi- smooth nonlinearity f(ρ) ∈ C0([0, ∞)), which has much better regularity near ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For fε, we have the following estimates Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Assume 0 < σ < 1, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20) |fε(ρ)| + |ρf ′ ε(ρ)| ≤ C1ρσ, ρ ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='21) |√ρf ′ ε(ρ)| + |ρ 3 2 f ′′ ε (ρ)| ≤ C2 � � � � � 1 ε1−2σ , 0 ≤ σ ≤ 1 2, ρσ− 1 2 , 1 2 < σ < 1, , ρ ≥ 0, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='22) |f ′ ε(ρ)| + |ρf ′′ ε (ρ)| + |ρ2f ′′′ ε (ρ)| ≤ C3 ε2−2σ , ρ ≥ 0, where C1, C2 and C3 depend exclusively on σ and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When ρ ≥ ε2, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17), we have f (k)(ρ) = f (k) ε (ρ) for 0 ≤ k ≤ 3, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='22) follows immediately from f(ρ) = βρσ and 0 < ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In the following, we assume that 0 ≤ ρ < ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19), we easily obtain that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='23) |Q(k) ε (ρ)| ≲ ε2σ−2−2k, 0 ≤ ρ < ε2, 0 ≤ k ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 7 From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17), using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='23), one gets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='24) |fε(ρ)| ≤ ρ|Qε(ρ)| ≲ ρε2σ−2 = ρσ � ρ ε2 �1−σ ≤ ρσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Similarly, one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='25) |ρf ′ ε(ρ)| ≤ |ρ2Q′ ε(ρ)| + |ρQε(ρ)| ≲ � ρ ε2 + 1 � ρε2σ−2 ≤ 2ρσ, which proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17), using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='23), one gets, when 0 < σ ≤ 1/2, |√ρf ′ ε(ρ)| ≲ ρ 1 2 � |Qε(ρ)| + ρ|Qε ′(ρ)| � ≤ ε � ε2σ−2 + ε2ε2σ−4� ≲ ε2σ−1, and when 1/2 < σ < 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='26) |√ρf ′ ε(ρ)| ≲ ρ 1 2 � |Qε(ρ)| + ρ|Qε ′(ρ)| � ≲ ρ 1 2 ε2σ−2 = ρσ− 1 2 � ρ ε2 �1−σ ≤ ρσ− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The estimate of |ρ 3 2 f ′′ ε (ρ)| can be obtained similarly, which completes the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='22), using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='23) again, one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='27) |f ′ ε(ρ)| ≤ |Qε(ρ)| + ρ|Q′ ε(ρ)| ≲ ε2σ−2 + ε2ε2σ−4 ≲ ε2σ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The estimate of |ρf ′′ ε (ρ)| and |ρ2f ′′′ ε (ρ)| can be obtained similarly, which completes the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Assume 0 < σ ≤ 1/2, we have ∥fε(|v|2)v∥L2 ≤ C1(∥v∥L∞)∥v∥L2, v ∈ L∞(Ω), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='28) ∥fε(|v|2)v∥H1 ≤ C2(∥v∥L∞)∥v∥H1, v ∈ H1(Ω) ∩ L∞(Ω), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='29) ∥fε(|v|2)v∥H2 ≤ C3 (∥v∥H2) ε1−2σ , v ∈ H2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='30) Assume 0 < σ < 1, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='31) ∥fε(|v|2)v∥H3 ≤ C4 (∥v∥H3) ε2−2σ , v ∈ H3(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20), one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='32) ∥fε(|v|2)v∥L2 ≤ ∥fε(|v|2)∥L∞∥v∥L2 ≲ ∥v∥2σ L∞∥v∥L2, which proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' By direct calculation, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20), one gets ��∇ � fε(|v|2)v ��� L2 = ��fε(|v|2)∇v + f ′ ε(|v|2)v(v∇v + v∇v) �� L2 ≤ � ∥fε(|v|2)∥L∞ + ∥f ′ ε(|v|2)v2∥L∞ + ∥f ′ ε(|v|2)|v|2∥L∞� ∥∇v∥L2 ≲ ∥v∥2σ L∞∥v∥H1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='33) which shows (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' To show (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='30), we note that ∂jk(fε(|v|2)v) = ∂j � (fε(|v|2) + f ′ ε(|v|2)|v|2)∂kv + f ′ ε(|v|2)v2∂kv � = (2f ′ ε(|v|2) + f ′′ ε (|v|2)|v|2)∂j|v|2∂kv + (fε(|v|2) + f ′ ε(|v|2)|v|2)∂jkv +f ′′ ε (|v|2)v2∂j|v|2∂kv + 2f ′ ε(|v|2)v∂jv∂kv + f ′ ε(|v|2)v2∂jkv, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='34) where ∂j = ∂xj and ∂jk = ∂xj∂xk for 1 ≤ j, k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Here we adopt the notations x = x1 (or x) when d = 1, x = (x1, x2)T (or (x, y)T ) when d = 2, and x = (x1, x2, x3)T 8 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG (or (x, y, z)T ) when d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='21) and noting that |∂j|v|2| ≤ 2|v| |∂jv|, one gets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='35) ��∂jk(fε(|v|2)v) �� ≲ � f ′ ε(|v|2)|v| + f ′′ ε (|v|2)|v|3� |∂jv| |∂kv| + � fε(|v|2) + f ′ ε(|v|2)|v|2� |∂jkv| ≲ |∂jv| |∂kv| ε1−2σ + |v|2σ|∂jkv|, which, by using H¨older’s inequality and Sobolev embedding, yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='36) ∥∂jk(fε(|v|2)v)∥L2 ≲ ∥∂jv∥L4∥∂kv∥L4 ε1−2σ + ∥v∥2σ L∞∥∂jkv∥L2 ≤ C(∥v∥H2) ε1−2σ , which implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Following Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2, noting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='35) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='36) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='22), we can similarly obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='31) and the details are omitted here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Assume 0 < σ < 1, we have |f(ρ) − fε(ρ)| ≤ Cε2σ1ρ<ε2, ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='37) |f(ρ) − fε(ρ)| = 0, ρ ≥ ε2, and, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='38) |f(ρ) − fε(ρ)| ≤ |f(ρ)| + |fε(ρ)| ≲ ρσ ≤ ε2σ, 0 ≤ ρ < ε2, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let Tmax be the maximal existing time for the solution of the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3) and take 0 < T < Tmax be a fixed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Based on the known existence and regularity results in [29, 18] for the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1), we make the assumption that the solution ψ satisfies ψ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' H1 0(Ω) ∩ H2(Ω)) ∩ C1([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' L2(Ω)) such that (A) ∥ψ∥L∞([0,T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='H2) + ∥∂tψ∥L∞([0,T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='L2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='39) M2 := max � ∥ψ∥L∞([0,T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='H2), ∥ψ∥L∞([0,T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='L∞), ∥V ∥H2� , and assume the following coupling condition between τ and h < 1 (B) τ ≲ � � � � � � � 1, d = 1, 1 | ln h|2 , d = 2, h, d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For the TSSP method (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='13), we can establish the following error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When 0 < σ ≤ 1/2, assume V ∈ H2(Ω) and under the assumption (A), for 0 < τ < 1 and 0 < h < 1, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='40) ∥ψ(·, tk) − INψk∥L2 ≲ τ 1/2+σ + h1+2σ, 0 ≤ k ≤ T τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 9 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When d = 1 and 0 < σ ≤ 1/2, under the following much weaker assumption V ∈ H1(Ω), ψ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' H1 0(Ω)), we have for 0 < τ < 1 and 0 < h < 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='41) ∥ψ(·, tk) − INψk∥L2 ≲ τ 1/2 + h, 0 ≤ k ≤ T τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1/2, assume V ∈ H2(Ω) ∩ W 1,∞(Ω) and under the assumption (A), there exist τ0 > 0 and h0 > 0 sufficiently small and depending on M2, ∥V ∥W 1,∞ and T such that for τ ≤ τ0 and h ≤ h0 satisfying the coupling condition (B), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='42) ∥ψ(·, tk) − INψk∥L2 ≲ τ + h2, ∥ψk∥l∞ ≤ 1 + M2 ∥ψ(·, tk) − INψk∥H1 ≲ τ 1 2 + h, 0 ≤ k ≤ T τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Moreover, when 1/2 < σ < 1, under the additional assumptions that V ∈ H3(Ω), ∇V ∈ H1 0(Ω) and ψ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' H3 ∗(Ω)) ∩ C1([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' H1(Ω)), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='43) ∥ψ(·, tk) − INψk∥H1 ≲ τ σ + h2σ, 0 ≤ k ≤ T τ , where H3 ∗(Ω) := {φ ∈ H3(Ω) | φ(x)|∂Ω = ∆φ(x)|∂Ω = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1, under the same assumptions as those for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='43), one can obtain the following error bound for the TSSP method (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='13) as ∥ψ(·, tk) − INψk∥H1 ≲ τ + h2, 0 ≤ k ≤ T τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5 for the case 0 < σ ≤ 1/2 Throughout this section, we assume that V ∈ H2(Ω), 0 < σ ≤ 1/2 and the assumption (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Some estimates for the operator B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For the operator B defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2), we have Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let v ∈ H1(Ω) such that ∥v∥L∞ ≤ M, then for σ > 0, we have ∥B(v)∥L2 ≤ C1(M, ∥V ∥L∞)∥v∥L2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) ∥B(v)∥H1 ≤ ∥v∥H1 � C2(M, ∥V ∥H1), d = 1, C2(M, ∥V ∥W 1,4), d = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From the definition of B in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3) ∥B(v)∥L2 ≤ ∥V ∥L∞∥v∥L2 + C(∥v∥L∞)∥v∥L2, which implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Introduce a continuous function G : C → C as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4) G(z) = � f ′(|z|2)z2 = βσ|z|2σ−2z2, z ̸= 0, 0, z = 0, z ∈ C, and identify f ′(|z|2)|z|2 with σf(|z|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5) f(|z|2) + |G(z)| ≲ |z|2σ, z ∈ C, σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 10 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG Direct calculation yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6) ∇B(v) = −i � V ∇v + v∇V + f(|v|2)∇v + f ′(|v|2)v(v∇v + v∇v) � = −i � V ∇v + v∇V + (1 + σ)f(|v|2)∇v + G(v)∇v � , where G(v)(x) := G(v(x)) for x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6), using H¨older’s inequality and noticing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7) ∥∇B(v)∥L2 ≲ ∥V ∥L∞∥∇v∥L2 + ∥v∥2σ L∞∥∇v∥L2 + � ∥v∥L∞∥∇V ∥L2, d = 1, ∥v∥L4∥∇V ∥L4, d = 2, 3, , where different estimates are used for v∇V for d = 1 and d = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Thus we have, by Sobolev embedding, ∥∇B(v)∥L2 ≤ C(∥v∥L∞)∥v∥H1 + ∥v∥H1 � C(∥V ∥H1), d = 1, C(∥V ∥W 1,4), d = 2, 3, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let v, w ∈ L∞(Ω) such that ∥v∥L∞ ≤ M and ∥w∥L∞ ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For any σ > 0, we have ∥B(v) − B(w)∥L2 ≤ C(M, ∥V ∥L∞)∥v − w∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8) ∥B(v) − B(w)∥L2 ≤ ∥V ∥L∞∥v − w∥L2 + ∥f(|v|2)v − f(|w|2)w∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For any z1, z2 ∈ C, let zθ = (1 − θ)z1 + θz2 and let γ(θ) = f(|zθ|2)zθ for 0 ≤ θ ≤ 1, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9) f(|z2|2)z2 − f(|z1|2)z1 = γ(1) − γ(0) = � 1 0 γ′(θ)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Recalling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='10) γ′(θ) = (1 + σ)f(|zθ|2)(z2 − z1) + G(zθ)(z2 − z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='10) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9), noticing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='11) |f(|z1|2)z1 − f(|z2|2)z2| ≤ sup 0≤θ≤1 |γ′(θ)| ≲ max{|z1|, |z2|}2σ|z1 − z2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Thus we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='12) ∥f(|v|2)v − f(|w|2)w∥L2 ≤ C(max{∥v∥L∞, ∥v∥L∞})∥v − w∥L2, which combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Let dB(·)[·] be the Gˆateaux derivative defined as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='13) dB(v)[w] := lim ε→0 B(v + εw) − B(v) ε , then we have Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let v ∈ L∞(Ω) such that ∥v∥L∞ ≤ M and w ∈ L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ > 0, we have ∥dB(v)[w]∥L2 ≤ C(M, ∥V ∥L∞)∥w∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Plugging (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='13), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='14) dB(v)[w] = −iV w + dB2(v)[w] = −i � V w + (1 + σ)f(|v|2)w + G(v)w � , where G is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='14), noting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5), we have ∥dB(v)[w]∥L2 ≤ ∥V ∥L∞∥w∥L2 + C(∥v∥L∞)∥w∥L2, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When 0 < σ ≤ 1/2, we have |Φτ B2(z1) − Φτ B2(z2)| ≤ (1 + Cτ) |z1 − z2|, ∀z1, z2 ∈ C, where C = 2σ|β| min{|z1|, |z2|}2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Recall that Φτ B2(z) = ze−iτf(|z|2) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Without loss of generality, we assume that |z2| ≤ |z1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' If z2 = 0, the conclusion follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In the following, we assume that z2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then, by noting that |1 − eiθ| ≤ |θ| for all θ ∈ R, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='15) |Φτ B2(z1) − Φτ B2(z2)| = |z1e−iτf(|z1|2) − z2e−iτf(|z2|2)| ≤ |z1 − z2| + |z2| ���1 − e−iτ(f(|z1|2)−f(|z2|2))��� ≤ |z1 − z2| + τ|z2| ��f(|z1|2) − f(|z2|2) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When 0 < σ ≤ 1/2, since 0 < |z2| ≤ |z1|, by the mean value theorem and the definition of f in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='16) ��f(|z1|2) − f(|z2|2) �� = |β| ��|z1|2σ − |z2|2σ�� ≤ 2σ|β||z1 − z2| min{|z1|, |z2|}1−2σ = 2σ|β||z1 − z2| |z2|1−2σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='16) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='15), we get the desired result immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Local truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let φ ∈ XN such that ∥φ∥H2 ≤ M and let 0 < τ < 1 and 0 < h < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Assume that V ∈ H2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When 0 < σ ≤ 1/2, we have ∥(I − eiτ∆)PNB(φ)∥L2 ≤ C1(M, ∥V ∥H2)τ 1/2+σ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17) ∥INB(φ) − PNB(φ)∥L2 ≤ C2(M, ∥V ∥H2)h1+2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Recalling the well-known facts that (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=', [42, 8, 10]) ∥v − PNv∥L2 ≲ h2|v|H2, ∥INv − PNv∥L2 ≲ h2|v|H2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19) ∥v − eit∆v∥L2 ≲ t∥v∥H2, v ∈ H1 0(Ω) ∩ H2(Ω), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20) noting that H2(Ω) is an algebra when 1 ≤ d ≤ 3, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='21) ∥(I − eiτ∆)(V φ)∥L2 ≲ τ∥V ∥H2∥φ∥H2, ∥(IN − PN)(V φ)∥L2 ≲ h2∥V ∥H2∥φ∥H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2), it remains to show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='18) with f(|φ|2)φ replacing B(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Using the regularized function fε defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17) with 0 < ε ≪ 1 and the 12 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG triangle inequality, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='22) ∥(I − eiτ∆)(f(|φ|2)φ)∥L2 ≤ ∥(I − eiτ∆)(f(|φ|2)φ − fε(|φ|2)φ)∥L2 + ∥(I − eiτ∆)(fε(|φ|2)φ)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='22), using ∥(I − eiτ∆)v∥L2 ≤ 2∥v∥L2 for the first term and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20) for the second term, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='23) ∥(I − eiτ∆)(f(|φ|2)φ)∥L2 ≤ 2∥f(|φ|2)φ − fε(|φ|2)φ∥L2 + τ∥fε(|φ|2)φ∥H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='30), we have ∥f(|φ|2)φ − fε(|φ|2)φ∥L2 ≲ ε2σ∥φ1|φ|<ε∥L2 ≤ |Ω| 1 2 ε1+2σ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='24) ∥fε(|φ|2)φ∥H2 ≤ C(M) ε1−2σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='25) Plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='24) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='25) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='23), we have ∥(I − eiτ∆)(f(|φ|2)φ)∥L2 ≤ C(M) inf 0<ε<1 � ε1+2σ + τ ε1−2σ � ≤ C(M)τ 1/2+σ, which combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='21) yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then we shall prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='23), using the triangle inequality, the L2-projection property of PN, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='24), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='30), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='26) ∥(IN − PN)(f(|φ|2)φ)∥L2 ≤ ∥(IN − PN)(f(|φ|2)φ − fε(|φ|2)φ)∥L2 + ∥(IN − PN)(fε(|φ|2)φ)∥L2 ≤ ∥IN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 + ∥PN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 + h2∥fε(|φ|2)φ∥H2 ≤ ∥IN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 + |Ω| 1 2 ε1+2σ + h2 C(M) ε1−2σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' By Parseval’s identity, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='27) ∥INv∥L2 = � � � �h N−1 � j=1 |v(xj)|2 ≤ � � � �h N−1 � j=1 ∥v∥2 l∞ ≤ |Ω| 1 2 ∥v∥l∞, v ∈ C0(Ω), which implies, by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4 again, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='28) ∥IN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 ≤ |Ω| 1 2 ∥(f(|φ|2) − fε(|φ|2))φ∥l∞ ≤ |Ω| 1 2 ε2σ∥φ1|φ|<ε∥l∞ ≤ |Ω| 1 2 ε1+2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='28) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='26), we have ∥(IN − PN)(f(|φ|2)φ)∥L2 ≤ C(M) inf 0<ε<1 � ε1+2σ + h2 ε1−2σ � ≤ C(M)h1+2σ, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Then we are able to show the local truncation error of the TSSP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6 (local truncation error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Assume that V ∈ H2 and under the assumption (A), for 0 ≤ k ≤ T/τ − 1, we have ∥PNψ(·, tk+1) − Φτ(PNψ(·, tk))∥L2(Ω) ≤ C(M2)τ � τ 1 2 +σ + h1+2σ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For the simplicity of notations, we define v(t) := ψ(·, tk + t) for 0 ≤ t ≤ τ and v0 := v(0) = ψ(·, tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' By the Sobolev embedding theorem, noting that eit∆ preserves the Hk-norm and PN doesn’t increase the Hk-norm for k ≥ 0, we have ∥eis∆v(t)∥L∞ ≲ ∥eis∆v(t)∥H2 = ∥v(t)∥H2 ≤ M2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='29) ∥PNv(t)∥L∞ ≲ ∥PNv(t)∥H2 ≤ ∥v(t)∥H2 ≤ M2, 0 ≤ s, t ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='30) It is well-known that (see [32, 11]) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='31) ψ(tk+1) = eiτ∆v0 + � τ 0 ei(τ−s)∆B(eis∆v0)ds + � τ 0 � s 0 ei(τ−s)∆dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))]dσds, where dB(·)[·] is the Gˆateaux derivative defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Applying PN on both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='31), one gets PNψ(tk+1) = eiτ∆PNv0 + � τ 0 ei(τ−s)∆PNB(eis∆v0)ds + � τ 0 � s 0 ei(τ−s)∆PN � dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] � dσds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='32) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6), recalling that v0 = ψ(tk), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='33) Φτ(PNψ(tk)) = eiτ∆INΦτ B(PNv0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Applying the first-order Taylor expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='34) Φτ B(w) = w + τB(w) + τ 2 � 1 0 (1 − θ)dB(Φθτ B (w))[B(Φθτ B (w))]dθ for w = PNv0 and plugging it into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='33), we have Φτ(PNψ(tk)) = eiτ∆PNv0 + τeiτ∆INB(PNv0) +τ 2eiτ∆IN �� 1 0 (1 − θ) � dB(Φθτ B (PNv0))[B(Φθτ B (PNv0))] � dθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='35) Subtracting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='35) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='32), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='36) PNψ(tk+1) − Φτ(PNψ(tk)) = e1 − e2 + e3, where e1 = � τ 0 � s 0 ei(τ−s)∆PN � dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] � dσds, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='37) e2 = τ 2eiτ∆IN �� 1 0 (1 − θ) � dB(Φθτ B (PNv0))[B(Φθτ B (PNv0))] � dθ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='38) e3 = � τ 0 ei(τ−s)∆PNB(eis∆v0)ds − τeiτ∆INB(PNv0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='39) 14 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG Next, we shall first estimate e1 and e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Noticing the property of eit∆ and PN, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='29), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='40) ���ei(τ−s)∆PN � dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] ���� L2 ≤ ∥dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))]∥L2 ≤ C(∥V ∥L∞, ∥ei(s−σ)∆v(σ)∥L∞)∥ei(s−σ)∆B(v(σ))∥L2 ≤ C(M2)∥B(v(σ))∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='37), using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='40) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1), we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='41) ∥e1∥L2 ≤ � τ 0 � s 0 ���ei(τ−s)∆PN � dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] ���� L2 dσds ≤ C(M2) � τ 0 � s 0 ∥B(v(σ))∥L2dσds ≤ C(M2)τ 2 max 0≤σ≤τ ∥B(v(σ))∥L2 ≤ C(M2)τ 2C(M2) max 0≤σ≤τ ∥v(σ)∥L2 ≤ C(M2)τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2), using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='30), one gets, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='42) ∥Φθτ B (PNv0)∥L∞ = ∥PNv0∥L∞ ≤ C(M2), 0 ≤ θ ≤ 1, ∥B(Φθτ B (PNv0))∥L∞ ≤ C(∥V ∥L∞, ∥PNv0∥L∞)∥PNv0∥L∞ ≤ C(M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='14), noticing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5), one easily gets (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='43) ∥dB(w1)[w2]∥L∞ ≤ C(∥V ∥L∞, ∥w1∥L∞)∥w2∥L∞, w1, w2 ∈ L∞(Ω), which combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='27) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='42), yields the estimate for e2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='38) as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='44) ∥e2∥L2 ≤ τ 2 ����IN �� 1 0 (1 − θ) � dB(Φθτ B (PNv0))[B(Φθτ B (PNv0))] � dθ ����� L2 ≤ τ 2|Ω| 1 2 max 0≤θ≤1 ��dB(Φθτ B (PNv0))[B(Φθτ B (PNv0))] �� l∞ ≤ τ 2|Ω| 1 2 C � ∥V ∥L∞, ∥Φθτ B (PNv0)∥L∞� ∥B(Φθτ B (PNv0))∥L∞ ≤ C(M2)τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then we shall estimate e3 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='39), which can be written as e3 = � τ 0 � ei(τ−s)∆PNB(eis∆v0) − eiτ∆INB(PNv0) � ds, which yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='45) ∥e3∥L2 ≤ τ max 0≤s≤τ ∥ei(τ−s)∆PNB(eis∆v0) − eiτ∆INB(PNv0)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 15 Using the standard property of eit∆ and PN, one gets (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='46) ∥ei(τ−s)∆PNB(eis∆v0) − eiτ∆INB(PNv0)∥L2 = ∥PNB(eis∆v0) − eis∆INB(PNv0)∥L2 ≤ ∥PNB(eis∆v0) − PNB(v0)∥L2 + ∥PNB(v0) − PNB(PNv0)∥L2 + ∥PNB(PNv0) − eis∆PNB(PNv0)∥L2 + ∥eis∆PNB(PNv0) − eis∆INB(PNv0)∥L2 ≤ ∥B(eis∆v0) − B(v0)∥L2 + ∥B(φ) − B(PNv0)∥L2 + ∥(I − eis∆)PNB(PNv0)∥L2 + ∥(PN − IN)B(PNv0)∥L2 =: ∥e1 3∥L2 + ∥e2 3∥L2 + ∥e3 3∥L2 + ∥e4 3∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For e1 3 and e2 3 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='46), using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2, recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='29), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='30), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='47) ∥e1 3∥L2 = ∥B(eis∆v0) − B(v0)∥L2 ≤ C(M2)∥(I − eis∆)v0∥L2 ≤ C(M2)τ, ∥e2 3∥L2 = ∥B(v0) − B(PNv0)∥L2 ≤ C(M2)∥v0 − PNv0∥L2 ≤ C(M2)h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For e3 3 and e4 3 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='46), using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5, we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='48) ∥e3 3∥L2 = ∥(I − eis∆)PNB(PNv0)∥L2 ≤ C(M2)τ 1+2σ 2 , ∥e4 3∥L2 = ∥(IN − PN)B(PNv0)∥L2 ≤ C(M2)h1+2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='47) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='48) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='46) and noticing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='45), we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='49) ∥e3∥L2 ≤ C(M2)τ � τ 1+2σ 2 + h1+2σ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Combing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='41), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='44), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='49) and noting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='36), we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6 can be generalized to 2D and 3D di- rectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Moreover, in 1D, under much weaker assumption that V ∈ H1(Ω) and ψ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' H1 0(Ω)), by using Sobolev embedding H1 �→ L∞ and the estimate (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=', [10, 11]) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='50) ∥v − eit∆v∥L2 ≲ √τ∥v∥H1, ∥v − PNv∥L2 ≲ h|v|H1, v ∈ H1 0(Ω), and following the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6, we can obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='51) ∥PNψ(tk+1) − Φτ(PNψ(tk))∥L2(Ω) ≤ Cτ �√τ + h � , where C depends on ∥V ∥H1 and ∥ψ∥L∞([0,T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Unconditional L2-stability and proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We shall show the unconditional L2-stability of the numerical flow by using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' With the estimate of the local truncation error and the unconditional L2-stability of the numerical flow, we are able to obtain the error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8 (unconditional L2-stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let v ∈ XN and w ∈ XN such that min{∥v∥L∞, ∥w∥L∞} ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When 0 < σ ≤ 1/2, we have ∥Φτ(v) − Φτ(w)∥L2 ≤ (1 + C(M)τ)∥v − w∥L2, where C(M) ∼ M 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 16 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='14), noting that eiτ∆ preserves the L2 norm, one gets (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='52) ∥Φτ(v) − Φτ(w)∥L2 = ∥eiτ∆INΦτ B(v) − eiτ∆INΦτ B(w)∥L2 = ∥INΦτ B(v) − INΦτ B(w)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='52), by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='27) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4, noting that IN is an identity on XN and recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='53) ∥INΦτ B(v) − INΦτ B(w)∥2 L2 = h N−1 � j=1 |Φτ B(v)(xj) − Φτ B(w)(xj)|2 = h N−1 � j=1 ���e−iτV (xj)Φτ B2(v)(xj) − e−iτV (xj)Φτ B2(w)(xj) ��� 2 = h N−1 � j=1 ��Φτ B2(v)(xj) − Φτ B2(w)(xj) ��2 ≤ (1 + C(M)τ)2h N−1 � j=1 |v(xj) − w(xj)|2 = (1 + C(M)τ)2∥INv − INw∥2 L2 = (1 + C(M)τ)2∥v − w∥2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In the error estimates, v and w in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8 are related to the exact solution and the numerical solution, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Hence, to control the constant C(M) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8, we can assume bound of the exact solution and thus get rid of the a priori estimate of the numerical solution, which explains why Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8 is called the unconditional L2-stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Under the assumption (A) and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19), one gets (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='54) ∥ψ(·, tk) − PNψ(·, tk)∥L2 ≤ C(M2)h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Hence it suffices to estimate ek := INψk − PNψ(·, tk) ∈ XN for 0 ≤ k ≤ T/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='15), for 0 ≤ k ≤ T/τ − 1, one has (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='55) ∥ek+1∥L2 = ∥INψk+1 − PNψ(·, tk+1)∥L2 = ∥Φτ(INψk) − PNψ(·, tk+1)∥L2 ≤ ∥Φτ(INψk) − Φτ(PNψ(·, tk))∥L2 + ∥Φτ(PNψ(·, tk)) − PNψ(·, tk+1)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' By Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' noting that ∥PNψ(·, tk)∥L∞ ≲ ∥PNψ(·, tk)∥H2 ≤ ∥ψ(·, tk)∥H2 ≤ M2, one has ∥ek∥L2(Ω) ≤ eC(M2)τ∥ek−1∥L2(Ω) + C(M2)τ � τ 1/2+σ + h1+2σ� , 1 ≤ k ≤ T/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' It follows from the discrete Gronwall’s inequality and ∥e0∥L2 = ∥INψ0 − PNψ0∥ ≤ C(M2)h2 that ∥ek∥L2(Ω) ≤ C(T, M2) � τ 1+2σ 2 + h1+2σ� , 0 ≤ k ≤ T/τ, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 17 The proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6 follows the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5 by replacing Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6 with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='51) and we shall omit it for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7 for the case σ ≥ 1/2 In this section, we assume that V ∈ H2(Ω) ∩ W 1,∞(Ω), σ ≥ 1/2 and the as- sumption (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The assumption V ∈ W 1,∞(Ω) is only used in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8 and can be obtained from V ∈ H2(Ω) in 1D or V ∈ H3(Ω) in 2D and 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Some estimates for the operator B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let v ∈ H2(Ω) such that ∥v∥H2 ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1/2, we have ∥B(v)∥H2(Ω) ≤ C(M, ∥V ∥H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2), noting that H2(Ω) is an algebra when 1 ≤ d ≤ 3, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) ∥B(v)∥H2 ≤ ∥V v∥H2 + ∥f(|v|2)v∥H2 ≤ ∥V ∥H2∥v∥H2 + ∥f(|v|2)v∥H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1/2, recalling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4), by similar calculation as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='34) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='35) and noting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5) as well as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2) ��f ′(|z|2)z ��+ ��f ′(|z|2)z ��+ ��f ′′(|z|2)z3��+ ��f ′′(|z|2)z2z �� ≲ |z|2σ−1, z ∈ C, σ ≥ 1 2, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3) ��∂jk(f(|v|2)v) �� ≲ |v|2σ|∂jkv| + |v|2σ−1|∂jv| |∂kv|, which yields, by Sobolev embedding, that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4) ∥∂jk(f(|v|2)v)∥L2 ≲ ∥v∥2σ L∞∥∂jkv∥L2 + ∥v∥2σ−1 L∞ ∥∇v∥2 L4 ≤ C(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Combing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1, noting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1), we obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let v, w ∈ H2(Ω) such that ∥v∥H2 ≤ M and ∥w∥H2 ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1/2, we have ∥B(v) − B(w)∥H1 ≤ C(M, ∥V ∥W 1,4)∥v − w∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6), one gets (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5) ∇ (B(v) − B(w)) = −i � ∇V (v − w) + i(1 + σ)(f(|v|2)∇v − f(|w|2)∇w) +G(v)∇v − G(w)∇w] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Using H¨older’s inequality and Sobolev embedding, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6) ∥∇(V (v − w))∥L2 ≤ ∥∇V ∥L4∥v − w∥L4 + ∥V ∥L∞∥∇(v − w)∥L2 ≲ ∥V ∥W 1,4∥v − w∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5), it remains to show that ∥f(|v|2)∇v − f(|w|2)∇w∥L2 ≤ C(M)∥v − w∥H1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7) ∥G(v)∇v − G(w)∇w∥L2 ≤ C(M)∥v − w∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8) When σ ≥ 1/2, following the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='11), we have, for z1, z2 ∈ C, |f(|z1|2) − f(|z2|2)| ≲ max{|z1|, |z2|}2σ−1|z1 − z2|, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9) |G(z1) − G(z2)| ≲ max{|z1|, |z2|}2σ−1|z1 − z2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='10) 18 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9) and Sobolev embedding, we have ∥f(|v|2)∇v − f(|w|2)∇w∥L2 ≤ ∥f(|v|2)∇(v − w)∥L2 + ∥(f(|v|2) − f(|w|2))∇w∥L2 ≤ C(∥v∥L∞)∥v − w∥H1 + C(max{∥v∥L∞, ∥w∥L∞})∥(v − w)∇w∥L2 ≤ C(M)∥v − w∥H1 + C(M)∥v − w∥L4∥∇w∥L4 ≤ C(M)∥v − w∥H1, which proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Similarly, we can prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let v, w ∈ H1(Ω) ∩ L∞(Ω) such that ∥v∥L∞ + ∥v∥H1 ≤ M and ∥w∥L∞ + ∥w∥H1 ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1/2, we have ∥dB(v)[w]∥H1 ≤ C(M, ∥V ∥W 1,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='14), using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='11) ∥dB(v)[w]∥H1 ≤ ∥V w∥H1 + (1 + σ)∥f(|v|2)w∥H1 + ∥G(v)w∥H1 ≲ ∥V ∥W 1,4∥w∥H1 + ∥f(|v|2)w∥H1 + ∥G(v)w∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1/2, recalling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='12) ∥f(|v|2)∥H1 = ∥f(|v|2)∥L2 + ∥∇f(|v|2)∥L2 ≲ ∥v∥2σ L∞ + ∥f ′(|v|2)v∇v∥L2 ≤ ∥v∥2σ L∞ + ∥v∥2σ−1 L∞ ∥∇v∥L2 ≤ C(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Similarly, one gets ∥G(v)∥H1 ≤ C(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='13) ∥u1u2∥H1 ≤ ∥u1∥L∞∥u2∥H1 + ∥u2∥L∞∥u1∥H1, u1, u2 ∈ H1(Ω) ∩ L∞(Ω), and recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5), we have ∥f(|v|2)w∥H1 ≤ ∥f(|v|2)∥L∞∥w∥H1 + ∥w∥L∞∥f(|v|2)∥H1 ≤ C(M), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='14) ∥G(v)w∥H1 ≤ ∥G(v)∥L∞∥w∥H1 + ∥w∥L∞∥G(v)∥H1 ≤ C(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='15) Plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='15) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='11) yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let v, w ∈ H2(Ω) such that ∥v∥H2 ≤ M and ∥w∥H2 ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' If |w(x)| ≤ C|v(x)| for all x ∈ Ω, when σ ≥ 1/2, we have ∥dB(v)[w]∥H2 ≤ C (M, ∥V ∥H2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The proof can be obtained similarly as the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 and we shall omit it here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let 0 < τ < 1 and v ∈ XN such that ∥v∥L∞ ≤ M and ∥v∥H2 ≤ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ > 0, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='16) ∥Φτ B(v)∥H1 ≤ (1 + C1(M, ∥V ∥W 1,4)τ) ∥v∥H1(Ω), and when σ ≥ 1/2, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17) ∥Φτ B(v)∥H2 ≤ C2(M1, ∥V ∥H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Recalling that Φτ B(v) = ve−iτ(V +f(|v|2)) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6), the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='16) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='17) follows similarly from the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 19 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let z1, z2 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1/2, one has |Φτ B2(z1) − Φτ B2(z2)| ≤ (1 + Cτ)|z1 − z2|, where C ∼ max{|z1|, |z2|}2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The proof follows from the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4 by replacing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='16) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Local truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7 (local truncation error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Assume that 0 < τ < 1, 0 < h < 1, V ∈ H2 and σ ≥ 1/2, under the assumption (A), for 0 ≤ k ≤ T/τ − 1, we have ∥PNψ(·, tk+1) − Φτ(PNψ(·, tk))∥L2(Ω) ≤ C1(M2)τ � τ + h2� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='18) ∥PNψ(·, tk+1) − Φτ(PNψ(·, tk))∥H1(Ω) ≤ C2(M2)τ � τ 1 2 + h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Following the notation in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6, we let v(t) = ψ(·, tk + t) for 0 ≤ t ≤ τ and v0 := v(0) = ψ(·, tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1/2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='29) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='30) are also valid and we have the same error decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1/2, the L2 estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='18) follows from the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6 by replacing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='48) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20) ∥e3 3∥L2 ≲ τ∥PNB(PNv0)∥H2 ≤ τ∥B(PNv0)∥H2 ≤ C(M2)τ, ∥e4 3∥L2 ≲ h2∥B(PNv0)∥H2 ≤ C(M2)h2, where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='20), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In the following, we shall show (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Using Sobolev embedding, the isometry property of eit∆ and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1, one gets (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='21) ∥ei(s−σ)∆B(v(σ))∥H1 = ∥B(v(σ))∥H1 ≤ C(M2), ∥ei(s−σ)∆B(v(σ))∥L∞ ≲ ∥ei(s−σ)∆B(v(σ))∥H2 = ∥B(v(σ))∥H2 ≤ C(M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Recalling the property of eit∆ and PN, using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3, noticing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='29) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='21), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='22) ���ei(τ−s)∆PN � dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] ���� H1 ≤ ∥dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))]∥H1 ≤ C(∥V ∥W 1,4, ∥ei(s−σ)∆v(σ)∥L∞∩H1, ∥ei(s−σ)∆B(v(σ))∥L∞∩H1) ≤ C(M2), which yields, for e1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='37), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='23) ∥e1∥H1 ≤ � τ 0 � s 0 ���ei(τ−s)∆PN � dB(ei(s−σ)∆v(σ))[ei(s−σ)∆B(v(σ))] ���� H1 dσds ≤ C(M2)τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For e2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='38), recalling the standard estimate of the interpolation operator [8, 42] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='24) ∥INφ∥H1 ≲ ∥φ∥H1 + h|φ|H2 ≲ ∥φ∥H2, φ ∈ H1 0(Ω) ∩ H2(Ω), one gets (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='25) ∥e2∥H1 = τ 2 ����IN �� 1 0 (1 − θ) � dB(Φθτ B (PNv0))[B(Φθτ B (PNv0))] � dθ ����� H1 ≲ τ 2∥dB(Φθτ B (PNv0))[B(Φθτ B (PNv0))]∥H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 20 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='25), noting that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='26) ��� B(Φθτ B (PNv0)) � (x) �� ≲ ��Φθτ B (PNv0)(x) �� = (PNv0)(x), x ∈ Ω, and using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='27) ∥e2∥H1 ≤ C(M2)τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then we shall estimate e3 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='45) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='46), it suffices for us to bound the H1-norm of the four terms ej 3(1 ≤ j ≤ 4) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Using the standard estimates, ∥φ − PNφ∥H1 ≲ h|φ|H2, ∥INφ − PNφ∥H1 ≲ h|φ|H2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='28) ∥φ − eit∆φ∥H1 ≲ √ t∥φ∥H2, φ ∈ H1 0(Ω) ∩ H2(Ω), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='29) and Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2, we have ∥e1 3∥H1 ≤ C(M2)∥eis∆v0 − v0∥H1 ≤ C(M2)√τ∥v0∥H2 ≤ C(M2)√τ, ∥e2 3∥H1 ≤ C(M2)∥v0 − PNv0∥H1 ≤ C(M2)h∥v0∥H2 ≤ C(M2)h, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='30) ∥e3 3∥H1 ≲ √τ∥PNB(PNv0)∥H2 ≤ √τ∥B(PNv0)∥H2 ≤ C(M2)√τ, ∥e4 3∥H1 ≲ h∥B(PNv0)∥H2 ≤ C(M2)h, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='31) which yields immediately (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='32) ∥e3∥H1 ≤ C(M2)τ �√τ + h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='23), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='27), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='32), we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' l∞-conditional L2- and H1-stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8 (l∞-conditional stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let 0 < τ < 1 and v, w ∈ XN such that ∥v∥l∞ ≤ M, ∥w∥l∞ ≤ M and ∥v∥H2 ≤ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When σ ≥ 1/2, ∥Φτ(v) − Φτ(w)∥L2 ≤ (1 + C1(M)τ)∥v − w∥L2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='33) ∥Φτ(v) − Φτ(w)∥H1 ≤ (1 + C2(M, M1, ∥V ∥W 1,∞)τ)∥v − w∥H1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='34) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The L2-stability (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='33) can be obtained from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='53) by using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6 instead of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In the following, we show the H1-stability (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6) and that eiτ∆ preserves the H1-norm, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='34) reduce to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='35) ∥INΦτ B(v) − INΦτ B(w)∥H1 ≤ (1 + C(M, M1, ∥V ∥W 1,∞)τ)∥v − w∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The proof is based on the following well-known equivalence relation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=', Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2 in [8]) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='36) ∥δ+ x φ∥l2 ≤ ∥∇INφ∥L2 ≤ π 2 ∥δ+ x φ∥l2, which implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='37) ∥INΦτ B(v) − INΦτ B(w)∥H1 ≤ ∥v − w∥H1 + ∥IN(Φτ B(v) − v) − IN(Φτ B(w) − w)∥H1 ≤ ∥v − w∥H1 + π 2 ∥δ+ x (Φτ B(v) − v) − δ+ x (Φτ B(w) − w)∥l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We define vθ j = (1 − θ)vj + θvj+1, wθ j = (1 − θ)wj + θwj+1, V θ j = (1 − θ)V (xj) + θV (xj+1), 0 ≤ θ ≤ 1, j = 0, · · · , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 21 By some elementary computation, recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4) and f ′(|z|2)|z|2 = σf(|z|2), one gets, for 0 ≤ j ≤ N − 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='38) δ+ x (Φτ B(v) − v)j = δ+ x � v(e−iτ(V +f(|v|2)) − 1) � j = 1 h � 1 0 d dθ � vθ j (e−iτ(V θ j +f(|vθ j |2)) − 1) � dθ = � 1 0 δ+ x vj(e−iτ(V θ j +f(|vθ j |2)) − 1)dθ − iτ � 1 0 e−iτV θ j vθ j δ+ x V (xj)e−iτf(|vθ j |2)dθ − iτ � 1 0 e−iτV θ j (σf(|vθ j |2)δ+ x vj + G(vθ j )δ+ x vj)e−iτf(|vθ j |2)dθ, Similarly, for 0 ≤ j ≤ N − 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='39) δ+ x (Φτ B(w) − w)j = � 1 0 δ+ x wj(e−iτ(V θ j +f(|wθ j |2)) − 1)dθ − iτ � 1 0 e−iτV θ j wθ jδ+ x V (xj)e−iτf(|wθ j |2)dθ − iτ � 1 0 e−iτV θ j (σf(|wθ j|2)δ+ x wj + G(wθ j)δ+ x wj)e−iτf(|wθ j |2)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We define the function e ∈ YN with ej = vj − wj, j = 0, · · · , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Subtracting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='39) from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='38), for 0 ≤ j ≤ N − 1, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='40) ��δ+ x (Φτ B(v) − v)j − δ+ x (Φτ B(w) − w)j �� ≤ � 1 0 ����δ+ x vj(e−iτ(V θ j +f(|vθ j |2)) − 1) − δ+ x wj(e−iτ(V θ j +f(|wθ j |2)) − 1) ��� + τ ��δ+ x V (xj) �� ���vθ j e−iτf(|vθ j |2) − wθ je−iτf(|wθ j |2)��� + στ ���δ+ x vjf(|vθ j |2)e−iτf(|vθ j |2) − δ+ x wjf(|wθ j|2)e−iτf(|wθ j |2)��� +τ ���δ+ x vjG(vθ j )e−iτf(|vθ j |2) − δ+ x wjG(wθ j)e−iτf(|wθ j |2)��� � dθ =: � 1 0 � J1 j + J2 j + J3 j + J4 j � dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For J1 j , by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9), one gets (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='41) J1 j ≤ ��δ+ x vj �� ���e−iτ(V θ j +f(|vθ j |2)) − e−iτ(V θ j +f(|wθ j |2))��� + ��δ+ x vj − δ+ x wj �� ���e−iτ(V θ j +f(|wθ j |2)) − 1 ��� ≤ τ ��δ+ x vj �� ��f(|vθ j |2) − f(|wθ j|2) �� + τ ��V θ j + f(|wθ j|2) �� ��δ+ x vj − δ+ x wj �� ≤ C(M)τ ��δ+ x vj �� (|ej| + |ej+1|) + C(M, ∥V ∥L∞)τ|δ+ x ej|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For J2 j , recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7), by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6 and 0 < τ < 1, one gets (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='42) J2 j = τ ��δ+ x V (xj) �� ��Φτ B2(vθ j ) − Φτ B2(wθ j) �� ≤ τ ��δ+ x V (xj) �� (1 + C(M)τ)(|ej| + |ej+1|) ≤ C(M)τ ��δ+ x V (xj) �� (|ej| + |ej+1|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 22 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG For J3 j , by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9) and 0 < τ < 1, one gets (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='43) J3 j ≲ τ ��δ+ x vj �� ���f(|vθ j |2)e−iτf(|vθ j |2) − f(|wθ j|2)e−iτf(|wθ j |2)��� + τ ��δ+ x vj − δ+ x wj �� ���f(|wθ j|2)e−iτf(|wθ j |2)��� ≤ τ ��δ+ x vj �� ���f(|vθ j |2) − f(|wθ j|2) �� + |f(|wθ j|2)| ���e−iτf(|vθ j |2) − e−iτf(|wθ j |2)��� � + τC(M) ��δ+ x vj − δ+ x wj �� ≤ τ ��δ+ x vj �� C(M)(1 + τ)|vθ j − wθ j| + τC(M) ��δ+ x vj − δ+ x wj �� ≤ C(M)τ ��δ+ x vj �� (|ej| + |ej+1|) + C(M)τ ��δ+ x ej �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='43), using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='10) instead of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9), one gets, for J4 j , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='44) J4 j ≤ C(M)τ ��δ+ x vj �� (|ej| + |ej+1|) + C(M)τ ��δ+ x ej �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='41)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='44) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='40), we have ��δ+ x (Φτ B(v) − v)j − δ+ x (Φτ B(w) − w)j �� ≤ C(M)τ ���δ+ x V (xj) �� + ��δ+ x vj ��� (|ej| + |ej+1|) + C(M, ∥V ∥L∞)τ ��δ+ x ej �� , which yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='45) ∥δ+ x (Φτ B(v) − v) − δ+ x (Φτ B(w) − w)∥2 l2 = h N−1 � j=0 ��δ+ x (Φτ B(v) − v)j − δ+ x (Φτ B(w) − w)j ��2 ≤ C(M)τ 2h N−1 � j=0 ���δ+ x V (xj) ��2 + ��δ+ x vj ��2� (|ej|2 + |ej+1|2) + C(M, ∥V ∥L∞)τ 2h N−1 � j=0 ��δ+ x ej ��2 ≤ C(M)τ 2 � �∥∇V ∥2 L∞∥e∥2 l2 + h N−1 � j=0 ��δ+ x vj ��2 (|ej|2 + |ej+1|2) � � + C(M, ∥V ∥L∞)τ 2∥δ+ x e∥2 l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When d = 1, one has |δ+ x vj| ≤ ∥∇v∥L∞ ≤ C(M1), which yields directly that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='46) h N−1 � j=0 ��δ+ x vj ��2 (|ej|2 + |ej+1|2) ≤ C(M1)∥e∥2 l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' However, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='46) cannot be directly generalized to 2D and 3D without assuming higher regularity on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Here, we present an alternative approach that can be gener- alized to 2D and 3D (see also Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Using the discrete Gargliardo-Nireberg inequality and the discrete Poincare’s inequality (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 of [7]), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='47) ∥φ∥l4 ≲ ∥φ∥ 3 4 l2∥δ+ x φ∥ 1 4 l2 ≲ ∥δ+ x φ∥l2, φ ∈ YN, ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 23 which implies, by first applying H¨older’s inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='46), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='48) h N−1 � j=0 ��δ+ x vj ��2 (|ej|2 + |ej+1|2) ≲ ∥δ+ x v∥2 l4∥e∥2 l4 ≲ ∥δ+ x v∥2 l4∥δ+ x e∥2 l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Using the following discrete version of the Sobolev embedding H2 �→ W 1,4 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='49) ∥δ+ x φ∥l4 ≲ ∥φ∥H2, φ ∈ XN, we have ∥δ+ x vj∥l4 ≲ ∥v∥H2 = M1, which yields from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='45) that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='50) ∥δ+ x (Φτ B(v) − v) − δ+ x (Φτ B(w) − w)∥2 l2 ≤ C(M, M1, ∥V ∥W 1,∞) � ∥e∥2 l2 + ∥δ+ x e∥2 l2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='50), using the discrete Poincare’s inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='27) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='36), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='51) ∥δ+ x (Φτ B(v) − v) − δ+ x (Φτ B(w) − w)∥2 l2 ≤ C(M, M1, ∥V ∥W 1,∞)∥δ+ x e∥2 l2 ≤ C(M, M1, ∥V ∥W 1,∞)∥∇e∥2 L2, which plugged into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='37) yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='35), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The 2D case follows exactly (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='47)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='49) in 2D follows from the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 in [7] with additional attention paid to the boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The 3D case follows (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='47)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='49) with slight modification: using H¨older’s inequality with index (3/2, 3) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then the discrete version of H1 �→ L6 and H2 �→ W 1,3 in 3D are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The proof of the first one can be found in [39] while the proof of the second one will follow the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='49) in 2D, which is the reason why we modify the estimates in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='42) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' With Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='8, we are able to obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='42) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Following the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7, we only need to estiamte ek = INψk − PNψ(·, tk) for 0 ≤ k ≤ T/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We shall first prove the error estimate in H1 norm by the standard argument of the mathematical induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Replacing ∥ · ∥L2 with ∥ · ∥H1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='55), one has for 0 ≤ k ≤ T/τ − 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='52) ∥ek+1∥H1 ≤ ∥Φτ(INψk) − Φτ(PNψ(·, tk))∥H1 + ∥Φτ(PNψ(·, tk)) − PNψ(·, tk+1)∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When k = 0, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='28), one gets ∥e0∥H1 = ∥INψ0 − PNψ0∥H1 ≲ h∥ψ0∥H2 ≤ C(M2)h, ∥ψ0∥l∞ ≤ ∥ψ0∥L∞ ≤ 1 + M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We assume that for 0 ≤ k ≤ m ≤ T/τ − 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='53) ∥ek∥H1 ≲ τ 1 2 + h, ∥ψk∥l∞ ≤ 1 + M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We shall prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='53) for m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='52), using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='34) and noting the assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='53), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='54) ∥em+1∥H1 ≤ (1 + C1τ)∥em∥H1 + C2τ � τ 1 2 + h � , where C1 and C2 are the costants in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='34) respectively, which depends exclusively on M2 and ∥V ∥W 1,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='54), standard discrete Gronwall’s in- equality yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='55) ∥em+1∥H1 ≤ 2eC0T C1 � τ 1 2 + h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 24 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG Recalling that ek = INψk − PNψ(tk) and ∥ψ(·, tk)∥L∞ ≤ M2, using the inverse inequality ∥φ∥L∞ ≲ h−1/2∥φ∥L2, ∀φ ∈ XN, we have ∥ψm+1∥l∞ = ∥INψm+1∥l∞ ≤ ∥em+1∥l∞ + ∥PNψ(·, tm+1)∥l∞ ≤ ∥em+1∥l∞ + ∥ψ(·, tm+1) − PNψ(·, tm+1)∥l∞ + ∥ψ(·, tm+1)∥l∞ ≤ ∥em+1∥l∞ + h− 1 2 ∥ψ(·, tm+1) − PNψ(·, tm+1)∥L2 + M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Hence, for τ ≤ τ0 and h ≤ h0 with τ0 > 0 and h0 > 0 depending on M2 and T, by Sobolev embedding and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='19), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='56) ∥ψm+1∥l∞ ≤ C∥em+1∥H1 + Ch2−1/2 + M2 ≤ 1 + M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='55) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='56), we proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='53) for k = m + 1 and thus for all 0 ≤ k ≤ T/τ by mathematical induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' With the l∞-bound of the numerical solution, the L2 estimate of ek follows the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5 by using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='33), which completes the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In 2D and 3D, we no longer have H1 �→ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' To obtain the l∞-bound of ψm+1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='56), we use the discrete Sobolev inequalities as in [5, 7, 8] ∥v∥l∞ ≤ C| ln h| ∥INv∥H1, ∥w∥l∞ ≤ Ch−1/2∥INw∥H1, where v and w are 2D and 3D mesh functions with zero at the boundary, respec- tively, and the interpolation operator IN can be defined similarly in 2D and 3D as in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Thus by requiring that the time step size τ satisfies the additional assumption (B), we can control the l∞-norm of the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='43) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In the following, we assume that 1/2 < σ < 1, V ∈ H3(Ω), ∇V ∈ H1 0(Ω), ψ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' H3 ∗(Ω)) ∩ C1([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' H1(Ω)) and let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='57) M3 := max � ∥ψ∥L∞([0,T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='H3), ∥ψ∥L∞([0,T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='L∞), ∥V ∥H3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We first show an analogous result of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let φ ∈ XN such that ∥φ∥H3 ≤ M and let 0 < τ < 1 and 0 < h < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Assume that V ∈ H3(Ω) and ∇V ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' When 1/2 < σ < 1, we have ∥(I − eiτ∆)PNB(φ)∥H1 ≤ C1(M, ∥V ∥H3)τ σ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='58) ∥INB(φ) − PNB(φ)∥H1 ≤ C2(M, ∥V ∥H3)h2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='59) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='21), noting that V φ ∈ H3 ∗(Ω), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='60) ∥(I − eiτ∆)(V φ)∥H1 ≲ τ∥V ∥H3∥φ∥H3, ∥(IN − PN)(V φ)∥H1 ≲ h2∥V ∥H3∥φ∥H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Following (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='23) with ∥ · ∥H1 replacing ∥ · ∥L2 and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='31), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='61) ∥(I − eiτ∆)(f(|φ|2)φ)∥H1 ≤ 2∥f(|φ|2)φ − fε(|φ|2)φ∥H1 + C(M) τ ε2−2σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6), one gets (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='62) ∇[f(|φ|2)φ − fε(|φ|2)φ] = (1 + σ)(f(|φ|2) − fε(|φ|2))∇φ + (G(φ) − f ′ ε(|φ|2)φ2)∇φ, from which, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4 and noting that G(z) = f ′(|z|2)z2 = f ′ ε(|z|2)z2 when |z| ≥ ε and |G(z)| + |f ′ ε(|z|2)z2| ≲ ε2σ when |z| < ε, one gets (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='63) ∥∇[f(|φ|2)φ − fε(|φ|2)φ]∥L2 ≲ ε2σ∥∇φ1|φ|<ε∥L2 ≤ ε2σ∥φ∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 25 Plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='63) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='61), we have ∥(I − eiτ∆)(f(|φ|2)φ)∥H1 ≤ C(M) inf 0<ε<1 � ε2σ + τ ε2−2σ � ≤ C(M)τ σ, which combined with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='60) yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Then we shall show (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Following (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='26) with ∥ · ∥H1 replacing ∥ · ∥L2, using the property of PN and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='63), one gets (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='64) ∥(IN − PN)(f(|φ|2)φ)∥H1 ≤ ∥IN(f(|φ|2)φ − fε(|φ|2)φ)∥H1 + ε2σ∥φ∥H1 + h2 C(M) ε2−2σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='36), one gets (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='65) ∥∇IN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 ≲ ∥δ+ x (f(|φ|2)φ − fε(|φ|2)φ)∥l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Let φθ j = (1 − θ)φj + θφj+1 for 0 ≤ θ ≤ 1 and 0 ≤ j ≤ N − 1, direct calculation gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='66) δ+ x � f(|φj|2)φj − fε(|φj|2)φj � = 1 h � 1 0 d dθ � f(|φθ j|2)φθ j − fε(|φθ j|2)φθ j � dθ = � 1 0 ��� f(|φθ j|2) + f ′(|φθ j|2)|φθ j|2� − � fε(|φθ j|2) + f ′ ε(|φθ j|2)|φθ j|2�� δ+ x φj +(G(φθ j) − f ′ ε(|φθ j|2)(φθ j)2)δ+ x φj � dθ, which implies, similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='63), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='67) ��δ+ x (f(|φj|2)φj − fε(|φj|2)φj) �� ≲ ε2σ|δ+ x φj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='65), using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='67) and recalling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='36) and φ ∈ XN, we obatin (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='68) ∥∇IN(f(|φ|2)φ − fε(|φ|2)φ)∥L2 ≲ ε2σ∥δ+ x φ∥l2 ≤ ε2σ∥∇φ∥L2, which plugged into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='64) yields ∥(IN − PN)(f(|φ|2)φ)∥H1 ≤ C(M) inf 0<ε<1 � ε2σ + h2 ε2−2σ � ≤ C(M)h2σ, which combined with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='60) yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='59) and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='12 (local truncation error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Assume that V ∈ H3(Ω), ∇V ∈ H1 0(Ω), ψ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' H3 ∗(Ω)) ∩ C1([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' H1(Ω)) and 1/2 < σ < 1, for 0 ≤ k ≤ T/τ − 1, we have ∥PNψ(·, tk+1) − Φτ(PNψ(·, tk))∥H1(Ω) ≤ C(M3)τ � τ σ + h2σ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Following the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7, we only need to modify the esti- mate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='30) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='31), which can be easily done by using the assumption ψ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' H3) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='11 and the standard estimates of the operators IN −PN, I − PN and I − eiτ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ Proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='43) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='12 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='34) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='52), and noting the l∞-bound of the numerical solution in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='42), then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='43) follows from the discrete Gronwall’s inequality immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' □ 26 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Numerical results In this section, we present some numerical examples for the NLSE with 0 < σ < 1 to confirm our error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In the following, we fix β = −1, V (x) ≡ 0, d = 1 and T = 1 and consider the following two initial set-ups: Type I: We consider the smooth initial datum (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) ψ0(x) = xe− x2 2 , x ∈ Ω = (−16, 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Type II: We consider the initial datum in H2(Ω) as in [30] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2) ψ0 = φ(1) ∥φ(1)∥L2 , φ(1)(x) = � l∈TN �φ(1) l sin(µl(x − a)), x ∈ Ω = (−1, 1) �φ(1) l = �φl |µl|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5 , �φl = � rand(−1, 1) + i rand(−1, 1), l even, 0, l odd, l ∈ TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' where rand(−1, 1) returns a uniformly distributed random number between −1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Note that both Types I and II initial data are chosen as odd functions to demon- strate the influence of the semi-smoothness of f at the origin since with an odd initial datum, the exact solution satisfies ψ(0, t) ≡ 0 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) is then solved by the TSSP method on the domain Ω with Type I and Type II initial setups for different σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' The ‘exact’ solution is obtained numerically by the Strang splitting sine pseudospectral method with a very fine mesh size he = 2−9 and a small time step size τe = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In our numerical experiments below, when testing the temporal convergence, we always fix the mesh size h = he.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' To quantify the error, we introduce the following error functions: ek L2 = ∥ψ(·, tk) − INψk∥L2, ek H1 = ∥ψ(·, tk) − INψk∥H1, 0 ≤ k ≤ n := T/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 exhibits the temporal and spatial errors in L2-norm of the TSSP (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='13) for the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) with Type I initial datum and different 0 < σ ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 (a) shows that the temporal convergence is first order in L2-norm for all the four σ and Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 (b) shows the spatial convergence is almost third order in L2-norm, which is also increasing with σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' These results are better than our error estimates in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5 and suggest that first order temporal convergence in L2-norm may hold for any σ > 0 and the spatial convergence may be of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' However, we remark that it is impossible to obtain the optimal temporal convergence and the high order spatial convergence by simply improving the local error estimates in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='6 and there must exist error cancellation between different steps, which require new techniques and in-depth analysis to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2 plots the temporal and spatial errors in L2- and H1-norm of the TSSP (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='13) for the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) with Type II H2 initial datum and fixed σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2 (a) shows that the temporal convergence is first order in L2-norm and half order in H1-norm and Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2 (b) shows the spatial convergence is second order in L2-norm and first order in H1-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' These results correspond with our error estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='42) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7 very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3 displays the temporal and spatial errors in H1-norm of the TSSP (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='13) for the NLSE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) with Type I smooth initial datum and different 0 < σ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3 (a) shows that the temporal convergence in H1-norm increases from half order to first order as σ increase from 0 to 1/2 and remains first order when σ ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' ERROR ESTIMATES FOR NLSE WITH SEMI-SMOOTH NONLINEARITY 27 10-4 10-3 10-2 10-1 10-5 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5 10-6 10-4 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Temporal errors (a) and spatial errors (b) in L2-norm for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='4 with Type I initial datum (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1) 10-4 10-3 10-2 10-4 10-2 10-3 10-2 10-5 10-2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Temporal errors (a) and spatial errors (b) in L2-norm and H1-norm for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5 with Type II initial data (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='2) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3 (b) shows the spatial convergence is almost 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5 order in H1-norm and is increasing with σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Similar to the observation of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1, these results are better than our error estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='43) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='7 and suggest that first order temporal convergence in H1-norm may hold for any σ ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' We would like to comment that the order reduction in H1-norm for 0 < σ < 1/2 is indeed resulted from the semi-smoothness of the nonlinearity instead of the regularity of the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Actually, we numerically checked that with the Type I smooth initial datum, the exact solution is roughly in H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5+2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Conclusion Error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schr¨odinger equation (NLSE) with semi-smooth nonlinearity f(ρ) = ρσ(σ > 0) were established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For 0 < σ ≤ 1 2, we prove error bounds at O(τ 1 2 +σ + h1+2σ) in L2-norm without any coupling conditions between τ and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' For σ ≥ 1 2, error bounds at O(τ +h2) in L2-norm and at O(τ 1 2 +h) in H1-norm are proved with mild coupling conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' In addition, when 1 2 < σ < 1 and under the assumption of H3-solution of the NLSE, we show an error bound at O(τ σ + h2σ) in H1-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Numerical results are reported to demonstrate our error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' 28 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' BAO AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' WANG 10-4 10-3 10-2 10-4 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='5 10-6 10-3 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content=' Temporal errors (a) and spatial errors (b) in H1- norm for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE1T4oBgHgl3EQfMwMt/content/2301.02992v1.pdf'} +page_content='25, 0.' metadata={'source': 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derivative +Tariq AlBanwa, Ahmed Al-Jamel, Eqab.M.Rabei and Mohamed.Al-Masaeed +Physics Department, Faculty of Science, Al al-Bayt University, +P.O. Box 130040, Mafraq 25113, Jordan +albanwatariq@gmail.com +aaljamel@aabu.edu.jo, aaljamel@gmail.com +eqabrabei@gmail.com +moh.almssaeed@gmail.com +January 10, 2023 +Abstract +In this work, the conformable Bateman Lagrangian for the damped harmonic +oscillator system is proposed using the conformable derivative concept. In other +words, the integer derivatives are replaced by conformable derivatives of order +α with 0 < α ≤ 1. +The corresponding conformable Euler-Lagrange equa- +tions of motion and fractional Hamiltonian are then obtained. The system is then +canonically quantized and the conformable Schrodinger equation is constructed. +The fractional-order dependence of the energy eigenvalues Eα +n and eigenfunctions +ψα +n are obtained using using suitable transformations and the extended fractional +Nikiforov-Uvarov method. The corresponding conformable continuity equation is +also derived and the probability density and probability current are thus suitably +defined. The probability density evolution as well as its dependence on α is plotted +and analyzed for various situations. It is found that the energy eigenvalues are real +and there are sort of gradual ordering in the behavior of the probability densities. +Keywords: Dissipative system, conformable derivative, canonical quantization, +damped quantum oscillator, Bateman system, conformable Lagrangian +1 +Introduction +Dissipation is an inescapable part in all real physical systems, from classical surfaces +in relative motion to quantum systems such as molecules, atoms, nuclei and radiating +point charge. Such non-conservative systems have been studied several decades ago +by researchers . Historically, Bateman [1] suggested a Lagrangian for the damped har- +monic oscillator that leads to the exact equations of motion. After that, the Hamiltonian +corresponding to Bateman’s Lagrangian is constructed independently by Cardirola and +1 + +Kanai [2, 3]. Because of the explicit time-dependence Lagrangian and Hamiltonian +of such systems, their quantization is not an easy task. This topic on the quantization +of non-conservative systems has attracted many researchers. In [4], the Lagrangian +for damped mechanical systems with various forms of dissipation are quantized using +the path integral formalism. In [5], they revisited the description of the damped har- +monic oscillator with an assessment of previous works mainly based on the Bateman- +Caldirola-Kanai model and a new model has been proposed that has better energy be- +havior and relate it to some existing open-systems approaches. In [6], the Feynman +path integral is applied and widened toward the calculation of the kernel of a quantum +damped harmonic oscillator. Besides, in [7], a suitable Hamiltonian that describes the +damped harmonic oscillator is constructed starting from Bateman Lagrangian. Also, +the Hamilton-Jacobi equation is written and the action function is obtained. Then, the +system is quantized using the WKB approximation and the canonical quantization. In +addition, K. Takahashi [8] performed quantization on the dual Bateman’s system (BDS) +by decomposing it into two effectively independent massless subsystems with reduced +degrees of freedom. The original massive BDS that satisfies the canonical quantization +condition is then rebuilt by superposing the two massless subsystems. +In the last two decades, the quantization of physical systems is extended to the +framework of fractional calculus by many researchers and has become of prime im- +portant in physics. The theory of fractional calculus is as old as classical calculus and +classified as generalized fractional integrals or derivatives. There are different defini- +tions of fractional derivative that are proposed, and the most popular definitions are +Riemann-Liouville, Riesz, and Caputo definitions. Each definition has some character- +istic properties; for general review see [9–13]. Researchers are paying attention to the +implementation of fractional calculus as they found that fractional order derivatives are +useful in the description of many physical phenomena in the real world. For instance, +Riewe [14, 15] , Rabei et al. [16, 17] and many others used the fractional calculus +techniques to construct the Lagrangian and Hamiltonian for the non conservative sys- +tems. Rabei et.al [18] discussed how to find the solution of Schrodinger equation for +some systems that have a fractional behavior in their Lagrangian and obey the WKB +approximation. Besides, the canonical quantization of a system with Brownian motion +is carried out using fractional calculus by Rabei et.al [19]. +In 2014, Khalil et al. [20] suggested a modern fashioned fractional derivative +termed as the conformable derivative (CD), which is defined using the usual funda- +mental limit definition of the classical derivative rather than in terms of integrals as in +the other definitions. Given a function f ∈ [0, ∞) → R. The conformable derivative +of f with order α is defined by +Dα[f(t)] = lim +ǫ→0 +f(t + ǫt1−α) − f(t) +ǫ +, +t > 0, +(1) +with 0 < α ≤ 1. This CD operator is linear and satisfies the general properties of +integer order derivatives, such as the formula of the derivative of the product or quo- +tient of two functions and the chain rule, which makes it favorable over the traditional +fractional derivatives [20]. In this paper, we adopt Dαf to denote the conformable +derivative (CD) of f of order α. +2 + +Many researchers considered this new definition as a more sophisticated candidate +to the extension of the classical derivative to the fractional domain. The CD has found +various applications in the physical sciences. For instance, the heat conformable dif- +ferential equation is investigated and the exact solutions are searched in [21]. Also, the +conformable Euler-Lagrange equation and Hamiltonian formulation were discussed in +[22]. The deformation of the ordinary quantum mechanics using the concept of con- +formable calculus is considered by Chung et.al [23]. The Authors defined two funda- +mental operators, namely, the α-position operator and α-momentum operator, and then +the α-Hamiltonian operator is constructed and the related conformable Schrodinger +equation is reached. They also presented a formulation for the conformable quantum +mechanics boosted by some illustrative physical applications. In [24], the search for +fractional ordering in the mass spectra of heavy quarkonia is investigated with a con- +formable derivative potential model. The saturation effects due to non-linearity effects +is discussed in these short-lived bound states. The CD is used by [25] to define the +conformable Schrodinger equation, and the conformable Nikiforov-Uvarov method is +introduced to obtain the energy eigenvalues and eigenfunctions. Also, it has been used +in [26] to study the sine-Gordon equation to obtain exact solitary wave solutions within +the frame of conformable calculus. In [27], CFD is used to study the conformable New- +tonian mechanics, where the conformable calculus of variations is introduced and the +conformable Euler–Lagrange equation is constructed. The CD is also used by [28] to +study the fractional singular Lagrangian system. They obtained the equations of mo- +tion and determined the action integral after. The fractional Christ-Lee model is then +discussed and quantized using WKB approximation to demonstrate the applicability +of their work. Using conformable calculus, the approximation methods employed in +quantum mechanics have recently been extended to become usable in conformable +quantum mechanics (Variational method [29] , Perturbation theory [30] and WKB ap- +proximation [31] ). In addition the conformable harmonic oscillator is quantized by +using α -creation and α -annihilation operators [32]. Furthermore, the deformation of +special relativity is articulated in the context of conformable derivatives [33]. Recently, +more than one equation has been solved and the behavior of the solution is studied using +conformable calculus such as (Laguerre differential equation [34], Angular equation of +the Schrodinger Equation [35] and Schrodinger equation with Hydrogen atom [36] ) +In this paper, we propose a conformable Lagrangian as a natural extension to +the classical Bateman Lagrangian [1]. The corresponding equations of motions and +conformable Hamiltonian will then be obtained. The canonical quantization for con- +formable systems as described in [23] will be used to obtain the possible energy eieg- +nvalues and eigenfunctions in terms of the α order. In the sequel, we review in section +2 the theoretical tools needed in our study. In section 3, formalism and quantization +procedure will be presented. Final in section 4 a summary and conclusions will be +given. +3 + +2 +The extended Nikiforov-Uvarov method with conformable +derivative (ENU-CD) +The extended Nikiforov-Uvarov method (ENU) is a generalization of the Nikiforov- +Uvarov method to obtain the eigenvalues and eigenfunctions of differential equations +that can be transformed into hypergeometric form [37]. It has been used in some re- +search articles in quantum mechanics to obtain the eigenvalues and eigenfunctions of +the wave equation [38, 39]. Consider the conformable differential equation of the stan- +dard form [25, 37]: +Dα[Dαψ(s)] + ˜τ(s) +σ Dαψ(s) + ˜σ(s) +σ2(s)ψ(s) = 0 +(2) +where ˜τ(s), σ(s) and ˜σ(s) are polynomials, at most second, third, and fourth degrees, +respectively, then it can be solved analytically. Using the key property of CFD [20, 37]: +Dαψ(s) = s1−αψ′(s) +(3) +and +Dα[Dαψ(s)] = (1 − α)s1−2αψ′(s) + s2−2αψ′′(s). +(4) +then Eq.(2) turned into +ψ′′(s) + (1 − α)σ(s)s−α + ˜τ(s) +s1−ασ(s) +ψ′(s) + +˜σ(s) +s2−2ασ2(s)ψ(s) = 0. +(5) +Introducing the conformable parameters ˜τf(s) = (1 − α)σ(s)s−α + ˜τ(s), σf(s) = +s2−2ασ2(s), and ˜σf(s) = σ(s), then we obtain the standard form of equation of the +conformable ENU [38] +ψ′′(s) + ˜τf(s) +σf(s)ψ′(s) + ˜σ(s) +σ2 +f(s)ψ(s) = 0 +(6) +The next natural step is to to propose the Ansatz +ψ(s) = φ(s)Y (s), +(7) +which reduces Eq.(6) to the hypergeometric form, +σf(s)Y ′′ + τ(s)Y ′ + h(s)Y (s) = 0. +(8) +Here φ(s) fulfills the first order differential equation +φ′(s) +φ(s) = πf(s) +σf(s), +(9) +and h(s) fulfills +h(s) = π′ +f(s) + G(s). +(10) +4 + +Here Y (s) is a type of hypergeometric functions whose polynomial solutions satisfy a +Rodriguez formula of the form +Yn(s) = Bn +ρ(s) +dn +dsn +� +σn +f (s)ρ(s) +� +, +(11) +where Bn is the normalization constant, and ρ is called the weight or density function +and must satisfy the condition +(σfρ)′ = τρ. +(12) +The function πf(s) and the function G(s) required for this method are defined through +the following relation +πf = +σ′ +f(s) − ˜τf(s) +2 +± +��σ′ +f(s) − ˜τf(s) +2 +�2 +− ˜σf(s) + G(s)σf(s), +(13) +and are chosen so that the function πf(s) is a polynomial of at most 2α degree. Also, +the function hn(s) is determined from the relation +hn(s) = −n +2 τ ′(s) − n(n − 1) +6 +σ′′ +f (s) + Cn, +(n = 0, 1, 2, ...) +(14) +where +τ(s) = ˜τf(s) + 2πf(s). +(15) +Then, the equality of Eq.(10) and Eq.(14) leads to the energy eigenvalues. +3 +Formalism +Consider Bateman’s Lagrangian [1] +L = m +2 +� +˙q2 − ω2q2� +eλt. +(16) +This Lagrangian describes the one dimensional damped harmonic oscillator. We define +the corresponding conformable Bateman Lagrangian as +L(qα, Dα +t qα, tα) = mα +2 ([Dα +t qα]2 − ω2αq2α)e +λtα +α +(17) +Introduce a new coordinate yα by +yα = qαe +λtα +2α +(18) +Then, +qα = +yα +e +λtα +α +(19) +Operating on both sides by Dα +t , we have +Dα +t qα = e +λtα +2α (Dα +t yα − yα λ +2 ) +e +λtα +α +(20) +5 + +Substituting this result in Eq.(17), and do some little algebra, we obtain +L(yα, Dα +t yα) = mα +2 ([Dα +t yα]2 + [λ2 +4 − ω2α]y2α − yαλDα +t yα). +(21) +The conformable Euler-Lagrange equation of motion can be obtained using +∂L +∂yα − Dα +t ( +∂L +∂[Dα +t yα]) = 0. +(22) +By noting that +∂L +∂yα += +mα[λ2 +4 − ω2α]yα − mαλDα +t yα +2 +(23) +∂L +∂ [Dα +t yα] += +mαDα +t yα − mαλyα +2 +Dα +t +∂L +∂[Dα +t yα] += +mαDα +t Dα +t yα − mαλDαyα +2 +The equation of motion reads as +Dα +t Dα +t yα + (λ2 +4 − ω2α)yα = 0 +(24) +The Hamiltonian is defined as +H(yα, P α +y ) = P α +y Dα +t yα − L. +(25) +The momentum operator is defined by [40] +P α +y = +∂L +∂[Dα +t yα] = mαDα +t yα − mαλyα +2 +(26) +Then, the Hamiltonian becomes +H(yα, P α +y ) = (P α +y )2 +2mα + 1 +2mαω2αy2α + 1 +2λyαP α +y . +(27) +To apply canonical quantization on the Hamiltonian Eq.(27), we first notice that +[P α +y + mαλyα +2 +, e−f(y)]ψ(y) += +[P α +y , e−f(y)]ψ(y) +(28) += +−iℏαDα +t [e−f(y), ψ(y)] + iℏαe−f(y)Dα +t ψ(y) += +−iℏαψ(y)Dα +t e−f(y) +And using +Dα +t e−f(y) = −y1−αf ′(y)e−f(y) +(29) +we obtain the commutator +[P α +y + mαλyα +2 +, e−f(y)] = iℏαy1−αf ′(y)e−f(y). +(30) +6 + +Expanding the LHS of this commutator +(P α +y + mαλyα +2 +)e−f(y) − e−f(y)(P α +y + mαλyα +2 +) = iℏαy1−αf ′(y)e−f(y) +(31) +Then multiplying both sides from the left by e+f(y), and with a little algebra, yields +e+f(y)(P α +y + mαλyα +2 +)e−f(y) = P α +y + mαλyα +2 ++ iℏαy1−αf ′(y) +(32) +Choose e+f(y) such as +f ′(y) = −mαλyα +2 +∗ +1 +iℏαy1−α +(33) +will give +f(y) = imαλy2α +4αℏα +(34) +Then +e ++imαλy2α +4αℏα +(P α +y + mαλyα +2 +)e +−imαλy2α +4αℏα += P α +y +(35) +Writing +η = e +imαλy2α +4αℏα +(36) +we obtain the gauge transformation +η(P α +y + mαλyα +2 +)η−1 = P α +y . +(37) +This result can be generalized to +η(P α +y + mαλyα +2 +)2η−1 = (P α +y )2. +(38) +Applying this on the Hamiltonian Eq.(27), we have +ηHη−1η = (P α +y )2 +2mα + 1 +2mα(ω2α − λ2 +4 )y2α +(39) +Thus, rather than applying canonical quantization on the Hamiltonian as given by +Eq.(27), we do this with the new Hamiltonian as given by the RHS of Eq.(39): +H = (P α +y )2 +2mα + 1 +2mα(ω2α − λ2 +4 )y2α. +(40) +Then, the corresponding conformable Schrodinger equation is +[(P α +y )2 +2mα + 1 +2mαΩ2y2α]ψ = Eαψ +(41) +7 + +where Ω2 = ω2α − λ2 +4 . On substituting for the conformable momentum operator +P α +y = −iℏDα +y , we find +−ℏ2α +2mα (Dα +y )2ψ + 1 +2mαΩ2y2αψ = Eαψ +(42) +Or +(Dα +y )2ψ = y2−2αψ′′ + (1 + α)y1−2αψ′ +(43) +Using Eq.(27), we find that +ψ′′ + (1 + α) +y +ψ′ + 2mα +y2ℏ2α (Eαy2α − mαΩ2 +2 +y4α)ψ = 0 +(44) +Then, we note that τf(y) = (1 + α), σf = y, and ˜σf(y) = 2mα +ℏ2α (Eαy2α − mαΩ2 +2 +y4α). +Choosing G(y) = S + Pyα−1 + Qy2α−1, then one can show that πf takes the form: +πf(y) = α +2 + − +� +(A + Byα + Fy2α)2 +(45) +with A = ± α +2 , S = 0, B = 0, P = 0, and Q = 2mαEα +ℏ2α +y2α + B2 + 2AF, where +F = mα +ℏα +� +ω2α − λ2 +4 . Then, using Eqs.(15),(10) and (14), we obtain: +τ(y) += +1 + −2(A + Fy2α) +(46) +hn(y) += +−n +2 τ ′ − n(n − 1) +6 +σ′′ +f + Cn +(47) +h(y) += +2nαFy2α−1 + Cn. +(48) +Equating Eq.(47) with Eq.(48), we obtain +Eα = αℏα +� +ω2α − λ2 +4 (n + 1 +2), +n = 0, 1, 2, 3, ... +(49) +This result shows that the energy eigenvalues are real. Setting α = 1, the traditional +damped harmonic oscillator energy eigenvalues are recovered, and in agreement with +the result obtained in [7]. This result coincides with that found in [30? ? ] using differ- +ent approach. The eigenfunctions can be found by finding firstly the needed functions +from Eqs.(7-11). The results are +φ(y) += +yαe +F +2α y2α +(50) +ρ(y) += +y−αe +F +α y2α +(51) +Yn(y) += +Bnyαe +F +α y2α dn +dyn +� +yn−αe +−F +α y2α� +(52) +Then, the eigenfunctions are +ψn(y) = Bnyαe +F +2α y2α dn +dyn +� +yn−αe +−F +α y2α� +, +(53) +8 + +where Bn is the normalization constant. To define the probability current density and +probability current correctly, we derive the continuity equation for Eq.(44). Multi- +plying Eq.(44) by e +−λtα +2 +, and then following analogous procedure for deriving the +continuity equation for traditional Schrodinger equation, we obtain +Dα +t [ψ∗ψe +−λtα +2 +]+Dα +y [ ℏα +2imα (ψ∗Dα +y ψ−ψDα +y ψ∗)e +−λtα +2 ++ λ +2 yαψ∗ψe +−λtα +2 +] = 0. (54) +We define the probability density ρ(y, t) and the probability current density j(y, t) as +ρ(y, t) = ψ∗ψe +−λtα +2 +, +(55) +and +j(y, t) = +ℏα +2imα (ψ∗Dα +y ψ − ψDα +y ψ∗)e +−λtα +2 ++ λ +2 yαψ∗ψe +−λtα +2 +, +(56) +respectively. Substituting from Eq.(53), we obtain +ρ(y, t) = B2 +ny2αe +F +α y2α � dn +dyn +� +yn−αe +−F +α y2α��2 +e− λtα +2 . +(57) +Due to dissipation, the eigenfunctions are normalized at the initial time t = 0 by the +relation: +� ∞ +0 +ρ(y, 0)dy = 1, +(58) +which fixes the normalization constant Bn at all times. To check the calculations, we +plotted in Figure (1) the time evolution of the probability density for the ground state +n = 0 and the first excited state n = 1 with parameter set λ = 0 and α = 1. This +assumes to represent the traditional quantum harmonic oscillator with no dissipation. +It is clear that we have demonstrated here the well-known behavior for this situation. +In Figure (2), we plotted the same situation α = 1 but with dissipation λ = 0.5. +It is found there is a decrease in the probability density as the system evolves with +time. This is due to dissipation as expected. However, the behavior of the probability +distribution decrease found in this work is such that not only the areas under the curves +decrease as time evolves, but also the peaks get lowered gradually. In comparison with +the results found in [7], their distributions decreases but keeping same peaks. Figure +(3) shows the behavior of the initial probability density distributions for the first three +states n = 0, 1, 2, 3 and the parameter F = 1, evaluated at different values of the +fractional order parameter α. It can be concluded that there is a trend toward a gradual +ordering in these distributions with α. This could suggest to use the conformable model +to describe or model nonlinear phenomena accompanied the original Bateman system. +9 + +t=0 +t=0.5 +t=1 +t=1.5 +-4 +-2 +0 +2 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +q +| +� +(q,t) +2 +Decrease in probability density (n=0, λ=0 and α=1) +(a) +t=0 +t=0.5 +t=1 +t=1.5 +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +q +|ψ(q,t) +2 +Decrease in probability density (n=1, λ=0 and α=1) +(b) +Figure 1: The probability density time evolution for the ground state n = 0 and the +first excited state n = 1 with parameter set λ = 0 and α = 1. +10 + +t=0 +t=0.5 +t=1 +t=1.5 +-4 +-2 +0 +2 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +q +|ψ(q,t) +2 +Decrease in probability density (n=0, λ=0.5 and α=1) +(a) +t=0 +t=0.5 +t=1 +t=1.5 +-4 +-2 +0 +2 +4 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +q +|ψ(q,t) +2 +Decrease in probability density (n=1, λ=0.5 and α=1) +(b) +Figure 2: The probability density time evolution for the ground state n = 0 and the +first excited state n = 1 with parameter set λ = 0.5 and α = 1. +11 + +α=1 +α�0.95 +α +�0.9 +α�0.85 +α�0.8 +-6 +-4 +-2 +0 +2 +4 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +y +|ψ(y) +2 +Normalized probability density (n=0) at t=0 +(a) +α=1 +α=0.95 +α=0.9 +α=0.85 +α=0.8 +-6 +-4 +-2 +0 +2 +4 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +y +|ψ(y) +2 +Normalized probability density (n=1) at t=0 +(b) +α=1 +α=0.95 +α=0.9 +α=0.85 +α=0.8 +-5 +0 +5 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +y +|ψ(y) +2 +Normalized probability density (n=2) at t=0 +(c) +α=1 +α=0.95 +α=0.9 +α=0.85 +α=0.8 +-5 +0 +5 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +y +|ψ(y) +2 +Normalized probability density (n=3) at t=0 +(d) +Figure 3: The behavior of the normalized probability density at time t = 0 in terms +of the fractional order α for the states n = 0, 1, 2, 3. The parameter values are chosen +such that F = mα +ℏα +� +ω2α − λ2 +4 = 1. +12 + +4 +Conclusion +In this work, the Bateman’s Lagrangian for the damped harmonic oscillator system is +formulated using the conformable derivative concept, from which we then obtained +the corresponding conformable Euler-Lagrange equations of motion and conformable +Hamiltonian. The system is then canonically quantized, which led to the appropri- +ate conformable Schrodinger equation. Using some transformations and the extended +conformable Nikiforov-Uvarov method, we were able to solve the equation and obtain +the energy eigenvalues and eigenfunctions as a function of the α order. The energy +eigenvalues were found to be real and equally spaced and the corresponding tradi- +tional damped harmonic oscillator was recovered by setting α = 1. We derived the +continuity equation to define the probability density and probability current. 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Al Horani, “Quantization of fractional singular Lagrangian +systems using WKB approximation,” International Journal of Modern Physics A, +vol. 33, no. 36, p. 1850222, 2018. +[29] M. Al-Masaeed, E. M. Rabei, and A. Al-Jamel, “Extension of the variational +method to conformable quantum mechanics,” Mathematical Methods in the +Applied Sciences, vol. 45, no. 5, pp. 2910–2920, 2022. [Online]. Available: +https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.7963 +[30] M. Al-Masaeed, E. M. Rabei, A. Al-Jamel, and D. Baleanu, “Extension of pertur- +bation theory to quantum systems with conformable derivative,” Modern Physics +Letters A, p. 2150228, 2021. +[31] M. Al-Masaeed, E. M. Rabei, and A. Al-Jamel, “Wkb approximation with +conformable operator,” Modern Physics Letters A, vol. 37, no. 22, p. 2250144, +2022. [Online]. Available: https://doi.org/10.1142/S0217732322501449 +[32] M. Al-Masaeed, E. M. Rabei, A. Al-Jamel, and D. Baleanu, “Quantization of +fractional harmonic oscillator using creation and annihilation operators,” Open +Physics, vol. 19, no. 1, pp. 395–401, 2021. +[33] A. Al-Jamel, M. Al-Masaeed, E. Rabei, and D. Baleanu, “The effect of +deformation of special relativity by conformable derivative,” Revista Mexicana +de F´ısica, vol. 68, no. 5 Sep-Oct, pp. 050 705 1–9, Aug. 2022, number: 5 Sep- +Oct. [Online]. Available: https://rmf.smf.mx/ojs/index.php/rmf/article/view/5877 +[34] E. M. Rabei, A. Al-Jamel, and M. Al-Masaeed, “The solution of conformable +laguerre differential equation using conformable laplace transform,” 2021. +[Online]. Available: https://arxiv.org/abs/2112.01322 +[35] E. M. Rabei, M. Al-Masaeed, and A. Al-Jamel, “Solution of the conformable +angular equation of the schrodinger equation,” 2022. [Online]. Available: +https://arxiv.org/abs/2203.11615 +15 + +[36] M. Al-Masaeed, +E. M. Rabei, +and A. Al-Jamel, +“Analytic study of +conformable schrodinger equation with hydrogen atom,” +2022. [Online]. +Available: https://arxiv.org/abs/2209.02699 +[37] H. Karayer, D. Demirhan, and F. B¨uy¨ukkılıc¸, “Extension of Nikiforov-Uvarov +method for the solution of Heun equation,” Journal of Mathematical Physics, +vol. 56, no. 6, p. 063504, Jun. 2015. +[38] A. Al-Jamel, “Saturation in heavy quarkonia spectra with energy-dependent con- +fining potential in N -dimensional space,” Modern Physics Letters A, vol. 33, +no. 32, p. 1850185, Oct. 2018. +[39] H. Karayer, D. Demirhan, and F. B¨uy¨ukkılıc¸, “Solution of Schr¨odinger equation +for two different potentials using extended Nikiforov-Uvarovmethod and polyno- +mial solutions of biconfluent Heun equation,” Journal of Mathematical Physics, +vol. 59, no. 5, p. 053501, May 2018. +[40] F. Mozaffari, H. Hassanabadi, H. Sobhani, and W. Chung, “On the conformable +fractional quantum mechanics,” Journal of the Korean Physical Society, vol. 72, +no. 9, pp. 980–986, 2018. +16 + diff --git a/idE0T4oBgHgl3EQf6wL5/content/tmp_files/load_file.txt b/idE0T4oBgHgl3EQf6wL5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..98d1663c5019fd1fce3221ebac7808352ee6f78b --- /dev/null +++ b/idE0T4oBgHgl3EQf6wL5/content/tmp_files/load_file.txt @@ -0,0 +1,550 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf,len=549 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='02769v1 [quant-ph] 7 Jan 2023 Quantization of the Bateman damping system with conformable derivative Tariq AlBanwa, Ahmed Al-Jamel, Eqab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='Rabei and Mohamed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='Al-Masaeed Physics Department, Faculty of Science, Al al-Bayt University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Box 130040, Mafraq 25113, Jordan albanwatariq@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='com aaljamel@aabu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='jo, aaljamel@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='com eqabrabei@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='com moh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='almssaeed@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='com January 10, 2023 Abstract In this work, the conformable Bateman Lagrangian for the damped harmonic oscillator system is proposed using the conformable derivative concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In other words, the integer derivatives are replaced by conformable derivatives of order α with 0 < α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The corresponding conformable Euler-Lagrange equa- tions of motion and fractional Hamiltonian are then obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The system is then canonically quantized and the conformable Schrodinger equation is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The fractional-order dependence of the energy eigenvalues Eα n and eigenfunctions ψα n are obtained using using suitable transformations and the extended fractional Nikiforov-Uvarov method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The corresponding conformable continuity equation is also derived and the probability density and probability current are thus suitably defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The probability density evolution as well as its dependence on α is plotted and analyzed for various situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' It is found that the energy eigenvalues are real and there are sort of gradual ordering in the behavior of the probability densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Keywords: Dissipative system, conformable derivative, canonical quantization, damped quantum oscillator, Bateman system, conformable Lagrangian 1 Introduction Dissipation is an inescapable part in all real physical systems, from classical surfaces in relative motion to quantum systems such as molecules, atoms, nuclei and radiating point charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Such non-conservative systems have been studied several decades ago by researchers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Historically, Bateman [1] suggested a Lagrangian for the damped har- monic oscillator that leads to the exact equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' After that, the Hamiltonian corresponding to Bateman’s Lagrangian is constructed independently by Cardirola and 1 Kanai [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Because of the explicit time-dependence Lagrangian and Hamiltonian of such systems, their quantization is not an easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' This topic on the quantization of non-conservative systems has attracted many researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In [4], the Lagrangian for damped mechanical systems with various forms of dissipation are quantized using the path integral formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In [5], they revisited the description of the damped har- monic oscillator with an assessment of previous works mainly based on the Bateman- Caldirola-Kanai model and a new model has been proposed that has better energy be- havior and relate it to some existing open-systems approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In [6], the Feynman path integral is applied and widened toward the calculation of the kernel of a quantum damped harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Besides, in [7], a suitable Hamiltonian that describes the damped harmonic oscillator is constructed starting from Bateman Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Also, the Hamilton-Jacobi equation is written and the action function is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Then, the system is quantized using the WKB approximation and the canonical quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In addition, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Takahashi [8] performed quantization on the dual Bateman’s system (BDS) by decomposing it into two effectively independent massless subsystems with reduced degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The original massive BDS that satisfies the canonical quantization condition is then rebuilt by superposing the two massless subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In the last two decades, the quantization of physical systems is extended to the framework of fractional calculus by many researchers and has become of prime im- portant in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The theory of fractional calculus is as old as classical calculus and classified as generalized fractional integrals or derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' There are different defini- tions of fractional derivative that are proposed, and the most popular definitions are Riemann-Liouville, Riesz, and Caputo definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Each definition has some character- istic properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' for general review see [9–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Researchers are paying attention to the implementation of fractional calculus as they found that fractional order derivatives are useful in the description of many physical phenomena in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' For instance, Riewe [14, 15] , Rabei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' [16, 17] and many others used the fractional calculus techniques to construct the Lagrangian and Hamiltonian for the non conservative sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Rabei et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='al [18] discussed how to find the solution of Schrodinger equation for some systems that have a fractional behavior in their Lagrangian and obey the WKB approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Besides, the canonical quantization of a system with Brownian motion is carried out using fractional calculus by Rabei et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='al [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In 2014, Khalil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' [20] suggested a modern fashioned fractional derivative termed as the conformable derivative (CD), which is defined using the usual funda- mental limit definition of the classical derivative rather than in terms of integrals as in the other definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Given a function f ∈ [0, ∞) → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The conformable derivative of f with order α is defined by Dα[f(t)] = lim ǫ→0 f(t + ǫt1−α) − f(t) ǫ , t > 0, (1) with 0 < α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' This CD operator is linear and satisfies the general properties of integer order derivatives, such as the formula of the derivative of the product or quo- tient of two functions and the chain rule, which makes it favorable over the traditional fractional derivatives [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In this paper, we adopt Dαf to denote the conformable derivative (CD) of f of order α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' 2 Many researchers considered this new definition as a more sophisticated candidate to the extension of the classical derivative to the fractional domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The CD has found various applications in the physical sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' For instance, the heat conformable dif- ferential equation is investigated and the exact solutions are searched in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Also, the conformable Euler-Lagrange equation and Hamiltonian formulation were discussed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The deformation of the ordinary quantum mechanics using the concept of con- formable calculus is considered by Chung et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='al [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The Authors defined two funda- mental operators, namely, the α-position operator and α-momentum operator, and then the α-Hamiltonian operator is constructed and the related conformable Schrodinger equation is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' They also presented a formulation for the conformable quantum mechanics boosted by some illustrative physical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In [24], the search for fractional ordering in the mass spectra of heavy quarkonia is investigated with a con- formable derivative potential model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The saturation effects due to non-linearity effects is discussed in these short-lived bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The CD is used by [25] to define the conformable Schrodinger equation, and the conformable Nikiforov-Uvarov method is introduced to obtain the energy eigenvalues and eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Also, it has been used in [26] to study the sine-Gordon equation to obtain exact solitary wave solutions within the frame of conformable calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In [27], CFD is used to study the conformable New- tonian mechanics, where the conformable calculus of variations is introduced and the conformable Euler–Lagrange equation is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The CD is also used by [28] to study the fractional singular Lagrangian system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' They obtained the equations of mo- tion and determined the action integral after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The fractional Christ-Lee model is then discussed and quantized using WKB approximation to demonstrate the applicability of their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Using conformable calculus, the approximation methods employed in quantum mechanics have recently been extended to become usable in conformable quantum mechanics (Variational method [29] , Perturbation theory [30] and WKB ap- proximation [31] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In addition the conformable harmonic oscillator is quantized by using α -creation and α -annihilation operators [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Furthermore, the deformation of special relativity is articulated in the context of conformable derivatives [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Recently, more than one equation has been solved and the behavior of the solution is studied using conformable calculus such as (Laguerre differential equation [34], Angular equation of the Schrodinger Equation [35] and Schrodinger equation with Hydrogen atom [36] ) In this paper, we propose a conformable Lagrangian as a natural extension to the classical Bateman Lagrangian [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The corresponding equations of motions and conformable Hamiltonian will then be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The canonical quantization for con- formable systems as described in [23] will be used to obtain the possible energy eieg- nvalues and eigenfunctions in terms of the α order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In the sequel, we review in section 2 the theoretical tools needed in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In section 3, formalism and quantization procedure will be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Final in section 4 a summary and conclusions will be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' 3 2 The extended Nikiforov-Uvarov method with conformable derivative (ENU-CD) The extended Nikiforov-Uvarov method (ENU) is a generalization of the Nikiforov- Uvarov method to obtain the eigenvalues and eigenfunctions of differential equations that can be transformed into hypergeometric form [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' It has been used in some re- search articles in quantum mechanics to obtain the eigenvalues and eigenfunctions of the wave equation [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Consider the conformable differential equation of the stan- dard form [25, 37]: Dα[Dαψ(s)] + ˜τ(s) σ Dαψ(s) + ˜σ(s) σ2(s)ψ(s) = 0 (2) where ˜τ(s), σ(s) and ˜σ(s) are polynomials, at most second, third, and fourth degrees, respectively, then it can be solved analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Using the key property of CFD [20, 37]: Dαψ(s) = s1−αψ′(s) (3) and Dα[Dαψ(s)] = (1 − α)s1−2αψ′(s) + s2−2αψ′′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (4) then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (2) turned into ψ′′(s) + (1 − α)σ(s)s−α + ˜τ(s) s1−ασ(s) ψ′(s) + ˜σ(s) s2−2ασ2(s)ψ(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (5) Introducing the conformable parameters ˜τf(s) = (1 − α)σ(s)s−α + ˜τ(s), σf(s) = s2−2ασ2(s), and ˜σf(s) = σ(s), then we obtain the standard form of equation of the conformable ENU [38] ψ′′(s) + ˜τf(s) σf(s)ψ′(s) + ˜σ(s) σ2 f(s)ψ(s) = 0 (6) The next natural step is to to propose the Ansatz ψ(s) = φ(s)Y (s), (7) which reduces Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (6) to the hypergeometric form, σf(s)Y ′′ + τ(s)Y ′ + h(s)Y (s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (8) Here φ(s) fulfills the first order differential equation φ′(s) φ(s) = πf(s) σf(s), (9) and h(s) fulfills h(s) = π′ f(s) + G(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (10) 4 Here Y (s) is a type of hypergeometric functions whose polynomial solutions satisfy a Rodriguez formula of the form Yn(s) = Bn ρ(s) dn dsn � σn f (s)ρ(s) � , (11) where Bn is the normalization constant, and ρ is called the weight or density function and must satisfy the condition (σfρ)′ = τρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (12) The function πf(s) and the function G(s) required for this method are defined through the following relation πf = σ′ f(s) − ˜τf(s) 2 ± ��σ′ f(s) − ˜τf(s) 2 �2 − ˜σf(s) + G(s)σf(s), (13) and are chosen so that the function πf(s) is a polynomial of at most 2α degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Also, the function hn(s) is determined from the relation hn(s) = −n 2 τ ′(s) − n(n − 1) 6 σ′′ f (s) + Cn, (n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=') (14) where τ(s) = ˜τf(s) + 2πf(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (15) Then, the equality of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (14) leads to the energy eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' 3 Formalism Consider Bateman’s Lagrangian [1] L = m 2 � ˙q2 − ω2q2� eλt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (16) This Lagrangian describes the one dimensional damped harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' We define the corresponding conformable Bateman Lagrangian as L(qα, Dα t qα, tα) = mα 2 ([Dα t qα]2 − ω2αq2α)e λtα α (17) Introduce a new coordinate yα by yα = qαe λtα 2α (18) Then, qα = yα e λtα α (19) Operating on both sides by Dα t , we have Dα t qα = e λtα 2α (Dα t yα − yα λ 2 ) e λtα α (20) 5 Substituting this result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (17), and do some little algebra, we obtain L(yα, Dα t yα) = mα 2 ([Dα t yα]2 + [λ2 4 − ω2α]y2α − yαλDα t yα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (21) The conformable Euler-Lagrange equation of motion can be obtained using ∂L ∂yα − Dα t ( ∂L ∂[Dα t yα]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (22) By noting that ∂L ∂yα = mα[λ2 4 − ω2α]yα − mαλDα t yα 2 (23) ∂L ∂ [Dα t yα] = mαDα t yα − mαλyα 2 Dα t ∂L ∂[Dα t yα] = mαDα t Dα t yα − mαλDαyα 2 The equation of motion reads as Dα t Dα t yα + (λ2 4 − ω2α)yα = 0 (24) The Hamiltonian is defined as H(yα, P α y ) = P α y Dα t yα − L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (25) The momentum operator is defined by [40] P α y = ∂L ∂[Dα t yα] = mαDα t yα − mαλyα 2 (26) Then, the Hamiltonian becomes H(yα, P α y ) = (P α y )2 2mα + 1 2mαω2αy2α + 1 2λyαP α y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (27) To apply canonical quantization on the Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (27), we first notice that [P α y + mαλyα 2 , e−f(y)]ψ(y) = [P α y , e−f(y)]ψ(y) (28) = −iℏαDα t [e−f(y), ψ(y)] + iℏαe−f(y)Dα t ψ(y) = −iℏαψ(y)Dα t e−f(y) And using Dα t e−f(y) = −y1−αf ′(y)e−f(y) (29) we obtain the commutator [P α y + mαλyα 2 , e−f(y)] = iℏαy1−αf ′(y)e−f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (30) 6 Expanding the LHS of this commutator (P α y + mαλyα 2 )e−f(y) − e−f(y)(P α y + mαλyα 2 ) = iℏαy1−αf ′(y)e−f(y) (31) Then multiplying both sides from the left by e+f(y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' and with a little algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' yields e+f(y)(P α y + mαλyα 2 )e−f(y) = P α y + mαλyα 2 + iℏαy1−αf ′(y) (32) Choose e+f(y) such as f ′(y) = −mαλyα 2 ∗ 1 iℏαy1−α (33) will give f(y) = imαλy2α 4αℏα (34) Then e +imαλy2α 4αℏα (P α y + mαλyα 2 )e −imαλy2α 4αℏα = P α y (35) Writing η = e imαλy2α 4αℏα (36) we obtain the gauge transformation η(P α y + mαλyα 2 )η−1 = P α y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (37) This result can be generalized to η(P α y + mαλyα 2 )2η−1 = (P α y )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (38) Applying this on the Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (27), we have ηHη−1η = (P α y )2 2mα + 1 2mα(ω2α − λ2 4 )y2α (39) Thus, rather than applying canonical quantization on the Hamiltonian as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (27), we do this with the new Hamiltonian as given by the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (39): H = (P α y )2 2mα + 1 2mα(ω2α − λ2 4 )y2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (40) Then, the corresponding conformable Schrodinger equation is [(P α y )2 2mα + 1 2mαΩ2y2α]ψ = Eαψ (41) 7 where Ω2 = ω2α − λ2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' On substituting for the conformable momentum operator P α y = −iℏDα y , we find −ℏ2α 2mα (Dα y )2ψ + 1 2mαΩ2y2αψ = Eαψ (42) Or (Dα y )2ψ = y2−2αψ′′ + (1 + α)y1−2αψ′ (43) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (27), we find that ψ′′ + (1 + α) y ψ′ + 2mα y2ℏ2α (Eαy2α − mαΩ2 2 y4α)ψ = 0 (44) Then, we note that τf(y) = (1 + α), σf = y, and ˜σf(y) = 2mα ℏ2α (Eαy2α − mαΩ2 2 y4α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Choosing G(y) = S + Pyα−1 + Qy2α−1, then one can show that πf takes the form: πf(y) = α 2 + − � (A + Byα + Fy2α)2 (45) with A = ± α 2 , S = 0, B = 0, P = 0, and Q = 2mαEα ℏ2α y2α + B2 + 2AF, where F = mα ℏα � ω2α − λ2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Then, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (15),(10) and (14), we obtain: τ(y) = 1 + −2(A + Fy2α) (46) hn(y) = −n 2 τ ′ − n(n − 1) 6 σ′′ f + Cn (47) h(y) = 2nαFy2α−1 + Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (48) Equating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (47) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (48), we obtain Eα = αℏα � ω2α − λ2 4 (n + 1 2), n = 0, 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (49) This result shows that the energy eigenvalues are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Setting α = 1, the traditional damped harmonic oscillator energy eigenvalues are recovered, and in agreement with the result obtained in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' This result coincides with that found in [30?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' ] using differ- ent approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The eigenfunctions can be found by finding firstly the needed functions from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='(7-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The results are φ(y) = yαe F 2α y2α (50) ρ(y) = y−αe F α y2α (51) Yn(y) = Bnyαe F α y2α dn dyn � yn−αe −F α y2α� (52) Then, the eigenfunctions are ψn(y) = Bnyαe F 2α y2α dn dyn � yn−αe −F α y2α� , (53) 8 where Bn is the normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' To define the probability current density and probability current correctly, we derive the continuity equation for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='(44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Multi- plying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (44) by e −λtα 2 , and then following analogous procedure for deriving the continuity equation for traditional Schrodinger equation, we obtain Dα t [ψ∗ψe −λtα 2 ]+Dα y [ ℏα 2imα (ψ∗Dα y ψ−ψDα y ψ∗)e −λtα 2 + λ 2 yαψ∗ψe −λtα 2 ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (54) We define the probability density ρ(y, t) and the probability current density j(y, t) as ρ(y, t) = ψ∗ψe −λtα 2 , (55) and j(y, t) = ℏα 2imα (ψ∗Dα y ψ − ψDα y ψ∗)e −λtα 2 + λ 2 yαψ∗ψe −λtα 2 , (56) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Substituting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (53), we obtain ρ(y, t) = B2 ny2αe F α y2α � dn dyn � yn−αe −F α y2α��2 e− λtα 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' (57) Due to dissipation, the eigenfunctions are normalized at the initial time t = 0 by the relation: � ∞ 0 ρ(y, 0)dy = 1, (58) which fixes the normalization constant Bn at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' To check the calculations, we plotted in Figure (1) the time evolution of the probability density for the ground state n = 0 and the first excited state n = 1 with parameter set λ = 0 and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' This assumes to represent the traditional quantum harmonic oscillator with no dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' It is clear that we have demonstrated here the well-known behavior for this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In Figure (2), we plotted the same situation α = 1 but with dissipation λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' It is found there is a decrease in the probability density as the system evolves with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' This is due to dissipation as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' However, the behavior of the probability distribution decrease found in this work is such that not only the areas under the curves decrease as time evolves, but also the peaks get lowered gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' In comparison with the results found in [7], their distributions decreases but keeping same peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Figure (3) shows the behavior of the initial probability density distributions for the first three states n = 0, 1, 2, 3 and the parameter F = 1, evaluated at different values of the fractional order parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' It can be concluded that there is a trend toward a gradual ordering in these distributions with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' This could suggest to use the conformable model to describe or model nonlinear phenomena accompanied the original Bateman system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' 9 t=0 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 t=1 t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='8 q | � (q,t) 2 Decrease in probability density (n=0, λ=0 and α=1) (a) t=0 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 t=1 t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='6 q |ψ(q,t) 2 Decrease in probability density (n=1, λ=0 and α=1) (b) Figure 1: The probability density time evolution for the ground state n = 0 and the first excited state n = 1 with parameter set λ = 0 and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' 10 t=0 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 t=1 t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='8 q |ψ(q,t) 2 Decrease in probability density (n=0, λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 and α=1) (a) t=0 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 t=1 t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='6 q |ψ(q,t) 2 Decrease in probability density (n=1, λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 and α=1) (b) Figure 2: The probability density time evolution for the ground state n = 0 and the first excited state n = 1 with parameter set λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' 11 α=1 α�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='95 α �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='9 α�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='85 α�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='8 6 4 2 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='6 y |ψ(y) 2 Normalized probability density (n=0) at t=0 (a) α=1 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='95 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='9 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='85 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='8 6 4 2 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 y |ψ(y) 2 Normalized probability density (n=1) at t=0 (b) α=1 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='95 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='9 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='85 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='8 5 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 y |ψ(y) 2 Normalized probability density (n=2) at t=0 (c) α=1 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='95 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='9 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='85 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='8 5 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content='5 y |ψ(y) 2 Normalized probability density (n=3) at t=0 (d) Figure 3: The behavior of the normalized probability density at time t = 0 in terms of the fractional order α for the states n = 0, 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The parameter values are chosen such that F = mα ℏα � ω2α − λ2 4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' 12 4 Conclusion In this work, the Bateman’s Lagrangian for the damped harmonic oscillator system is formulated using the conformable derivative concept, from which we then obtained the corresponding conformable Euler-Lagrange equations of motion and conformable Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The system is then canonically quantized, which led to the appropri- ate conformable Schrodinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Using some transformations and the extended conformable Nikiforov-Uvarov method, we were able to solve the equation and obtain the energy eigenvalues and eigenfunctions as a function of the α order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' The energy eigenvalues were found to be real and equally spaced and the corresponding tradi- tional damped harmonic oscillator was recovered by setting α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' We derived the continuity equation to define the probability density and probability current.' metadata={'source': 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W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' Chung, “On the conformable fractional quantum mechanics,” Journal of the Korean Physical Society, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' 72, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' 980–986, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE0T4oBgHgl3EQf6wL5/content/2301.02769v1.pdf'} diff --git a/idFMT4oBgHgl3EQf4zEN/vector_store/index.faiss 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Institute of Automation, Chinese Academy of Sciences +3 College of Computer Science, Chongqing University +4 Center for Advanced Intelligence Project, RIKEN +5 Department of Automation, Tsinghua University +{xumingyu2021, lianzheng2016}@ia.ac.cn, lfeng@cqu.edu.cn, +liubin@nlpr.ia.ac.cn, jhtao@tsinghua.edu.cn +Abstract +Noisy partial label learning (noisy PLL) is an im- +portant branch of weakly supervised learning. Un- +like PLL where the ground-truth label must reside +in the candidate set, noisy PLL relaxes this con- +straint and allows the ground-truth label may not +be in the candidate set. To address this problem, +existing works attempt to detect noisy samples and +estimate the ground-truth label for each noisy sam- +ple. However, detection errors are inevitable, and +these errors will accumulate during training and +continuously affect model optimization. To address +this challenge, we propose a novel framework for +noisy PLL, called “Dynamically Adjusted Label +Importance (DALI)”. It aims to reduce the negative +impact of detection errors by trading off the ini- +tial candidate set and model outputs with theoret- +ical guarantees. Experimental results on multiple +datasets demonstrate that our DALI succeeds over +existing state-of-the-art approaches on noisy PLL. +Our code will soon be publicly available. +1 +Introduction +Partial label learning (PLL) [Feng and An, 2018; Yan and +Guo, 2020] (also called ambiguous label learning [Chen et +al., 2014; Chen et al., 2017] and superset label learning [Liu +and Dietterich, 2012; Liu and Dietterich, 2014]) is a typical +type of weakly supervised learning. Unlike supervised learn- +ing where each sample is associated with a ground-truth label, +PLL requires identifying the ground-truth label from a set of +candidate labels. Due to the low annotation cost of partially +labeled samples, PLL has attracted increasing attention from +researchers and applied in many tasks, such as object recog- +nition [Chen et al., 2017], web mining [Huiskes and Lew, +2008], and ecological informatics [Briggs et al., 2012]. +The basic assumption of PLL is that the ground-truth la- +bel must reside in the candidate set [Jin and Ghahramani, +2002]. However, this assumption may not be satisfied due +to the unprofessional judgment of the annotators [Cid-Sueiro, +∗Equal Contribution +2012]. Recently, some researchers have relaxed this assump- +tion and focused on a more practical setting called noisy +PLL [Lv et al., 2021]. In noisy PLL, the ground-truth la- +bel may not conceal in the candidate set. To deal with this +task, [Lv et al., 2021] leveraged noise-tolerant loss functions +to avoid overemphasizing noisy samples in the learning pro- +cess. However, they cannot fully exploit the useful informa- +tion in noisy samples. To address this challenge, [Lian et +al., 2022] and [Wang et al., 2022b] proposed to detect noisy +samples and estimate the pseudo label for each noisy sample. +However, detection errors are unavoidable. These errors can +accumulate during training and continuously affect model op- +timization, thereby limiting their performance on noisy PLL. +To this end, we propose a novel framework for noisy PLL, +called “Dynamically Adjusted Label Importance (DALI)”. +Although we may make mistakes in noisy sample detection, +DALI allows us to leverage the initial candidate set and restart +correction. Meanwhile, we propose a strategy to automati- +cally determine the weighting coefficient in the learning pro- +cess. To further improve the performance, we incorporate +DALI with co-training [Blum and Mitchell, 1998] and mixup +[Zhang et al., 2018], which are powerful in noisy label learn- +ing [Li et al., 2020]. We also perform theoretical analysis +and prove the feasibility of our proposed method. To verify +the effectiveness of DALI, we conduct experiments on mul- +tiple benchmark datasets. Experimental results demonstrate +that our method outperforms currently advanced approaches, +setting the new state-of-the-art records. The main contribu- +tions of this paper can be summarized as follows: +• We propose a novel framework for noisy PLL with the- +oretical guarantees. Our DALI can reduce the negative +impact of prediction errors in noisy sample detection by +trading off the initial candidate set and model outputs. +• We further propose an automatic parameter selection +strategy for weighting coefficients. Combining DALI +with mixup and co-training, we can achieve better per- +formance under noisy conditions. +• Experimental results on multiple datasets demonstrate +the effectiveness of our proposed method. DALI is su- +perior to currently advanced approaches on noisy PLL. +The remainder of this paper is organized as follows: In Sec- +arXiv:2301.12077v1 [cs.CV] 28 Jan 2023 + +tion 2, we briefly review some recent works. In Section 3, we +formalize the problem statement and describe our proposed +method. In Section 4, we present our experimental datasets +and setup in detail. In Section 5, we illustrate the experimen- +tal results and analysis on benchmark datasets. Finally, we +conclude this paper and discuss our future work in Section 6. +2 +Related Work +The ground-truth label of PLL is concealed in the candi- +date set. Therefore, the core of dealing with this task is to +disambiguate candidate labels. In this section, we first in- +troduce two typical disambiguation strategies, i.e., average- +based methods and identification-based methods. After that, +we briefly review some recent works on noisy PLL. +2.1 +Average-based PLL +The most intuitive solution is the average-based method, +which assumes that each candidate label has an equal proba- +bility of being the ground-truth label. Typically, [H¨ullermeier +and Beringer, 2006] leveraged k-nearest neighbors for label +disambiguation. For each sample, they treated all candidate +labels of its neighborhood equally and predicted the ground- +truth label through voting strategies. Differently, [Cour et al., +2009] maximized the average output of candidate and non- +candidate labels in parametric models. However, average- +based methods can be severely affected by false positive la- +bels in the candidate set [Jin and Ghahramani, 2002]. +2.2 +Identification-based PLL +To address the shortcomings of average-based methods, re- +searchers have focused on identification-based methods to +deal with PLL. Different from average-based methods that +treat all candidate labels equally, identification-based meth- +ods treat the ground-truth label as a latent variable and maxi- +mize its estimated probability via maximum margin criterion +[Yu and Zhang, 2016] or maximum likelihood criterion [Liu +and Dietterich, 2012]. +Recently, deep learning has been applied in identification- +based methods and achieved promising performance on mul- +tiple datasets. +For example, [Lv et al., 2020] proposed a +self-training strategy that disambiguated candidate labels via +model outputs. +[Feng et al., 2020] introduced classifier- +consistent and risk-consistent algorithms under the uniform +candidate label generation assumption. Furthermore, [Wen +et al., 2021] relaxed this assumption and proposed a family +of loss functions for label disambiguation. More recently, +[Wang et al., 2022a] introduced contrastive learning into +PLL. This model was able to learn discriminative representa- +tions and achieved promising results under varying ambiguity +levels. The above methods rely on a fundamental assump- +tion that the ground-truth label must conceal in the candidate +set. However, this assumption may not be satisfied due to the +unprofessional judgment of the annotators, thereby limiting +their applications in real-world scenarios. +2.3 +Noisy PLL +Due to its more practical setting, noisy PLL has attracted in- +creasing attention from researchers. The core challenge in +noisy PLL is how to deal with noisy samples. +Typically, +[Lv et al., 2021] utilized the noise-tolerant loss functions to +avoid overemphasizing noisy samples during training. [Lian +et al., 2022] proposed an iterative refinement network to pu- +rify noisy samples and reduce the noise level of the dataset. +[Wang et al., 2022b] detected clean samples by a distance- +based sample selection mechanism and dealt with noisy sam- +ples by a semi-supervised contrastive framework. Existing +noisy PLL methods need to detect noisy samples, but detec- +tion errors are unavoidable. These errors can accumulate and +continuously influence the training process. In this paper, we +propose DALI to reduce the negative impact of prediction er- +rors in noisy sample detection by trading off the initial can- +didate set and model outputs. Experimental results on multi- +ple benchmark datasets demonstrate the effectiveness of our +method under noisy conditions. +3 +Method +In this section, we first formalize the problem statement for +noisy PLL. After that, we discuss the motivation and intro- +duce our proposed method in detail. +3.1 +Problem Definition +Let X be the input space and Y = {1, 2, · · · , c} be the la- +bel space with c distinct categories. We consider a partially +labeled dataset D = {(xi, S(xi))}N +i=1 where N is the num- +ber of samples and S(xi) ∈ {0, 1}c is the candidate set for +the sample xi ∈ X. We denote the jth element of S(xi) as +Sj(xi). Here, Sj(xi) is equal to 1 if the label j is a candidate +label for xi, and otherwise 0. +The goal of noisy PLL is to learn a c-class classifier that +minimizes the classification risk on the dataset. Unlike PLL +where the ground-truth label must reside in the candidate set, +noisy PLL relaxes this constraint and allows the ground-truth +label may not be in the candidate set. For a fair compari- +son, we adopt the same data generation procedure as previous +work [Lian et al., 2022]. To generate candidate labels, we +first flip incorrect labels to false positive labels with a proba- +bility q and aggregate the flipped ones with the ground-truth +label. After that, we assume that each sample has a probabil- +ity η of being the noisy sample. For each noisy sample, we +further select a label from the non-candidate set, move it into +the candidate set, and move the ground-truth label out of the +candidate set. In this paper, we denote the probability q as the +ambiguity level and the probability η as the noise level. +3.2 +Motivation +Let us consider our exam experience. When we are unfamil- +iar with a test, we believe that the correct answer must be +in the candidate set. Even if every option is wrong, we still +choose the most likely answer from the candidate set to finish +the test. But as we become familiar with the exam, we learn +to question the correctness of the candidate answers. If we +believe every option is wrong, we will consider options out- +side the candidate set. Applying the same strategy to neural +networks gives us the ability to estimate the correct label for +noisy samples. + +user A: Horse +Ground Truth: Donkey +user C: Deer +user B: Mule +Image +Origin PLL +DALI +Classification +Pseudo Label Generation +Figure 1: The overall structure of DALI. The network receives input x and produces softmax prediction probabilities P(x). Different from +traditional PLL that completely believe the candidate set, DALI can deal with noisy samples by the weighting mechanism. +3.3 +DALI Framework +Motivated by the above idea, we propose a simple yet effec- +tive framework for noisy PLL. The overall structure of DALI +is shown in Figure 1. Specifically, we first predict the soft- +max prediction probabilities P(x) for each sample x. In tra- +ditional PLL, we totally trust the candidate set S(x) and gen- +erate the pseudo label P old(x) via the following formula: +P old(x) = Normalize (S(x)P(x)) , +(1) +where Normalize(·) is a normalization function that ensures +the sum of probabilities is equal to 1. We discuss two nor- +malization functions: Onehot(·) and Scale(·). Specifically, +Onehot(·) sets the maximum value to 1 and other values to +0. Scale(z) = z1/K +i +/� +j z1/K +j +, where zi is the ith element +of z and K > 0 is a scaling factor. In this paper, we choose +Onehot(·) as the default normalization function. The classifi- +cation performance of Scale(·) is left for our future work. +Different from traditional PLL, DALI introduces a weight- +ing coefficient λ to control the reliability of the candidate set: +˜S(x) = S(x) + λ (1 − S(x)) , +(2) +P new(x) = Normalize +� +˜S(x)P(x) +� +, +(3) +where λ = 0 means that we fully trust the given candidate +set. λ = 1 means that we do not trust the candidate set but +believe our judgment P(x). In a particular exam, when we +are unfamiliar with the test, we believe the correct answer +should conceal in candidate labels. As we become more fa- +miliar with the test, we gradually question candidate labels. +To mimic such a process, we adjust the value of weighting +coefficient λ during training. Specifically, we set λ = 0 at the +beginning and gradually increase λ in the learning process. +3.4 +Theoretical Analysis +We further conduct a theoretical analysis to demonstrate the +feasibility of DALI. Suppose P(x), S(x), and w(x) are the +prediction probabilities, the candidate set, and the pseudo la- +bel of the sample x. Let Si, Pi and wi be the ith element of +S(x), P(x) and w(x). In this section, we provide proofs for +two normalization functions: Onehot(·) and Scale(·). +Proof for Onehot Normalization +During training, we should optimize the following objectives: +1) w(x) should satisfy 0 ≤ wi ≤ 1 and �c +i=1 wi = 1. 2) We +should minimize the classification loss on w(x) and P(x). +3) w(x) should be small at non-candidate labels. The final +objective function is shown as follows: +max +c +� +i=1 +wi log Pi + M +� c +� +i=1 +wiSi − 1 +� +s.t. +c +� +i +wi = 1, wi ≥ 0, +(4) +where M is a penalty factor. Introduce the Lagrange multi- +pliers δi, i ∈ [1, c] and γ into Eq. 4, we have: +L = +c +� +i=1 +wi log Pi + M +� c +� +i=1 +wiSi − 1 +� ++ γ +� +1 − +c +� +i=1 +wi +� ++ +c +� +i=1 +δiwi. +(5) +Combined with the Karush-Kuhn-Tucker (KKT) condi- +tions, the optimal point should satisfy: +log Pi + MSi − γ + δi = 0, +(6) +c +� +i=1 +wi = 1, δi ≥ 0, wi ≥ 0, δiwi = 0. +(7) +Since Si ∈ {0, 1}, we have MSi = log(eMSi + (1 − Si)). +Therefore, the equivalent equation of Eq. 6 is: +δi = γ − log(eMSi + (1 − Si))Pi. +(8) + +1.00 +0.80 +0.60 +0.40 +0.20 +0.00 +mule +deer +horse +donkey1.00 +0.80 +0.60 +0.40 +0.20 +0.00 +mule +deer +horse +donkey1.00 +0.80 +0.60 +0.40 +0.20 +0.00 +horse +mule +deer +donkey1.00 +0.80 +0.60 +0.40 +0.20 +0.00 +mule +deer +horse +donkeyCombined with δi ≥ 0 in Eq. 7, we have: +γ ≥ max +i +� +log(eMSi + (1 − Si))Pi +� +. +(9) +δi > 0 is true if γ > maxi +� +log(eMSi + (1 − Si))Pi +� +. Ac- +cording to δiwi = 0, we always have wi = 0, which conflicts +with �c +i=1 wi = 1. Therefore, we have: +γ = max +i +� +log(eMSi + (1 − Si))Pi +� +. +(10) +We assume that only one i0 ∈ [1, c] reaches the maximum +(i.e., γ), then we have wi = 0, i ∈ [1, c]/i0. Combined with +�c +i=1 wi = 1, we can get wi0 = 1. Therefore, w(x) should +satisfy the following equation: +w(x) = Onehot +� +log(eMS(x) + (1 − S(x)))P(x) +� +. (11) +We mark λ = e−M. Then Eq. 11 can be converted to an +equivalent version, which is same as our DALI in Eq. 3. +w(x) = Onehot (S(x) + λ(1 − S(x))P(x)) . +(12) +Proof for Scale Normalization +Besides the above optimization objectives in Eq. 4, we further +integrate entropy regularization on w(x) to avoid overconfi- +dence of pseudo labels. The final objective function is shown +as follows: +max +c +� +i=1 +wi log Pi + M +� c +� +i=1 +wiSi − 1 +� +− K +c +� +i=1 +wi log wi +s.t. +c +� +i +wi = 1, +(13) +where M and K are penalty factors. Introduce the Lagrange +multiplier γ in Eq. 13, we have: +L = +c +� +i=1 +wi log Pi + M +� c +� +i=1 +wiSi − 1 +� +− K +c +� +i=1 +wi log wi + γ +� +1 − +c +� +i +wi +� +. +(14) +Since the optimal point should satisfy ∇wL = 0, we have: +log Pi + MSi − K (1 + log wi) − γ = 0. +(15) +Since Si ∈ {0, 1}, we have MSi = log(eMSi + (1 − Si)). +Therefore, the equivalent equation of Eq. 15 is: +log(eMSi + (1 − Si))Pi − (K + γ) − K log wi = 0, (16) +wK +i = +� +eMSi + (1 − Si) +� +Pi +eK+γ +. +(17) +We mark λ = e−M. Then Eq. 17 can be converted to: +wi = ((Si + λ(1 − Si))Pi)1/K +e1+(γ−M)/K +. +(18) +Since �c +i wi = 1, then we have: +c +� +i=1 +((Si + λ(1 − Si))Pi)1/K +e1+(γ−M)/K += 1, +(19) +e1+(γ−M)/K = +c +� +i=1 +((Si + λ (1 − Si))Pi)1/K . +(20) +Combined Eq. 18 and Eq. 20, we have: +wi = +((Si + λ(1 − Si))Pi)1/K +�c +i=1 ((Si + λ(1 − Si))Pi)1/K . +(21) +Since Scale(z) = z1/K +i +/� +j z1/K +j +, the equivalent equation +of Eq. 21 is shown as follows: +w(x) = Scale ((S(x) + λ(1 − S(x)))P(x)) . +(22) +3.5 +Implementation Detail +Although the implementation process of DALI is simple, +it requires several key components to make it effective for +noisy PLL. These include methods for adaptively adjusting +the value of λ during training, employing co-training to avoid +accumulated errors, and incorporating mixup to further im- +prove the classification performance. +Adaptively Adjusted Importance +An appropriate λ is important for DALI. As described in Sec- +tion 3.3, different epochs need distinct values of λ. It is time- +consuming and needs lots of manual effort to select the appro- +priate λ for each epoch. Therefore, we propose an adaptively +adjusted strategy in this section. +Specifically, we first estimate the noise level of the dataset +before training. Intuitively, we can randomly select a sub- +set of training samples, annotate the ground-truth labels by +professional annotators, and estimate the noise level of the +dataset. Or we can automatically estimate noisy rate via ex- +isting approaches, such as Gaussian mixture model [Arazo et +al., 2019; Song et al., 2021] or the cross-validation [Liu and +Tao, 2015; Song et al., 2019]. Through experimental analy- +sis, we observe that DALI can still achieve good performance +even if the estimated noise rate is slightly different from the +actual noise rate. +Suppose y ∈ [1, c] is the ground-truth label for the sam- +ple x. Since η controls the noise level of the dataset, we +have P(Sy(x) = 0) = η. We assume that the pseudo label +generated by DALI ˆy = arg max1≤i≤c P new(x) is accurate, +i.e., ˆy = y. Then we have P(Sˆy(x) = 0) = η. To esti- +mate the value of λ, we first study the equivalent meaning of +Sˆy(x) = 0, which is: +max +Sj(x)=0 (Sj(x) + λ (1 − Sj(x))) Pj(x) +≥ +max +Sj(x)=1 (Sj(x) + λ (1 − Sj(x))) Pj(x). +(23) +We simplify the left and right sides of Eq.23 as follows: +max +Sj(x)=0(Sj(x) + λ(1 − Sj(x)))Pj(x) += max +Sj(x)=0 λ(1 − Sj(x))Pj(x) += max +j +λ(1 − Sj(x))Pj(x), +(24) + +max +Sj(x)=1(Sj(x) + λ(1 − Sj(x)))Pj(x) += max +Sj(x)=1 Sj(x)Pj(x) += max +j +Sj(x)Pj(x). +(25) +Then, we have: +max +j +λ(1 − Sj(x))Pj(x) ≥ max +j +Sj(x)Pj(x), +(26) +λ ≥ +maxj Sj(x)Pj(x) +maxj(1 − Sj(x))Pj(x), +(27) +Therefore, P(Sˆy(x) = 0) = η can be converted to: +P +� +λ ≥ +maxj Sj(x)Pj(x) +maxj (1 − Sj(x))Pj(x) +� += η. +(28) +It means that λ is the η-quantile of +� +maxj Sj(x)Pj(x) +maxj(1 − Sj(x))Pj(x) +� +x∈D +. +(29) +To adaptively adjust λ at each epoch, we store the value of +maxj Sj(x)Pj(x) +maxj(1−Sj(x))Pj(x) for all samples into a list and compute the +η-quantile of the list as λ. +Co-training Strategy +In noisy label learning, self-training suffers from the accumu- +lated errors caused by the sample-selection bias [Jiang et al., +2018]. To address this issue, researchers propose co-training +[Blum and Mitchell, 1998], which contains multiple branches +and cross-updates these branches. In this section, we incor- +porate co-training to improve the noise robustness. +Among all PLL methods, PiCO [Wang et al., 2022a] is a +natural co-training network with two branches: one for clas- +sification and one for clustering. The output of the classifica- +tion branch guides the model to update clustering prototypes. +While the output of the clustering branch affects the update +of the pseudo label for classification. Therefore, we combine +DALI with PiCO to deal with noisy PLL. +Mixup Training +Mixup training prevents the model from overfitting noisy +samples [Zhang et al., 2018; Li et al., 2020]. Therefore, we +further incorporate mixup into DALI. Consider a pair of sam- +ples xi and xj. Suppose ˆyi and ˆyj are pseudo labels of xi and +xj, respectively. We create a virtual training sample by linear +interpolation: +xmix = αxi + (1 − α)xj, +(30) +ˆymix = αˆyi + (1 − α)ˆyj, +(31) +where α ∼ Beta(ζ, ζ) and ζ is a hyper-parameter. We define +the mixup loss Lmix as the cross-entropy loss on xmix and ˆymix. +During model optimization, we combine the mixup loss Lmix +and the PLL loss Lpll into the joint loss function: +L = Lpll + λmixLmix, +(32) +where λmix is the hyper-parameter that controls the trade-off +between different losses. +4 +Experimental Databases and Setup +4.1 +Corpus Description +We conduct experiments on two popular benchmark datasets +for noisy PLL, i.e., CIFAR-10 [Krizhevsky, 2009] and +CIFAR-100 [Krizhevsky, 2009]. +Since CIFAR-100 has a +large number of categories, we consider q ∈ {0.1, 0.3, 0.5} +for CIFAR-10 and q ∈ {0.01, 0.03, 0.05} for CIFAR-100. +We select η ∈ {0.1, 0.2, 0.3} and consider strong noisy rate +in Section 5.2. Meanwhile, we examine our method on fine- +grained datasets in Section 5.3. +4.2 +Baselines +To evaluate the performance of DALI, we implement the fol- +lowing state-of-the-art PLL methods and the latest noisy PLL +methods as baselines: 1) CC [Feng et al., 2020]: a classifier- +consistent PLL method under the uniform candidate label +generation assumption. 2) RC [Feng et al., 2020]: a risk- +consistent PLL method under the same generation assump- +tion as CC. 3) LWC [Wen et al., 2021]: a PLL method that +considers the trade-off between losses on candidate and non- +candidate sets, and exploits cross-entropy as the loss func- +tion. 4) LWS [Wen et al., 2021]: a PLL method that inte- +grates the weighted loss with the sigmoid function. 5) PiCO +[Wang et al., 2022a]: a PLL method using contrastive learn- +ing. 6) IRNet [Lian et al., 2022]: a noisy PLL method that +progressively purifies noisy samples by moving the estimated +ground-truth label into the candidate set. 7) PiCO+ [Wang et +al., 2022b]: a noisy PLL method that detects clean samples by +a distance-based sample selection mechanism and deals with +noisy samples by a semi-supervised contrastive framework. +4.3 +Implementation Details +There are mainly three user-specific parameters in DALI, i.e., +λ, λmix, and e0. Here, λ is a weighting coefficient that con- +trols the trade-off between initial candidate set and model out- +puts, λmix controls the trade-off between the PLL loss and +the mixup loss, and e0 is the start epoch of DALI. Besides +dynamically adjusted λ, we further compare with fixed λ +from {0.0, 0.1, 0.4, 0.5, 0.7}. Meanwhile, we select e0 from +{80, 100, 140} and set λmix = 1.0 as the default parameter. +Following the standard experimental setup in PLL [Wen et +al., 2021; Wang et al., 2022a], we split a clean validation set +from the training set to determine hyper-parameters. After +that, we transform the validation set back to its original PLL +form and incorporate it into the training set to accomplish the +final model training. +For mixup, we set ζ = 4 as the default parameter. We +choose Onehot(·) as the default normalization function. The +classification performance of Scale(·) is left of our future +work. We uses 18-layer ResNet [He et al., 2016] to predict +the prediction probabilities. To optimize all trainable param- +eters, we choose the SGD optimizer with a momentum of 0.9 +and set weight decay to 1e-3. We set the initial learning rate +to 0.01 and adjust it using the cosine scheduler. To eliminate +randomness of the result, we run each experiment three times +with different seeds and report the average results on the test +set. All experiments are implemented with PyTorch and car- +ried out with NVIDIA Tesla V100 GPU. + +Dataset +Method +q = 0.1 +q = 0.3 +q = 0.5 +η = 0.1 +η = 0.2 +η = 0.3 +η = 0.1 +η = 0.2 +η = 0.3 +η = 0.1 +η = 0.2 +η = 0.3 +CIFAR-10 +♦CC +79.81 +77.06 +73.87 +74.09 +71.43 +68.08 +69.87 +59.35 +48.93 +♦RC +80.87 +78.22 +75.24 +79.69 +75.69 +71.01 +72.46 +59.72 +49.74 +♦LWC +79.13 +76.15 +74.17 +77.47 +74.02 +69.10 +70.59 +57.42 +48.93 +♦LWS +82.97 +79.46 +74.28 +80.93 +76.07 +69.70 +70.41 +58.26 +39.42 +♦PiCO +90.78 +87.27 +84.96 +89.71 +85.78 +82.25 +88.11 +82.41 +68.75 +♦PiCO+ +93.64 +93.13 +92.18 +92.32 +92.22 +89.95 +91.07 +89.68 +84.08 +♦IRNet +93.44 +92.57 +92.38 +92.81 +92.18 +91.35 +91.51 +90.76 +86.19 +♦DALI +94.15 +94.04 +93.77 +93.44 +93.25 +92.42 +92.67 +91.83 +89.80 +♥PiCO+ +94.58 +94.74 +94.43 +94.02 +94.03 +92.94 +93.56 +92.65 +88.21 +♥DALI +95.83 +95.86 +95.75 +95.52 +95.41 +94.67 +95.19 +93.89 +92.26 +Dataset +Method +q = 0.01 +q = 0.03 +q = 0.05 +η = 0.1 +η = 0.2 +η = 0.3 +η = 0.1 +η = 0.2 +η = 0.3 +η = 0.1 +η = 0.2 +η = 0.3 +CIFAR-100 +♦CC +53.63 +48.84 +45.50 +51.85 +47.48 +43.37 +50.64 +45.87 +40.87 +♦RC +52.73 +48.59 +45.77 +52.15 +48.25 +43.92 +46.62 +45.46 +40.31 +♦LWC +53.16 +48.64 +45.51 +51.69 +47.60 +43.39 +50.55 +45.85 +39.83 +♦LWS +56.05 +50.66 +45.71 +53.59 +48.28 +42.20 +45.46 +39.63 +33.60 +♦PiCO +68.27 +62.24 +58.97 +67.38 +62.01 +58.64 +67.52 +61.52 +58.18 +♦PiCO+ +71.42 +70.22 +66.14 +70.89 +69.03 +64.22 +69.40 +66.67 +62.24 +♦IRNet +71.17 +70.10 +68.77 +71.01 +70.15 +68.18 +70.73 +69.33 +68.09 +♦DALI +72.26 +71.98 +71.04 +71.43 +70.79 +69.14 +72.28 +71.35 +70.05 +♥PiCO+ +75.04 +74.31 +71.79 +74.68 +73.65 +69.97 +73.06 +71.37 +67.56 +♥DALI +76.52 +76.55 +76.09 +77.27 +76.29 +75.29 +76.87 +75.23 +74.49 +Table 1: Inductive performance of different PLL methods. Here, ♦ denotes the model without mixup and ♥ denotes the model with mixup. +5 +Results and Discussion +5.1 +Main Results +For a fair comparison, we reproduce all baselines and report +results under the same data generation procedure described in +Section 3.1. Experimental results are shown in Table 1. We +observe that a large portion of the performance gain is due +to mixup rather than model innovation. To exclude the effect +of mixup, we compare different methods under the same aug- +mentation strategy. Experimental results demonstrate that our +DALI succeeds over all baselines on all datasets. In particu- +lar, the performance gap between our method and baselines +further widens as the noise level increases. +To exclude the impact of data generation strategies, we re- +port the results of DALI under another generation procedure +in [Wang et al., 2022b]. Specifically, any incorrect label has +a probability q of being the candidate label and the ground- +truth label has a probability 1 − η of being an element in the +candidate set. Experimental results are shown in Table 2. In +this table, we further compare with noise-tolerant loss func- +tions in [Lv et al., 2021], i.e., MSE and GCE. Experimental +results demonstrate that our DALI can achieve state-of-the-art +performance under different data generation strategies. +The above results demonstrate the noise robustness of our +proposed method. On the one hand, existing PLL methods are +mainly designed for clean samples, but ignore the presence of +noisy samples. Our DALI is able to deal with noisy samples +by trading off the initial candidate set and model outputs. On +the other hand, existing noisy PLL methods need to detect +noisy samples, but detection errors are unavoidable. These +errors can accumulate and continuously influence the training +Method +q = 0.05 +q = 0.1 +♦CC +37.90±3.27 +22.28±6.18 +♦RC +46.11±0.38 +38.03±1.79 +♦LWS +42.31±1.05 +17.76±4.47 +♦GCE +31.65±0.71 +24.21±1.67 +♦MSE +27.36±0.40 +22.98±1.74 +♦PiCO +59.81±0.24 +45.32±0.89 +♥PiCO+ +72.98±0.22 +62.24±0.97 +♥DALI +75.62±0.34 +74.75±0.38 +Table 2: Accuracy comparisons on CIFAR-100 (η = 0.2) with an- +other data generation strategy in [Wang et al., 2022c]. +Method +η = 0.3 +η = 0.4 +η = 0.5 +♦RC +39.19±0.20 +33.64±0.82 +26.91±0.83 +♦PiCO +52.18±0.52 +44.17±0.08 +35.51±1.14 +♥PiCO+ +70.46±0.51 +66.41±0.58 +60.50±0.99 +♦DALI +70.06±0.32 +68.82±0.37 +63.39±0.82 +♥DALI +75.07±0.23 +71.76±0.56 +69.59±0.62 +Table 3: Performance on CIFAR-100 (q = 0.05) with severe noise. +process. Differently, DALI allows us to take advantage of +the initial candidate set and restart correction to deal with the +problem of accumulated errors. +5.2 +Robustness with Severe Noise +In this section, we conduct experiments on the datasets with +severe noise to demonstrate the noise robustness of DALI. In +particular, we choose η ∈ {0.3, 0.4, 0.5}. Table 3 compares +DALI with the three most competitive baselines, i.e., RC, + +Method +CUB-200 +CIFAR-100H +(q = 0.05, η = 0.2) +(q = 0.5, η = 0.2) +♦CC +26.98±1.16 +34.57±0.99 +♦RC +44.74±2.47 +48.03±0.47 +♦LWS +18.65±2.15 +22.18±6.12 +♦GCE +5.13±38.65 +33.21±2.03 +♦MSE +20.92±1.20 +35.20±1.03 +♦PiCO +53.05±2.03 +59.81±0.25 +♥PiCO+ +60.65±0.79 +68.31±0.47 +♥DALI +63.91±0.35 +72.36±0.20 +Table 4: Classification performance on fine-grained datasets. +(a) CIFAR-10 (q = 0.3) +(b) CIFAR-100 (q = 0.05) +Figure 2: Classification performance of fixed λ and dynamically +adjusted λ. The noise level of these datasets is fixed to η = 0.3. We +mark the results of dynamically adjusted λ with red lines. +PiCO, and PiCO+. We observe that our method succeeds over +these baselines under varying noise levels. Taking the results +on η = 0.5 as an example, DALI outperforms the currently +advanced approaches by 9.09%. These results show that our +DALI is more robust to noise than existing algorithms. +5.3 +Fine-Grained Partial Label Learning +Considering that similar categories are more likely to be +added to the candidate set, we conduct experiments on +more challenging fine-grained datasets including CIFAR- +100H [Wang et al., 2022a] and CUB-200 [Welinder et al., +2011]. Different from previous strategies that generate can- +didate labels from the entire label space, we generate candi- +date labels belonging to the same superclass. In our imple- +mentation, we leverage a pre-trained encoder for CUB-200, +otherwise the model will not converge. Experimental results +are shown in Table 4. We observe that DALI outperforms +all baselines on all datasets. Therefore, we conclude that our +method is also effective on fine-grained classification tasks. +5.4 +Ablation Study +Dynamically adjusted λ vs. fixed λ. Proper λ is impor- +tant for DALI. Figure 2 compares the performance of dynam- +ically adjusted λ and fixed λ. We observe that well-chosen λ +can achieve similar performance to our dynamically adjusted +strategy, but most fixed λ performs worse than dynamically +adjusted λ. Furthermore, adaptively adjusted λ requires less +manual effort. Therefore, we prefer to use the dynamically +adjusted strategy in DALI. +Self-training vs. co-training. Table 5 compares the per- +formance of self-training and co-training. In this table, DALI +η +DALI-CT +DALI +CIFAR-10 +(q = 0.3) +0.1 +91.11 +93.44 +0.2 +90.66 +93.25 +0.3 +88.38 +92.42 +CIFAR-100 +(q = 0.05) +0.1 +66.76 +72.28 +0.2 +65.10 +71.35 +0.3 +59.19 +70.05 +Table 5: Classification performance of self-training and co-training. +(a) λmix on CIFAR-10 (e0 = 80) +(b) e0 on CIFAR-10 (λmix = 1) +(c) λmix on CIFAR-100 (e0 = 80) +(d) e0 on CIFAR-100 (λmix = 1) +Figure 3: Parameter sensitivity analysis on CIFAR-10 (q = 0.3, η = +0.3) and CIFAR-100 (q = 0.05, η = 0.3). +is our proposed method and DALI-CT denotes the model +without co-training. These models are implemented without +mixup to reveal the real impact of co-training. From Table +5, we observe that DALI outperforms DALI-CT in all cases. +These results demonstrate the effectiveness of co-training. +Parameter sensitivity analysis. As described in Section +4.3, we choose e0 from {80, 100, 140} and set λmix = 1.0 +as the default parameter. Besides these values, we further +investigate the impact of e0 and λmix on a larger range of +values. +Experimental results are shown in Figure 3. +We +observe that different datasets require distinct λmix. Choos- +ing an appropriate λmix can further improve the performance +of our method. Meanwhile, as e0 gradually increases, the +classification performance improves first and then decreases. +Therefore, warm-up training is necessary for DALI. But too +large e0 causes the model to overfit noise samples. Therefore, +a good choice of hyper-parameters can remarkably improve +performance under noisy conditions. +6 +Conclusion +In this paper, we propose a novel framework with theoretical +guarantees for noisy PLL, called DALI. It exploits weight- +ing coefficients to dynamically adjust the importance of the +initial candidate set and model outputs. Experimental results + +92.42 +9203-----91:94 +92.03 +90.20 +90.07 +accuracy +88.12 +86.16 +te: +84.21 +82.69 +82.25 +82.25 +80.29 +0.0 +0.1 +0.4 +0.5 +0.7 +入70.56 +70.56 +70.05 +68.67 +68.07 +accuracy +65.91 +65.58 +63.96 +test +63.08 +60.59 +58.10 +58.10 +55.61 +0.0 +0.1 +0.4 +0.5 +0.7 +入94.91 +94.91 +94.80 +94.83 +94.59 +94.43 +94.41 +accuracy +93.91 +93.57 +test +93.42 +92.92 +92.42 +92.42 +91.92 +0.0 +0.1 +0.5 +1.0 +2.0 +5.0 +8.0 +入mix94.67 +94.67 +94.56 +94.59 +94.54 +94.53 +94.40 +94.41 +accuracy +94.14 +94.13 +test +93.88 +93.61 +93.35 +93.35 +93.09 +20 +40 +80 +100 +140 +300 +400 +500 +eo73.88 +73.88 +73.15 +73.11 +accuracy +72.50 +72.35 +test +71.58 +71.05 +70.96 +70.82 +70.59 +70.05 +70.05 +69.28 +0.0 +0.1 +0.5 +1.0 +2.0 +5.0 +8.0 +Λmix74.40 +074.49 +74.49 +73.88 +73.65 +73.29 +accuracy +72.81 +72.58 +test +71.96 +71.84 +71.12 +70.28 +70.39 +70.28 +69.44 +20 +40 +80 +100 +140 +300 +400 +500 +eoon multiple benchmark datasets demonstrate the effectiveness +of our method. DALI succeeds over currently advanced ap- +proaches on noisy PLL. Meanwhile, our method shows strong +competitiveness in PLL with severe noise and fine-grained +PLL. Through ablation studies, we verify the importance of +each component in the framework, including the dynamically +adjusted strategy, co-training and mixup. +This paper chooses Onehot(·) as the normalization func- +tion. In the future, we will systematically investigate the in- +fluence of other normalization functions, such as Scale(·). +Acknowledgments +This work is founded by the National Natural Science Foun- +dation of China (NSFC) under Grants 62201572, 61831022, +62276259 and U21B2010. +References +[Arazo et al., 2019] Eric Arazo, Diego Ortego, Paul Albert, +Noel O’Connor, and Kevin McGuinness. Unsupervised la- +bel noise modeling and loss correction. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='cn, liubin@nlpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='cn, jhtao@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='cn Abstract Noisy partial label learning (noisy PLL) is an im- portant branch of weakly supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Un- like PLL where the ground-truth label must reside in the candidate set, noisy PLL relaxes this con- straint and allows the ground-truth label may not be in the candidate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To address this problem, existing works attempt to detect noisy samples and estimate the ground-truth label for each noisy sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' However, detection errors are inevitable, and these errors will accumulate during training and continuously affect model optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To address this challenge, we propose a novel framework for noisy PLL, called “Dynamically Adjusted Label Importance (DALI)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' It aims to reduce the negative impact of detection errors by trading off the ini- tial candidate set and model outputs with theoret- ical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Experimental results on multiple datasets demonstrate that our DALI succeeds over existing state-of-the-art approaches on noisy PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Our code will soon be publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 1 Introduction Partial label learning (PLL) [Feng and An, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Yan and Guo, 2020] (also called ambiguous label learning [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2017] and superset label learning [Liu and Dietterich, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Liu and Dietterich, 2014]) is a typical type of weakly supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Unlike supervised learn- ing where each sample is associated with a ground-truth label, PLL requires identifying the ground-truth label from a set of candidate labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Due to the low annotation cost of partially labeled samples, PLL has attracted increasing attention from researchers and applied in many tasks, such as object recog- nition [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2017], web mining [Huiskes and Lew, 2008], and ecological informatics [Briggs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The basic assumption of PLL is that the ground-truth la- bel must reside in the candidate set [Jin and Ghahramani, 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' However, this assumption may not be satisfied due to the unprofessional judgment of the annotators [Cid-Sueiro, ∗Equal Contribution 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Recently, some researchers have relaxed this assump- tion and focused on a more practical setting called noisy PLL [Lv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In noisy PLL, the ground-truth la- bel may not conceal in the candidate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To deal with this task, [Lv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2021] leveraged noise-tolerant loss functions to avoid overemphasizing noisy samples in the learning pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' However, they cannot fully exploit the useful informa- tion in noisy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To address this challenge, [Lian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022] and [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022b] proposed to detect noisy samples and estimate the pseudo label for each noisy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' However, detection errors are unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' These errors can accumulate during training and continuously affect model op- timization, thereby limiting their performance on noisy PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To this end, we propose a novel framework for noisy PLL, called “Dynamically Adjusted Label Importance (DALI)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Although we may make mistakes in noisy sample detection, DALI allows us to leverage the initial candidate set and restart correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Meanwhile, we propose a strategy to automati- cally determine the weighting coefficient in the learning pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To further improve the performance, we incorporate DALI with co-training [Blum and Mitchell, 1998] and mixup [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2018], which are powerful in noisy label learn- ing [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We also perform theoretical analysis and prove the feasibility of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To verify the effectiveness of DALI, we conduct experiments on mul- tiple benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Experimental results demonstrate that our method outperforms currently advanced approaches, setting the new state-of-the-art records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The main contribu- tions of this paper can be summarized as follows: We propose a novel framework for noisy PLL with the- oretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Our DALI can reduce the negative impact of prediction errors in noisy sample detection by trading off the initial candidate set and model outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We further propose an automatic parameter selection strategy for weighting coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Combining DALI with mixup and co-training, we can achieve better per- formance under noisy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Experimental results on multiple datasets demonstrate the effectiveness of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' DALI is su- perior to currently advanced approaches on noisy PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The remainder of this paper is organized as follows: In Sec- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='12077v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='CV] 28 Jan 2023 tion 2, we briefly review some recent works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In Section 3, we formalize the problem statement and describe our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In Section 4, we present our experimental datasets and setup in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In Section 5, we illustrate the experimen- tal results and analysis on benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Finally, we conclude this paper and discuss our future work in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 2 Related Work The ground-truth label of PLL is concealed in the candi- date set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, the core of dealing with this task is to disambiguate candidate labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In this section, we first in- troduce two typical disambiguation strategies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', average- based methods and identification-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' After that, we briefly review some recent works on noisy PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 Average-based PLL The most intuitive solution is the average-based method, which assumes that each candidate label has an equal proba- bility of being the ground-truth label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Typically, [H¨ullermeier and Beringer, 2006] leveraged k-nearest neighbors for label disambiguation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' For each sample, they treated all candidate labels of its neighborhood equally and predicted the ground- truth label through voting strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Differently, [Cour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2009] maximized the average output of candidate and non- candidate labels in parametric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' However, average- based methods can be severely affected by false positive la- bels in the candidate set [Jin and Ghahramani, 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 Identification-based PLL To address the shortcomings of average-based methods, re- searchers have focused on identification-based methods to deal with PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Different from average-based methods that treat all candidate labels equally, identification-based meth- ods treat the ground-truth label as a latent variable and maxi- mize its estimated probability via maximum margin criterion [Yu and Zhang, 2016] or maximum likelihood criterion [Liu and Dietterich, 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Recently, deep learning has been applied in identification- based methods and achieved promising performance on mul- tiple datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' For example, [Lv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2020] proposed a self-training strategy that disambiguated candidate labels via model outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' [Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2020] introduced classifier- consistent and risk-consistent algorithms under the uniform candidate label generation assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Furthermore, [Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2021] relaxed this assumption and proposed a family of loss functions for label disambiguation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' More recently, [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022a] introduced contrastive learning into PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' This model was able to learn discriminative representa- tions and achieved promising results under varying ambiguity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The above methods rely on a fundamental assump- tion that the ground-truth label must conceal in the candidate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' However, this assumption may not be satisfied due to the unprofessional judgment of the annotators, thereby limiting their applications in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 Noisy PLL Due to its more practical setting, noisy PLL has attracted in- creasing attention from researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The core challenge in noisy PLL is how to deal with noisy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Typically, [Lv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2021] utilized the noise-tolerant loss functions to avoid overemphasizing noisy samples during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' [Lian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022] proposed an iterative refinement network to pu- rify noisy samples and reduce the noise level of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022b] detected clean samples by a distance- based sample selection mechanism and dealt with noisy sam- ples by a semi-supervised contrastive framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Existing noisy PLL methods need to detect noisy samples, but detec- tion errors are unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' These errors can accumulate and continuously influence the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In this paper, we propose DALI to reduce the negative impact of prediction er- rors in noisy sample detection by trading off the initial can- didate set and model outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Experimental results on multi- ple benchmark datasets demonstrate the effectiveness of our method under noisy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 3 Method In this section, we first formalize the problem statement for noisy PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' After that, we discuss the motivation and intro- duce our proposed method in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 Problem Definition Let X be the input space and Y = {1, 2, · · · , c} be the la- bel space with c distinct categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We consider a partially labeled dataset D = {(xi, S(xi))}N i=1 where N is the num- ber of samples and S(xi) ∈ {0, 1}c is the candidate set for the sample xi ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We denote the jth element of S(xi) as Sj(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Here, Sj(xi) is equal to 1 if the label j is a candidate label for xi, and otherwise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The goal of noisy PLL is to learn a c-class classifier that minimizes the classification risk on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Unlike PLL where the ground-truth label must reside in the candidate set, noisy PLL relaxes this constraint and allows the ground-truth label may not be in the candidate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' For a fair compari- son, we adopt the same data generation procedure as previous work [Lian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To generate candidate labels, we first flip incorrect labels to false positive labels with a proba- bility q and aggregate the flipped ones with the ground-truth label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' After that, we assume that each sample has a probabil- ity η of being the noisy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' For each noisy sample, we further select a label from the non-candidate set, move it into the candidate set, and move the ground-truth label out of the candidate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In this paper, we denote the probability q as the ambiguity level and the probability η as the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 Motivation Let us consider our exam experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' When we are unfamil- iar with a test, we believe that the correct answer must be in the candidate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Even if every option is wrong, we still choose the most likely answer from the candidate set to finish the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' But as we become familiar with the exam, we learn to question the correctness of the candidate answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' If we believe every option is wrong, we will consider options out- side the candidate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Applying the same strategy to neural networks gives us the ability to estimate the correct label for noisy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' user A: Horse Ground Truth: Donkey user C: Deer user B: Mule Image Origin PLL DALI Classification Pseudo Label Generation Figure 1: The overall structure of DALI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The network receives input x and produces softmax prediction probabilities P(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Different from traditional PLL that completely believe the candidate set, DALI can deal with noisy samples by the weighting mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 DALI Framework Motivated by the above idea, we propose a simple yet effec- tive framework for noisy PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The overall structure of DALI is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Specifically, we first predict the soft- max prediction probabilities P(x) for each sample x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In tra- ditional PLL, we totally trust the candidate set S(x) and gen- erate the pseudo label P old(x) via the following formula: P old(x) = Normalize (S(x)P(x)) , (1) where Normalize(·) is a normalization function that ensures the sum of probabilities is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We discuss two nor- malization functions: Onehot(·) and Scale(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Specifically, Onehot(·) sets the maximum value to 1 and other values to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Scale(z) = z1/K i /� j z1/K j , where zi is the ith element of z and K > 0 is a scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In this paper, we choose Onehot(·) as the default normalization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The classifi- cation performance of Scale(·) is left for our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Different from traditional PLL, DALI introduces a weight- ing coefficient λ to control the reliability of the candidate set: ˜S(x) = S(x) + λ (1 − S(x)) , (2) P new(x) = Normalize � ˜S(x)P(x) � , (3) where λ = 0 means that we fully trust the given candidate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' λ = 1 means that we do not trust the candidate set but believe our judgment P(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In a particular exam, when we are unfamiliar with the test, we believe the correct answer should conceal in candidate labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' As we become more fa- miliar with the test, we gradually question candidate labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To mimic such a process, we adjust the value of weighting coefficient λ during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Specifically, we set λ = 0 at the beginning and gradually increase λ in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='4 Theoretical Analysis We further conduct a theoretical analysis to demonstrate the feasibility of DALI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Suppose P(x), S(x), and w(x) are the prediction probabilities, the candidate set, and the pseudo la- bel of the sample x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Let Si, Pi and wi be the ith element of S(x), P(x) and w(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In this section, we provide proofs for two normalization functions: Onehot(·) and Scale(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Proof for Onehot Normalization During training, we should optimize the following objectives: 1) w(x) should satisfy 0 ≤ wi ≤ 1 and �c i=1 wi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 2) We should minimize the classification loss on w(x) and P(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 3) w(x) should be small at non-candidate labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The final objective function is shown as follows: max c � i=1 wi log Pi + M � c � i=1 wiSi − 1 � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' c � i wi = 1, wi ≥ 0, (4) where M is a penalty factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Introduce the Lagrange multi- pliers δi, i ∈ [1, c] and γ into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 4, we have: L = c � i=1 wi log Pi + M � c � i=1 wiSi − 1 � + γ � 1 − c � i=1 wi � + c � i=1 δiwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (5) Combined with the Karush-Kuhn-Tucker (KKT) condi- tions, the optimal point should satisfy: log Pi + MSi − γ + δi = 0, (6) c � i=1 wi = 1, δi ≥ 0, wi ≥ 0, δiwi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (7) Since Si ∈ {0, 1}, we have MSi = log(eMSi + (1 − Si)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, the equivalent equation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 6 is: δi = γ − log(eMSi + (1 − Si))Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (8) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='00 mule deer horse donkey1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='00 mule deer horse donkey1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='00 horse mule deer donkey1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='00 mule deer horse donkeyCombined with δi ≥ 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 7, we have: γ ≥ max i � log(eMSi + (1 − Si))Pi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (9) δi > 0 is true if γ > maxi � log(eMSi + (1 − Si))Pi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Ac- cording to δiwi = 0, we always have wi = 0, which conflicts with �c i=1 wi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, we have: γ = max i � log(eMSi + (1 − Si))Pi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (10) We assume that only one i0 ∈ [1, c] reaches the maximum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', γ), then we have wi = 0, i ∈ [1, c]/i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Combined with �c i=1 wi = 1, we can get wi0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, w(x) should satisfy the following equation: w(x) = Onehot � log(eMS(x) + (1 − S(x)))P(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (11) We mark λ = e−M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 11 can be converted to an equivalent version, which is same as our DALI in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' w(x) = Onehot (S(x) + λ(1 − S(x))P(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (12) Proof for Scale Normalization Besides the above optimization objectives in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 4, we further integrate entropy regularization on w(x) to avoid overconfi- dence of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The final objective function is shown as follows: max c � i=1 wi log Pi + M � c � i=1 wiSi − 1 � − K c � i=1 wi log wi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' c � i wi = 1, (13) where M and K are penalty factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Introduce the Lagrange multiplier γ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 13, we have: L = c � i=1 wi log Pi + M � c � i=1 wiSi − 1 � − K c � i=1 wi log wi + γ � 1 − c � i wi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (14) Since the optimal point should satisfy ∇wL = 0, we have: log Pi + MSi − K (1 + log wi) − γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (15) Since Si ∈ {0, 1}, we have MSi = log(eMSi + (1 − Si)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, the equivalent equation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 15 is: log(eMSi + (1 − Si))Pi − (K + γ) − K log wi = 0, (16) wK i = � eMSi + (1 − Si) � Pi eK+γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (17) We mark λ = e−M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 17 can be converted to: wi = ((Si + λ(1 − Si))Pi)1/K e1+(γ−M)/K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (18) Since �c i wi = 1, then we have: c � i=1 ((Si + λ(1 − Si))Pi)1/K e1+(γ−M)/K = 1, (19) e1+(γ−M)/K = c � i=1 ((Si + λ (1 − Si))Pi)1/K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (20) Combined Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 18 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 20, we have: wi = ((Si + λ(1 − Si))Pi)1/K �c i=1 ((Si + λ(1 − Si))Pi)1/K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (21) Since Scale(z) = z1/K i /� j z1/K j , the equivalent equation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 21 is shown as follows: w(x) = Scale ((S(x) + λ(1 − S(x)))P(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (22) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='5 Implementation Detail Although the implementation process of DALI is simple, it requires several key components to make it effective for noisy PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' These include methods for adaptively adjusting the value of λ during training, employing co-training to avoid accumulated errors, and incorporating mixup to further im- prove the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Adaptively Adjusted Importance An appropriate λ is important for DALI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' As described in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3, different epochs need distinct values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' It is time- consuming and needs lots of manual effort to select the appro- priate λ for each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, we propose an adaptively adjusted strategy in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Specifically, we first estimate the noise level of the dataset before training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Intuitively, we can randomly select a sub- set of training samples, annotate the ground-truth labels by professional annotators, and estimate the noise level of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Or we can automatically estimate noisy rate via ex- isting approaches, such as Gaussian mixture model [Arazo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2021] or the cross-validation [Liu and Tao, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Through experimental analy- sis, we observe that DALI can still achieve good performance even if the estimated noise rate is slightly different from the actual noise rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Suppose y ∈ [1, c] is the ground-truth label for the sam- ple x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Since η controls the noise level of the dataset, we have P(Sy(x) = 0) = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We assume that the pseudo label generated by DALI ˆy = arg max1≤i≤c P new(x) is accurate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', ˆy = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Then we have P(Sˆy(x) = 0) = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To esti- mate the value of λ, we first study the equivalent meaning of Sˆy(x) = 0, which is: max Sj(x)=0 (Sj(x) + λ (1 − Sj(x))) Pj(x) ≥ max Sj(x)=1 (Sj(x) + λ (1 − Sj(x))) Pj(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (23) We simplify the left and right sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='23 as follows: max Sj(x)=0(Sj(x) + λ(1 − Sj(x)))Pj(x) = max Sj(x)=0 λ(1 − Sj(x))Pj(x) = max j λ(1 − Sj(x))Pj(x), (24) max Sj(x)=1(Sj(x) + λ(1 − Sj(x)))Pj(x) = max Sj(x)=1 Sj(x)Pj(x) = max j Sj(x)Pj(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (25) Then, we have: max j λ(1 − Sj(x))Pj(x) ≥ max j Sj(x)Pj(x), (26) λ ≥ maxj Sj(x)Pj(x) maxj(1 − Sj(x))Pj(x), (27) Therefore, P(Sˆy(x) = 0) = η can be converted to: P � λ ≥ maxj Sj(x)Pj(x) maxj (1 − Sj(x))Pj(x) � = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (28) It means that λ is the η-quantile of � maxj Sj(x)Pj(x) maxj(1 − Sj(x))Pj(x) � x∈D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (29) To adaptively adjust λ at each epoch, we store the value of maxj Sj(x)Pj(x) maxj(1−Sj(x))Pj(x) for all samples into a list and compute the η-quantile of the list as λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Co-training Strategy In noisy label learning, self-training suffers from the accumu- lated errors caused by the sample-selection bias [Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To address this issue, researchers propose co-training [Blum and Mitchell, 1998], which contains multiple branches and cross-updates these branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In this section, we incor- porate co-training to improve the noise robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Among all PLL methods, PiCO [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022a] is a natural co-training network with two branches: one for clas- sification and one for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The output of the classifica- tion branch guides the model to update clustering prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' While the output of the clustering branch affects the update of the pseudo label for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, we combine DALI with PiCO to deal with noisy PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Mixup Training Mixup training prevents the model from overfitting noisy samples [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, we further incorporate mixup into DALI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Consider a pair of sam- ples xi and xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Suppose ˆyi and ˆyj are pseudo labels of xi and xj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We create a virtual training sample by linear interpolation: xmix = αxi + (1 − α)xj, (30) ˆymix = αˆyi + (1 − α)ˆyj, (31) where α ∼ Beta(ζ, ζ) and ζ is a hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We define the mixup loss Lmix as the cross-entropy loss on xmix and ˆymix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' During model optimization, we combine the mixup loss Lmix and the PLL loss Lpll into the joint loss function: L = Lpll + λmixLmix, (32) where λmix is the hyper-parameter that controls the trade-off between different losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 4 Experimental Databases and Setup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 Corpus Description We conduct experiments on two popular benchmark datasets for noisy PLL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', CIFAR-10 [Krizhevsky, 2009] and CIFAR-100 [Krizhevsky, 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Since CIFAR-100 has a large number of categories, we consider q ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='5} for CIFAR-10 and q ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05} for CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We select η ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3} and consider strong noisy rate in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Meanwhile, we examine our method on fine- grained datasets in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 Baselines To evaluate the performance of DALI, we implement the fol- lowing state-of-the-art PLL methods and the latest noisy PLL methods as baselines: 1) CC [Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2020]: a classifier- consistent PLL method under the uniform candidate label generation assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 2) RC [Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2020]: a risk- consistent PLL method under the same generation assump- tion as CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 3) LWC [Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2021]: a PLL method that considers the trade-off between losses on candidate and non- candidate sets, and exploits cross-entropy as the loss func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 4) LWS [Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2021]: a PLL method that inte- grates the weighted loss with the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 5) PiCO [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022a]: a PLL method using contrastive learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 6) IRNet [Lian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022]: a noisy PLL method that progressively purifies noisy samples by moving the estimated ground-truth label into the candidate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 7) PiCO+ [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022b]: a noisy PLL method that detects clean samples by a distance-based sample selection mechanism and deals with noisy samples by a semi-supervised contrastive framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 Implementation Details There are mainly three user-specific parameters in DALI, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', λ, λmix, and e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Here, λ is a weighting coefficient that con- trols the trade-off between initial candidate set and model out- puts, λmix controls the trade-off between the PLL loss and the mixup loss, and e0 is the start epoch of DALI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Besides dynamically adjusted λ, we further compare with fixed λ from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Meanwhile, we select e0 from {80, 100, 140} and set λmix = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='0 as the default parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Following the standard experimental setup in PLL [Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022a], we split a clean validation set from the training set to determine hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' After that, we transform the validation set back to its original PLL form and incorporate it into the training set to accomplish the final model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' For mixup, we set ζ = 4 as the default parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We choose Onehot(·) as the default normalization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The classification performance of Scale(·) is left of our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We uses 18-layer ResNet [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2016] to predict the prediction probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To optimize all trainable param- eters, we choose the SGD optimizer with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='9 and set weight decay to 1e-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We set the initial learning rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='01 and adjust it using the cosine scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To eliminate randomness of the result, we run each experiment three times with different seeds and report the average results on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' All experiments are implemented with PyTorch and car- ried out with NVIDIA Tesla V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Dataset Method q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='5 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 CIFAR-10 ♦CC 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='81 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='06 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='87 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='09 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='43 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='08 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='87 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='35 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='93 ♦RC 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='87 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='22 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='24 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='69 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='69 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='01 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='46 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='72 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='74 ♦LWC 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='13 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='15 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='17 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='47 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='02 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='10 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='59 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='42 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='93 ♦LWS 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='97 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='46 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='28 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='93 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='07 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='70 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='41 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='26 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='42 ♦PiCO 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='78 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='27 84.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='68 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='08 ♦IRNet 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='44 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='57 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='38 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='81 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='18 91.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='21 ♥DALI 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='83 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='86 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='75 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='52 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='41 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='67 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='19 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='89 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='26 Dataset Method q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='01 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='03 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 η = 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='64 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='87 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='87 ♦RC 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='73 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='59 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='77 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='15 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='25 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='92 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='62 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='46 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='31 ♦LWC 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='16 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='64 45.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='28 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='35 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05 ♥PiCO+ 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='04 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='31 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='79 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='68 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='65 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='97 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='06 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='37 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='56 ♥DALI 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='52 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='55 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='09 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='27 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='29 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='29 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='87 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='23 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='49 Table 1: Inductive performance of different PLL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Here, ♦ denotes the model without mixup and ♥ denotes the model with mixup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 5 Results and Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 Main Results For a fair comparison, we reproduce all baselines and report results under the same data generation procedure described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Experimental results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We observe that a large portion of the performance gain is due to mixup rather than model innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To exclude the effect of mixup, we compare different methods under the same aug- mentation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Experimental results demonstrate that our DALI succeeds over all baselines on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In particu- lar, the performance gap between our method and baselines further widens as the noise level increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' To exclude the impact of data generation strategies, we re- port the results of DALI under another generation procedure in [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Specifically, any incorrect label has a probability q of being the candidate label and the ground- truth label has a probability 1 − η of being an element in the candidate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Experimental results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In this table, we further compare with noise-tolerant loss func- tions in [Lv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2021], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', MSE and GCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Experimental results demonstrate that our DALI can achieve state-of-the-art performance under different data generation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The above results demonstrate the noise robustness of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' On the one hand, existing PLL methods are mainly designed for clean samples, but ignore the presence of noisy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Our DALI is able to deal with noisy samples by trading off the initial candidate set and model outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' On the other hand, existing noisy PLL methods need to detect noisy samples, but detection errors are unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' These errors can accumulate and continuously influence the training Method q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 ♦CC 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='90±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='27 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='28±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='18 ♦RC 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='38 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='03±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='79 ♦LWS 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='31±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='76±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='47 ♦GCE 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='71 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='21±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='67 ♦MSE 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='40 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='98±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='74 ♦PiCO 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='24 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='89 ♥PiCO+ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='98±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='22 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='97 ♥DALI 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='34 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='38 Table 2: Accuracy comparisons on CIFAR-100 (η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2) with an- other data generation strategy in [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Method η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='4 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='5 ♦RC 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='20 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='82 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='91±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='83 ♦PiCO 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='52 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='08 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='51±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='14 ♥PiCO+ 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='51 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='58 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='50±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='99 ♦DALI 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='06±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='32 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='37 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='82 ♥DALI 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='07±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='23 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='56 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='62 Table 3: Performance on CIFAR-100 (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05) with severe noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Differently, DALI allows us to take advantage of the initial candidate set and restart correction to deal with the problem of accumulated errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 Robustness with Severe Noise In this section, we conduct experiments on the datasets with severe noise to demonstrate the noise robustness of DALI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In particular, we choose η ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Table 3 compares DALI with the three most competitive baselines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', RC, Method CUB-200 CIFAR-100H (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2) (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='5, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2) ♦CC 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='98±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='16 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='99 ♦RC 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='74±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='47 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='47 ♦LWS 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='65±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='15 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='18±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='12 ♦GCE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='13±38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='65 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='21±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='03 ♦MSE 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='92±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='20 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='20±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='03 ♦PiCO 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='03 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='25 ♥PiCO+ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='79 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='47 ♥DALI 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='91±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='35 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='20 Table 4: Classification performance on fine-grained datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (a) CIFAR-10 (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3) (b) CIFAR-100 (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05) Figure 2: Classification performance of fixed λ and dynamically adjusted λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' The noise level of these datasets is fixed to η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We mark the results of dynamically adjusted λ with red lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' PiCO, and PiCO+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We observe that our method succeeds over these baselines under varying noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Taking the results on η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='5 as an example, DALI outperforms the currently advanced approaches by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='09%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' These results show that our DALI is more robust to noise than existing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 Fine-Grained Partial Label Learning Considering that similar categories are more likely to be added to the candidate set, we conduct experiments on more challenging fine-grained datasets including CIFAR- 100H [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2022a] and CUB-200 [Welinder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Different from previous strategies that generate can- didate labels from the entire label space, we generate candi- date labels belonging to the same superclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In our imple- mentation, we leverage a pre-trained encoder for CUB-200, otherwise the model will not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Experimental results are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We observe that DALI outperforms all baselines on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, we conclude that our method is also effective on fine-grained classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='4 Ablation Study Dynamically adjusted λ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' fixed λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Proper λ is impor- tant for DALI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Figure 2 compares the performance of dynam- ically adjusted λ and fixed λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We observe that well-chosen λ can achieve similar performance to our dynamically adjusted strategy, but most fixed λ performs worse than dynamically adjusted λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Furthermore, adaptively adjusted λ requires less manual effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, we prefer to use the dynamically adjusted strategy in DALI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Self-training vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' co-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Table 5 compares the per- formance of self-training and co-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' In this table, DALI η DALI-CT DALI CIFAR-10 (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='11 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='66 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='38 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='42 CIFAR-100 (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='1 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='76 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='10 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='19 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05 Table 5: Classification performance of self-training and co-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' (a) λmix on CIFAR-10 (e0 = 80) (b) e0 on CIFAR-10 (λmix = 1) (c) λmix on CIFAR-100 (e0 = 80) (d) e0 on CIFAR-100 (λmix = 1) Figure 3: Parameter sensitivity analysis on CIFAR-10 (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3) and CIFAR-100 (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='05, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' is our proposed method and DALI-CT denotes the model without co-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' These models are implemented without mixup to reveal the real impact of co-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' From Table 5, we observe that DALI outperforms DALI-CT in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' These results demonstrate the effectiveness of co-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Parameter sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' As described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='3, we choose e0 from {80, 100, 140} and set λmix = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='0 as the default parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Besides these values, we further investigate the impact of e0 and λmix on a larger range of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Experimental results are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' We observe that different datasets require distinct λmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Choos- ing an appropriate λmix can further improve the performance of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Meanwhile, as e0 gradually increases, the classification performance improves first and then decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, warm-up training is necessary for DALI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' But too large e0 causes the model to overfit noise samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Therefore, a good choice of hyper-parameters can remarkably improve performance under noisy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' 6 Conclusion In this paper, we propose a novel framework with theoretical guarantees for noisy PLL, called DALI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' It exploits weight- ing coefficients to dynamically adjust the importance of the initial candidate set and model outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Experimental results 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='42 9203-----91:94 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='03 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='20 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='07 accuracy 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='12 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='16 te: 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='21 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='69 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='96 test 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='08 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='59 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='10 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='10 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='0 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='84 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='12 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='28 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='39 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='28 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content='44 20 40 80 100 140 300 400 500 eoon multiple benchmark datasets demonstrate the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' DALI succeeds over currently advanced ap- proaches on noisy PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Meanwhile, our method shows strong competitiveness in PLL with severe noise and fine-grained PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' Through ablation studies, we verify the importance of each component in the framework, including the dynamically adjusted strategy, co-training and mixup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itFLT4oBgHgl3EQfbC-l/content/2301.12077v1.pdf'} +page_content=' This paper chooses Onehot(·) as the normalization func- tion.' metadata={'source': 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index 0000000000000000000000000000000000000000..ceae581c7d134ad0d0afada168fb382f147d015e --- /dev/null +++ b/mtE5T4oBgHgl3EQfHw5U/content/tmp_files/2301.05442v1.pdf.txt @@ -0,0 +1,1472 @@ +arXiv:2301.05442v1 [physics.atom-ph] 13 Jan 2023 +Application of the correlated B-spline basis functions to the leading relativistic and +QED corrections of helium +Hao Fang1,2, Yong-Hui Zhang1, Pei-Pei Zhang1, and Ting-Yun Shi1,† +1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, +Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, +Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China and +2University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China +(Dated: January 16, 2023) +B-spline functions have been widely used in computational atomic physics. Different from the +traditional B-spline basis (a simple product of two B-splines), the recently developed correlated +B-spline basis functions(C-BSBF), in which the interelectronic coordinate r12 is included explicitly, +have greatly improved the computational accuracy of polarizability [S. J. Yang et al., Phys. Rev. +A 95, 062505 (2017)] and bethe logarithm [ S. J. Yang et al., Phys. Rev. A 100, 042509 (2019)] for +singlet states of helium. Here, we report the extension of the C-BSBF to the leading relativistic and +QED correction calculations for energy levels of the 1 1S, 2 1S, 2 3S, and 3 3S states of helium. The +relativistic kinetic term p4 +1, contact potential δ3(r1), δ3(r12) and Araki-Sucher correction ⟨1/r3 +12⟩ are +calculated by using the global operator method, in which rn +12 and rn +12 ln r12 involved are calculated +with the generalization of Laplace’s expansions. +The obtained values for the ground state are +δErel/α2 = −1.951 754 7(2) and δEQED/α3 =57.288 165(2), consistent with previous results, which +opens the possibility of calculating higher-order relativistic and QED effects using the C-BSBF. +PACS numbers: +I. +INTRODUCTION +The measured precision of helium atomic spectroscopy +has approached the part-per-trillion level [1, 2], which +allows the test of quantum electrodynamics (QED) and +the determination of the fine-structure constant α and +the nuclear charge radius [1–7] by combination with the +high-accuracy atomic structure calculations [8–11]. +In +addition, from the theoretical point of view, as the sim- +plest many-electron system, traditionally helium is an +ideal testing ground for different methods of the descrip- +tion of atomic structure. +It is known that finite basis set variational calcula- +tions are the most powerful tool for solving the Coulomb +three-body bound-state problem exactly, such as helium, +in which their basis functions included explicitly the in- +terelectron separation are particularly important. +For +example, using the explicitly correlated exponential basis +with nonlinear parameters, Pachucki et al. have accom- +plished complete α7m Lamb shift of helium triplet states, +which improved the theoretical accuracy of ionization en- +ergies by more than an order of magnitude [8]. Hylleraas +variation technique is employed to finish the calculations +of the hyperfine structure of the 2 3PJ state in 7Li+ up +to order mα6, which has improved previous calculations +by one order of magnitude [12]. However, in order to get +rid of loss of stability when the number of basis functions +increases, these high-precision calculations must be sup- +plemented by applying multiprecision package as well as +variational optimized nonlinear parameters. +†Email Address: tyshi@wipm.ac.cn +B-splines have the property of being ‘complete enough’ +and linear independence even for a large basis, which has +been widely used in computational atomic physics [13– +22]. +With the development of high-resolution atomic +spectroscopy, calculations of highly accurate energies are +required. However, high-accuracy computational results +are difficult to achieve with the traditional B-spline basis +functions, for systems with strong electron correlations. +For example, Lin et al.[23] gave a nonrelativistic ground +state energy of −2.903 582 0 for the helium by using the +B-spline basis, which had four accurate figures at the cost +of a large number of the configurations. Also the rela- +tivistic energy for the 2 1S0 state of helium given by us- +ing the partial wave ℓmax=15 was only with six accurate +figures [24]. So it is necessary to introduce the interelec- +tronic coordinate into the traditional B-spline basis. +Recently, Tang et al. developed a method to calcu- +late the Bethe logarithm, the dominant part of QED, +of the hydrogen atom using the B-spline basis set [18], +which not only can calculate low-lying states with high +precision using relatively small basis sets, but also can +calculate highly-excited Rydberg states. Then Zhang et +al. extended it to calculate the Bethe logarithms for the +S state of the helium atom [22], in which the Bethe log- +arithms for the triplet state with weak electron correla- +tion can be reached with five to eight accurate figures, +but the precision is limited for the single state as the +electron correlation effect is not included in the basis set. +Therefore, Yang et al. have developed the explicitly cor- +related B-spline basis method and successfully applied it +to the calculation of energy levels, static dipole polariz- +abilities [25], and Bethe logarithms [26] for the singlet +states of the helium atom. The nonrelativistic ground +state energy has reached −2.903 724 377 1(2) [25], which +is six orders of magnitude better than the result of Lin et + +2 +al. [23]. Moreover, they have been able to obtain static +dipole polarizabilities with a relative error of 10−9 and +Bethe logarithms with a relative error of 10−7, respec- +tively, which shows that the correlated B-spline basis +functions(C-BSBF) can describe well the electronic cor- +relation of the singlet states and effectively improve the +numerical convergence rates. +This work will employ the C-BSBF to evaluate the +leading relativistic and QED corrections to energy levels +of the helium atom. The global operator method will be +used to improve the numerical convergence for the rela- +tivistic kinetic term p4 +1, contact potential δ3(r1), δ3(r12) +and Araki-Sucher correction ⟨1/r3 +12⟩, which will expand +the scope of using of the C-BSBF and present a mani- +festation that the C-BSBF can be effectively applied to +numerical calculations of the expectation values of singu- +lar operators as well. +This paper is organized as follows. The theoretical for- +mulas and methods used in our calculations are presented +in section II. In section III we calculate the leading rela- +tivistic and QED corrections to energy levels for the 1 1S, +2 1S, 2 3S and 3 3S states of helium. Comparisons with +results of available literature are made as well. Conclu- +sions are given in section IV. Atomic units (a.u.) are used +throughout this paper. +II. +THEORY AND METHOD +A. +Correlated B-spline basis functions(C-BSBF) +The nonrelativistic Hamiltonian for a two-electron +atom with an infinite mass nucleus has the form of +H = +2 +� +i=1 +�p2 +i +2 − Z +ri +� ++ 1 +r12 +, +(1) +where pi = −i∇i is the momentum operator of the ith +electron, ri is the coordinate of the ith electron to the +atomic nucleus, r12 is the interelectronic coordinate, and +the nuclear charge Z = 2 for the helium atom. +The two-electron wave function is expanded by the fol- +lowing C-BSBF in which the interelectronic coordinate +r12 is included explicitly, +φij,c,ℓ1ℓ2 = A +� +rc +12Bk +i (r1) Bk +j (r2) YLM +ℓ1ℓ2 (ˆr1, ˆr2) +� +, +(2) +where the operator A ensures the antisymmetry of the +basis function with respect to the exchange of the two +electrons, Bk +i (r) is the ith of N B-spline functions with +the order of k and constrained to a spherical cavity [14], +c is the power of the r12 coordinate, and the coupled +spherical harmonic function is given by +YLM +ℓ1ℓ2 (ˆr1, ˆr2) = +� +m1m2 +⟨ℓ1ℓ2m1m2 | LM⟩ +× Yℓ1m1 (ˆr1) Yℓ2m2 (ˆr2) , +(3) +with ⟨ℓ1ℓ2m1m2 | LM⟩ being the Clebsch-Gordan coef- +ficient. In the present calculations, the cavity radius of +R0 is chosen appropriately, the r12 power c is restricted +to be 0 or 1 without making integral evaluations overly +complicated, and the orbital angular momentum ℓ1 and +ℓ2 are less than the maximum partial wave ℓmax. +B. +Leading relativistic and QED corrections +The leading relativistic correction to the nonrelativis- +tic energy of the two-electron atom is given by the ex- +pectation value of the Breit-Pauli Hamiltonian with the +nonrelativistic wave function ψ, +δErel = ⟨ψ|HBP |ψ⟩ , +(4) +where +HBP = α2 +� +−1 +8 +� +p4 +1 + p4 +2 +� ++ πδ3 (r12) + Zπ +2 +� +δ3 (r1) ++δ3 (r2) +� +− 1 +2r12 +� +p1 · p2 + r12 · (r12 · p1)p2 +r2 +12 +�� +, (5) +for S-state [11, 27, 28], where α +=7.297 352 569 +3(11)×10−3 [29] is the fine structure constant, δ3(r12), +δ3(r1), and δ3(r2) represent the Dirac delta functions. +The last term of Eq. (5) is a retardation term, since +this correction is due to the retardation of the elec- +tromagnetic field produced by an electron [30], and +− +� +p1 · p2 + r12 · (r12 · p1)p2/r2 +12 +� +/2r12 is labelled as H2. +The leading QED correction can be expressed as an +expectation value of the following effective operators [11, +31, 32], +δEQED = α3 +�4Z +3 +�19 +30 − 2 ln α − ln k0 +� +⟨ψ|δ3(r1) ++δ3(r2)|ψ⟩ + +�164 +15 + 14 +3 ln α +� +⟨ψ|δ3(r12)|ψ⟩ +− 7 +6π ⟨ψ|r−3 +12 |ψ⟩ +� +. +(6) +Here ln k0 is the Bethe logarithm, and the last term in +TABLE I: Bethe logarithm for the 1 1S, 2 1S, 2 3S and 3 3S +states of helium. +State +Zhang[22] and Yang[26] +Korobov[33] +11S +4.370 160 22(5) +4.370 160 223 070 3(3) +21S +4.366 412 71(1) +4.366 412 726 417(1) +23S +4.364 036 7(2) +4.364 036 820 476(1) +33S +4.368 666 7(1) +4.368 666 996 159(2) +Eq. (6) is usually called Araki-Sucher correction [31, 34, +35], and the expectation of ⟨ψ|r−3 +12 |ψ⟩ is defined as +⟨ψ|r−3 +12 |ψ⟩ = lim +a→0⟨r−3 +12 Θ(r12 − a) ++ 4π(γ + ln a)δ3(r12)⟩ , +(7) + +3 +where Θ(x) and γ are the step function and the Euler +constant, respectively. +Compared with the relativistic +correction, the more difficult to calculate in the leading +QED correction are Bethe logarithm and Araki-sucher +correction. The Bethe logarithms for the 1 1S, 2 1S, 2 3S +and 3 3S state of the helium atom are summarized in Ta- +ble I calculated by Zhang et al.[22] using traditional B- +spline function and Yang et al.[26] using the C-BSBF, re- +spectively, which based on the Drake-Goldman’s method. +The Korobov’s results listed in the last column of Table I +based on the integral representation method of Schwartz +are the benchmarks. The value of the Bethe logarithms +from Zhang et al. and Yang et al. are used in this work, +which will achieve the complete calculation of the lead- +ing relativistic and QED correction using the B-spline +function. +Drachman proposed the global operator method to +evaluate the two-particle contact potential δ3(r1) and +δ3(r12), which achieved significant improvements over the +direct evaluations [36]. We employ the equivalent form +containing global operators Drachman given to calculate +the expectation value of δ3(r1) and δ3(r12), +4π +� +ψ +��δ3(ri) +�� ψ +� +=4⟨ψ|r−1 +i +(Eψ − V )|ψ⟩ +− 2 +2 +� +s=1 +⟨∇sψ|r−1 +i +|∇sψ⟩ , +(8) +4π +� +ψ +��δ3(r12) +�� ψ +� +=2⟨ψ|r−1 +12 (Eψ − V )|ψ⟩ +− +2 +� +s=1 +⟨∇sψ|r−1 +12 |∇sψ⟩ , +(9) +where Eψ is the corresponding eigenvalue of the two- +electron wave function ψ, and V = −Z/r1−Z/r2+1/r12. +It will result in a slow convergence for the kinetic term +p4 +1 + p4 +2 in the relativistic correction if we calculate its +expectation value directly in the C-BSBF. Pachucki and +Komasa also used a similar way to transform both the +kinetic term and the Araki-Sucher correction to much +more regular forms and obtained much better numerical +convergence on that account [37]. In the present calcu- +lations, as Pachucki and Komasa have done, we use the +following expression to evaluate ⟨p4 +1 + p4 +2⟩, +2 +� +i=1 +� +ψ +��p4 +i +�� ψ +� += 4 +� +ψ +��(Eψ − V )2�� ψ +� +− 2 +� +∇2 +1ψ|∇2 +2ψ +� +.(10) +The integration of ⟨ψ|r−2 +12 |ψ⟩ will be involved in Eq. (10), +and it is also evaluated to be as following by using the +global operator method, +� +ψ +��r−2 +12 +�� ψ +� +=2⟨ψ| ln r12(V − Eψ)|ψ⟩ ++ +2 +� +i=1 +⟨∇iψ| ln r12|∇iψ⟩ , +(11) +since we find that ∇2 +1 ln r12 = ∇2 +2 ln r12 = r−2 +12 . The com- +plete expansion of Eq. (10) is written as +2 +� +i=1 +� +ψ +��p4 +i +�� ψ +� += 4E2 +ψ − 8Eψ +� +ψ +����−2Z +r1 ++ 1 +r12 +���� ψ +� ++4 +� +ψ +���� +2Z2 +r2 +1 +− 2Z2 +r1r2 +− +2Z +r1r12 ++ 1 +r2 +12 +���� ψ +� +−2 +� +∇2 +1ψ|∇2 +2ψ +� +. +(12) +The Araki-Sucher correction is converted to the regular +form as well so as to facilitate the present numerical eval- +uations, +� +ψ +��r−3 +12 +�� ψ +� += − +2 +� +i=1 +� +∇iψ +��r−1 +12 ln r12 +�� ∇iψ +� ++ +� +ψ +����2 (Eψ − V ) ln r12 +r12 ++4π(1 + γ)δ3 (r12) +�� ψ +� +. +(13) +where rn +12 ln r12 (n = −2, −1, 0, 1) will be involved in inte- +gration. In addition to the above three terms, the expec- +tation values of other operators appearing in Eqs. (5)-(6) +will be calculated in the C-BSBF directly. +C. +Laplace’s expansion of rn +12 and rn +12 ln r12 +The integration of rn +12 and rn +12 ln r12 are involved in the +computation of Breit-Pauli operators and Araki-Sucher +corrections. It is crucial to process this type of the in- +tegration in spherical coordinates, which requires sepa- +rating their radial and angular dimensions. The gener- +alization of Laplace’s expansion to arbitrary powers and +functions of r12 given by Sack [38] is used to calculate the +integration in which different powers of r12 is involved. +rn +12 can be expanded in the form +rn +12 = +∞ +� +ℓ=0 +Rnℓ(r1, r2)Pℓ(cos θ12) , +(14) +where +the +Legendre +polynomials +of +cos θ12 +is +ex- +pressed +by +using +the +identity +as +Pℓ(cos θ12) += +4π/(2ℓ+1) +m=ℓ +� +m=−ℓ +Y ∗ +ℓm(ˆr1)Yℓm(ˆr2), and the radial function +Rnℓ(r1, r2) has been formulated by Sack [38] as following +Rnℓ(r1, r2) = +� +− 1 +2n +� +ℓ +� 1 +2 +� +ℓ +rn +> +�r< +r> +�ℓ +× +2F1 +� +l − 1 +2n, −1 +2 − 1 +2n; l + 3 +2; r2 +< +r2> +� +. +(15) +In Eq. (15), r< = min(r1, r2), r> = max(r1, r2), and the +hypergeometric function has the form of + +4 +2F1(α, β; γ; x) = 1 + +∞ +� +1 +(α)s(β)s +(γ)ss! xs , +(16) +where the Pochhammer symbol is defined as +(α)s = +� +1 +if s = 0 +α(α + 1) · · · (α + s − 1) if s > 0 . +(17) +The hypergeometric function is finite series if either α +or β is zero or a negative integer, which implies that for all +positive odd integer values of n, the series of Rnℓ break +off; and for n = −1, they consists of the leading term +only. For positive even n, the summation is truncated to +ℓ = n +2 , since the factor (− 1 +2n)ℓ ensures that Rnℓ vanishes +when ℓ > n +2 . In addition, the individual functions Rnℓ +are divergent for n ≤ −2, but they remain integrable as +long as n > −3 [35, 39]. Present calculations involve the +integrations of ⟨ψ|r−2 +12 |ψ⟩ and ⟨ψ|r−3 +12 |ψ⟩. So giving ap- +propriate radial expansions of r−2 +12 and r−3 +12 is important +in the computation of radial and angular integrations. +Substituting n = −2 , ℓ = 0 and n = −2 , ℓ = 1 sepa- +rately into Eq.(15), and summation of the series, as a re- +sult the following specific expressions in terms of reverse +hyperbolic tangent function tanh−1(x) are achieved, +R−2,0 (r1, r2) = tanh−1(x) +xr2 +> +, +(18) +R−2,1 (r1, r2) = +3 +2x2r2 +> +× +�� +x2 + 1 +� +tanh−1(x) − 1 +� +, +(19) +where x = r; then the recurrence relation +r2 +1 + r2 +2 +r1r2 +Rn,ℓ − ℓ + 2 + 1 +2n +ℓ + 3 +2 +Rn,ℓ+1 +−ℓ − 1 − 1 +2n +ℓ − 1 +2 +Rn,ℓ−1 = 0 , +(20) +can be used to calculate the radial functions for other +values of ℓ. For n = −3, the expansion coefficients of the +hypergeometric functions are cancelled, and the hyper- +geometric functions are reduced to a series summation of +xn. the hypergeometric function can be expressed as an- +alytic functions that is independent of ℓ, correspondingly +the radial expansion of R−3,l can be written as [40] +R−3,l (r1, r2) = (2ℓ + 1)xℓ +(1 − x2) r3> +. +(21) +Next we will give the explicit formula for the product +of r12 with different powers and ln r12. Differentiation of +Eq. (14), the expansion for rn +12 ln r12 can be expressed as +rn +12 ln r12 = +� +ℓ +Rn ln,ℓ(r1, r2)Pℓ (cos θ12) , +(22) +where Rn ln,ℓ(r1, r2) represents the radial function of +rn +12 ln r12, and Rn ln,ℓ(r1, r2) = ∂Rnℓ(r1,r2) +∂n +. Similarly, the +following recurrence relation for Rn ln,ℓ(r1, r2) can be de- +rived by taking the derivative of Eq. (20), +1 +2ℓ + 3Rn,ℓ+1 − +1 +2ℓ − 1Rn,ℓ−1 = r2 +1 + r2 +2 +r1r2 +Rn ln,ℓ +− 2ℓ + 4 + n +2ℓ + 3 +Rn ln,ℓ+1 − 2ℓ − 2 − n +2ℓ − 1 +Rn ln,ℓ−1 . +(23) +Then +we +can +calculate +the +integration +with +the +rn +12 ln r12 (n ≥ −2) operator in the present paper. For +example, for n = −2 , ℓ = 0 and n = −2 , ℓ = 1, +R−2 ln,0 = tanh−1(x) ln(r2 +> − r2 +<) +2r2 +>x +, +(24) +R−2 ln,1 =3 +� +ln(r2 +> − r2 +<) − 1 +� +4r2 +>x2 +× +�� +x2 + 1 +� +tanh−1(x) − x +� +, +(25) +and the estimations of R−2 ln,ℓ for other values of ℓ > 1 +can be obtained according to the recurrence relation of +Eq. (23). +III. +RESULTS AND DISCUSSIONS +The C-BSBF on an exponential grid [14] are gener- +ated using B-splines constrained to a spherical cavity. +The cavity radius of R0 = 20 a.u. is for the 1 1S state, +R0 = 40 a.u. is for the 2 1S state, and R0 = 70 a.u. is +for both the 2 3S and 3 3S states. Yang et al.[25] have +implemented the correlated B-splines to calculate the he- +lium atomic energy level and their non-relativistic ground +state energy is −2.903 724 377 1(2) a.u.. A knot distribu- +tion optimization was performed for any individual states +and present values of energies for the 1 1S, 2 1S, 2 3S and +3 3S states are listed in Table II. The optimized result of +−2.903 724 377 034 0(2) a.u. is obtained for the ground +state, which has thirteen significant digits in agreement +with Drake’s. The 2 1S, 2 3S, and 3 3S states also reached +fourteen significant digits in agreement with Drake. +TABLE II: Energies for the 1 1S, 2 1S, 2 3S and 3 3S states of +helium. +State +This work +Ref.[41] +11S +−2.903 724 377 034 0(2) −2.903 724 377 034 119 5 +21S +−2.145 974 046 054 4(2) −2.145 974 046 054 419(6) +23S +−2.175 229 378 236 7(2) −2.175 229 378 236 791 30 +33S +−2.068 689 067 472 4(2) −2.068 689 067 472 457 19 +It can be seen from Eq. (12) that the computation of +⟨p4 +1⟩ involves many operators, which are classified into +two categories for dealing with. One type is the general + +5 +TABLE III: The expectation values of other operators needed for evaluating the relativistic kinetic terms for the 1 1S, 2 1S, +2 3S, and 3 3S states of helium. +Operater +11S +21S +23S +33S +⟨1/r1⟩ +1.688 316 800 717 1(2) +1.135 407 686 126 1(2) +1.154 664 152 972 0(1) +1.063 674 075 760 7(2) +1.688 316 800 717a +1.135 407 686 125 609(6)b +1.154 664 152 972 107 60(20)b 1.063 674 075 760 76(10)b +1.688 316 800 635c +1.135 407 686c +1.154 664 152c +1.063 674 075 7c +⟨1/r2 +1⟩ +6.017 408 867 0(3) +4.146 939 019 80(6) +4.170 445 551 31(2) +4.042 948 747 4(3) +6.017 408 867 0(1)a +4.146 939 019 0(12)b +4.170 445 551 336 2(4)b +4.042 948 747 477(4)b +⟨1/r1r2⟩ +2.708 655 474 480(4) +0.561 861 467 461(2) +0.560 729 635 682 9(3) +0.240 684 804 629 3(2) +2.708 655 474 480a +0.561 861 467 459 6(7)b +0.560 729 635 682 926 40(20)b 0.240 684 804 629 353(11)b +⟨1/r12⟩ +0.945 818 448 799 95(5) +0.249 682 652 394 3(6) +0.268 197 855 414 82(5) +0.117 318 168 097 65(4) +0.945 818 448 800a +0.249 682 652 393 566 7(19)b 0.268 197 855 414 847 80(20)b 0.117 318 168 097 636(6)b +0.945 818 448 705 9c +0.249 682 652 3c +0.268 197 855 3c +0.117 318 168 0c +⟨1/r1r12⟩ +1.920 943 921 900 0(5) +0.340 633 845 861 2(8) +0.322 696 221 719 8(2) +0.131 426 560 051 19(5) +1.920 943 921 900a +0.340 633 845 861 0(19)b +0.322 696 221 719 854 32(8)b +0.131 426 560 051 184(5)b +a Drake [41]. +b Drake [42]. +c Yu et al. [43]. +operators that are relatively simple to compute, includ- +ing 1/r1, 1/r2 +1, 1/r1r2, 1/r12 and 1/r1r12. We give the +final convergence values directly in Table III, and there +are at least ten significant digits of our results that are +consistent with Drake’s. This also demonstrates the high +accuracy of the wave function obtained for the C-BSBF. +The other type is the operators 1/r2 +12 and ∇2 +1∇2 +2 that +are more difficult to calculate. The numerical results of +⟨1/r2 +12⟩, ⟨∇2 +1∇2 +2⟩ and ⟨p4 +1⟩ as the number of B-splines N +increased are given in the last three columns of Table IV. +Good convergent values of ⟨1/r2 +12⟩ under the C-BSBF +are achieved with the global operator method. For the +ground state, the present result of 1.464 770 923 3(5) is +obtained, which has eleven significant figures and agrees +well with reference values with the explicitly correlated +exponential basis [10] and the Hylleraas basis [41]. Our +expectation values of 1/r2 +12 for the 2 1S, 2 3S and 3 3S +states of the helium atom at least have eight convergent +digits, which are all in good agreement with results in +available literatures [9, 10, 42]. For the ⟨∇2 +1∇2 +2⟩ opera- +tor, no suitable treatment could be found to make it con- +verge faster, for which the direct calculation method was +used. Therefore, the convergent accuracy of ⟨∇2 +1∇2 +2⟩ is +relatively lower, which is also the main reason to limit the +numerical precision of ⟨p4 +1⟩. The present result of ⟨p4 +1⟩ for +the 1 1S state from the C-BSBF has nine digits, consis- +tent with Drake’s Hylleraas results [41, 42]. Present nu- +merical convergence for the triplet states are better than +for the singlet states by one to two significant figures, +and our values are both good agreement with Hylleraas +results [42]. +We also calculated ⟨1/r2 +12⟩ and ⟨∇2 +1∇2 +2⟩ using the tradi- +tional B-spline basis set, and results for the ground state +are 1.463 697 and 7.079, respectively. Since these singu- +larity operators only have one to three significant digits, +which are difficult to use in high-precision calculations +at the atomic energy level. It is convenient to find that +the primary explanation for this is that the traditional +B-spline basis set makes it difficult to describe the lo- +cal properties of the wave function with high accuracy +without including the electron correlation effect. +The expectation values of other three components from +HBP and the singular electron-electron ⟨1/r3 +12⟩ from the +leading QED corrections are shown in Table V. The ex- +pectation values of δ3(r12) for the triplet states equal +zero, so they are not listed in Table V. Yu et al. [43] +employed the same C-BSBF to give numerical results +of δ3(r1) by direct calculation when the power of r12 is +c = 5, which are also shown in Table V. The direct calcu- +lation of δ3(r1) is highly dependent on the origin value of +the wave function, and the global operator method can +be used to further improve the calculation accuracy. The +result of the δ3(r1) of the ground state using the global +operator method is 1.810 429 32(2), one can see that nu- +merical accuracy of the δ3(r1) can reach a precision of +eight to twelve significant digits. It can be seen that our +computational accuracy with c = 1 is completely compa- +rable to theirs [43], with the except for the ground state +with relatively sensitive electron correlations. They also +tried to improve the direct calculation accuracy of δ3(r1) +by increasing the power of r12, but the global operator +method is still necessary to effectively improve the nu- +merical convergence. For example, our result of ⟨δ3(r12)⟩ +for the 1 1S state is 0.106 345 370 66(4), that is more +accurate than 0.106 346 068 of Yu et al. [43] by five or- +ders of magnitude and is well consistent with Drake’s +Hylleraas value of 0.106 345 370 636 3(12) [42] as well. +Present results for the retardation term H2 have at least + +6 +TABLE IV: Convergence of the relativistic kinetic terms for the 1 1S, 2 1S, 2 3S and 3 3S states of helium as the number of +B-splines N increased. The expectation values of 1/r2 +12 and ∇2 +1∇2 +2 are also listed in the second and third columns. The partial +wave is ℓmax = 4. +N +⟨1/r2 +12⟩ +⟨∇2 +1∇2 +2⟩ +⟨p4 +1⟩ +1 1S +50 +1.464 770 923 579 +7.133 709 835 +54.088 067 177 +60 +1.464 770 923 463 +7.133 709 771 +54.088 067 242 +70 +1.464 770 923 406 +7.133 709 763 +54.088 067 251 +Extrap. +1.464 770 923 3(5) +7.133 709 7(2) +54.088 067 2(2) +Ref. [10] +1.464 771 +7.133 710 +Ref. [41] +1.464 770 923 350(1) +54.088 067 230(2) +2 1S +50 +0.143 724 814 027 +1.428 212 689 1 +41.118 675 563 8 +60 +0.143 724 814 013 +1.428 212 706 4 +41.118 675 546 0 +70 +0.143 724 814 008 +1.428 212 705 8 +41.118 675 546 6 +Extrap. +0.143 724 814 00(5) +1.428 212 70(4) +41.118 675 54(4) +Ref. [10] +0.143 725 +1.428 213 +Ref. [42] +0.143 724 814 00(7) +41.118 675 544(19) +2 3S +50 +0.088 906 004 870 +0.488 197 568 41 +41.835 540 798 28 +60 +0.088 906 004 913 +0.488 197 569 31 +41.835 540 797 46 +70 +0.088 906 004 921 +0.488 197 569 91 +41.835 540 796 85 +Extrap. +0.088 906 004 9(2) +0.488 197 570(4) +41.835 540 796(4) +Ref.[9] +0.088 906 +0.488 198 +Ref.[42] +0.088 906 004 932 625(5) +41.835 540 797 348(6) +3 3S +50 +0.023 097 669 645 +0.329 220 596 46 +40.475 439 870 27 +60 +0.023 097 669 653 +0.329 220 596 68 +40.475 439 868 42 +70 +0.023 097 669 655 +0.329 220 596 89 +40.475 439 868 25 +Extrap. +0.023 097 669 65(3) +0.329 220 597(2) +40.475 439 868(5) +Ref.[42] +0.023 097 669 656 893(13) +40.475 439 868 127 2(3) +seven convergent figures and agree with Drake’s [42]. The +expectation of singular electron-electron ⟨1/r3 +12⟩ are com- +puted with the global operator method by the C-BSBF +and confronted with previous results obtained from dif- +ferent basis functions as well. Present the C-BSBF re- +sult of 0.989 272(2) with an accuracy of five decimals is +achieved for the ground state, which is comparable to re- +sults of 0.989 273 5 and 0.989 272 4(13) with explicitly +correlated Gaussian (ECG) functions [44] and exponen- +tial basis functions [45], respectively, in numerical pre- +cision. Employed Hylleraas basis and exponential basis +respectively, Drake [42] improved reference values with +three additional exact digits. Our result for the ground +state is expected to recover more figures of Drake’s re- +sult if adopting higher power of r12. For the 2 1S and +2 3S states, our values are in agreement with previous +values obtained by Hylleraas basis and exponential ba- +sis [28]. There are five convergent figures in our result +⟨1/r3 +12⟩=0.008 922 57(2) for the 3 3S state. +The singular electron-electron ⟨1/r3 +12⟩ expectation +value is also computed using the traditional B-spline ba- +sis set, and the ground state result is 1.197(N = 70, +ℓmax = 4). It can be seen that the traditional B-spline is +entirely inaccurate in calculating ⟨1/r3 +12⟩, and this type +of operator for divergence require a more accurate de- +scription of the local properties of the wave function [44] +than 1/r2 +12 and ∇2 +1∇2 +2. As a result, the B-spline basis set +containing electron correlation is essential. +The final relativistic corrections are presented in the +top half of Table VI. Comparisons are made with results +obtained using the explicitly correlated exponential ba- +sis [11] and Hylleraas basis [42]. Our relativistic correc- +tions results are completely consistent with most precise +previous calculations [11, 42] and can reach eight to ten +significant figures. The leading QED corrections for the +S states to the energy level are summarized in the bottom +half of Table VI, which used the Bethe logarithm values +obtained from B-splines [22, 26], and Korobov’s Bethe +logarithm values [33] as a benchmark, respectively. +It +can be seen that our calculated results are in good agree- +ment with the significant figures listed by Yerokhin et +al.[11], where the results of the singlet state calculated +using Korobov’s Bethe logarithm values are almost iden- +tical to the results from B-splines, which are mainly ex- +plained by the relatively low accuracy of δ3(r1) and 1/r3 +12, +and the improved accuracy of the triplet state is the re- + +7 +TABLE V: The expectation values of δ3(r1), δ3(r12), H2 and 1/r3 +12 for the 1 1S, 2 1S, 2 3S and 3 3S states of helium. Comparisons +with results obtained in available literatures are also made. The partial wave is ℓmax = 4. +N +⟨δ3(r1)⟩ +⟨δ3(r12)⟩ +⟨H2⟩ +⟨1/r3 +12⟩ +11S +50 +1.810 429 325 97 +0.106 345 370 649 3 +−0.139 094 671 8 +0.989 271 57 +60 +1.810 429 323 14 +0.106 345 370 658 3 +−0.139 094 675 1 +0.989 271 98 +70 +1.810 429 321 51 +0.106 345 370 646 3 +−0.139 094 677 3 +0.989 272 26 +Extrap. +1.810 429 32(2) +0.106 345 370 66(4) +−0.139 094 67(2) +0.989 272(2) +Ref.[43] +1.810 429 318 371 521 8 +0.106 346 068 +Ref.[42] +1.810 429 318 499 0(6) +0.106 345 370 636 3(12) +−0.139 094 690 539 20(20) +0.989 273 544 768(13) +Ref.[44] +0.989 273 5 +Ref.[45] +0.989 272 4(13) +21S +50 +1.309 460 780 907 +0.008 648 433 612 1 +−0.009 253 044 67 +0.067 946 402 +60 +1.309 460 780 719 +0.008 648 433 588 4 +−0.009 253 044 78 +0.067 946 439 +70 +1.309 460 780 607 +0.008 648 433 587 3 +−0.009 253 044 97 +0.067 946 465 +Extrap. +1.309 460 780 5(8) +0.008 648 433 58(5) +−0.009 253 045(2) +0.067 946 4(2) +Ref.[43] +1.309 460 780 3 +0.008 648 6 +Ref.[42] +1.309 460 780 1(4) +0.008 648 433 6(14) +−0.009 253 046 05(4) +Ref.[46] +0.067 946 32 +23S +50 +1.320 355 082 933 78 +−0.001 628 430 082 9 +0.038 861 479 8 +60 +1.320 355 082 931 58 +−0.001 628 430 067 4 +0.038 861 479 6 +70 +1.320 355 082 931 10 +−0.001.628 430 064 8 +0.038 861 481 0 +Extrap. +1.320 355 082 930(6) +−0.001 628 430 06(4) +0.038 861 46(3) +Ref.[43] +1.320 355 082 9 +Ref.[42] +1.320 355 082 934 92(9) +−0.001 628 430 061 553(3) +Ref.[46] +0.038 861 485 631 95 +33S +50 +1.285 060 253 969 23 +−0.000 504 504 232 33 +0.008 922 569 5 +60 +1.285 060 253 936 06 +−0.000 504 504 228 95 +0.008 922 569 6 +70 +1.285 060 253 938 13 +−0.000 504 504 228 33 +0.008 922 569 9 +Extrap. +1.285 060 253 93(7) +−0.000 504 504 228(9) +0.008 922 57(2) +Ref.[43] +1.285 060 253 9 +Ref.[42] +1.285 060 253 932 1(13) +−0.000 504 504 227 201(9) +sult of the limited accuracy of Bethe logarithm. +The +overall computational accuracy of the leading QED cor- +rection is determined mainly by the contribution of the +Araki-Sucher term and δ3(r1) for the ground state, by +the contribution of the Bethe logarithms for other states. +It can be seen that the leading QED corrections results +can reach at least seven significant digits, which already +reaches the accuracy level of the contribution of the lead- +ing relativistic correction in this work. In addition, the +numerical accuracy of the singlet is expected to improve +with increasing power c of r12 in basis function. +IV. +SUMMARY AND OUTLOOK +In this work, we have calculated the leading relativis- +tic and QED corrections of the energy levels of the he- +lium atom using the C-BSBF. The expectation values of +the relativistic kinetic term p4 +1, contact potential δ3(r1), +δ3(r12) and Araki-Sucher correction ⟨1/r3 +12⟩, which are +more difficult to calculate directly, were treated by a +global operator method to improve their numerical con- +vergence, and the two-electron distance function is also +introduced to deal with the Laplace expansion method +proposed by Sack [38]. Together with the high-precision +calculation of the Bethe logarithms [26], the C-BSBF is +able to achieve the high-precision calculation of the lead- +ing relativistic and QED corrections for the energy levels +of the helium atom. +It is emphasized that the corre- +lated factor r12 in the C-BSBF is crucial to calculate +p4 +1, δ3(r12) and ⟨1/r3 +12⟩, without this factor, these oper- +ators have a very slow convergence. The C-BSBF can +provide stable numerical convergence based on its ap- +proximate linear independence and sufficient considera- +tion of the electronic correlation. It can be seen from +Table VII that the C-BSBF can determine the accuracy +of the 23S − 21S transition frequency (up to mα5-order +correction) to the kHz level, which is consistent with the + +8 +TABLE VI: The leading relativistic and QED corrections, δErel and δEQED for the 1 1S, 2 1S, 2 3S and 3 3S states of helium. +The corresponding comparison data given in available literatures are also listed. +11S +21S +23S +33S +the leading relativistic correction +δErel/α2 +−1.951 754 7(2) +−2.034 167 33(2) +−2.164 477 971(2) +−2.045 092 764(2) +Ref.[42] +−1.951 754 767 +−2.034 167 342 +−2.164 477 972 +−2.045 092 764 +the leading QED correction +δEQED/α3(BL with B-splines) +57.288 165(2) +42.523 605 2(2) +43.010 017(2) +41.839 303 4(7) +δEQED/α3(BL from Korobov) +57.288 165(1) +42.523 605 10(8) +43.010 017 06(2) +41.839 301 459(9) +Ref.[11] +57.288 165 2 +42.523 605 1 +43.010 016 8 +results of Pachucki et al., reaching a level similar to the +latest experiment [1]. A further improvement in our re- +sults is expected, if adopting higher power of r12. The +calculations are carried out applying double precision, no +multi-precision is needed. This method provides a new +approach to the calculation of atom energy levels. +TABLE VII: The 23S − 21S transition frequency for the he- +lium atom along the leading relativistic and QED corrections, +in KHz. +∆E(23S − 21S) +Ref.[47] +NR +192 490 838 748(2) +192 490 838 756 +mα4 +45 657 862(8) +45 657 859 +mα5 +−1 243 669(6) +−1 243 671 +Expt. [1] +192 510 702 148.72(20) +Recently, Mitroy and Tang suggested testing the QED +theory using tune-out wavelength, which opens a new +way to test fundamental atomic structure theory [48]. +The 413 nm tune-out wavelengths for the helium atom +23S1 state discrepancies in the latest experiments by +Baldwin’s team and theoretical values by Drake based +on the Hylleraas basis set with the NRQED method [49], +in which the calculation only estimates the electric-field +dependence of the Bethe logarithm [50]. The precision of +the experiment is expected to improve further, and the +QED theory will be tested at higher precision. The ab- +initio calculation of the electric field dependence of the +Bethe logarithm is important for further improving the +theoretical prediction accuracy of the 413 nm tune-out +wavelength to a level of ppb. The successful application +of the C-BSBF in singular operator calculations in this +work suggests that the C-BSBF is expected to be used +to calculate the electric field dependence of Bethe loga- +rithms to improve the theoretical calculation accuracy of +the 413 nm tune-out wavelength. In addition to the C- +BSBF is also expected to be extended to the second-order +perturbation of the Breit–Pauli operators [51] and rela- +tivistic corrections to the Bethe logarithm [52] of helium +atom in the future. +V. +ACKONWLEDGEMENT +This work is supported by the National Natural Sci- +ence Foundation of China under Grants No. 12274423 +and No. 12274417, by the Chinese Academy of Sciences +Project for Young Scientists in Basic Research under +Grant No. YSBR-055. +[1] R. Rengelink, Y. van der Werf, R. Notermans, R. Jan- +nin, K. Eikema, M. Hoogerland, and W. Vassen, Preci- +sion spectroscopy of helium in a magic wavelength optical +dipole trap, Nature Physics 14, 1132 (2018). +[2] X. Zheng, Y. 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A 98, 032503 (2018). + diff --git a/mtE5T4oBgHgl3EQfHw5U/content/tmp_files/load_file.txt b/mtE5T4oBgHgl3EQfHw5U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cde76b224432347f972f884878174e5282fb6ba6 --- /dev/null +++ b/mtE5T4oBgHgl3EQfHw5U/content/tmp_files/load_file.txt @@ -0,0 +1,969 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf,len=968 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='05442v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='atom-ph] 13 Jan 2023 Application of the correlated B-spline basis functions to the leading relativistic and QED corrections of helium Hao Fang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Yong-Hui Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Pei-Pei Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' and Ting-Yun Shi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='† 1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Wuhan Institute of Physics and Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Innovation Academy for Precision Measurement Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Wuhan 430071,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' People’s Republic of China and 2University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' People’s Republic of China (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' 2023) B-spline functions have been widely used in computational atomic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Different from the traditional B-spline basis (a simple product of two B-splines), the recently developed correlated B-spline basis functions(C-BSBF), in which the interelectronic coordinate r12 is included explicitly, have greatly improved the computational accuracy of polarizability [S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' A 95, 062505 (2017)] and bethe logarithm [ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' A 100, 042509 (2019)] for singlet states of helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Here, we report the extension of the C-BSBF to the leading relativistic and QED correction calculations for energy levels of the 1 1S, 2 1S, 2 3S, and 3 3S states of helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The relativistic kinetic term p4 1, contact potential δ3(r1), δ3(r12) and Araki-Sucher correction ⟨1/r3 12⟩ are calculated by using the global operator method, in which rn 12 and rn 12 ln r12 involved are calculated with the generalization of Laplace’s expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The obtained values for the ground state are δErel/α2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='951 754 7(2) and δEQED/α3 =57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='288 165(2), consistent with previous results, which opens the possibility of calculating higher-order relativistic and QED effects using the C-BSBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' PACS numbers: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' INTRODUCTION The measured precision of helium atomic spectroscopy has approached the part-per-trillion level [1, 2], which allows the test of quantum electrodynamics (QED) and the determination of the fine-structure constant α and the nuclear charge radius [1–7] by combination with the high-accuracy atomic structure calculations [8–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' In addition, from the theoretical point of view, as the sim- plest many-electron system, traditionally helium is an ideal testing ground for different methods of the descrip- tion of atomic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' It is known that finite basis set variational calcula- tions are the most powerful tool for solving the Coulomb three-body bound-state problem exactly, such as helium, in which their basis functions included explicitly the in- terelectron separation are particularly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' For example, using the explicitly correlated exponential basis with nonlinear parameters, Pachucki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' have accom- plished complete α7m Lamb shift of helium triplet states, which improved the theoretical accuracy of ionization en- ergies by more than an order of magnitude [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Hylleraas variation technique is employed to finish the calculations of the hyperfine structure of the 2 3PJ state in 7Li+ up to order mα6, which has improved previous calculations by one order of magnitude [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' However, in order to get rid of loss of stability when the number of basis functions increases, these high-precision calculations must be sup- plemented by applying multiprecision package as well as variational optimized nonlinear parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' †Email Address: tyshi@wipm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='cn B-splines have the property of being ‘complete enough’ and linear independence even for a large basis, which has been widely used in computational atomic physics [13– 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' With the development of high-resolution atomic spectroscopy, calculations of highly accurate energies are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' However, high-accuracy computational results are difficult to achieve with the traditional B-spline basis functions, for systems with strong electron correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' For example, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [23] gave a nonrelativistic ground state energy of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='903 582 0 for the helium by using the B-spline basis, which had four accurate figures at the cost of a large number of the configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Also the rela- tivistic energy for the 2 1S0 state of helium given by us- ing the partial wave ℓmax=15 was only with six accurate figures [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' So it is necessary to introduce the interelec- tronic coordinate into the traditional B-spline basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Recently, Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' developed a method to calcu- late the Bethe logarithm, the dominant part of QED, of the hydrogen atom using the B-spline basis set [18], which not only can calculate low-lying states with high precision using relatively small basis sets, but also can calculate highly-excited Rydberg states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Then Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' extended it to calculate the Bethe logarithms for the S state of the helium atom [22], in which the Bethe log- arithms for the triplet state with weak electron correla- tion can be reached with five to eight accurate figures, but the precision is limited for the single state as the electron correlation effect is not included in the basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Therefore, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' have developed the explicitly cor- related B-spline basis method and successfully applied it to the calculation of energy levels, static dipole polariz- abilities [25], and Bethe logarithms [26] for the singlet states of the helium atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The nonrelativistic ground state energy has reached −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='903 724 377 1(2) [25], which is six orders of magnitude better than the result of Lin et 2 al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Moreover, they have been able to obtain static dipole polarizabilities with a relative error of 10−9 and Bethe logarithms with a relative error of 10−7, respec- tively, which shows that the correlated B-spline basis functions(C-BSBF) can describe well the electronic cor- relation of the singlet states and effectively improve the numerical convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' This work will employ the C-BSBF to evaluate the leading relativistic and QED corrections to energy levels of the helium atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The global operator method will be used to improve the numerical convergence for the rela- tivistic kinetic term p4 1, contact potential δ3(r1), δ3(r12) and Araki-Sucher correction ⟨1/r3 12⟩, which will expand the scope of using of the C-BSBF and present a mani- festation that the C-BSBF can be effectively applied to numerical calculations of the expectation values of singu- lar operators as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The theoretical for- mulas and methods used in our calculations are presented in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' In section III we calculate the leading rela- tivistic and QED corrections to energy levels for the 1 1S, 2 1S, 2 3S and 3 3S states of helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Comparisons with results of available literature are made as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Conclu- sions are given in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Atomic units (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=') are used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' THEORY AND METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Correlated B-spline basis functions(C-BSBF) The nonrelativistic Hamiltonian for a two-electron atom with an infinite mass nucleus has the form of H = 2 � i=1 �p2 i 2 − Z ri � + 1 r12 , (1) where pi = −i∇i is the momentum operator of the ith electron, ri is the coordinate of the ith electron to the atomic nucleus, r12 is the interelectronic coordinate, and the nuclear charge Z = 2 for the helium atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The two-electron wave function is expanded by the fol- lowing C-BSBF in which the interelectronic coordinate r12 is included explicitly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' φij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='ℓ1ℓ2 = A � rc 12Bk i (r1) Bk j (r2) YLM ℓ1ℓ2 (ˆr1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' ˆr2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (2) where the operator A ensures the antisymmetry of the basis function with respect to the exchange of the two electrons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Bk i (r) is the ith of N B-spline functions with the order of k and constrained to a spherical cavity [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' c is the power of the r12 coordinate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' and the coupled spherical harmonic function is given by YLM ℓ1ℓ2 (ˆr1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' ˆr2) = � m1m2 ⟨ℓ1ℓ2m1m2 | LM⟩ × Yℓ1m1 (ˆr1) Yℓ2m2 (ˆr2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (3) with ⟨ℓ1ℓ2m1m2 | LM⟩ being the Clebsch-Gordan coef- ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' In the present calculations, the cavity radius of R0 is chosen appropriately, the r12 power c is restricted to be 0 or 1 without making integral evaluations overly complicated, and the orbital angular momentum ℓ1 and ℓ2 are less than the maximum partial wave ℓmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Leading relativistic and QED corrections The leading relativistic correction to the nonrelativis- tic energy of the two-electron atom is given by the ex- pectation value of the Breit-Pauli Hamiltonian with the nonrelativistic wave function ψ, δErel = ⟨ψ|HBP |ψ⟩ , (4) where HBP = α2 � −1 8 � p4 1 + p4 2 � + πδ3 (r12) + Zπ 2 � δ3 (r1) +δ3 (r2) � − 1 2r12 � p1 · p2 + r12 · (r12 · p1)p2 r2 12 �� , (5) for S-state [11, 27, 28], where α =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='297 352 569 3(11)×10−3 [29] is the fine structure constant, δ3(r12), δ3(r1), and δ3(r2) represent the Dirac delta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The last term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (5) is a retardation term, since this correction is due to the retardation of the elec- tromagnetic field produced by an electron [30], and − � p1 · p2 + r12 · (r12 · p1)p2/r2 12 � /2r12 is labelled as H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The leading QED correction can be expressed as an expectation value of the following effective operators [11, 31, 32], δEQED = α3 �4Z 3 �19 30 − 2 ln α − ln k0 � ⟨ψ|δ3(r1) +δ3(r2)|ψ⟩ + �164 15 + 14 3 ln α � ⟨ψ|δ3(r12)|ψ⟩ − 7 6π ⟨ψ|r−3 12 |ψ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (6) Here ln k0 is the Bethe logarithm, and the last term in TABLE I: Bethe logarithm for the 1 1S, 2 1S, 2 3S and 3 3S states of helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' State Zhang[22] and Yang[26] Korobov[33] 11S 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='370 160 22(5) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='370 160 223 070 3(3) 21S 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='366 412 71(1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='366 412 726 417(1) 23S 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='364 036 7(2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='364 036 820 476(1) 33S 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='368 666 7(1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='368 666 996 159(2) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (6) is usually called Araki-Sucher correction [31, 34, 35], and the expectation of ⟨ψ|r−3 12 |ψ⟩ is defined as ⟨ψ|r−3 12 |ψ⟩ = lim a→0⟨r−3 12 Θ(r12 − a) + 4π(γ + ln a)δ3(r12)⟩ , (7) 3 where Θ(x) and γ are the step function and the Euler constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Compared with the relativistic correction, the more difficult to calculate in the leading QED correction are Bethe logarithm and Araki-sucher correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The Bethe logarithms for the 1 1S, 2 1S, 2 3S and 3 3S state of the helium atom are summarized in Ta- ble I calculated by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [22] using traditional B- spline function and Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [26] using the C-BSBF, re- spectively, which based on the Drake-Goldman’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The Korobov’s results listed in the last column of Table I based on the integral representation method of Schwartz are the benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The value of the Bethe logarithms from Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' and Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' are used in this work, which will achieve the complete calculation of the lead- ing relativistic and QED correction using the B-spline function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Drachman proposed the global operator method to evaluate the two-particle contact potential δ3(r1) and δ3(r12), which achieved significant improvements over the direct evaluations [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' We employ the equivalent form containing global operators Drachman given to calculate the expectation value of δ3(r1) and δ3(r12), 4π � ψ ��δ3(ri) �� ψ � =4⟨ψ|r−1 i (Eψ − V )|ψ⟩ − 2 2 � s=1 ⟨∇sψ|r−1 i |∇sψ⟩ , (8) 4π � ψ ��δ3(r12) �� ψ � =2⟨ψ|r−1 12 (Eψ − V )|ψ⟩ − 2 � s=1 ⟨∇sψ|r−1 12 |∇sψ⟩ , (9) where Eψ is the corresponding eigenvalue of the two- electron wave function ψ, and V = −Z/r1−Z/r2+1/r12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' It will result in a slow convergence for the kinetic term p4 1 + p4 2 in the relativistic correction if we calculate its expectation value directly in the C-BSBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Pachucki and Komasa also used a similar way to transform both the kinetic term and the Araki-Sucher correction to much more regular forms and obtained much better numerical convergence on that account [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' In the present calcu- lations, as Pachucki and Komasa have done, we use the following expression to evaluate ⟨p4 1 + p4 2⟩, 2 � i=1 � ψ ��p4 i �� ψ � = 4 � ψ ��(Eψ − V )2�� ψ � − 2 � ∇2 1ψ|∇2 2ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (10) The integration of ⟨ψ|r−2 12 |ψ⟩ will be involved in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (10), and it is also evaluated to be as following by using the global operator method, � ψ ��r−2 12 �� ψ � =2⟨ψ| ln r12(V − Eψ)|ψ⟩ + 2 � i=1 ⟨∇iψ| ln r12|∇iψ⟩ , (11) since we find that ∇2 1 ln r12 = ∇2 2 ln r12 = r−2 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The com- plete expansion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (10) is written as 2 � i=1 � ψ ��p4 i �� ψ � = 4E2 ψ − 8Eψ � ψ ����−2Z r1 + 1 r12 ���� ψ � +4 � ψ ���� 2Z2 r2 1 − 2Z2 r1r2 − 2Z r1r12 + 1 r2 12 ���� ψ � −2 � ∇2 1ψ|∇2 2ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (12) The Araki-Sucher correction is converted to the regular form as well so as to facilitate the present numerical eval- uations, � ψ ��r−3 12 �� ψ � = − 2 � i=1 � ∇iψ ��r−1 12 ln r12 �� ∇iψ � + � ψ ����2 (Eψ − V ) ln r12 r12 +4π(1 + γ)δ3 (r12) �� ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (13) where rn 12 ln r12 (n = −2, −1, 0, 1) will be involved in inte- gration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' In addition to the above three terms, the expec- tation values of other operators appearing in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (5)-(6) will be calculated in the C-BSBF directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Laplace’s expansion of rn 12 and rn 12 ln r12 The integration of rn 12 and rn 12 ln r12 are involved in the computation of Breit-Pauli operators and Araki-Sucher corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' It is crucial to process this type of the in- tegration in spherical coordinates, which requires sepa- rating their radial and angular dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The gener- alization of Laplace’s expansion to arbitrary powers and functions of r12 given by Sack [38] is used to calculate the integration in which different powers of r12 is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' rn 12 can be expanded in the form rn 12 = ∞ � ℓ=0 Rnℓ(r1, r2)Pℓ(cos θ12) , (14) where the Legendre polynomials of cos θ12 is ex- pressed by using the identity as Pℓ(cos θ12) = 4π/(2ℓ+1) m=ℓ � m=−ℓ Y ∗ ℓm(ˆr1)Yℓm(ˆr2), and the radial function Rnℓ(r1, r2) has been formulated by Sack [38] as following Rnℓ(r1, r2) = � − 1 2n � ℓ � 1 2 � ℓ rn > �r< r> �ℓ × 2F1 � l − 1 2n, −1 2 − 1 2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' l + 3 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' r2 < r2> � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (15) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (15), r< = min(r1, r2), r> = max(r1, r2), and the hypergeometric function has the form of 4 2F1(α, β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' x) = 1 + ∞ � 1 (α)s(β)s (γ)ss!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' xs , (16) where the Pochhammer symbol is defined as (α)s = � 1 if s = 0 α(α + 1) · · · (α + s − 1) if s > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (17) The hypergeometric function is finite series if either α or β is zero or a negative integer, which implies that for all positive odd integer values of n, the series of Rnℓ break off;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' and for n = −1, they consists of the leading term only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' For positive even n, the summation is truncated to ℓ = n 2 , since the factor (− 1 2n)ℓ ensures that Rnℓ vanishes when ℓ > n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' In addition, the individual functions Rnℓ are divergent for n ≤ −2, but they remain integrable as long as n > −3 [35, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Present calculations involve the integrations of ⟨ψ|r−2 12 |ψ⟩ and ⟨ψ|r−3 12 |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' So giving ap- propriate radial expansions of r−2 12 and r−3 12 is important in the computation of radial and angular integrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Substituting n = −2 , ℓ = 0 and n = −2 , ℓ = 1 sepa- rately into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (15), and summation of the series, as a re- sult the following specific expressions in terms of reverse hyperbolic tangent function tanh−1(x) are achieved, R−2,0 (r1, r2) = tanh−1(x) xr2 > , (18) R−2,1 (r1, r2) = 3 2x2r2 > × �� x2 + 1 � tanh−1(x) − 1 � , (19) where x = r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' then the recurrence relation r2 1 + r2 2 r1r2 Rn,ℓ − ℓ + 2 + 1 2n ℓ + 3 2 Rn,ℓ+1 −ℓ − 1 − 1 2n ℓ − 1 2 Rn,ℓ−1 = 0 , (20) can be used to calculate the radial functions for other values of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' For n = −3, the expansion coefficients of the hypergeometric functions are cancelled, and the hyper- geometric functions are reduced to a series summation of xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' the hypergeometric function can be expressed as an- alytic functions that is independent of ℓ, correspondingly the radial expansion of R−3,l can be written as [40] R−3,l (r1, r2) = (2ℓ + 1)xℓ (1 − x2) r3> .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (21) Next we will give the explicit formula for the product of r12 with different powers and ln r12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Differentiation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (14), the expansion for rn 12 ln r12 can be expressed as rn 12 ln r12 = � ℓ Rn ln,ℓ(r1, r2)Pℓ (cos θ12) , (22) where Rn ln,ℓ(r1, r2) represents the radial function of rn 12 ln r12, and Rn ln,ℓ(r1, r2) = ∂Rnℓ(r1,r2) ∂n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Similarly, the following recurrence relation for Rn ln,ℓ(r1, r2) can be de- rived by taking the derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (20), 1 2ℓ + 3Rn,ℓ+1 − 1 2ℓ − 1Rn,ℓ−1 = r2 1 + r2 2 r1r2 Rn ln,ℓ − 2ℓ + 4 + n 2ℓ + 3 Rn ln,ℓ+1 − 2ℓ − 2 − n 2ℓ − 1 Rn ln,ℓ−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (23) Then we can calculate the integration with the rn 12 ln r12 (n ≥ −2) operator in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' For example, for n = −2 , ℓ = 0 and n = −2 , ℓ = 1, R−2 ln,0 = tanh−1(x) ln(r2 > − r2 <) 2r2 >x , (24) R−2 ln,1 =3 � ln(r2 > − r2 <) − 1 � 4r2 >x2 × �� x2 + 1 � tanh−1(x) − x � , (25) and the estimations of R−2 ln,ℓ for other values of ℓ > 1 can be obtained according to the recurrence relation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS The C-BSBF on an exponential grid [14] are gener- ated using B-splines constrained to a spherical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The cavity radius of R0 = 20 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' is for the 1 1S state, R0 = 40 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' is for the 2 1S state, and R0 = 70 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' is for both the 2 3S and 3 3S states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [25] have implemented the correlated B-splines to calculate the he- lium atomic energy level and their non-relativistic ground state energy is −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='903 724 377 1(2) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='. A knot distribu- tion optimization was performed for any individual states and present values of energies for the 1 1S, 2 1S, 2 3S and 3 3S states are listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The optimized result of −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='903 724 377 034 0(2) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' is obtained for the ground state, which has thirteen significant digits in agreement with Drake’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The 2 1S, 2 3S, and 3 3S states also reached fourteen significant digits in agreement with Drake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' TABLE II: Energies for the 1 1S, 2 1S, 2 3S and 3 3S states of helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' State This work Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [41] 11S −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='903 724 377 034 0(2) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='903 724 377 034 119 5 21S −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='145 974 046 054 4(2) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='145 974 046 054 419(6) 23S −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='175 229 378 236 7(2) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='175 229 378 236 791 30 33S −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='068 689 067 472 4(2) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='068 689 067 472 457 19 It can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' (12) that the computation of ⟨p4 1⟩ involves many operators, which are classified into two categories for dealing with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' One type is the general 5 TABLE III: The expectation values of other operators needed for evaluating the relativistic kinetic terms for the 1 1S, 2 1S, 2 3S, and 3 3S states of helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Operater 11S 21S 23S 33S ⟨1/r1⟩ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='688 316 800 717 1(2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='135 407 686 126 1(2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='154 664 152 972 0(1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='063 674 075 760 7(2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='688 316 800 717a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='135 407 686 125 609(6)b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='154 664 152 972 107 60(20)b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='063 674 075 760 76(10)b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='688 316 800 635c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='135 407 686c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='154 664 152c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='063 674 075 7c ⟨1/r2 1⟩ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='017 408 867 0(3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='146 939 019 80(6) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='170 445 551 31(2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='042 948 747 4(3) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='017 408 867 0(1)a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='146 939 019 0(12)b 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='170 445 551 336 2(4)b 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='042 948 747 477(4)b ⟨1/r1r2⟩ 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='920 943 921 900a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='340 633 845 861 0(19)b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='322 696 221 719 854 32(8)b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='131 426 560 051 184(5)b a Drake [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' b Drake [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' c Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' operators that are relatively simple to compute, includ- ing 1/r1, 1/r2 1, 1/r1r2, 1/r12 and 1/r1r12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' We give the final convergence values directly in Table III, and there are at least ten significant digits of our results that are consistent with Drake’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' This also demonstrates the high accuracy of the wave function obtained for the C-BSBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The other type is the operators 1/r2 12 and ∇2 1∇2 2 that are more difficult to calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The numerical results of ⟨1/r2 12⟩, ⟨∇2 1∇2 2⟩ and ⟨p4 1⟩ as the number of B-splines N increased are given in the last three columns of Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Good convergent values of ⟨1/r2 12⟩ under the C-BSBF are achieved with the global operator method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' For the ground state, the present result of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='464 770 923 3(5) is obtained, which has eleven significant figures and agrees well with reference values with the explicitly correlated exponential basis [10] and the Hylleraas basis [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Our expectation values of 1/r2 12 for the 2 1S, 2 3S and 3 3S states of the helium atom at least have eight convergent digits, which are all in good agreement with results in available literatures [9, 10, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' For the ⟨∇2 1∇2 2⟩ opera- tor, no suitable treatment could be found to make it con- verge faster, for which the direct calculation method was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Therefore, the convergent accuracy of ⟨∇2 1∇2 2⟩ is relatively lower, which is also the main reason to limit the numerical precision of ⟨p4 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The present result of ⟨p4 1⟩ for the 1 1S state from the C-BSBF has nine digits, consis- tent with Drake’s Hylleraas results [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Present nu- merical convergence for the triplet states are better than for the singlet states by one to two significant figures, and our values are both good agreement with Hylleraas results [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' We also calculated ⟨1/r2 12⟩ and ⟨∇2 1∇2 2⟩ using the tradi- tional B-spline basis set, and results for the ground state are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='463 697 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='079, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Since these singu- larity operators only have one to three significant digits, which are difficult to use in high-precision calculations at the atomic energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' It is convenient to find that the primary explanation for this is that the traditional B-spline basis set makes it difficult to describe the lo- cal properties of the wave function with high accuracy without including the electron correlation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The expectation values of other three components from HBP and the singular electron-electron ⟨1/r3 12⟩ from the leading QED corrections are shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The ex- pectation values of δ3(r12) for the triplet states equal zero, so they are not listed in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [43] employed the same C-BSBF to give numerical results of δ3(r1) by direct calculation when the power of r12 is c = 5, which are also shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The direct calcu- lation of δ3(r1) is highly dependent on the origin value of the wave function, and the global operator method can be used to further improve the calculation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The result of the δ3(r1) of the ground state using the global operator method is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='810 429 32(2), one can see that nu- merical accuracy of the δ3(r1) can reach a precision of eight to twelve significant digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' It can be seen that our computational accuracy with c = 1 is completely compa- rable to theirs [43], with the except for the ground state with relatively sensitive electron correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' They also tried to improve the direct calculation accuracy of δ3(r1) by increasing the power of r12, but the global operator method is still necessary to effectively improve the nu- merical convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' For example, our result of ⟨δ3(r12)⟩ for the 1 1S state is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='106 345 370 66(4), that is more accurate than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='106 346 068 of Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [43] by five or- ders of magnitude and is well consistent with Drake’s Hylleraas value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='106 345 370 636 3(12) [42] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Present results for the retardation term H2 have at least 6 TABLE IV: Convergence of the relativistic kinetic terms for the 1 1S, 2 1S, 2 3S and 3 3S states of helium as the number of B-splines N increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The expectation values of 1/r2 12 and ∇2 1∇2 2 are also listed in the second and third columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The partial wave is ℓmax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' N ⟨1/r2 12⟩ ⟨∇2 1∇2 2⟩ ⟨p4 1⟩ 1 1S 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='464 770 923 579 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='133 709 835 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='088 067 177 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='464 770 923 463 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='133 709 771 54.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='835 540 797 348(6) 3 3S 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='023 097 669 645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='329 220 596 46 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='475 439 870 27 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='023 097 669 653 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='329 220 596 68 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='475 439 868 42 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='023 097 669 655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='329 220 596 89 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='475 439 868 25 Extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='023 097 669 65(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='329 220 597(2) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='475 439 868(5) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [42] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='023 097 669 656 893(13) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='475 439 868 127 2(3) seven convergent figures and agree with Drake’s [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The expectation of singular electron-electron ⟨1/r3 12⟩ are com- puted with the global operator method by the C-BSBF and confronted with previous results obtained from dif- ferent basis functions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Present the C-BSBF re- sult of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='989 272(2) with an accuracy of five decimals is achieved for the ground state, which is comparable to re- sults of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='989 273 5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='989 272 4(13) with explicitly correlated Gaussian (ECG) functions [44] and exponen- tial basis functions [45], respectively, in numerical pre- cision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Employed Hylleraas basis and exponential basis respectively, Drake [42] improved reference values with three additional exact digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Our result for the ground state is expected to recover more figures of Drake’s re- sult if adopting higher power of r12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' For the 2 1S and 2 3S states, our values are in agreement with previous values obtained by Hylleraas basis and exponential ba- sis [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' There are five convergent figures in our result ⟨1/r3 12⟩=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='008 922 57(2) for the 3 3S state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The singular electron-electron ⟨1/r3 12⟩ expectation value is also computed using the traditional B-spline ba- sis set, and the ground state result is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='197(N = 70, ℓmax = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' It can be seen that the traditional B-spline is entirely inaccurate in calculating ⟨1/r3 12⟩, and this type of operator for divergence require a more accurate de- scription of the local properties of the wave function [44] than 1/r2 12 and ∇2 1∇2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' As a result, the B-spline basis set containing electron correlation is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The final relativistic corrections are presented in the top half of Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Comparisons are made with results obtained using the explicitly correlated exponential ba- sis [11] and Hylleraas basis [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Our relativistic correc- tions results are completely consistent with most precise previous calculations [11, 42] and can reach eight to ten significant figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The leading QED corrections for the S states to the energy level are summarized in the bottom half of Table VI, which used the Bethe logarithm values obtained from B-splines [22, 26], and Korobov’s Bethe logarithm values [33] as a benchmark, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' It can be seen that our calculated results are in good agree- ment with the significant figures listed by Yerokhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [11], where the results of the singlet state calculated using Korobov’s Bethe logarithm values are almost iden- tical to the results from B-splines, which are mainly ex- plained by the relatively low accuracy of δ3(r1) and 1/r3 12, and the improved accuracy of the triplet state is the re- 7 TABLE V: The expectation values of δ3(r1), δ3(r12), H2 and 1/r3 12 for the 1 1S, 2 1S, 2 3S and 3 3S states of helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Comparisons with results obtained in available literatures are also made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The partial wave is ℓmax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' N ⟨δ3(r1)⟩ ⟨δ3(r12)⟩ ⟨H2⟩ ⟨1/r3 12⟩ 11S 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='810 429 325 97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='106 345 370 649 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='139 094 671 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='989 271 57 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='810 429 323 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='106 345 370 658 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='139 094 675 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='989 271 98 70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='810 429 321 51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='106 345 370 646 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='139 094 677 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='989 272 26 Extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='810 429 32(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='106 345 370 66(4) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='139 094 67(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='989 272(2) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [43] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='810 429 318 371 521 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='106 346 068 Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [42] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='810 429 318 499 0(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='106 345 370 636 3(12) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='139 094 690 539 20(20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='989 273 544 768(13) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [44] 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The overall computational accuracy of the leading QED cor- rection is determined mainly by the contribution of the Araki-Sucher term and δ3(r1) for the ground state, by the contribution of the Bethe logarithms for other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' It can be seen that the leading QED corrections results can reach at least seven significant digits, which already reaches the accuracy level of the contribution of the lead- ing relativistic correction in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' In addition, the numerical accuracy of the singlet is expected to improve with increasing power c of r12 in basis function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' SUMMARY AND OUTLOOK In this work, we have calculated the leading relativis- tic and QED corrections of the energy levels of the he- lium atom using the C-BSBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The expectation values of the relativistic kinetic term p4 1, contact potential δ3(r1), δ3(r12) and Araki-Sucher correction ⟨1/r3 12⟩, which are more difficult to calculate directly, were treated by a global operator method to improve their numerical con- vergence, and the two-electron distance function is also introduced to deal with the Laplace expansion method proposed by Sack [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' Together with the high-precision calculation of the Bethe logarithms [26], the C-BSBF is able to achieve the high-precision calculation of the lead- ing relativistic and QED corrections for the energy levels of the helium atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' It is emphasized that the corre- lated factor r12 in the C-BSBF is crucial to calculate p4 1, δ3(r12) and ⟨1/r3 12⟩, without this factor, these oper- ators have a very slow convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The C-BSBF can provide stable numerical convergence based on its ap- proximate linear independence and sufficient considera- tion of the electronic correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' It can be seen from Table VII that the C-BSBF can determine the accuracy of the 23S − 21S transition frequency (up to mα5-order correction) to the kHz level, which is consistent with the 8 TABLE VI: The leading relativistic and QED corrections, δErel and δEQED for the 1 1S, 2 1S, 2 3S and 3 3S states of helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The corresponding comparison data given in available literatures are also listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' 11S 21S 23S 33S the leading relativistic correction δErel/α2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='951 754 7(2) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='034 167 33(2) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='164 477 971(2) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='045 092 764(2) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [42] −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='951 754 767 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='034 167 342 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='164 477 972 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='045 092 764 the leading QED correction δEQED/α3(BL with B-splines) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='288 165(2) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='523 605 2(2) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='010 017(2) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='839 303 4(7) δEQED/α3(BL from Korobov) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='288 165(1) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='523 605 10(8) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='010 017 06(2) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='839 301 459(9) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [11] 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='288 165 2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='523 605 1 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='010 016 8 results of Pachucki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=', reaching a level similar to the latest experiment [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' A further improvement in our re- sults is expected, if adopting higher power of r12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The calculations are carried out applying double precision, no multi-precision is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' This method provides a new approach to the calculation of atom energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' TABLE VII: The 23S − 21S transition frequency for the he- lium atom along the leading relativistic and QED corrections, in KHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' ∆E(23S − 21S) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [47] NR 192 490 838 748(2) 192 490 838 756 mα4 45 657 862(8) 45 657 859 mα5 −1 243 669(6) −1 243 671 Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' [1] 192 510 702 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content='72(20) Recently, Mitroy and Tang suggested testing the QED theory using tune-out wavelength, which opens a new way to test fundamental atomic structure theory [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The 413 nm tune-out wavelengths for the helium atom 23S1 state discrepancies in the latest experiments by Baldwin’s team and theoretical values by Drake based on the Hylleraas basis set with the NRQED method [49], in which the calculation only estimates the electric-field dependence of the Bethe logarithm [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The precision of the experiment is expected to improve further, and the QED theory will be tested at higher precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The ab- initio calculation of the electric field dependence of the Bethe logarithm is important for further improving the theoretical prediction accuracy of the 413 nm tune-out wavelength to a level of ppb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' The successful application of the C-BSBF in singular operator calculations in this work suggests that the C-BSBF is expected to be used to calculate the electric field dependence of Bethe loga- rithms to improve the theoretical calculation accuracy of the 413 nm tune-out wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' In addition to the C- BSBF is also expected to be extended to the second-order perturbation of the Breit–Pauli operators [51] and rela- tivistic corrections to the Bethe logarithm [52] of helium atom in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' ACKONWLEDGEMENT This work is supported by the National Natural Sci- ence Foundation of China under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' 12274423 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' 12274417, by the Chinese Academy of Sciences Project for Young Scientists in Basic Research under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE5T4oBgHgl3EQfHw5U/content/2301.05442v1.pdf'} +page_content=' YSBR-055.' metadata={'source': 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Liang,1 Zhi-Lin Chen,1 Xiao-Yan Li,1 +Xiang-Gao Wang,1 and En-Wei Liang1 +1Laboratory for Relativistic Astrophysics, Department of Physics, Guangxi University, Nanning 530004, China +2Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210034, +China +3School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China +ABSTRACT +Long-duration GRB 200829A was detected by Fermi-GBM and Swift-BAT/XRT, and then rapidly +observed by other ground-based telescopes. It has a weak γ-ray emission in the very early phase and +followed by a bright spiky γ-ray emission pulse. The radiation spectrum of the very early emission +is best fitted by a power-law function with index ∼ −1.7. However, the bright spiky γ-ray pulse, +especially the time around the peak, exhibits a distinct two-component radiation spectra, i.e., Band +function combined with a blackbody radiation spectrum. We infer the photospheric properties and +reveal a medium magnetization at photospheric position by adopting the initial size of the outflow as +r0 = 109 cm. It implies that Band component in this pulse may be formed during the dissipation of +magnetic field. The power-law radiation spectra found in the very early prompt emission may imply +the external-shock origination of this phase. Then, we perform Markov Chain Monte Carlo method +fitting on the light-curves of this burst, where the jet corresponding to the γ-ray pulses at around +20 s is used to refresh the external-shock. It is shown that the light-curves of very early phase and +X-ray afterglow after 40 s, involving the X-ray bump at around 100 s, can be well modelled in the +external-shock scenario. For the obtained initial outflow, we estimate the minimum magnetization +factor of the jet based on the fact that the photospheric emission of this jet is missed in the very early +phase. +Keywords: Gamma-ray bursts (629) +1. INTRODUCTION +Theoretically, it is generally believed that gamma-ray bursts (GRBs) originated from collapse of massive stars or +mergers of double compact stars (e.g., Colgate 1974; Paczynski 1986; Eichler et al. 1989; Narayan et al. 1992; Woosley +1993; MacFadyen & Woosley 1999; Piran 2004; Zhang & M´esz´aros 2004; Woosley & Bloom 2006; Kumar & Zhang +2015). Observationally, GRBs generally appear as a brief and intense γ-rays followed by a long-lived afterglow emission. +The prompt γ-rays are highly variable with a duration from millisecond to thousands of seconds. The observational +spectra are usually well fitted by an empirical function, characterized by a smoothly joint broken power-law function, +the so-called Band function (Band et al. 1993) or a quasi-thermal spectral component appear in the spectra of some +GRBs. The previous observations demonstrated that thermal components exhibit different observational properties. +They either can be detected during the entire duration of the prompt emission (e.g., Ghirlanda et al. 2013) or may +be only found at the beginning of the burst duration, and subsequently appear with a nonthermal component. The +detection of a diversified spectral characteristic shows that GRB ejecta may have a diverse jet composition. It may be +neither fully matter-dominated ejecta nor fully magnetized outflows. More realistically, GRB outflows are likely to be +a hybrid jet, which carries the two components simultaneously and launches at the central engine (e.g., Gao & Zhang +2015). The light-curves of afterglow emission usually can be decomposed into four power-law segments, i.e., an initial +steep decay, a shallow decay, a normal decay, and a late steeper decay, sometimes accompanied by one or several +Corresponding author: Da-Bin Lin, Rui-Jing Lu +lindabin@gxu.edu.cn, luruijing@gxu.edu.cn + +2 +flares (Zhang et al. 2006; Nousek et al. 2006). It is commonly believed that the multi-wavelength afterglow is mainly +from the external shock, which is formed during a relativistic jet propagating in the circum-burst medium (e.g., +M´esz´aros & Rees 1997). However, the origin of the prompt γ-rays is not well understood. The prompt γ-rays may +be from the internal shock in an erratic relativistic fireball, a dissipative photosphere, a Poynting-flux dominated jet, +or even an external shock (e.g., Rees & Meszaros 1992; Meszaros & Rees 1993; Rees & Meszaros 1994; Giannios 2008; +Beloborodov 2010; Vurm et al. 2011; Zhang & Yan 2011; Burgess et al. 2016; Huang et al. 2018). +It is not a new idea that the prompt γ-rays of GRBs originate from the external shock. Burgess et al. (2016) have +shown that the prompt emission of GRB 141028A is very likely originated from an external shock. Huang et al. (2018) +suggested that GRB 120729A is an external shock origin for both the prompt γ-ray emission and afterglow. They +also systematically investigate single pulse GRBs in the Swift’s GRBs, and find that a small fraction of GRBs (GRBs +120729A, 051111, and 070318) are likely to originate from an external shock for both the prompt γ-ray emission and +afterglow. However, Huang et al. (2018) focuses on the bursts appearing as a single pulse from the prompt emission +to its afterglow. In fact, the central engine of GRBs may re-activity and launch relativistic ejecta several times. The +late launched ejecta may be observed as flares in the afterglow and interact with the external shock at a later period. +The burst GRB 200829A maybe in the above scenarios. GRB 200829A was detected by Fermi-GBM and Swift- +BAT/XRT, and the light-curve of prompt emission is composed of an initial very early weak emission (with a duration +∼ 5 s) followed by a bright spiky γ-ray pulse with a duration ∼ 10 s. We find that the spectra in the γ-ray pulse of +GRB 200829A exhibits a distinct two-component, i.e., Band function combined with a blackbody radiation spectrum, +especially in the peak time. It means that the thermal component should be indeed existence, and GRB 200829A +outflows are likely to be a hybrid jet. What’s more, the radiation spectrum in its very early phase can be fitted with +power-law spectral model with index ∼ −1.7, which may be an indication of the origin of an external-forward shock. +The central engine of GRB 200829A may re-activity and launch jets at different times, resulting in the bright spiky +γ-ray pulses when jets collide with each other. +In this paper, we present a detailed analysis of γ-rays and X-ray emission from the long GRB 200829A detected +by Fermi and Swift. The paper is organized as follows. In Section 2, we introduce the observations and light-curves +features of GRB 200829A. In Section 3, the detailed analysis and results of GRB 200829A are performed. In this +section, we also analyzed the other properties of GRB 200829A in different phase. In Section 4, the summary and +discussions are presented. +2. OBSERVATIONS AND DATA REDUCTION +The long GRB 200829A was first detected by Fermi Gamma-Ray Burst Monitor (GBM) at 13 : 58 : 14.66 UT (T0) on +2020 August 29 with duration T90 ∼ 6.9 s (Lesage et al. 2020). In addition to the Fermi-GBM, Swift-BAT triggered the +burst at 13 : 59 : 34 UT on 2020 August 29 (Palmer et al. 2020) and Swift-XRT began to observe the burst at 128.7 s +after the BAT trigger (Gropp et al. 2020). Oates et al. (2020) created a SED at 900 s after the BAT trigger and found a +photometric redshift of z = 1.25±0.02 for this burst. The optical afterglow is detected on first two days after the GRB +trigger (Pozanenko et al. 2020b). In the left panels of Figure 1, we show the light-curves of prompt γ-rays and afterglows +of GRB 200829A with respect to the Fermi trigger. The inset in the upper part of this panel shows the light-curves of +prompt emission based on the Fermi observation in the linear spaces. Here, the Fermi data are from the Fermi Science +Support Center1 and a GBM light-curve and source spectra are extracted from the TTE (Time-Tagged-Events) data by +using a python source package named gtBurst2, the BAT/XRT data are taken from the UK Swift Science Data Center3, +and the optical data of GRB 200829A are from Siegel et al. (2020); Pozanenko et al. (2020a); Lipunov et al. (2020b); +Kuin et al. (2020); Lipunov et al. (2020a); Hentunen & Nissinen (2020); Moskvitin et al. (2020b); Zhu et al. (2020b); +Moskvitin et al. (2020a); Pankov et al. (2020); Zhu et al. (2020a); Izzo (2020); Volnova et al. (2020); De Pasquale +(2020); Pozanenko et al. (2020b). +Based on the light-curves in the left panels of Figure 1, one can find that the prompt γ-rays is dominated by a +bright spiky γ-ray pulses in the period of tobs ∼ [15, 30] s based on GBM observation, which is preceded by a small +γ-ray pulse in the period of tobs ∼ [6, 10] s based on BAT observation. However, it should be noted that the small +γ-ray pulse in the period of tobs ∼ [6, 10] s is not significantly in the light-curve of GBM observation. Except these +two γ-ray episodes, there is a significant γ-ray emission in the very early phase of the prompt emission (tobs < 6 s) +1 https://fermi.gsfc.nasa.gov/ssc/data/access/ +2 https://github.com/giacomov/gtburst +3 http://www.swift.ac.uk/burst analyser/00993768/ + +3 +based on BAT observation. This can also be found in the right panels of Figure 1, which shows the GBM light-curve +of GRB 200829A without background subtracted (upper panel) and the signal significance (bottom panel). One can +find that the signal significance in the period of ∼ [0, 10] s is higher than ∼ 4σ, which reveal a significant γ-ray photons +in this period. In the following section, we present the detailed studies on the spectra and the corresponding physical +implications for the very early phase and the bright spiky γ-ray pulses. +3. DETAILED ANALYSIS OF GRB 200829A AND RESULTS +3.1. Very early prompt gamma-ray emission +For the very early phase of the prompt emission, the spectral fitting with Band function4 reports the values of +α = −1.75 ± 0.09, E0 = 9976.67 ± 51113.36, and β = −2.42 ± 5.08 (see the third line of Table 1). The values of +E0 and β could not be well constrained from the spectral fitting. Then, we perform the spectral analysis of the very +early phase with the power-law (PL) function5 or cutoff power-law (CPL) function. Here, the spectral fitting with PL +function reports the power-law index ˆΓ = −1.79 ± 0.06 (see the second line of Table 1), and the spectral fitting with +CPL function could not present a well fitting and the corresponding result is not reported. The spectral fitting results +for the very early prompt emission with Band function (left panel) and PL function (middle panel) are also shown in +Figure 2. We note that the values of α = −1.75 ± 0.09 and ˆΓ = −1.75 ± 0.06 from the spectral fittings are almost the +same. Here, the value of α can be well constrained in the spectral fitting with Band function. This fact may imply +that the intrinsic radiation spectrum in this period may be consistent with a PL spectral model with ˆΓ ∼ −1.7 or a +Band function with a break at ∼ 10 MeV and power-law index ∼ −1.7 in its low-energy regime (E ≲ 10 MeV)6. +The reasons are as follows. Firstly, the spectral fitting on such kind of intrinsic radiation spectrum with a Band +function would not provide a well constraint on the value of E0 and thus β. This is consistent with our spectral fitting +result for this period based on a Band function. In addition, a Band function with a break at ∼ 10 MeV, the power- +law index ∼ −1.7 in its low-energy regime (E ≲ 10 MeV), and the power-law index ≳ −2.5 in its high-energy regime +(E ≳ 10 MeV) can be modelled with a PL function and ˆΓ = −1.79 in Fermi-GBM energy band (8 keV-40 MeV). This +is also consistent with our spectral fitting result for this period based on a PL function. Secondly and importantly, +we perform the spectral fitting on the Swift-BAT observation for this period based on a PL model and the value of +ˆΓ = −1.71 is reported. +We note that such kind of intrinsic radiation spectrum is very different from the general Band radiation component +of GRBs’ prompt emission, of which the value of α is around −1 and the break energy E0 is around 400 keV. The right +panel of Figure 2 shows the relation of Ep and α based on the spectral fitting results with a Band function, where +the blue symbols are from the figure 8 of Poolakkil et al. (2021) and represent the GOOD sample for time-integrated +spectral fits with Band function. In this panel, the spectral analysis result for the very early phase of the prompt +emission based on Band function is also tentatively shown with pink “⋆” even though the value of E0 could not be +well constrained, and the spectral fitting results of the small γ-ray pulse ([5, 10] s) or the bright spiky γ-ray pulses +([16, 26] s) with Band function are also showed. One can find that such kind of radiation spectrum is very different +from the general Band radiation component of GRBs’ prompt emission, involved that of the bright spiky γ-ray pulses +or the small γ-ray pulse. Then, we would like to believe that the very early phase of the prompt emission in this burst +may be originated from the other channel rather than that for the bright spiky γ-ray pulses or the small γ-ray pulse. +3.2. Bright spiky γ-ray pulse and deriving physical parameters +There is a bright spiky γ-ray pulse appearing at tobs ∼ [16, 26] s after the Fermi trigger. In order to perform detailed +analysis of this pulse, we divide this pulse into several time intervals with 1 s time span and perform the spectral fitting +on these time intervals with Band function. The spectral fitting results are reported in Table 2 and shown in the left +panels of Figure 3. A distinct multi-component of radiation spectrum is found in several time intervals of this pulse, +e.g., [18, 19] s. Then, we also perform the spectral analysis together with Band function and a blackbody radiation +component (BB) 7, i.e., “Band+BB”. The spectral fitting results based on Band+BB model are also reported in +Table 2 and shown in the right panels of Figure 3. We also estimate the Bayesian Information Criterion (BIC; Schwarz +4 Band function is described as N(E) = N0(E/100keV)α exp(−E/E0) for E ≤ (α − β)E0 and N(E) = N0[(α − β)E0/100keV]α−β exp(β − +α)(E/100keV)β for E ≥ (α − β)E0, where N0 is the normalization, and α, β, and E0 are parameters in the spectral fittings. The peak +photon energy of E2N(E) is Ep = (α + 2)E0. +5 The PL function is described as N(E) = N0(E/1keV)ˆΓ with ˆΓ being the photon spectral index. +6 Please see Appendix A for a comprehensive analysis about the radiation spectrum in this period. +7 NBB(E) = +8.0525×KE2 +(kT )4(e(E/kT )−1) , where kT is the blackbody temperature keV; K is the L39/ D2 +10, where L39 is the source luminosity in units +of 1039 erg/s and D10 is the distance to the source in units of 10 kpc. + +4 +1978) for the spectral fitting with Band function and that with Band+BB model. The values of BIC from the spectral +analysis are also reported in Table 2. The BIC is adopted to evaluate the goodness of the model fitting, taking into +account the model complexity and the different numbers of free parameters. Generally, the model with a lowest BIC +is preferred. By comparing the values of BIC from the spectral analysis, one can find that the Band+BB model is +preferred for the radiation spectrum of the time intervals around the peak of the bright spiky γ-ray pulse. Since the +value of ∆BIC=BICBand − BICBand+BB is in the range of 12-25, it is strong to support a blackbody component in +these time intervals8. +The temperature and flux of the blackbody component, together with the radius of the jet base (size of the central +engine) r0 and z, can provide useful information about the physics of the photosphere. Meanwhile, due to the presence +of Band energy spectrum component, the jet compositions of GRB 200829A maybe hybrid. +Therefore, following +Gao & Zhang (2015), we estimate the radius and Lorentz factor of the photosphere based on the blackbody component +found in the period of [18,22] s by assuming the hybrid outflow of GRB 200829A. In the calculations, we assume that +there is no dissipation below the photosphere and the radiation efficiency ∼ 52.1% (please see Section 4). The results +are shown in the left panels of Figure 4, the blue and olive symbols are the physical quantities calculated based on +r0 = 108 cm and r0 = 109 cm, solid and hollow “⋆” represent the physical parameters rph, Γph, respectively. It +indicates that the values of rph increases with time and Γph remains constant for low value of r0 and when r0 is large, +it increases and eventually declines. We also infer the dimensionless entropy η and the magnetization factor σ, where +σ0 and σph are the magnetization factor of the outflow at r0 and rph, respectively. The results are shown in the middle +panels of Figure 4, the blue and olive symbols are the same as those in the left panels of Figure 4, and solid and hollow +“⋆” represent the physical parameters η, 1 + σph, and 1 + σ0, respectively. It is shown that the dimensionless entropy +η fluctuates in the range of 100 to 300. In addition, the values of 1 + σph can be around 5 if r0 = 109 cm is adopted +and around 1 if r0 = 108 cm is adopted. Together with the Band and BB components found in this burst, the initial +radius of the outflow producing the bright spiky γ-rays should be around or larger than 109 cm, i.e., r0 ≳ 109 cm. This +result is consistent with that found in GRBs with identified photospheric emission, e.g., GRB 120323A, GRB 131014A +and GRB 220426A (e.g., Guiriec et al. 2013, 2015; Deng et al. 2022). The non-thermal component in the bright spiky +γ-rays, i.e., Band component, seem to be formed during the dissipation of the magnetic energy. +3.3. Afterglow analysis and a self-consistent Paradigm for bursting +Following the prompt γ-ray emission in this burst, a late bump appears at tobs > 40 s with a rising in the period +of tobs ∼ [40, 100] s and a decaying after tobs ∼ 100 s. It is reasonable to believe that the decaying phase of the late +bump is the normal decay of the external-forward shock. For the X-ray emission in this phase, the closure relation +(Zhang & M´esz´aros 2004) of α ≈ 3β/2 with F ∝ ν−βt−α can be found, where the value of α = 1.30 ± 0.03 and +β = 0.80 ± 0.05 are obtained based on the observations of Swift. It reveals that the X-ray emission in this phase is in +the spectral regime of νm < ν < νc for an external-forward shock in the interstellar medium. +The very early phase of the prompt emission may be originated from the external shock. The reasons are as follows. +Firstly, we have performed a joint spectral analysis by combining the observations of Swift-BAT and Fermi-GBM +for the very early phase of the prompt emission in Section 3.1. The spectral analysis reveals that the very early +phase of the prompt emission in this burst may be originated from the other channel rather than that for the small +γ-ray pulse or the bright spiky γ-ray pulses. Secondly, the radiation spectrum in this phase is strongly reminiscent +of the GRB 120729A, of which the radiation spectrum in the prompt emission for Fermi-GBM energy band can be +well modelled with a PL function and photon spectral index ˆΓ ∼ −1.479(Huang et al. 2018). Since the light-curve +of the prompt emission in GRB 120729A appears as a single long and smooth pulse, which extends continuously to +the X-rays, it is suggested that both the prompt emission and the afterglows are originated from an external-forward +shock (Huang et al. 2018). Thirdly, the spectral index of the very early prompt emission based on Swift-BAT and +Fermi-GBM observations is almost the same as that of the decaying phase in the late bump based on the Swift-XRT +observation (see Table 1 and Table 3). This is different from that in GRB 120729A, of which the spectral index in the +X-ray energy band evolves from -1.47 in the early phase of the prompt emission to -1.83 in the late phase of afterglow. +It may reveal that the X-rays may in the same spectral regime in GRB 200829A but in different spectral regime in +8 In the spirit of Burnham & Anderson (2004), the value of ∆BIC can be used as the strength of the evidence to allow a quick comparison and +ranking of candidate hypotheses or models. For ∆BIC = BICA −BICB with BICA > BICB, the strength of the evidence can be summarized +as follows: the situation with ∆BIC ⩽ 2 provides no evidence against the model-A; the situation with 4 ⩽ ∆BIC ⩽ 7 provides positive +evidence against the model-A; the situation with ∆BIC ⩾ 10 provides very strong evidence against the model-A (Burnham & Anderson +2004). +9 By performing joint spectral fitting of the Swift-BAT and Fermi-GBM observations for GRB 120729A, we obtain ˆΓ ∼ −1.47 and ˆΓ ∼ −1.49 +for the period of [0, 10] s and [1, 2] s after the Fermi trigger, respectively. + +5 +GRB 120729A for the very early prompt emission and the late phase of afterglow. Then, we would like to believe that +the early phase of prompt emission (tobs < 6 s) has a same origination as that of the decaying phase of the late bump, +i.e., they all stem from the external-forward shock. In addition, the two γ-ray pulses in the period of ∼ [6, 26] s should +reflect the re-activity of the central engine of GRB 200829A. +Then, we suggest that the central engine of GRB 200829A may be intermittent and launch several episode of ejecta +separated by a long quiescent interval (Lin et al. 2018). +The very early phase of the prompt emission originates +from the external shock, which is formed during the propagation of the first launched ejecta in the circum-burst +medium. The later launched ejecta, of which the internal dissipation is responsible for the two γ-ray pulses, collide +with the formed external shock in the period of tobs ∼ [60, 100] s. Then, the energy injection into the external shock +is presented in this period and correspondingly a rising phase appears in the period of tobs ∼ [60, 100] s. Based on the +above paradigm, we fit the very early prompt emission and the late bump with an external-forward shock in the ISM +(see Appendix B for detail modeling), of which the free parameters are the isotropic kinetic energy Ek,0, the initial +Lorentz factor Γ0, the fraction of shock energy to electron energy ǫe, the fraction of shock energy to magnetic field +energy ǫB, the interstellar medium density n0, the jet opening angle θj, and δ. Here, the energy injection rate of the +external-forward shock in the period of [ts, te] = [20, 100] s is described as dEinj/dtobs = Ek,0δ/(te − ts) with δ being a +free parameter in out fitting. In our fitting, a Markov Chain Monte Carlo method based on the emcee Python package +(Foreman-Mackey et al. 2013) is adopted to search for the best-fit parameter set. The optimal result is shown in the +left panel of Figure 1 with wine line for X-ray data and blue line for optical data, and the obtained parameters at the +1σ confidence level are log10 Ek,0 = 53.65+0.07 +−0.07 erg, log10 Γ0 = 3.17+0.05 +−0.01, log10 ǫe = −0.31+0.01 +−0.01, log10 ǫB = −5.15+0.17 +−0.19, +log10 n0 = 1.27+0.19 +−0.18 cm−3, p = 2.001+0.002 +−0.001, θj = 0.09+0.01 +−0.01, log10 δ = 0.81+0.04 +−0.03. The corresponding posterior probability +density functions for the physical parameters are presented in Figure 5. From the left panel of Figure 1, one can find +that the external-forward shock with a refreshed phase can well describe both the very early prompt emission and the +late bump in the afterglows for GRB 200829A. +4. SUMMARY AND DISCUSSIONES +Observationally, GRB 200829A appears with a weak γ-ray emission in the very early phase, followed by a small +γ-ray pulse at around 6 s and a bright spiky γ-ray pulse at around 20 s after the Fermi trigger. After the bright +spiky γ-ray pulse, a smooth bump in the X-ray bands appears. We perform detail spectral analysis on the very early +prompt emission and the bright spiky γ-ray pulse. It reveals that the very early prompt emission can be well fitted +by a power-law spectral model with index ∼ −1.7. However, the bright spiky γ-ray pulse, especially the time around +the pulse peak, exhibits a distinct two-component, i.e., Band function combined with a blackbody radiation spectrum. +This indicate that the origination of the very early prompt emission and the bright spiky γ-ray pulse may be different. +The power-law spectral index of the very early prompt emission is almost the same as that of the normal decay phase +in the X-ray smooth bump, which is suggested to be originated from the external-forward shock. Then, we suggest +that the central engine of GRB 200829A may be intermittent and launch several episode of ejecta separated by a long +quiescent interval. The very early phase of the prompt emission originates from the external shock, which is formed +during the propagation of the first launched ejecta in the circum-burst medium. The later launched ejecta, of which +the internal dissipation is responsible for the two γ-ray pulses, collide with the formed external shock in the period of +tobs ∼ [60, 100] s. Then, the energy injection into the external shock is presented in this period and correspondingly +a rising phase appears in the period of tobs ∼ [60, 100] s. Based on the above paradigm, we fit the very early prompt +emission and the late bump with an external-forward shock in the ISM based on Markov Chain Monte Carlo method. +It is shown that the light-curves of the very early prompt emission, X-ray afterglow after 40 s involving the X-ray +bump at around 100 s, and the later optical afterglow can be well modelled in the above paradigm. +We also perform detail study on the jet producing the bright spiky γ-ray pulse. Based on the blackbody radiation +component found in this pulse, the magnetization of the jet at the photosphere is estimated to be ∼ 4 if the initial +size of the fireball r0 = 109 cm is adopted. Then, the non-thermal component in the bright spiky γ-rays, i.e., Band +component, seems to be formed during the dissipation of the magnetic energy. This may lead to a high radiation +efficiency of the jet. With the energy injection in the period of [20, 100] s, the radiation efficiency of the bright spiky +γ-ray pulse is estimated as ηγ = Eγ/(Eγ +Einj) ∼ 52.1%, where Einj = dEinj/dtobs ×(te −ts) and Eγ ≈ 1.41×1054 erg +is the isotropic energy of the bright spiky γ-ray pulse. The obtained high value of radiation efficiency is consistent +with the scenario that the non-thermal component in this pulse is formed during the dissipation of the magnetic +energy in the jet. Besides, the Lorentz factor of the jet at the photosphere is estimated to be around 500 (400) if + +6 +r0 = 108 cm (r0 = 109 cm) is adopted. The Lorentz factor of the jet can also be estimated as follows. The distance of +the jet dissipation location rdis relative to the central engine of the burst and the Lorentz factor Γdis of the dissipation +region may be related to the pulse duration ∆tpulse as ∆tpulse = Rdis/(2Γ2 +disc) ∼ 4 s (full-width at half maximum). +In addition, the dissipation location should be less than the location of the external shock at the same observer time, +i.e., Rdis ≲ Res,20 s ∼ 4 × 1016 cm, where Res,20 s is the location of the external shock at the observer time 20 s +and obtained based on the initial fireball (without energy injection) and Equations (B1)-(B5). Then, one can have +Γdis ≲ 408. Interesting, the Lorentz factor of the jet producing the bright spiky γ-ray pulse can be estimated based +on the blackbody radiation component. We find that the Lorentz factor of the jet is consistent with that estimated +based on the blackbody radiation component in the bright spiky γ-ray pulse. Please see the left panel of Figure 4, +where Γph ∼ 400 is obtained if r0 = 109 cm is adopted. +The magnetization of the outflow would affect its photospheric emission (e.g., Zhang & Pe’er 2009; Gao & Zhang +2015). Since the emission of the initial fireball, involving the photospheric emission, missed in the observation, the +magnetization of the initial fireball would be high. In the spirit of Zhang & Pe’er (2009), the outflow with magnetization +σ ≳ 125 (σ ≳ 162) is required if r0 = 108 cm (r0 = 109 cm) is adopted. Here, the luminosity of the initial fireball +Lw is estimated as Lw ∼ Ek,0/2.5 s. Corresponding, the related photosphere emission is plotted in the right panel of +Figure 5, where the observed power-law radiation spectrum in the period of tobs ∼ [0, 5] s is shown with a black solid. +ACKNOWLEDGMENTS +We thank the anonymous referee of this work for useful comments and suggestions that improved the paper. +We acknowledge the use of the Fermi archive’s public data. +We appreciate Xing Yang for his help in this work. +This work is supported by the National Natural Science Foundation of China (grant Nos. +12273005, 11673006, +U1938116, U1938201, U1731239, and U1938106), the Guangxi Science Foundation (grant Nos. 2018GXNSFFA281010, +2017AD22006, 2018GXNSFGA281007, and 2018GXNSFDA281033), and China Manned Spaced Project (CMS-CSST- +2021-B11). + +7 +10 +4 +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +-10 +10 +-8 +10 +-6 +10 +-4 +10 +-2 +0 +10 +20 +30 +40 +5000 +10000 +15000 +GRB 200829A +GBM n4 + + +Counts +BAT 10keV +XRT 10keV +XRT 10keV + +X-ray Flux density [Jy] +Time Since Fermi Trigger (t +obs +) [s] +10 +0 +10 +2 +10 +4 +10 +6 +10 +8 +R-GCN +R-GCN +Optical Flux density [ +Jy] + + +Figure 1. Left panel—light-curves of GRB 200829A from prompt emission to its afterglows and the BAT/XRT data are the +flux density at 10 keV extrapolated from BAT/XRT observation, where the inset of the upper-right panel shows the prompt +γ-rays in the linear spaces. The MCMC fitting result based on the model in Appendix B is shown with wine line and blue line for +X-ray and optical data, respectively. Here, the data showed with gray “×” and “+” symbols are not used in our fittings. Right +panel— GBM light-curve of GRB 200829A without background subtracted (upper panel) and the signal significance (bottom +panel), where the background were estimated by fitting the light-curve before and after the burst with polynomial model. It +reveals that there is significantly photons in the period of [0, 10] s from GRB 200829A. +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +10 +100 +1000 +10000 + 0-5 s + 5-10 s + 16-26 s + 16-17 s + 17-18 s + 18-19 s + 19-20 s + 20-21 s + 21-22 s + 22-23 s + 23-24 s + 24-25 s + 25-26 s + + +E +p + [keV] +Low-energy Index, + +Figure 2. Spectral fitting results of the very early prompt emission (tobs ∈ [0, 5] s) in GRB 200829A. Here, the joint spectral +fitting by combining Swift-BAT and Fermi-GBM observations based on the Band function (left panel) or power-law function +(middle panel) are performed. In addition, the relation of Ep and α based on the spectral fitting results with Band function +are plotted in right panel with “⋆” symbols, where the blue symbols are from Poolakkil et al. (2021). Here, the different green +hollow symbols are the time-resolved spectral fitting results in the period of [16, 26] s. + +bn200829582 0.000-5.000 s +100 +正 +10 +Spectral fitting with PL model + s-1 keV-1 +0.1 +0.01 +Photons cm-2 +10-3 +10-4 +10-5 +10-5 +10-7 + sign(data-model) × A Total Statistic +10-8 +10 +0 +10 +100 +1000 +104 +105 +Energy (keV)bn2008295820.000-5.000s +100 +10 +Spectral fitting with Band function +s-1 keV-1 +0.1 +0.01 +cm-2 +10-3 +L +Photons +10-4 +10-5 +10-6 +10-7 + Statis tic +10-8 + sign(data-model) × A Total : +10 +L +10 +100 +1000 +104 +Energy (keV) +1012000 +10000 +8000 +6000 +4000 - +2000 +10 +8- +6 - +4. +2 +0 +-2 +-100 +-50 +0 +50 +100 +Time since triggel8 +0.1 +1 +10 +100 +1000 +104 +105 +keV2 (Photons cm−2 s−1 keV−1) +GRB 200829A t:18.000−19.000 s +10 +100 +1000 +104 +105 +−10 +−5 +0 +5 +10 +(data−model)/error +Energy (keV) +0.1 +1 +10 +100 +1000 +104 +105 +keV2 (Photons cm−2 s−1 keV−1) +GRB 200829A t:18.000−19.000 s +10 +100 +1000 +104 +105 +−10 +0 +10 +(data−model)/error +Energy (keV) +0.1 +1 +10 +100 +1000 +104 +105 +keV2 (Photons cm−2 s−1 keV−1) +GRB 200829A t:19.000−20.000 s +10 +100 +1000 +104 +105 +−4 +−2 +0 +2 +4 +(data−model)/error +Energy (keV) +0.1 +1 +10 +100 +1000 +104 +105 +keV2 (Photons cm−2 s−1 keV−1) +GRB 200829A t:19.000−20.000 s +10 +100 +1000 +104 +105 +−4 +−2 +0 +2 +4 +(data−model)/error +Energy (keV) +0.1 +1 +10 +100 +1000 +104 +105 +keV2 (Photons cm−2 s−1 keV−1) +GRB 200829A t:20.000−21.000 s +10 +100 +1000 +104 +105 +−4 +−2 +0 +2 +4 +(data−model)/error +Energy (keV) +0.1 +1 +10 +100 +1000 +104 +105 +keV2 (Photons cm−2 s−1 keV−1) +GRB 200829A t:20.000−21.000 s +10 +100 +1000 +104 +105 +−4 +−2 +0 +2 +(data−model)/error +Energy (keV) +0.1 +1 +10 +100 +1000 +104 +105 +keV2 (Photons cm−2 s−1 keV−1) +GRB 200829A t:21.000−22.000 s +10 +100 +1000 +104 +105 +−4 +−2 +0 +2 +4 +(data−model)/error +Energy (keV) +0.1 +1 +10 +100 +1000 +104 +105 +keV2 (Photons cm−2 s−1 keV−1) +GRB 200829A t:21.000−22.000 s +10 +100 +1000 +104 +105 +−4 +−2 +0 +2 +4 +(data−model)/error +Energy (keV) +Figure 3. Spectral fitting results of the bright spiky γ-ray pulse in the period of tobs ∈ [18, 22] s based on Band function (left +panel) or Band+BB model (right panel). + +9 +18 +19 +20 +21 +22 +10 +10 +10 +11 +10 +12 +10 +13 + + r +ph + (r +0 +=10^8 cm) + r +ph + (r +0 +=10^9 cm) +r +ph +(cm) +Time Since Fermi Trigger [s] +0 +200 +400 +600 +800 +1000 +1200 + +ph + (r +0 +=10^8 cm) + +ph + (r +0 +=10^9 cm) +ph +18 +19 +20 +21 +22 +50 +100 +150 +200 +250 +300 + +(r +0 +=10^8 cm) + +(r +0 +=10^9 cm) + +Time Since Fermi Trigger [s] +0 +4 +8 +12 +16 +20 +24 +28 + 1+ +ph + (r +0 +=10^8 cm) + 1+ +ph +(r +0 +=10^9 cm) + 1+ +(r +0 +=10^8 cm) + 1+ +(r +0 +=10^9 cm) +1+ +ph + , 1+ +10 +1 +10 +2 +10 +3 +10 +4 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 + PL + r +0 +=10^8 cm + r +0 +=10^9 cm + + +keV(photons cm +-2 +s +-1 +keV +-1 +) +Time Since Fermi Trigger [s] +Figure 4. Left and middle panels—Temporal evolution of derived properties (rph, Γph, η, 1 + σph, and 1 + σ0) based on the +blackbody radiation component found in the bright spiky γ-ray pulse. Right panel—Power-law radiation spectrum found in the +period of tobs ∈ [0, 5] s (solid line) and the predicted lower limits of the photospheric emission (magenta and purple solid lines) +for different parameters. + +10 +Figure 5. Posterior probability density functions for the physical parameters of the external-forward shock in GRB 200829A +from MCMC simulations. + +11 +Table 1. Spectral fitting results of the very early prompt emission in GRB 200829A. +Time interval (s) +Model +α (or ˆΓ) a +β +E0(keV) +N0b +χ2 +r +[0, 5] +PL +−1.79 ± 0.06 +- +- +21.59 ± 5.98 +1.08 +[0, 5] +Band +−1.75 ± 0.09 +−2.42 ± 5.08 +9976.67 ± 51113.36 +0.006 ± 0.0006 +1.08 +[5, 10] +Band +−0.17 ± 0.79 +−2.25 ± 0.27 +54.94 ± 41.55 +0.05 ± 0.06 +0.99 +aThe photon spectral index ˆΓ is for PL model and α is for Band function model. +b N0 is in unit of photons · cm−2 · s−1 · keV−1. + +12 +Table 2. Spectral fitting results of the bright spicky γ-ray pulse in GRB 200829A. +Time interval (s) +Band +Band + BB +α +E0(keV) +β +N0a +BIC +α +E0(keV) +β +N0a +kT(keV) +Ka +BIC +∆BICb +[16, 26] +-0.47 ± 0.01 +231.41 ± 4.11 +-2.47 ± 0.02 +0.41 ± 0.00 +948.83 +-0.52 ± 0.02 +286.22 ± 9.53 +-2.56 ± 0.02 +0.32 ± 0.01 +32.82 ± 1.68 +16.34 ± 1.74 +841.76 +107.07 +[16, 17] +-0.53 ± 0.18 +225.71 ± 60.30 +-2.29 ± 0.19 +0.06 ± 0.01 +510.41 +-0.80 ± 0.20 +599.36 ± 420.69 +-3.40 ± 2.15 +0.02 ± 0.00 +35.86 ± 6.62 +7.19 ± 1.95 +517.64 +-7.23 +[17, 18] +-0.40 ± 0.04 +283.01 ± 16.61 +-2.83 ± 0.13 +0.23 ± 0.01 +550.50 +-0.44 ± 0.07 +350.38 ± 34.98 +-3.15 ± 0.24 +0.18 ± 0.01 +42.30 ± 5.88 +17.22 ± 5.21 +548.53 +1.96 +[18, 19] +-0.25 ± 0.03 +216.73 ± 7.72 +-2.86 ± 0.07 +0.58 ± 0.01 +595.01 +-0.31 ± 0.05 +273.54 ± 17.43 +-3.16 ± 0.12 +0.41 ± 0.02 +40.05 ± 3.53 +39.41 ± 7.56 +572.53 +22.49 +[19, 20] +-0.25 ± 0.03 +221.12 ± 7.34 +-2.32 ± 0.02 +0.92 ± 0.02 +536.65 +-0.36 ± 0.05 +297.34 ± 26.24 +-2.38 ± 0.03 +0.65 ± 0.05 +42.56 ± 3.85 +57.77 ± 14.49 +523.84 +12.81 +[20, 21] +-0.32 ± 0.03 +207.24 ± 6.69 +-2.42 ± 0.02 +1.04 ± 0.02 +658.42 +-0.40 ± 0.05 +264.04 ± 18.63 +-2.50 ± 0.03 +0.76 ± 0.05 +35.26 ± 3.28 +47.93 ± 10.16 +636.37 +22.05 +[21, 22] +-0.41 ± 0.03 +188.50 ± 7.07 +-2.59 ± 0.04 +0.90 ± 0.03 +601.88 +-0.45 ± 0.05 +232.22 ± 15.55 +-2.71 ± 0.06 +0.66 ± 0.04 +27.89 ± 2.56 +33.86 ± 5.88 +576.20 +25.67 +[22, 23] +-0.60 ± 0.06 +139.79 ± 13.33 +-2.31 ± 0.05 +0.41 ± 0.03 +497.37 +-0.84 ± 0.11 +254.12 ± 56.51 +-2.46 ± 0.11 +0.22 ± 0.04 +24.06 ± 3.31 +13.07 ± 4.12 +501.53 +-4.17 +[23, 24] +-1.03 ± 0.09 +195.52 ± 37.43 +-2.45 ± 0.17 +0.14 ± 0.02 +490.10 +-1.20 ± 0.18 +340.63 ± 181.20 +-2.48 ± 0.31 +0.18 ± 0.23 +23.79 ± 3.84 +1.04 ± 9.77 +495.21 +-5.11 +[24, 25] +-0.59 ± 0.23 +80.59 ± 24.47 +-2.23 ± 0.10 +0.22 ± 0.08 +547.41 +-1.35 ± 0.15 +582.80 ± 366.04 +-2.50 ± 1.23 +0.04 ± 0.01 +22.00 ± 2.69 +7.76 ± 1.46 +559.13 +-11.72 +[25, 26] +-0.86 ± 0.30 +102.82 ± 56.82 +-2.19 ± 0.16 +0.09 ± 0.05 +513.84 +-1.19 ± 1.09 +203.83 ± 657.78 +-2.13 ± 0.24 +0.07 ± 0.21 +21.75 ± 10.54 +0.71 ± 4.79 +526.66 +-12.82 +aN0 is in unit of photons · cm−2 · s−1 · keV−1; K is the L39/ D2 +10, where L39 is the source luminosity in units of 1039 erg/s and +D10 is the distance to the source in units of 10 kpc. +b The ∆BIC is the value of BICBand − BICBand+BB. + +13 +Table 3. Results of spectral fits for tobs ∈ [230, 52000] s of GRB 200829A. +GRB +Interval(s) +Band +χ2 +r +ˆΓ +GRB 200829A +230-700 +BAT+XRT +1.00 +−2.05 ± 0.04 +700-2000 +XRT +1.11 +−1.75 ± 0.01 +5116-7428 +XRT +0.94 +−1.76 ± 0.05 +12119-13162 +XRT +1.09 +−1.83 ± 0.06 +28067-52000 +XRT +1.19 +−1.89 ± 0.06 + +14 +APPENDIX +A. DISCUSSION ABOUT THE PROMPT EMISSION OF GRB 200829A IN THE PERIOD OF [0, 5] S +In this section, we present a comprehensive discussion about the radiation spectrum in the prompt emission of +GRB 200829A in the period of [0, 5] s. We would like to conclude that the intrinsic radiation spectrum in this period +may be consistent with a PL spectral model with ˆΓ ∼ −1.7 or a Band function with a break at ∼ 10 MeV and power- +law index ∼ −1.7 in its low-energy regime (E ≲ 10 MeV), rather than a Band function with α ∼ −1, β ∼ −3, and +Ep ∼ 200 keV. This conclusion is made based on the comprehensive comparison between the spectral fitting results +on the observational data and those on the synthetic data of Fermi observation. Here, the synthetic data of Fermi +observation is generated based on the python source package threeML10 (Vianello et al. 2015) and the Band function +with α = −1, β = −3, and Ep = 200 keV is adopted as the intrinsic radiation spectrum to produce synthetic data. In +addition, the signal significance of the synthetic data is set as that of the observational data of GRB 200829A in the +period of [0, 5] s. The spectral fittings in this section are performed based on the MCMC method to produce posterior +predictions for the model parameters11 and the python source package emcee (Foreman-Mackey et al. 2013) is used +for our MCMC sampling. The spectral fitting results are reported in Table 4. +The reasons for our above conclusion are as follows. +1. In the spectral fitting, the values of “Residuals (σ)” (see the bottom part in each panel of Figure 6) provides the +most important information to confront the spectral model with the observed data. A good spectral model for +the observational data should provide a well distribution of “Residuals (σ)”, such as that shown in the bottom +part of the upper-right panel in Figure 6. In Figure 6, the upper-left and upper-right panels show the spectral +fitting results on the synthetic data with a PL model and a Band function, respectively. Since the intrinsic +radiation spectrum of the synthetic data is a Band function with Ep = 200 keV, the spectral fitting on the +synthetic data with a Band function should provide an optimal fitting. Actually, the values of the corresponding +“Residuals (σ)” are indeed well distributed around zero. In the spectral fitting on the synthetic data with a PL +model, however, the values of “Residuals (σ)” appear as positive around Ep and negative below/above ∼ Ep. It +reveals that even though the Band function with α = −1, β = −3, and Ep = 200 keV can be described as a PL +model with ˆΓ ∼ −1.65 (see second line of Table 4), the observational data would exceed the PL model around Ep +and fail to reach the PL model below/above ∼ Ep. This behavior is consistent with the theoretical expectation. +In the bottom-left and bottom-right panels of Figure 6, we show the spectral fitting results on the observational +data of GRB 200829A in the period of [0, 5] s with a PL spectral model and a Band function, respectively. The +spectral fitting results are also reported in the fourth and fifth lines of Table 4. One can find that “Residuals +(σ)” in these two panels are well distributed around zero, which is very similar to that in the upper-right panel. +It implies that the intrinsic radiation spectrum of this period should be consistent with a PL spectral model +with ˆΓ ∼ −1.7 or a Band function with a break at ∼ 10 MeV and power-law index ∼ −1.7 in its low-energy +regime (E ≲ 10 MeV), rather than a Band function with α ∼ −1, β ∼ −3, and Ep ∼ 200 keV. This is because +that if the intrinsic radiation spectrum of the observational data is a Band function with α ∼ −1, β ∼ −3, and +Ep = 200 keV, the values of “Residuals (σ)” would be positive ∼ 200 keV and negative below/above ∼ 200 keV +on average. However, this behavior could not be evidently found in the bottom-left panel of Figure 6. +2. If the intrinsic radiation spectrum in this period is the Band function with E0 ∼ 200 keV, the spectral fitting +results on the low-energy regime, e.g., the energy band of Swift-BAT (15-150 keV), with a PL spectral model +would be very different from that on the energy band of Fermi-GBM instrument (8 keV-40 MeV). Then, we +perform the spectral fittings on the data in the 15-150 keV energy band. The posterior probability density +functions for the physical parameters of the spectral model are shown in Figure 7, where the upper and bottom +panels are the spectral fitting results on the synthetic data and the observational data in the 15-150 keV energy +band, respectively. A PL spectral model and Band function are adopted in the spectral fittings for the left and +10 https://github.com/threeML/threeML +11 This method is different from that used in the main text of the present paper. In the main text, the spectral model parameters are obtained +based on the package Xspec by maximizing the likelihood. However, one can find that the model parameters are consistent with each other +in these two fitting methods. + +15 +right panels, respectively. It is shown that the spectral fittings on the synthetic data with a PL spectral model +for different energy regime are indeed presented very different values of power-law index ˆΓ, i.e., ˆΓ = −1.65+0.04 +−0.04 +for the 8 keV-40 MeV energy band and ˆΓ = −1.44+0.10 +−0.10 for the 15-150 keV energy band. Interestingly, the +spectral fittings on the synthetic data with a Band function almost report the same values of α, β, and E0 for +the 15-150 keV energy band and the 8 keV-40 MeV energy band. According to the fitting results reported in +Table 4, one can find that the spectral fittings on the observational data in the 15-150 keV energy band and +those in the 8 keV-40 MeV energy band are almost presented the same fitting results. Please comparing the +eighth line with the fourth line, or the ninth line with the fifth line in Table 4. It implies that the radiation +spectrum in this period should be consistent with a PL spectral model with ˆΓ ∼ −1.7 or a Band function with +a break at ∼ 10 MeV and power-law index ∼ −1.7 in its low-energy regime (E ≲ 10 MeV), rather than a Band +function with α ∼ −1, β ∼ −3, and Ep ∼ 200 keV. +In summary, by comparing the spectral fitting results on the observational data to those on the synthetic data, +we can conclude that the intrinsic radiation spectrum in this period should be consistent with a PL spectral model +with ˆΓ ∼ −1.7 or a Band function with a break at ∼ 10 MeV and power-law index ∼ −1.7 in its low-energy regime +(E ≲ 10 MeV). +B. MODEL +In this section, the dynamics and the emission of the external-forward shock are presented as follows. The dynamics +of the external-forward shock can be described with the following equations (e.g., Sari et al. 1998; Huang et al. 1999): +dΓ +dtobs += +1 +M ′ +� 1 +c2 +dEinj +dtobs +− (Γ2 − 1) dm +dtobs +� +, +(B1) +dm +dtobs += 4πρR2 dR +dtobs +, +(B2) +dU ′ +dtobs += (1 − ǫ)(Γ − 1)c2 dm +dtobs +, +(B3) +dR +dtobs += +cβ +1 − β (1 + z), +(B4) +β = +� +1 − 1/Γ2, +(B5) +where Γ, dEinj/dtobs, R, ǫ, and cβ are the Lorentz factor, the energy injection rate (with respect to the observer time +tobs), location, the radiation efficiency, and the velocity of the external-forward shock, and M ′ = M ′ +ej + m + U ′/c2 +is the total mass, including the initial mass M ′ +ej = Ek,0/[(Γ0 − 1)c2] of the ejecta, the sweep-up mass m from the +circum-burst medium, and the internal energy U ′ of the shocked material from the external shock. Here, Ek,0 is the +initial isotropic kinetic energy of the fireball, Γ0 = Γ(tobs = 0) is the initial bulk Lorentz factor of the fireball, c is the +velocity of light, z is the redshift of the burst, and ρ is the density of the circum-burst environment. Two cases of +circum-burst medium, i.e., interstellar medium (ISM) and wind, are generally studied. Correspondingly, we take (e.g., +Chevalier & Li 2000) +ρ = +� +5 × 1011A∗R−2 g · cm−1, wind, +n0mp cm−3, +ISM, +(B6) +with mp being the proton mass, A∗ is a dimensionless constant. For simplicity, the energy injection into the external +shock due to the late activity of the central engine is assumed with a constant energy injection rate over the period of +tobs ∈ [ts, te], where ts and te are the beginning and the end of the energy injection, respectively. By describing Einj +as Einj = Ek,0δ, one thus can have dEinj/dtobs = Ek,0δ/(te − ts). +The main radiation mechanism of the external-forward shock in GRBs is the synchrotron radiation of the sweep-up +electrons (Sari et al. 1998; Sari & Piran 1999). ǫe and ǫB are introduced to represent the fractions of the shock energy +used to accelerate electrons and contributing to the magnetic energy, respectively. Then, the magnetic field behind the +shock is B′ = (32πǫBρ/mp)1/2Γc. The sweep-up electrons are accelerated to a power-law distribution of Lorentz factor +γe, i.e., Q ∝ γ′ +e +−p for γ′ +e,min ⩽ γe ⩽ γ′ +e,max, where p(> 2) is the power-law index, γe,min = ǫe(p − 2)mpΓ/[(p − 1)me] +(Sari et al. 1998), and γe,max = +� +9m2ec4/(8B′q3e) with qe being the electron charge (e.g., Kumar et al. 2012). Then, one + +16 +can have ǫ = ǫradǫe with ǫrad = min{1, (γe,min/γe,c)(p−2)} (Fan & Piran 2006), where γe,c = 6πmec(1+z)/(σTΓB′2tobs) +is the efficient cooling Lorentz factor of electrons. +Equations (B1)-(B5) describe the evolution of hydrodynamic blastwave approximately. A more rigorous treatment +can be found in Nava et al. (2013) and Zhang (2018) (see Eq. (8.66) in this book). For our studied burst, the blastwave +is affected by the energy injection and thus its evolution could not be simply estimated with hydrodynamic equations +in Nava et al. (2013) and Zhang (2018). A more complicated equations are required. For the phase without energy +injection, we also present the light curve of afterglows based on the hydrodynamic equations in Nava et al. (2013) and +Zhang (2018). It is found that the obtained light-curves of afterglows are almost the same as those obtained with +Equations (B1)-(B5). + +17 +Figure 6. Fitting results of the synthetic data (upper panels) and the observational data (bottom panels) in the 8 keV-40 MeV +energy band, where a PL spectral model and Band function are adopted in the left and right panels, respectively. + +18 +Figure 7. Posterior probability density functions for the physical parameters of the spectral fitting on the synthetic data +(upper panels) and the observational data (bottom panels) in the 15-150 keV energy band, where a PL spectral model and Band +function are adopted in the left and right panels, respectively. + +19 +Table 4. Spectral fitting results of simulation and observation of [0, 5] s in GRB 200829A. +Model +α (or ˆΓ) +β +E0(keV) +N0 +Data sources +PL +−1.65+0.04 +−0.04 +- +- +31.52+6.14 +−5.32 +synthetic data (8 keV-40 MeV) +Band +−1.16+0.14 +−0.11 +−3.71+0.88 +−0.83 +276.67+91.20 +−71.06 +0.03+0.01 +−0.00 +synthetic data (8 keV-40 MeV) +PL +−1.73+0.08 +−0.09 +- +- +15.65+11.32 +−7.15 +observational data (8 keV-40 MeV) +Band +−1.61+0.10 +−0.11 +−2.94+0.85 +−1.21 +11021.76+13229.91 +−5533.27 +0.00 ± 0.00 +observational data (8 keV-40 MeV) +PL +−1.44+0.10 +−0.10 +- +- +13.49+7.00 +−4.68 +synthetic data (15-150 keV) +Band +−1.09+0.17 +−0.15 +−3.25+1.19 +−1.26 +268.28+141.98 +−110.57 +0.03+0.01 +−0.00 +synthetic data (15-150 keV) +PL +−1.71+0.14 +−0.15 +- +- +19.06+17.86 +−9.79 +observational data (15-150 keV) +Band +−1.62+0.15 +−0.17 +−3.47+1.14 +−0.98 +18813.06+34951.87 +−11705.06 +0.00 ± 0.00 +observational data (15-150 keV) + +20 +REFERENCES +Band, D., Matteson, J., Ford, L., et al. 1993, ApJ, 413, 281 +Beloborodov, A. 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P., Y Fu, S., Liu, X., et al. 2020a, GRB +Coordinates Network, 28330, 1 +—. 2020b, GRB Coordinates Network, 28324, 1 + diff --git a/otAzT4oBgHgl3EQfAfoQ/content/tmp_files/load_file.txt b/otAzT4oBgHgl3EQfAfoQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e0ca3bd27a2d9c6a478977b80b04a330fce5acdd --- /dev/null +++ b/otAzT4oBgHgl3EQfAfoQ/content/tmp_files/load_file.txt @@ -0,0 +1,1172 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf,len=1171 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00925v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='HE] 3 Jan 2023 Draft version January 4, 2023 Typeset using LATEX default style in AASTeX63 GRB 200829A: External Shock Origin of the Very Early Prompt Emission?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Jing Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 Da-Bin Lin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 Rui-Jing Lu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 Lu-Yao Jiang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 3 Wen-Qiang Liang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 Zhi-Lin Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 Xiao-Yan Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 Xiang-Gao Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 and En-Wei Liang1 1Laboratory for Relativistic Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Guangxi University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Nanning 530004,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' China 2Key Laboratory of Dark Matter and Space Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Purple Mountain Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Nanjing 210034,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' China 3School of Astronomy and Space Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' China ABSTRACT Long-duration GRB 200829A was detected by Fermi-GBM and Swift-BAT/XRT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' and then rapidly observed by other ground-based telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It has a weak γ-ray emission in the very early phase and followed by a bright spiky γ-ray emission pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The radiation spectrum of the very early emission is best fitted by a power-law function with index ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' However, the bright spiky γ-ray pulse, especially the time around the peak, exhibits a distinct two-component radiation spectra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Band function combined with a blackbody radiation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We infer the photospheric properties and reveal a medium magnetization at photospheric position by adopting the initial size of the outflow as r0 = 109 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It implies that Band component in this pulse may be formed during the dissipation of magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The power-law radiation spectra found in the very early prompt emission may imply the external-shock origination of this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, we perform Markov Chain Monte Carlo method fitting on the light-curves of this burst, where the jet corresponding to the γ-ray pulses at around 20 s is used to refresh the external-shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It is shown that the light-curves of very early phase and X-ray afterglow after 40 s, involving the X-ray bump at around 100 s, can be well modelled in the external-shock scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' For the obtained initial outflow, we estimate the minimum magnetization factor of the jet based on the fact that the photospheric emission of this jet is missed in the very early phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Keywords: Gamma-ray bursts (629) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' INTRODUCTION Theoretically, it is generally believed that gamma-ray bursts (GRBs) originated from collapse of massive stars or mergers of double compact stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Colgate 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Paczynski 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Eichler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Woosley 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' MacFadyen & Woosley 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Piran 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Zhang & M´esz´aros 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Woosley & Bloom 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Kumar & Zhang 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Observationally, GRBs generally appear as a brief and intense γ-rays followed by a long-lived afterglow emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The prompt γ-rays are highly variable with a duration from millisecond to thousands of seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The observational spectra are usually well fitted by an empirical function, characterized by a smoothly joint broken power-law function, the so-called Band function (Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 1993) or a quasi-thermal spectral component appear in the spectra of some GRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The previous observations demonstrated that thermal components exhibit different observational properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' They either can be detected during the entire duration of the prompt emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Ghirlanda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2013) or may be only found at the beginning of the burst duration, and subsequently appear with a nonthermal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The detection of a diversified spectral characteristic shows that GRB ejecta may have a diverse jet composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It may be neither fully matter-dominated ejecta nor fully magnetized outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' More realistically, GRB outflows are likely to be a hybrid jet, which carries the two components simultaneously and launches at the central engine (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Gao & Zhang 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The light-curves of afterglow emission usually can be decomposed into four power-law segments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', an initial steep decay, a shallow decay, a normal decay, and a late steeper decay, sometimes accompanied by one or several Corresponding author: Da-Bin Lin, Rui-Jing Lu lindabin@gxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='cn, luruijing@gxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='cn 2 flares (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Nousek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It is commonly believed that the multi-wavelength afterglow is mainly from the external shock, which is formed during a relativistic jet propagating in the circum-burst medium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', M´esz´aros & Rees 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' However, the origin of the prompt γ-rays is not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The prompt γ-rays may be from the internal shock in an erratic relativistic fireball, a dissipative photosphere, a Poynting-flux dominated jet, or even an external shock (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Rees & Meszaros 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Meszaros & Rees 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Rees & Meszaros 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Giannios 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Beloborodov 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Vurm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Zhang & Yan 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It is not a new idea that the prompt γ-rays of GRBs originate from the external shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2016) have shown that the prompt emission of GRB 141028A is very likely originated from an external shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2018) suggested that GRB 120729A is an external shock origin for both the prompt γ-ray emission and afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' They also systematically investigate single pulse GRBs in the Swift’s GRBs, and find that a small fraction of GRBs (GRBs 120729A, 051111, and 070318) are likely to originate from an external shock for both the prompt γ-ray emission and afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' However, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2018) focuses on the bursts appearing as a single pulse from the prompt emission to its afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In fact, the central engine of GRBs may re-activity and launch relativistic ejecta several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The late launched ejecta may be observed as flares in the afterglow and interact with the external shock at a later period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The burst GRB 200829A maybe in the above scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' GRB 200829A was detected by Fermi-GBM and Swift- BAT/XRT, and the light-curve of prompt emission is composed of an initial very early weak emission (with a duration ∼ 5 s) followed by a bright spiky γ-ray pulse with a duration ∼ 10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We find that the spectra in the γ-ray pulse of GRB 200829A exhibits a distinct two-component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Band function combined with a blackbody radiation spectrum, especially in the peak time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It means that the thermal component should be indeed existence, and GRB 200829A outflows are likely to be a hybrid jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' What’s more, the radiation spectrum in its very early phase can be fitted with power-law spectral model with index ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7, which may be an indication of the origin of an external-forward shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The central engine of GRB 200829A may re-activity and launch jets at different times, resulting in the bright spiky γ-ray pulses when jets collide with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In this paper, we present a detailed analysis of γ-rays and X-ray emission from the long GRB 200829A detected by Fermi and Swift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In Section 2, we introduce the observations and light-curves features of GRB 200829A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In Section 3, the detailed analysis and results of GRB 200829A are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In this section, we also analyzed the other properties of GRB 200829A in different phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In Section 4, the summary and discussions are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' OBSERVATIONS AND DATA REDUCTION The long GRB 200829A was first detected by Fermi Gamma-Ray Burst Monitor (GBM) at 13 : 58 : 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='66 UT (T0) on 2020 August 29 with duration T90 ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='9 s (Lesage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In addition to the Fermi-GBM, Swift-BAT triggered the burst at 13 : 59 : 34 UT on 2020 August 29 (Palmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2020) and Swift-XRT began to observe the burst at 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 s after the BAT trigger (Gropp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Oates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020) created a SED at 900 s after the BAT trigger and found a photometric redshift of z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='02 for this burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The optical afterglow is detected on first two days after the GRB trigger (Pozanenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In the left panels of Figure 1, we show the light-curves of prompt γ-rays and afterglows of GRB 200829A with respect to the Fermi trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The inset in the upper part of this panel shows the light-curves of prompt emission based on the Fermi observation in the linear spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Here, the Fermi data are from the Fermi Science Support Center1 and a GBM light-curve and source spectra are extracted from the TTE (Time-Tagged-Events) data by using a python source package named gtBurst2, the BAT/XRT data are taken from the UK Swift Science Data Center3, and the optical data of GRB 200829A are from Siegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Pozanenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Lipunov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Kuin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Lipunov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Hentunen & Nissinen (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Moskvitin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Moskvitin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Pankov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Izzo (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Volnova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' De Pasquale (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Pozanenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Based on the light-curves in the left panels of Figure 1, one can find that the prompt γ-rays is dominated by a bright spiky γ-ray pulses in the period of tobs ∼ [15, 30] s based on GBM observation, which is preceded by a small γ-ray pulse in the period of tobs ∼ [6, 10] s based on BAT observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' However, it should be noted that the small γ-ray pulse in the period of tobs ∼ [6, 10] s is not significantly in the light-curve of GBM observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Except these two γ-ray episodes, there is a significant γ-ray emission in the very early phase of the prompt emission (tobs < 6 s) 1 https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='gov/ssc/data/access/ 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='com/giacomov/gtburst 3 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='swift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='uk/burst analyser/00993768/ 3 based on BAT observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This can also be found in the right panels of Figure 1, which shows the GBM light-curve of GRB 200829A without background subtracted (upper panel) and the signal significance (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' One can find that the signal significance in the period of ∼ [0, 10] s is higher than ∼ 4σ, which reveal a significant γ-ray photons in this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In the following section, we present the detailed studies on the spectra and the corresponding physical implications for the very early phase and the bright spiky γ-ray pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' DETAILED ANALYSIS OF GRB 200829A AND RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Very early prompt gamma-ray emission For the very early phase of the prompt emission, the spectral fitting with Band function4 reports the values of α = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='09, E0 = 9976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='67 ± 51113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='36, and β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='42 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='08 (see the third line of Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The values of E0 and β could not be well constrained from the spectral fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, we perform the spectral analysis of the very early phase with the power-law (PL) function5 or cutoff power-law (CPL) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Here, the spectral fitting with PL function reports the power-law index ˆΓ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 (see the second line of Table 1), and the spectral fitting with CPL function could not present a well fitting and the corresponding result is not reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The spectral fitting results for the very early prompt emission with Band function (left panel) and PL function (middle panel) are also shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We note that the values of α = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='09 and ˆΓ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 from the spectral fittings are almost the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Here, the value of α can be well constrained in the spectral fitting with Band function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This fact may imply that the intrinsic radiation spectrum in this period may be consistent with a PL spectral model with ˆΓ ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 or a Band function with a break at ∼ 10 MeV and power-law index ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 in its low-energy regime (E ≲ 10 MeV)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The reasons are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Firstly, the spectral fitting on such kind of intrinsic radiation spectrum with a Band function would not provide a well constraint on the value of E0 and thus β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This is consistent with our spectral fitting result for this period based on a Band function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In addition, a Band function with a break at ∼ 10 MeV, the power- law index ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 in its low-energy regime (E ≲ 10 MeV), and the power-law index ≳ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='5 in its high-energy regime (E ≳ 10 MeV) can be modelled with a PL function and ˆΓ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='79 in Fermi-GBM energy band (8 keV-40 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This is also consistent with our spectral fitting result for this period based on a PL function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Secondly and importantly, we perform the spectral fitting on the Swift-BAT observation for this period based on a PL model and the value of ˆΓ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='71 is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We note that such kind of intrinsic radiation spectrum is very different from the general Band radiation component of GRBs’ prompt emission, of which the value of α is around −1 and the break energy E0 is around 400 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The right panel of Figure 2 shows the relation of Ep and α based on the spectral fitting results with a Band function, where the blue symbols are from the figure 8 of Poolakkil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2021) and represent the GOOD sample for time-integrated spectral fits with Band function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In this panel, the spectral analysis result for the very early phase of the prompt emission based on Band function is also tentatively shown with pink “⋆” even though the value of E0 could not be well constrained, and the spectral fitting results of the small γ-ray pulse ([5, 10] s) or the bright spiky γ-ray pulses ([16, 26] s) with Band function are also showed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' One can find that such kind of radiation spectrum is very different from the general Band radiation component of GRBs’ prompt emission, involved that of the bright spiky γ-ray pulses or the small γ-ray pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, we would like to believe that the very early phase of the prompt emission in this burst may be originated from the other channel rather than that for the bright spiky γ-ray pulses or the small γ-ray pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Bright spiky γ-ray pulse and deriving physical parameters There is a bright spiky γ-ray pulse appearing at tobs ∼ [16, 26] s after the Fermi trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In order to perform detailed analysis of this pulse, we divide this pulse into several time intervals with 1 s time span and perform the spectral fitting on these time intervals with Band function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The spectral fitting results are reported in Table 2 and shown in the left panels of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' A distinct multi-component of radiation spectrum is found in several time intervals of this pulse, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', [18, 19] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, we also perform the spectral analysis together with Band function and a blackbody radiation component (BB) 7, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', “Band+BB”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The spectral fitting results based on Band+BB model are also reported in Table 2 and shown in the right panels of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We also estimate the Bayesian Information Criterion (BIC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Schwarz 4 Band function is described as N(E) = N0(E/100keV)α exp(−E/E0) for E ≤ (α − β)E0 and N(E) = N0[(α − β)E0/100keV]α−β exp(β − α)(E/100keV)β for E ≥ (α − β)E0, where N0 is the normalization, and α, β, and E0 are parameters in the spectral fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The peak photon energy of E2N(E) is Ep = (α + 2)E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 5 The PL function is described as N(E) = N0(E/1keV)ˆΓ with ˆΓ being the photon spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 6 Please see Appendix A for a comprehensive analysis about the radiation spectrum in this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 7 NBB(E) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='0525×KE2 (kT )4(e(E/kT )−1) , where kT is the blackbody temperature keV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' K is the L39/ D2 10, where L39 is the source luminosity in units of 1039 erg/s and D10 is the distance to the source in units of 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 4 1978) for the spectral fitting with Band function and that with Band+BB model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The values of BIC from the spectral analysis are also reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The BIC is adopted to evaluate the goodness of the model fitting, taking into account the model complexity and the different numbers of free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Generally, the model with a lowest BIC is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' By comparing the values of BIC from the spectral analysis, one can find that the Band+BB model is preferred for the radiation spectrum of the time intervals around the peak of the bright spiky γ-ray pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Since the value of ∆BIC=BICBand − BICBand+BB is in the range of 12-25, it is strong to support a blackbody component in these time intervals8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The temperature and flux of the blackbody component, together with the radius of the jet base (size of the central engine) r0 and z, can provide useful information about the physics of the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Meanwhile, due to the presence of Band energy spectrum component, the jet compositions of GRB 200829A maybe hybrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Therefore, following Gao & Zhang (2015), we estimate the radius and Lorentz factor of the photosphere based on the blackbody component found in the period of [18,22] s by assuming the hybrid outflow of GRB 200829A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In the calculations, we assume that there is no dissipation below the photosphere and the radiation efficiency ∼ 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1% (please see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The results are shown in the left panels of Figure 4, the blue and olive symbols are the physical quantities calculated based on r0 = 108 cm and r0 = 109 cm, solid and hollow “⋆” represent the physical parameters rph, Γph, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It indicates that the values of rph increases with time and Γph remains constant for low value of r0 and when r0 is large, it increases and eventually declines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We also infer the dimensionless entropy η and the magnetization factor σ, where σ0 and σph are the magnetization factor of the outflow at r0 and rph, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The results are shown in the middle panels of Figure 4, the blue and olive symbols are the same as those in the left panels of Figure 4, and solid and hollow “⋆” represent the physical parameters η, 1 + σph, and 1 + σ0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It is shown that the dimensionless entropy η fluctuates in the range of 100 to 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In addition, the values of 1 + σph can be around 5 if r0 = 109 cm is adopted and around 1 if r0 = 108 cm is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Together with the Band and BB components found in this burst, the initial radius of the outflow producing the bright spiky γ-rays should be around or larger than 109 cm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', r0 ≳ 109 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This result is consistent with that found in GRBs with identified photospheric emission, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', GRB 120323A, GRB 131014A and GRB 220426A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Guiriec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2013, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The non-thermal component in the bright spiky γ-rays, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Band component, seem to be formed during the dissipation of the magnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Afterglow analysis and a self-consistent Paradigm for bursting Following the prompt γ-ray emission in this burst, a late bump appears at tobs > 40 s with a rising in the period of tobs ∼ [40, 100] s and a decaying after tobs ∼ 100 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It is reasonable to believe that the decaying phase of the late bump is the normal decay of the external-forward shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' For the X-ray emission in this phase, the closure relation (Zhang & M´esz´aros 2004) of α ≈ 3β/2 with F ∝ ν−βt−α can be found, where the value of α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='03 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='05 are obtained based on the observations of Swift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It reveals that the X-ray emission in this phase is in the spectral regime of νm < ν < νc for an external-forward shock in the interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The very early phase of the prompt emission may be originated from the external shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The reasons are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Firstly, we have performed a joint spectral analysis by combining the observations of Swift-BAT and Fermi-GBM for the very early phase of the prompt emission in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The spectral analysis reveals that the very early phase of the prompt emission in this burst may be originated from the other channel rather than that for the small γ-ray pulse or the bright spiky γ-ray pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Secondly, the radiation spectrum in this phase is strongly reminiscent of the GRB 120729A, of which the radiation spectrum in the prompt emission for Fermi-GBM energy band can be well modelled with a PL function and photon spectral index ˆΓ ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='479(Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Since the light-curve of the prompt emission in GRB 120729A appears as a single long and smooth pulse, which extends continuously to the X-rays, it is suggested that both the prompt emission and the afterglows are originated from an external-forward shock (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Thirdly, the spectral index of the very early prompt emission based on Swift-BAT and Fermi-GBM observations is almost the same as that of the decaying phase in the late bump based on the Swift-XRT observation (see Table 1 and Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This is different from that in GRB 120729A, of which the spectral index in the X-ray energy band evolves from -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='47 in the early phase of the prompt emission to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='83 in the late phase of afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It may reveal that the X-rays may in the same spectral regime in GRB 200829A but in different spectral regime in 8 In the spirit of Burnham & Anderson (2004), the value of ∆BIC can be used as the strength of the evidence to allow a quick comparison and ranking of candidate hypotheses or models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' For ∆BIC = BICA −BICB with BICA > BICB, the strength of the evidence can be summarized as follows: the situation with ∆BIC ⩽ 2 provides no evidence against the model-A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' the situation with 4 ⩽ ∆BIC ⩽ 7 provides positive evidence against the model-A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' the situation with ∆BIC ⩾ 10 provides very strong evidence against the model-A (Burnham & Anderson 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 9 By performing joint spectral fitting of the Swift-BAT and Fermi-GBM observations for GRB 120729A, we obtain ˆΓ ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='47 and ˆΓ ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='49 for the period of [0, 10] s and [1, 2] s after the Fermi trigger, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 5 GRB 120729A for the very early prompt emission and the late phase of afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, we would like to believe that the early phase of prompt emission (tobs < 6 s) has a same origination as that of the decaying phase of the late bump, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', they all stem from the external-forward shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In addition, the two γ-ray pulses in the period of ∼ [6, 26] s should reflect the re-activity of the central engine of GRB 200829A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, we suggest that the central engine of GRB 200829A may be intermittent and launch several episode of ejecta separated by a long quiescent interval (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The very early phase of the prompt emission originates from the external shock, which is formed during the propagation of the first launched ejecta in the circum-burst medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The later launched ejecta, of which the internal dissipation is responsible for the two γ-ray pulses, collide with the formed external shock in the period of tobs ∼ [60, 100] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, the energy injection into the external shock is presented in this period and correspondingly a rising phase appears in the period of tobs ∼ [60, 100] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Based on the above paradigm, we fit the very early prompt emission and the late bump with an external-forward shock in the ISM (see Appendix B for detail modeling), of which the free parameters are the isotropic kinetic energy Ek,0, the initial Lorentz factor Γ0, the fraction of shock energy to electron energy ǫe, the fraction of shock energy to magnetic field energy ǫB, the interstellar medium density n0, the jet opening angle θj, and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Here, the energy injection rate of the external-forward shock in the period of [ts, te] = [20, 100] s is described as dEinj/dtobs = Ek,0δ/(te − ts) with δ being a free parameter in out fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In our fitting, a Markov Chain Monte Carlo method based on the emcee Python package (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2013) is adopted to search for the best-fit parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The optimal result is shown in the left panel of Figure 1 with wine line for X-ray data and blue line for optical data, and the obtained parameters at the 1σ confidence level are log10 Ek,0 = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='07 erg, log10 Γ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='17+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01, log10 ǫe = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='31+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01, log10 ǫB = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='19, log10 n0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='27+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='18 cm−3, p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='001+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='002 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='001, θj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01, log10 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='81+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The corresponding posterior probability density functions for the physical parameters are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' From the left panel of Figure 1, one can find that the external-forward shock with a refreshed phase can well describe both the very early prompt emission and the late bump in the afterglows for GRB 200829A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' SUMMARY AND DISCUSSIONES Observationally, GRB 200829A appears with a weak γ-ray emission in the very early phase, followed by a small γ-ray pulse at around 6 s and a bright spiky γ-ray pulse at around 20 s after the Fermi trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' After the bright spiky γ-ray pulse, a smooth bump in the X-ray bands appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We perform detail spectral analysis on the very early prompt emission and the bright spiky γ-ray pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It reveals that the very early prompt emission can be well fitted by a power-law spectral model with index ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' However, the bright spiky γ-ray pulse, especially the time around the pulse peak, exhibits a distinct two-component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Band function combined with a blackbody radiation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This indicate that the origination of the very early prompt emission and the bright spiky γ-ray pulse may be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The power-law spectral index of the very early prompt emission is almost the same as that of the normal decay phase in the X-ray smooth bump, which is suggested to be originated from the external-forward shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, we suggest that the central engine of GRB 200829A may be intermittent and launch several episode of ejecta separated by a long quiescent interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The very early phase of the prompt emission originates from the external shock, which is formed during the propagation of the first launched ejecta in the circum-burst medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The later launched ejecta, of which the internal dissipation is responsible for the two γ-ray pulses, collide with the formed external shock in the period of tobs ∼ [60, 100] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, the energy injection into the external shock is presented in this period and correspondingly a rising phase appears in the period of tobs ∼ [60, 100] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Based on the above paradigm, we fit the very early prompt emission and the late bump with an external-forward shock in the ISM based on Markov Chain Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It is shown that the light-curves of the very early prompt emission, X-ray afterglow after 40 s involving the X-ray bump at around 100 s, and the later optical afterglow can be well modelled in the above paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We also perform detail study on the jet producing the bright spiky γ-ray pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Based on the blackbody radiation component found in this pulse, the magnetization of the jet at the photosphere is estimated to be ∼ 4 if the initial size of the fireball r0 = 109 cm is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, the non-thermal component in the bright spiky γ-rays, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Band component, seems to be formed during the dissipation of the magnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This may lead to a high radiation efficiency of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' With the energy injection in the period of [20, 100] s, the radiation efficiency of the bright spiky γ-ray pulse is estimated as ηγ = Eγ/(Eγ +Einj) ∼ 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1%, where Einj = dEinj/dtobs ×(te −ts) and Eγ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='41×1054 erg is the isotropic energy of the bright spiky γ-ray pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The obtained high value of radiation efficiency is consistent with the scenario that the non-thermal component in this pulse is formed during the dissipation of the magnetic energy in the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Besides, the Lorentz factor of the jet at the photosphere is estimated to be around 500 (400) if 6 r0 = 108 cm (r0 = 109 cm) is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The Lorentz factor of the jet can also be estimated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The distance of the jet dissipation location rdis relative to the central engine of the burst and the Lorentz factor Γdis of the dissipation region may be related to the pulse duration ∆tpulse as ∆tpulse = Rdis/(2Γ2 disc) ∼ 4 s (full-width at half maximum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In addition, the dissipation location should be less than the location of the external shock at the same observer time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Rdis ≲ Res,20 s ∼ 4 × 1016 cm, where Res,20 s is the location of the external shock at the observer time 20 s and obtained based on the initial fireball (without energy injection) and Equations (B1)-(B5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, one can have Γdis ≲ 408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Interesting, the Lorentz factor of the jet producing the bright spiky γ-ray pulse can be estimated based on the blackbody radiation component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We find that the Lorentz factor of the jet is consistent with that estimated based on the blackbody radiation component in the bright spiky γ-ray pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Please see the left panel of Figure 4, where Γph ∼ 400 is obtained if r0 = 109 cm is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The magnetization of the outflow would affect its photospheric emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Zhang & Pe’er 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Gao & Zhang 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Since the emission of the initial fireball, involving the photospheric emission, missed in the observation, the magnetization of the initial fireball would be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In the spirit of Zhang & Pe’er (2009), the outflow with magnetization σ ≳ 125 (σ ≳ 162) is required if r0 = 108 cm (r0 = 109 cm) is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Here, the luminosity of the initial fireball Lw is estimated as Lw ∼ Ek,0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Corresponding, the related photosphere emission is plotted in the right panel of Figure 5, where the observed power-law radiation spectrum in the period of tobs ∼ [0, 5] s is shown with a black solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank the anonymous referee of this work for useful comments and suggestions that improved the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We acknowledge the use of the Fermi archive’s public data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We appreciate Xing Yang for his help in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This work is supported by the National Natural Science Foundation of China (grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 12273005, 11673006, U1938116, U1938201, U1731239, and U1938106), the Guangxi Science Foundation (grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2018GXNSFFA281010, 2017AD22006, 2018GXNSFGA281007, and 2018GXNSFDA281033), and China Manned Spaced Project (CMS-CSST- 2021-B11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 7 10 4 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 10 10 8 10 6 10 4 10 2 0 10 20 30 40 5000 10000 15000 GRB 200829A GBM n4 Counts BAT 10keV XRT 10keV XRT 10keV X-ray Flux density [Jy] Time Since Fermi Trigger (t obs ) [s] 10 0 10 2 10 4 10 6 10 8 R-GCN R-GCN Optical Flux density [ Jy] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Left panel—light-curves of GRB 200829A from prompt emission to its afterglows and the BAT/XRT data are the flux density at 10 keV extrapolated from BAT/XRT observation, where the inset of the upper-right panel shows the prompt γ-rays in the linear spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The MCMC fitting result based on the model in Appendix B is shown with wine line and blue line for X-ray and optical data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Here, the data showed with gray “×” and “+” symbols are not used in our fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Right panel— GBM light-curve of GRB 200829A without background subtracted (upper panel) and the signal significance (bottom panel), where the background were estimated by fitting the light-curve before and after the burst with polynomial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It reveals that there is significantly photons in the period of [0, 10] s from GRB 200829A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='0 10 100 1000 10000 0-5 s 5-10 s 16-26 s 16-17 s 17-18 s 18-19 s 19-20 s 20-21 s 21-22 s 22-23 s 23-24 s 24-25 s 25-26 s E p [keV] Low-energy Index, Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Spectral fitting results of the very early prompt emission (tobs ∈ [0, 5] s) in GRB 200829A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Here, the joint spectral fitting by combining Swift-BAT and Fermi-GBM observations based on the Band function (left panel) or power-law function (middle panel) are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In addition, the relation of Ep and α based on the spectral fitting results with Band function are plotted in right panel with “⋆” symbols, where the blue symbols are from Poolakkil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Here, the different green hollow symbols are the time-resolved spectral fitting results in the period of [16, 26] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' bn200829582 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000 s 100 正 10 Spectral fitting with PL model s-1 keV-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 Photons cm-2 10-3 10-4 10-5 10-5 10-7 sign(data-model) × A Total Statistic 10-8 10 0 10 100 1000 104 105 Energy (keV)bn2008295820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000s 100 10 Spectral fitting with Band function s-1 keV-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 cm-2 10-3 L Photons 10-4 10-5 10-6 10-7 Statis tic 10-8 sign(data-model) × A Total : 10 L 10 100 1000 104 Energy (keV) 1012000 10000 8000 6000 4000 - 2000 10 8- 6 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2 0 2 100 50 0 50 100 Time since triggel8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 1 10 100 1000 104 105 keV2 (Photons cm−2 s−1 keV−1) GRB 200829A t:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000−19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000 s 10 100 1000 104 105 −10 −5 0 5 10 (data−model)/error Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 1 10 100 1000 104 105 keV2 (Photons cm−2 s−1 keV−1) GRB 200829A t:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000−19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000 s 10 100 1000 104 105 −10 0 10 (data−model)/error Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 1 10 100 1000 104 105 keV2 (Photons cm−2 s−1 keV−1) GRB 200829A t:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000−20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000 s 10 100 1000 104 105 −4 −2 0 2 4 (data−model)/error Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 1 10 100 1000 104 105 keV2 (Photons cm−2 s−1 keV−1) GRB 200829A t:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000−20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000 s 10 100 1000 104 105 −4 −2 0 2 4 (data−model)/error Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 1 10 100 1000 104 105 keV2 (Photons cm−2 s−1 keV−1) GRB 200829A t:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000−21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000 s 10 100 1000 104 105 −4 −2 0 2 4 (data−model)/error Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 1 10 100 1000 104 105 keV2 (Photons cm−2 s−1 keV−1) GRB 200829A t:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000−21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000 s 10 100 1000 104 105 −4 −2 0 2 (data−model)/error Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 1 10 100 1000 104 105 keV2 (Photons cm−2 s−1 keV−1) GRB 200829A t:21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000−22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000 s 10 100 1000 104 105 −4 −2 0 2 4 (data−model)/error Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='1 1 10 100 1000 104 105 keV2 (Photons cm−2 s−1 keV−1) GRB 200829A t:21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000−22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='000 s 10 100 1000 104 105 −4 −2 0 2 4 (data−model)/error Energy (keV) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Spectral fitting results of the bright spiky γ-ray pulse in the period of tobs ∈ [18, 22] s based on Band function (left panel) or Band+BB model (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 9 18 19 20 21 22 10 10 10 11 10 12 10 13 r ph (r 0 =10^8 cm) r ph (r 0 =10^9 cm) r ph (cm) Time Since Fermi Trigger [s] 0 200 400 600 800 1000 1200 ph (r 0 =10^8 cm) ph (r 0 =10^9 cm) ph 18 19 20 21 22 50 100 150 200 250 300 (r 0 =10^8 cm) (r 0 =10^9 cm) Time Since Fermi Trigger [s] 0 4 8 12 16 20 24 28 1+ ph (r 0 =10^8 cm) 1+ ph (r 0 =10^9 cm) 1+ (r 0 =10^8 cm) 1+ (r 0 =10^9 cm) 1+ ph ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 1+ 10 1 10 2 10 3 10 4 10 1 10 2 10 3 10 4 10 5 PL r 0 =10^8 cm r 0 =10^9 cm keV(photons cm 2 s 1 keV 1 ) Time Since Fermi Trigger [s] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Left and middle panels—Temporal evolution of derived properties (rph, Γph, η, 1 + σph, and 1 + σ0) based on the blackbody radiation component found in the bright spiky γ-ray pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Right panel—Power-law radiation spectrum found in the period of tobs ∈ [0, 5] s (solid line) and the predicted lower limits of the photospheric emission (magenta and purple solid lines) for different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 10 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Posterior probability density functions for the physical parameters of the external-forward shock in GRB 200829A from MCMC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 11 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Spectral fitting results of the very early prompt emission in GRB 200829A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Time interval (s) Model α (or ˆΓ) a β E0(keV) N0b χ2 r [0, 5] PL −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='59 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='08 [0, 5] Band −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='09 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='42 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='08 9976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='67 ± 51113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='006 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='0006 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='08 [5, 10] Band −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='79 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='27 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='94 ± 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='99 aThe photon spectral index ˆΓ is for PL model and α is for Band function model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' b N0 is in unit of photons · cm−2 · s−1 · keV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 12 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Spectral fitting results of the bright spicky γ-ray pulse in GRB 200829A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Time interval (s) Band Band + BB α E0(keV) β N0a BIC α E0(keV) β N0a kT(keV) Ka BIC ∆BICb [16, 26] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='41 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00 948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='02 286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='22 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='82 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='68 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='34 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='74 841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='76 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='07 [16, 17] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='18 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='71 ± 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='20 599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='36 ± 420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='69 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='40 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='86 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='62 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='19 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='95 517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='64 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='23 [17, 18] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='61 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 550.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='56 572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='53 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='49 [19, 20] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='03 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='12 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='34 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='34 ± 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='05 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='56 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='85 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='77 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='49 523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='84 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='81 [20, 21] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='03 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='24 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='02 658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='42 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='05 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='26 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='28 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='93 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='16 636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='37 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='05 [21, 22] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='03 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='50 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='03 601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='05 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='22 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='89 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='56 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='86 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='88 576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='20 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='67 [22, 23] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='79 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='03 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='11 254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='12 ± 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='31 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='07 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='12 501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='17 [23, 24] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='09 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='52 ± 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='02 490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='18 340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='63 ± 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='23 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='79 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='77 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='11 [24, 25] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='23 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='59 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='47 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='08 547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='15 582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='80 ± 366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='50 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='69 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='76 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='46 559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='13 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='72 [25, 26] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='30 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='82 ± 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='05 513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='19 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='09 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='83 ± 657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='21 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='75 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='71 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='79 526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='66 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='82 aN0 is in unit of photons · cm−2 · s−1 · keV−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' K is the L39/ D2 10, where L39 is the source luminosity in units of 1039 erg/s and D10 is the distance to the source in units of 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' b The ∆BIC is the value of BICBand − BICBand+BB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 13 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Results of spectral fits for tobs ∈ [230, 52000] s of GRB 200829A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' GRB Interval(s) Band χ2 r ˆΓ GRB 200829A 230-700 BAT+XRT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 700-2000 XRT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='11 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 5116-7428 XRT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='94 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='05 12119-13162 XRT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='09 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 28067-52000 XRT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='19 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 14 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' DISCUSSION ABOUT THE PROMPT EMISSION OF GRB 200829A IN THE PERIOD OF [0, 5] S In this section, we present a comprehensive discussion about the radiation spectrum in the prompt emission of GRB 200829A in the period of [0, 5] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' We would like to conclude that the intrinsic radiation spectrum in this period may be consistent with a PL spectral model with ˆΓ ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 or a Band function with a break at ∼ 10 MeV and power- law index ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 in its low-energy regime (E ≲ 10 MeV), rather than a Band function with α ∼ −1, β ∼ −3, and Ep ∼ 200 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This conclusion is made based on the comprehensive comparison between the spectral fitting results on the observational data and those on the synthetic data of Fermi observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Here, the synthetic data of Fermi observation is generated based on the python source package threeML10 (Vianello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2015) and the Band function with α = −1, β = −3, and Ep = 200 keV is adopted as the intrinsic radiation spectrum to produce synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In addition, the signal significance of the synthetic data is set as that of the observational data of GRB 200829A in the period of [0, 5] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The spectral fittings in this section are performed based on the MCMC method to produce posterior predictions for the model parameters11 and the python source package emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2013) is used for our MCMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The spectral fitting results are reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The reasons for our above conclusion are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In the spectral fitting, the values of “Residuals (σ)” (see the bottom part in each panel of Figure 6) provides the most important information to confront the spectral model with the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' A good spectral model for the observational data should provide a well distribution of “Residuals (σ)”, such as that shown in the bottom part of the upper-right panel in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In Figure 6, the upper-left and upper-right panels show the spectral fitting results on the synthetic data with a PL model and a Band function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Since the intrinsic radiation spectrum of the synthetic data is a Band function with Ep = 200 keV, the spectral fitting on the synthetic data with a Band function should provide an optimal fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Actually, the values of the corresponding “Residuals (σ)” are indeed well distributed around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In the spectral fitting on the synthetic data with a PL model, however, the values of “Residuals (σ)” appear as positive around Ep and negative below/above ∼ Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It reveals that even though the Band function with α = −1, β = −3, and Ep = 200 keV can be described as a PL model with ˆΓ ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='65 (see second line of Table 4), the observational data would exceed the PL model around Ep and fail to reach the PL model below/above ∼ Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This behavior is consistent with the theoretical expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In the bottom-left and bottom-right panels of Figure 6, we show the spectral fitting results on the observational data of GRB 200829A in the period of [0, 5] s with a PL spectral model and a Band function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The spectral fitting results are also reported in the fourth and fifth lines of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' One can find that “Residuals (σ)” in these two panels are well distributed around zero, which is very similar to that in the upper-right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It implies that the intrinsic radiation spectrum of this period should be consistent with a PL spectral model with ˆΓ ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 or a Band function with a break at ∼ 10 MeV and power-law index ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 in its low-energy regime (E ≲ 10 MeV), rather than a Band function with α ∼ −1, β ∼ −3, and Ep ∼ 200 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' This is because that if the intrinsic radiation spectrum of the observational data is a Band function with α ∼ −1, β ∼ −3, and Ep = 200 keV, the values of “Residuals (σ)” would be positive ∼ 200 keV and negative below/above ∼ 200 keV on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' However, this behavior could not be evidently found in the bottom-left panel of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' If the intrinsic radiation spectrum in this period is the Band function with E0 ∼ 200 keV, the spectral fitting results on the low-energy regime, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', the energy band of Swift-BAT (15-150 keV), with a PL spectral model would be very different from that on the energy band of Fermi-GBM instrument (8 keV-40 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, we perform the spectral fittings on the data in the 15-150 keV energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The posterior probability density functions for the physical parameters of the spectral model are shown in Figure 7, where the upper and bottom panels are the spectral fitting results on the synthetic data and the observational data in the 15-150 keV energy band, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' A PL spectral model and Band function are adopted in the spectral fittings for the left and 10 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='com/threeML/threeML 11 This method is different from that used in the main text of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In the main text, the spectral model parameters are obtained based on the package Xspec by maximizing the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' However, one can find that the model parameters are consistent with each other in these two fitting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 15 right panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It is shown that the spectral fittings on the synthetic data with a PL spectral model for different energy regime are indeed presented very different values of power-law index ˆΓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', ˆΓ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 for the 8 keV-40 MeV energy band and ˆΓ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='44+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='10 for the 15-150 keV energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Interestingly, the spectral fittings on the synthetic data with a Band function almost report the same values of α, β, and E0 for the 15-150 keV energy band and the 8 keV-40 MeV energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' According to the fitting results reported in Table 4, one can find that the spectral fittings on the observational data in the 15-150 keV energy band and those in the 8 keV-40 MeV energy band are almost presented the same fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Please comparing the eighth line with the fourth line, or the ninth line with the fifth line in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It implies that the radiation spectrum in this period should be consistent with a PL spectral model with ˆΓ ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 or a Band function with a break at ∼ 10 MeV and power-law index ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 in its low-energy regime (E ≲ 10 MeV), rather than a Band function with α ∼ −1, β ∼ −3, and Ep ∼ 200 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' In summary, by comparing the spectral fitting results on the observational data to those on the synthetic data, we can conclude that the intrinsic radiation spectrum in this period should be consistent with a PL spectral model with ˆΓ ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 or a Band function with a break at ∼ 10 MeV and power-law index ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='7 in its low-energy regime (E ≲ 10 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' MODEL In this section, the dynamics and the emission of the external-forward shock are presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The dynamics of the external-forward shock can be described with the following equations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Sari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 1999): dΓ dtobs = 1 M ′ � 1 c2 dEinj dtobs − (Γ2 − 1) dm dtobs � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (B1) dm dtobs = 4πρR2 dR dtobs ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (B2) dU ′ dtobs = (1 − ǫ)(Γ − 1)c2 dm dtobs ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (B3) dR dtobs = cβ 1 − β (1 + z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (B4) β = � 1 − 1/Γ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (B5) where Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' dEinj/dtobs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' and cβ are the Lorentz factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' the energy injection rate (with respect to the observer time tobs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' location,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' the radiation efficiency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' and the velocity of the external-forward shock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' and M ′ = M ′ ej + m + U ′/c2 is the total mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' including the initial mass M ′ ej = Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='0/[(Γ0 − 1)c2] of the ejecta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' the sweep-up mass m from the circum-burst medium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' and the internal energy U ′ of the shocked material from the external shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Here, Ek,0 is the initial isotropic kinetic energy of the fireball, Γ0 = Γ(tobs = 0) is the initial bulk Lorentz factor of the fireball, c is the velocity of light, z is the redshift of the burst, and ρ is the density of the circum-burst environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Two cases of circum-burst medium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', interstellar medium (ISM) and wind, are generally studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Correspondingly, we take (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Chevalier & Li 2000) ρ = � 5 × 1011A∗R−2 g · cm−1, wind, n0mp cm−3, ISM, (B6) with mp being the proton mass, A∗ is a dimensionless constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' For simplicity, the energy injection into the external shock due to the late activity of the central engine is assumed with a constant energy injection rate over the period of tobs ∈ [ts, te], where ts and te are the beginning and the end of the energy injection, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' By describing Einj as Einj = Ek,0δ, one thus can have dEinj/dtobs = Ek,0δ/(te − ts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The main radiation mechanism of the external-forward shock in GRBs is the synchrotron radiation of the sweep-up electrons (Sari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Sari & Piran 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' ǫe and ǫB are introduced to represent the fractions of the shock energy used to accelerate electrons and contributing to the magnetic energy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, the magnetic field behind the shock is B′ = (32πǫBρ/mp)1/2Γc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' The sweep-up electrons are accelerated to a power-law distribution of Lorentz factor γe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Q ∝ γ′ e −p for γ′ e,min ⩽ γe ⩽ γ′ e,max, where p(> 2) is the power-law index, γe,min = ǫe(p − 2)mpΓ/[(p − 1)me] (Sari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 1998), and γe,max = � 9m2ec4/(8B′q3e) with qe being the electron charge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Then, one 16 can have ǫ = ǫradǫe with ǫrad = min{1, (γe,min/γe,c)(p−2)} (Fan & Piran 2006), where γe,c = 6πmec(1+z)/(σTΓB′2tobs) is the efficient cooling Lorentz factor of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Equations (B1)-(B5) describe the evolution of hydrodynamic blastwave approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' A more rigorous treatment can be found in Nava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2013) and Zhang (2018) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='66) in this book).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' For our studied burst, the blastwave is affected by the energy injection and thus its evolution could not be simply estimated with hydrodynamic equations in Nava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2013) and Zhang (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' A more complicated equations are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' For the phase without energy injection, we also present the light curve of afterglows based on the hydrodynamic equations in Nava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' (2013) and Zhang (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' It is found that the obtained light-curves of afterglows are almost the same as those obtained with Equations (B1)-(B5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 17 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Fitting results of the synthetic data (upper panels) and the observational data (bottom panels) in the 8 keV-40 MeV energy band, where a PL spectral model and Band function are adopted in the left and right panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 18 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Posterior probability density functions for the physical parameters of the spectral fitting on the synthetic data (upper panels) and the observational data (bottom panels) in the 15-150 keV energy band, where a PL spectral model and Band function are adopted in the left and right panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 19 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Spectral fitting results of simulation and observation of [0, 5] s in GRB 200829A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' Model α (or ˆΓ) β E0(keV) N0 Data sources PL −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='04 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='52+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='14 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='32 synthetic data (8 keV-40 MeV) Band −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='11 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='71+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='83 276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='67+91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='20 −71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00 synthetic data (8 keV-40 MeV) PL −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='09 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='65+11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='32 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='15 observational data (8 keV-40 MeV) Band −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='61+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='11 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='85 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='21 11021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='76+13229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='91 −5533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00 observational data (8 keV-40 MeV) PL −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='44+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='10 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='49+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='68 synthetic data (15-150 keV) Band −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='15 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='25+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='19 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='26 268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='28+141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='98 −110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='03+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00 synthetic data (15-150 keV) PL −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='71+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='15 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06+17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='86 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='79 observational data (15-150 keV) Band −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='62+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='17 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='47+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='98 18813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06+34951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='87 −11705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content='00 observational data (15-150 keV) 20 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2020a, GRB Coordinates Network, 28330, 1 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} +page_content=' 2020b, GRB Coordinates Network, 28324, 1' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfAfoQ/content/2301.00925v1.pdf'} diff --git a/pNE0T4oBgHgl3EQfqgGa/vector_store/index.pkl b/pNE0T4oBgHgl3EQfqgGa/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..4b138a6c2e4cdecc6079476c4e81251f63395e7b --- /dev/null +++ b/pNE0T4oBgHgl3EQfqgGa/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:458e55abee5e46a15100aff2df28bb6b7138303295e76052ac00a49aa91b95ab +size 228516 diff --git a/ptE1T4oBgHgl3EQfPQMl/content/tmp_files/2301.03024v1.pdf.txt b/ptE1T4oBgHgl3EQfPQMl/content/tmp_files/2301.03024v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6885629fb4fe7f01077c4164e3c40a839da1601c --- /dev/null +++ b/ptE1T4oBgHgl3EQfPQMl/content/tmp_files/2301.03024v1.pdf.txt @@ -0,0 +1,2011 @@ +arXiv:2301.03024v1 [hep-ph] 8 Jan 2023 +Oblique corrections from leptoquarks +Francisco Albergaria‡ and Lu´ıs Lavoura‖ +Universidade de Lisboa, Instituto Superior T´ecnico, CFTP, +Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal +January 10, 2023 +Abstract +We present general formulas for the oblique-correction parameters S, T, U, V , +W, and X in a model of New Physics having arbitrary numbers of scalar leptoquarks +of the five permissible types. We allow for a general mixing among the scalars of +the various electric charges, viz. −4/3, −1/3, 2/3, and 5/3. +1 +Introduction and notation +Leptoquarks: +In this paper we consider a model of New Physics (NP), i.e. an extension +of the Standard Model (SM), that has arbitrary numbers of scalars placed in triplets of +colour-SU(3) which are +SU(2) singlets with weak hypercharge1 −1/3 +σ0,−2; +(1) +SU(2) singlets with weak hypercharge −4/3 +σ0,−8; +(2) +SU(2) doublets with weak hypercharge 7/6 +� +δ1,7 +δ−1,7 +� +; +(3) +‡E-mail: francisco.albergaria@tecnico.ulisboa.pt +‖E-mail: balio@cftp.tecnico.ulisboa.pt +1We use the normalization Y = Q − T3, where Y is the weak hypercharge, Q is the electric charge, +and T3 is the third component of weak isospin. +1 + +SU(2) doublets with weak hypercharge 1/6 +� +δ1,1 +δ−1,1 +� +; +(4) +SU(2) triplets with weak hypercharge −1/3 + + +τ2,−2 +τ0,−2 +τ−2,−2 + + . +(5) +In Eqs. (1)–(5), +the letter σ denotes singlets of gauge SU(2), the letter δ stands for doublets, and the +letter τ means triplets; +the first number in the subscript is two times the third component of weak isospin; +the second number in the subscript is six times the weak hypercharge. +In the notation of the recent Ref. [1], which refers to the original Ref. [2], +the scalars in Eq. (1) are leptoquarks of type Φ1, viz. scalars placed in the representation +(3, 1, −1/3) of the gauge group SU(3) × SU(2) × U(1); +the scalars in Eq. (2) are leptoquarks of type Φ˜1, viz. scalars placed in the representation +(3, 1, −4/3) of SU(3) × SU(2) × U(1); +the scalars in Eq. (3) are leptoquarks of type Φ2, viz. scalars placed in the representation +(3, 2, 7/6) of the gauge group; +the scalars in Eq. (4) are leptoquarks of type Φ˜2, viz. scalars placed in the representation +(3, 2, 1/6); +the scalars in Eq. (5) are leptoquarks of type Φ3, viz. scalars placed in the representation +(3, 3, −1/3) of SU(3) × SU(2) × U(1). +Leptoquarks are scalars that are in multiplets of SU(3)×SU(2)×U(1) such that they have +(renormalizable) Yukawa couplings to one lepton multiplet and one quark multiplet of the +SM. Leptoquarks have recently been much used in models that seek to explain one or more +unexpected experimental results like those on the muon magnetic moment, the W ± mass, +the decays b → cτν, and the decays b → sℓ+ℓ−; see for instance Refs. [3, 4, 5, 6, 7, 8, 9, 10]. +2 + +Oblique parameters: +The purpose of this paper is to compute the oblique parameters +for this generic NP model. The oblique parameters are defined as [11]2,3 +S += +16πc2 +W +g2 +� +AZZ (m2 +Z) − AZZ (0) +m2 +Z +− ∂Aγγ (q2) +∂q2 +���� +q2=0 ++c2 +W − s2 +W +cWsW +∂AγZ (q2) +∂q2 +���� +q2=0 +� +, +(6a) +T += +4π +g2s2 +W +�AW W (0) +m2 +W +− AZZ (0) +m2 +Z +� +, +(6b) +U += +16π +g2 +�AW W (m2 +W) − AW W (0) +m2 +W +− c2 +W +AZZ (m2 +Z) − AZZ (0) +m2 +Z +−s2 +W +∂Aγγ (q2) +∂q2 +���� +q2=0 ++ 2cWsW +∂AγZ (q2) +∂q2 +���� +q2=0 +� +, +(6c) +V += +4π +g2s2 +W +� +∂AZZ (q2) +∂q2 +���� +q2=m2 +Z +− AZZ (m2 +Z) − AZZ (0) +m2 +Z +� +, +(6d) +W += +4π +g2s2 +W +� +∂AW W (q2) +∂q2 +���� +q2=m2 +W +− AW W (m2 +W) − AW W (0) +m2 +W +� +, +(6e) +X += +4πcW +g2sW +� +∂AγZ (q2) +∂q2 +���� +q2=0 +− AγZ (m2 +Z) − AγZ (0) +m2 +Z +� +, +(6f) +where g is the SU(2) gauge coupling constant and cW and sW are the cosine and the sine, +respectively, of the Weinberg angle. The functions AV V ′ (q2) are the coefficients of the +metric tensor gµν in the vacuum-polarization tensor +Πµν +V V ′ +� +q2� += gµν AV V ′ � +q2� ++ qµqν BV V ′ � +q2� +(7) +between gauge bosons Vµ and V ′ +ν carrying four-momentum q. In AV V ′ (q2) +one only takes into account the dispersive part—one discards the absorptive part; +one subtracts the SM contribution from the full result.4 +This paper generalizes earlier partial results in Refs. [17, 18, 19, 20, 21, 22]. We believe +that some results of this paper may be generalized easily to the computation of the +oblique parameters arising from scalars with any exotic electric charges; we thus complete +2We use the sign conventions in Ref. [12]. Those conventions differ from the ones used in many other +papers, viz. in Ref. [11]. For a resource paper on sign conventions, see Ref. [13]; using the notation of +that paper, our convention has ηe = ηZ = 1 and η = −1. +3The definitions (6) build on, and generalize, previous work in Refs. [14, 15, 16]. They are appropriate +for the case where the functions AV V ′ � +q2� +are not linear in the range 0 < q2 < m2 +Z. This means that, +using those definitions, NP does not need to be much above the Fermi scale. +4In this paper we do not have to perform this subtraction, since we are dealing with NP scalars that +do not mix with the SM scalars. +3 + +the earlier computation [23] of the oblique parameters for scalars of electric charges either +0 or 1. +This paper is organized as follows. In the next section we write the Lagrangian for the +gauge interactions of the scalars, carefully defining the mixing matrices that appear in that +Lagrangian. Section 3 performs the computation of the relevant diagrams. Sections 4–7 +give the results for the oblique parameters. In Appendix A we explicitly demonstrate that +S and U are finite as they should. +2 +Interactions +Numbers of scalars: +There are in our NP model nσ,−2 multiplets of type (1), nσ,−8 +multiplets of type (2), nδ,7 multiplets of type (3), nδ,1 multiplets of type (4), and nτ,−2 +multiplets of type (5). All these five numbers nσ,−2, nσ,−8, nδ,7, nδ,1, and nτ,−2 must be +multiples of three, since all the scalars are in triplets of SU(3); otherwise, those numbers +are free. +Our model has fractionary-charge scalars that we call +h-type scalars, viz. the ones that have electric charge Qh = 5/3; +u-type scalars, viz. the ones that have electric charge Qu = 2/3; +d-type scalars, viz. the ones that have electric charge Qd = −1/3; +l-type scalars, viz. the ones that have electric charge Ql = −4/3. +The total numbers of h-type scalars, u-type scalars, d-type scalars, and l-type scalars are +nh += +nδ,7, +(8a) +nu += +nδ,7 + nδ,1 + nτ,−2, +(8b) +nd += +nσ,−2 + nδ,1 + nτ,−2, +(8c) +nl += +nσ,−8 + nτ,−2. +(8d) +respectively. +Scalar mixing: +One bi-diagonalizes the leptoquark mass matrices by making +δ1,7 += +H1h, +(9a) + + +δ−1,7 +δ1,1 +τ2,−2 + + += + + +U1 +U2 +U3 + + u, +(9b) + + +σ0,−2 +δ−1,1 +τ0,−2 + + += + + +D1 +D2 +D3 + + d, +(9c) +� σ0,−8 +τ−2,−2 +� += +� L1 +L2 +� +l, +(9d) +4 + +where h in Eq. (9a) stands for a column matrix containing the nh h-type scalars; and +analogously for u, d, and l in the other three Eqs. (9). The matrices H1, U1, U2, U3, D1, +D2, D3, L1, and L2 have dimensions nδ,7 × nh, nδ,7 × nu, nδ,1 × nu, nτ,−2 × nu, nσ,−2 × nd, +nδ,1 × nd, nτ,−2 × nd, nσ,−8 × nl, and nτ,−2 × nl, respectively. The matrices +H1, + + +U1 +U2 +U3 + + , + + +D1 +D2 +D3 + + , +and +� +L1 +L2 +� +(10) +are unitary.5 +Mixing matrices: +We next define the mixing matrices that appear in the charged- +current interactions of the scalars. They are +N += +H† +1U1, +(11a) +V += +U† +2D2 + +√ +2 U† +3D3, +(11b) +Q += +√ +2 D† +3L2, +(11c) +respectively. The factors +√ +2 in Eqs. (11) arise because the charged gauge interactions of +the triplets are +√ +2 times stronger than those of the doublets. The mixing matrices that +appear in the neutral-current interactions of the scalars are +¯H += +H − 2Qhs2 +W × +1nh, +(12a) +¯U += +U − 2Qus2 +W × +1nu, +(12b) +¯D += +D + 2Qds2 +W × +1nd, +(12c) +¯L += +L + 2Qls2 +W × +1nl, +(12d) +where +1m denotes the m × m unit matrix and the matrices H, U, D, and L are related +to the mixing matrices in Eqs. (11) through +H += +NN†, +(13a) +U += +V V † − N†N, +(13b) +D += +V †V − QQ†, +(13c) +L += +Q†Q, +(13d) +respectively. +5Without loss of generality, we might have chosen a basis where H1 would be equal to the unit matrix. +But we refrain from doing that, in order to keep the notation as general as possible. +5 + +Gauge interactions: +The gauge interactions of the scalars are given by the following +pieces of the Lagrangian: +LASS += +−ieAθ +� +Qh +� +h +� +h∗∂θh − h∂θh∗� ++ Qu +� +u +� +u∗∂θu − u∂θu∗� ++Qd +� +d +� +d∗∂θd − d∂θd∗� ++ Ql +� +l +� +l∗∂θl − l∂θl∗� +� +, +(14a) +LAASS += +e2AθAθ +� +Q2 +h +� +h +hh∗ + Q2 +u +� +u +uu∗ + Q2 +d +� +d +dd∗ + Q2 +l +� +l +ll∗ +� +, +(14b) +LW SS += +i g +√ +2 W + +θ +�� +h,u +Nhu +� +h∗∂θu − u∂θh∗� ++ +� +u,d +Vud +� +u∗∂θd − d∂θu∗� ++ +� +d,l +Qdl +� +d∗∂θl − l∂θd∗� +� ++ H.c., +(14c) +LW W SS += +g2 +2 W + +θ W −θ +�� +h,h′ +� +NN†� +hh′ h∗h′ + +� +u,u′ +� +N†N + V V †� +uu′ u∗u′ ++ +� +d,d′ +� +V †V + QQ†� +dd′ d∗d′ + +� +l,l′ +� +Q†Q +� +ll′ l∗l′ +� +, +(14d) +LZSS += +i +g +2cW +Zθ +�� +h +¯Hhh′ � +h∗∂θh′ − h′∂θh∗� ++ +� +u,u′ +¯Uuu′ � +u∗ ∂θu′ − u′ ∂θu∗� +− +� +d,d′ +¯Ddd′ � +d∗ ∂θd′ − d′ ∂θd∗� +− +� +l,l′ +¯Lll′ � +l∗ ∂θl′ − l′ ∂θl∗� +� +, +(14e) +LZZSS += +g2 +4c2 +W +ZθZθ +�� +h,h′ +� ¯H2� +hh′ h∗h′ + +� +u,u′ +� ¯U2� +uu′ u∗u′ ++ +� +d,d′ +� ¯D2� +dd′ d∗d′ + +� +l,l′ +�¯L2� +ll′ l∗l′ +� +, +(14f) +LAZSS += +− eg +cW +AθZθ +� +Qh +� +h,h′ +¯Hhh′ h∗h′ + Qu +� +u,u′ +¯Uuu′ u∗u′ +−Qd +� +d,d′ +¯Ddd′ d∗d′ − Ql +� +l,l′ +¯Lll′ l∗l′ +� +. +(14g) +Notice the presence in Eq. (14c) of the matrices defined in Eqs. (11) and the presence in +Eqs. (14e)–(14g) of the matrices defined in Eqs. (12). Also notice that, out of the four +mixing matrices in Eq. (14d), NN† = H and Q†Q = L coincide with two matrices in +Eqs. (13). +6 + +3 +Tools for the computation +Suppose the vertices VµS1S2 and VµV ′ +νS1S2 in our model have the Feynman rules +k +p +S2 +S1 +Vµ += iX (k − p)µ , +(15a) +S2 +S1 +V ′ +ν +Vµ += iY gµν, +(15b) +where V and V ′ are gauge bosons (v.g. either A, Z, W +, or W −) and S1 and S2 are +scalars. Using those Feynman rules we have +AV V ′S1S2 � +q2� += X1X2 +4π2 B00 +� +q2, m2 +1, m2 +2 +� +(16) +for diagrams of the form in Fig. 1, +S2 +S1 +V +V ′ +Figure 1: One type of diagram for vacuum polarization. V and V ′ are gauge bosons, S1 +is a scalar with mass m1, and S2 is a scalar with mass m2. +and +AV V ′S1 � +q2� += − Y +16π2 A0 +� +m2 +1 +� +(17) +for diagrams of the form in Fig. 2. +7 + +S1 +V +V ′ +Figure 2: Another type of diagram for vacuum polarization. V and V ′ are gauge bosons +and S1 is a scalar with mass m1. +In Eqs. (16) and (17), q is the four-momentum of the gauge bosons. The Passarino– +Veltman (PV) functions [24] B00 (q2, m2 +1, m2 +2) and A0 (m2 +1) are defined by6 +µǫ +� +d4−ǫk +(2π)4−ǫ kθkψ +1 +k2 − m2 +1 +1 +(k + q)2 − m2 +2 += +i +16π2 +� +gθψ B00 +� +q2, m2 +1, m2 +2 +� ++qθqψ B11 +� +q2, m2 +1, m2 +2 +�� +, +(18a) +µǫ +� +d4−ǫk +(2π)4−ǫ +1 +k2 − m2 +1 += +i +16π2 A0 +� +m2 +1 +� +, +(18b) +respectively, where µ is an arbitrary quantity with mass dimension. The quantities q2, m2 +1, +and m2 +2 are assumed to be non-negative. The PV function B11 (q2, m2 +1, m2 +2) in Eq. (18a) +is not needed in this paper; on the other hand, we need +B′ +00 +� +q2, m2 +1, m2 +2 +� +≡ +∂B00 (q2, m2 +1, m2 +2) +∂q2 +, +(19a) +¯B00 +� +q2, m2 +1, m2 +2 +� +≡ +B00 (q2, m2 +1, m2 +2) − B00 (0, m2 +1, m2 +2) +q2 +. +(19b) +The PV functions may be numerically evaluated by using softwares like LoopTools [25] +and COLLIER [26].7 +They may also be evaluated through analytic formulas given in +Ref. [27]. In particular, from those analytic formulas one gathers that +B00 +� +0, m2 +1, m2 +2 +� += A0 (m2 +1) + A0 (m2 +2) +4 ++ θ+ (m2 +1, m2 +2) +8 +, +(20) +where [28] +θ+ (a, b) ≡ + + + +a + b − 2ab +a − b ln a +b +⇐ +a ̸= b, +0 +⇐ +a = b. +(21) +6We use the definitions of Ref. [25] for the PV functions. +7For the present purposes, in the results given by those codes one should take only the real, viz. +dispersive, parts of the PV functions, while dropping the absorptive parts, which have no relevance for +the oblique parameters. +8 + +4 +T +For the oblique parameter T we obtain +T += +1 +8πs2 +Wm2 +W +� +4 +�� +h,u +|Nhu|2 B00(0, m2 +h, m2 +u) + +� +u,d +|Vud|2 B00(0, m2 +u, m2 +d) ++ +� +d,l +|Qdl|2 B00(0, m2 +d, m2 +l ) +� +(22a) +− +� +h +� +NN†� +hh A0 +� +m2 +h +� +− +� +u +� +N†N + V V †� +uu A0 +� +m2 +u +� +− +� +d +� +V †V + QQ†� +dd A0 +� +m2 +d +� +− +� +l +� +Q†Q +� +ll A0 +� +m2 +l +� +(22b) +−2 +�� +h,h′ +�� ¯Hhh′ +��2 B00 +� +0, m2 +h, m2 +h′ +� ++ +� +u,u′ +�� ¯Uuu′ +��2 B00 +� +0, m2 +u, m2 +u′ +� ++ +� +d,d′ +�� ¯Ddd′ +��2 B00 +� +0, m2 +d, m2 +d′ +� ++ +� +l,l′ +��¯Lll′ +��2 B00 +� +0, m2 +l , m2 +l′ +� +� +(22c) ++ +� +h +� ¯H2� +hh A0 +� +m2 +h +� ++ +� +u +� ¯U2� +uu A0 +� +m2 +u +� ++ +� +d +� ¯D2� +dd A0 +� +m2 +d +� ++ +� +l +�¯L2� +ll A0 +� +m2 +l +� +� +. +(22d) +where lines (22a) and (22b) originate in AW W (0) while lines (22c) and (22d) originate in +AZZ (0). Utilizing Eq. (20) on Eq. (22), the latter becomes much simpler: +T += +1 +16πs2 +Wm2 +W +�� +h,u +|Nhu|2 θ+ +� +m2 +h, m2 +u +� ++ +� +u,d +|Vud|2 θ+ +� +m2 +u, m2 +d +� ++ +� +d,l +|Qdl|2 θ+ +� +m2 +d, m2 +l +� +− +� +h · · · > t1 is +p(xn, . . . , x1) ≡ tr{ΠxnUn . . . Πx1U1ρ0U † +1Πx1 . . . U† +n} +(1) +with Uk ≡ e−iH(tk−tk−1) the unitary time evolution operator between two times (ℏ ≡ 1). Note +that Eq. (1) can be experimentally reconstructed by performing n repeated measurements on +a quantum system and by repeating this procedure many times to create sufficient statistics. +Next, pick some k ∈ {1, . . . , n−1} and define p(xn, . . . , �� +xk , . . . , x1) to be the same probability +as in Eq. (1) except that no measurement is performed at time tk (and thus no outcome xk +is recorded), which is indicated with the notation �� +xk and obtained from Eq. (1) by dropping +the two projectors Πxk. Then, the process is classical if the following “probability sum rule” +is satisfied for all k < n and all n > 1 (up to some error much smaller than the considered +probabilities): +� +xk +p(xn, . . . , xk, . . . , x1) = p(xn, . . . , �� +xk , . . . , x1). +(2) +In words, a process is classical if marginalizing over measurement results is identical to not +measuring at any given time tk. Since measurements can be disturbing in quantum mechan- +ics, even on average, the validity of Eq. (2) signifies the absence of quantum effects from the +perspective of measuring X. An example violating Eq. (2) is the famous double slit experi- +ment, see Fig. 1. The following facts further support the idea that this is a good definition of +classicality (though, as emphasized above, not the only one). +First, observe that Eq. (2) defines a classical stochastic process [38], where it is also known +as the Kolmogorov consistency condition. Classicality as considered here therefore has a clear +operational meaning, which was also used in Ref. [7–15]: a process is classical if (at least +3 + +SciPost Physics +Submission +Figure 1: The double slit experiment where a coherent source of particles ρ0 hits a +detection screen at position x2 after passing a wall with two holes (the double slit). +(a) The particle’s location x1 is also measured at the double slit, allowing to decide +through which slit it passed (corresponding trajectories indicated by dashed lines). +No interference pattern is seen on the detection screen, also not after averaging over +x1. +(b) There is no measurement of the particle’s location at the double slit, it +thus retains its coherent wave-like properties and an interference pattern emerges. +Clearly, Eq. (2) is violated: the dynamics is non-classical. +in principle) a classical stochastic process can be used to generate the same measurement +statistics. The idea of defining classicality in this way is rooted in a “black-box-mentality”: +there might be some very expensive quantum computer in front of you, but if the available +measurement statistics can be simulated, or emulated, with a classical stochastic processes, +then the measurement statistics alone do not allow you to draw the conclusion that there +is anything quantum going on in the computer. Furthermore, this definition of classicality +also has a clear practical motivation because classical stochastic processes are much easier to +analyse and simulate than quantum stochastic processes. +Moreover, Eq. (2) implies the validity of Leggett-Garg inequalities [39] and it is closely +related but not equivalent to the conditions imposed in the consistent or decoherent histories +formalism [5, 6], which we review below in Sec. 2.3. Importantly, however, the definition of +classicality used here does not hinge on any specific interpretation of quantum mechanics. +Confirming Eq. (2) experimentally only requires measurements of X, no further hidden as- +sumption is contained in its definitions. Clearly, classicality is defined with respect to some +observable X, i.e., a system that behaves classical with respect to X can behave non-classical +with respect to a different observable Y . +Finally, notice that Eq. (2) could be also vio- +lated in a classical context, for instance, whenever an external agent (e.g., some observer or +experimenter) actively intervenes in the process, e.g., by performing feedback control opera- +tions [13,14,40]. We exclude these scenarios here by definition of the probabilities in Eq. (1), +which in the classical limit (replacing projectors on Hilbert space by characteristic functions +on phase space) clearly obey Eq. (2). +In the remainder of this section, we first review the well known decoherence approach and +ask whether it explains classicality according to the definition used here (Sec. 2.2). Afterwards, +we comment on the relation to the perhaps less well known consistent or decoherent histories +approach (Sec. 2.3). Finally, Sec. 2.4 concludes with an intuitive explanation why Eq. (2) can +be satisfied for an isolated quantum system. +4 + +SciPost Physics +Submission +2.2 +The decoherence approach for open quantum systems +We consider an open quantum system (OQS) S coupled to some environment or bath B. The +total Hilbert space is thus a tensor product HS⊗HB of the system and bath Hilbert spaces HS +and HB, respectively. The dynamics in the full system-bath space is unitary and generated +by a Hamiltonian HSB = HS +HB +VSB with HS (HB) the system (bath) Hamiltonian alone +(suppressing tensor products with the identity in the notation) and VSB their interaction. +The reduced system state ρS(t) = trB{ρSB(t)} is obtained from a partial trace of the full +system-bath state over the bath degrees of freedom. In contrast to ρSB(t), ρS(t) does not +evolve in a unitary way. +Decoherence happens whenever it is possible to identify a fixed special basis {|ψx⟩}, which +is called the pointer basis. The special role of this basis is to ensure that any initial OQS state +ρS(0) becomes after a characteristic (and typically very short) decoherence time tdec diagonal +in that basis. In equations, +ρS(0) = +� +x,y +cx,y(0)|ψx⟩⟨ψy| +−→ +t≥tdec +ρS(t) ≈ +� +x +px(t)|ψx⟩⟨ψx|. +(3) +Here, the cx,y(0) = ⟨ψx|ρS(0)|ψy⟩ are complex numbers, which ensure positivity and normal- +ization of ρS(0) but are otherwise arbitrary, and the px(t) ≈ cx,x(0) are close to the initial +probabilities to be in state |ψx⟩. Equation (3) is indeed a remarkable robust prediction of +OQS theory [1–4,41,42]. In particular, we repeat that the pointer basis is fixed, i.e., it does +not depend on the initial system state, but it is determined by the system-bath Hamiltonian +and the initial bath state (though the dependence on the latter should be mild in realistic +situations). The pointer basis is also often described as “stable”, “robust” or “objective” [1–4] +and we come back to these properties below. +We further add some clarifications. +First, for the sake of generality one should stress +that the pointer basis might not be a basis of pure states |ψx⟩, but rather a complete set of +orthonormal projectors {ΠS +x} acting on the system Hilbert space, where certain projectors +can have a rank greater than one [43]. In that case, there exist “decoherence-free subspaces” +(caused, e.g., by additional conservation laws), but they do not change the fundamental point +of our discussion and we continue to call {ΠS +x} the pointer basis for simplicity. Moreover, +we here assume pointer states to be orthogonal, which is typically the case for finite dimen- +sional OQS, but pointer states can be non-orthogonal (e.g., coherent states of a harmonic +oscillator [44]). Second, in Eq. (3) we allowed the probabilities px(t) to be time-dependent. +Their change, however, typically happens on a time scale much slower than the decoherence +time scale (see, e.g., Ref. [45]) and is called “dissipation”—a phenomenon already known from +classical open systems. Third, we remark that a more nuanced presentation of decoherence +is possible. For instance, in order to determine the measurement basis, Zurek in his seminal +paper was actually interested in the decoherence of the measurement apparatus, which was +in turn coupled to the system to be measured and an environment [46]. However, also the +measurement apparatus is an OQS, and for the remainder of this paper it is not necessary +to explicitly distinguish between system and measurement apparatus. In the following we +call the phenenology explained above OQS decoherence to distinguish it from the decoherent +histories mentioned later. +Next, we ask whether decoherence explains the emergence of classicality according to +Eq. (2) if applied to a system observable XS = �M +x=1 λxΠS +x, i.e., an observable commuting +with the pointer basis and acting trivially on the bath space. To this end, we first confirm +5 + +SciPost Physics +Submission +that for all times larger than the decoherence time tdec we have +DρS ≡ +� +x +ΠS +xρS(t)ΠS +x = ρS(t), +(4) +where D is a dephasing operation in the pointer basis.1 Thus, measuring and averaging is +identical to not measuring. Next, let us additionally assume that Eq. (4) holds for the full +system-bath state: +DρSB(tk) = +� +x +ΠS +xρSB(tk)ΠS +x = ρSB(tk). +(5) +If that is the case, one can confirm our definition of classicality, i.e., the validity of Eq. (2). +However, Eq. (4) does not imply Eq. (5), even though the converse is true. +Thus, OQS +decoherence does not imply classical measurement statistics according to Eq. (2). +It is thus worth thinking about which condition on top of decoherence could imply classi- +cality, and it seems that two fundamentally different strategies are conceivable. +The first strategy takes a more detailed look at the environmental degrees of freedom. +Indeed, the validity of Eq. (5) is equivalent to having vanishing quantum discord [47] in the +pointer basis, and testing Eq. (4) has been suggested as a tool to probe non-classical system- +bath correlations [48]. However, deciding whether the system-bath state has zero quantum +discord or not requires knowledge of the full system-bath state. This knowledge is unavailable +experimentally and therefore the condition of zero quantum discord is inaccessible from an +operational perspective of OQS theory. Moreover, the idea of having zero quantum discord is +problematic from the perspective of having a unitarily evolving “universe” consisting of the +system and the bath as explained later in Sec. 2.3. +However, a refinement of this idea is possible and has lead to the recently much studied +approach of quantum Darwinism, see Ref. [36, 37] and references therein. +In a nutshell, +quantum Darwinism starts by dividing the bath into many different “fragments” F ⊂ B and +asserts that most fragments, even those of small size, have (close) to zero quantum discord +with respect to the pointer basis, i.e., Eq. (5) applies to most states ρSF (tk) = trB\F {ρSB(tk)}, +where B \ F denotes all bath degrees of freedom except those of the fragment. The resulting +classical correlations between the system S and most fragments F allow external observers +to learn about the system state even by only looking at a small fragment F of the bath, and +different observers looking at different small fragments will agree about the state of S. Thus, +an objective world emerges. +For the important mechanism of photons scattered off some material object the idea +of quantum Darwinism is indeed intuitively appealing because the scattered photons allow +different observers, by looking at different narrow angles at the object, to infer, e.g., the same +colour or position of it. Moreover, since photons are non-interacting and scatter off to infinity, +it becomes clear that their detection does not change the future evolution of the object. As +a consequence, Eq. (2) follows. +Quantum Darwinism thus provides a sufficient criterium for objectivity and classical mea- +surement statistics, but it is questionable whether it is always necessary. In condensed matter +and other situations, the bath does not split into non-interacting fragments and perturbations +might not be able to escape to infinity. In this case, quantum Darwinism will generically +1In principle, Eq. (3) implies only an approximate equality (≈) in Eq. (4). However, since classical behaviour +should be always understood as some approximation, we replace ≈ by = whenever we mean “equal up to some +irrelevant measurement error”. +6 + +SciPost Physics +Submission +hold at most for transient times [49], yet objectivity and Kolmogorov consistency might nev- +ertheless arise—as this paper will indeed confirm now within and later also without OQS +decoherence. +Within the paradigm of OQS decoherence, this brings us to the second strategy imply- +ing classical behaviour. This strategy is different from the first by rejecting the idea that +information about (fragments of) the bath is directly accessible. Instead, solely the degrees +of freedom of the OQS are deemed operationally accessible, and it then becomes necessary +to think about how could one locally decide whether the pointer states are stable, robust +or objective. Historically, Zurek introduced for this purpose the “predictability sieve” [50], +which requires to compute the change in von Neumann entropy of the OQS state ρS(t) as a +function of a pure initial state ρS(0) = |ψ(0)⟩⟨ψ(0)|S. If it changes very slowly, the dynamics +are predictable as the state remains approximately pure, but if it changes very rapidly, the +dynamics are unpredictable as the state becomes very mixed. Now, from what we said above, +we see that the initial pointer states are characterized by a slow change in von Neumann +entropy, whereas superpositions of pointer states quickly decohere into a mixture on a time +scale tdec. The predictability sieve thus selects out the pointer states. +Although the predictability sieve has appealing properties, it is ultimatively not satisfac- +tory for the following reason. If we want to find out whether something is predictable (or +stable, robust or objective), it is best to really “take a look at it”. For instance, the memories +in our computers are stable because we can repeatedly read them out without changing their +state. How can this idea be formalized mathematically? Clearly, one way to test this prop- +erty is to measure the OQS in the pointer basis, say at some time t1 ≥ tdec, and then to look +whether this measurement influences the future evolution of the OQS at some time t2 > t1, +for instance, by checking whether the future probabilities of the pointer states depend on the +measurement at time t1. We now notice that this idea to check for predictability is exactly +equal to testing Eq. (2), i.e., our definition of classicality. Clearly, other ways are possible, +but within this second strategy they should always be related to watching the response to +some form of external perturbation or intervention on the OQS: do the pointer states remain +stable if we shake them a bit? +After having spelled out the basic idea, it remains mostly a technical problem to realize +that the condition of Markovianity as defined in Ref. [16] (for introductions see Refs. [13,14]) +is sufficient to imply classical measurement statistics. In short, this definition of Markovianity +is based on the idea that local operations on the system performed by an external agent do +not influence the OQS dynamics generated by the environment. Importantly, the property of +Markovianity can be checked by local system operations only (no knowledge of the bath state +is required) [13, 14, 16, 17]. However, since the definition of Markovianity requires to check +multi-time correlations (in complete analogy to the classical definition), knowledge of the time +evolution of ρS(t) alone is insufficient to check for Markovianity (a discussion focused on this +point can be found in Ref. [18]). +The connection to Markovianity now becomes transparent by realizing that “shaking a bit +the system” is an external intervention that will sooner or later also influence the environment. +Can this influence of the environment cause a different behaviour of the system? If the answer +is no, then this precisely means that the dynamics is Markovian. In that case, we can conclude +the following. First, we found above that OQS decoherence implies that DρS(t1) = ρS(t1) +for t1 ≥ tdec. Obviously, one also has IρS(t1) = ρS(t1) where I is the identity operation +which, operationally speaking, literally means “do nothing!” Now, according to the definition +of Markovianity explained above, the dynamics induced by the environment is insensitive to +7 + +SciPost Physics +Submission +Figure 2: Overview over the relations between different concepts defined in the text. +The arrows mean strict mathematical implications. The equivalence between global +decoherence and zero discord has to be understood for local observables (discord +is undefined without a system-bath tensor product structure). +Moreover, global +decoherence and zero discord are here understood as applying repeatedly for all +times tk considered in Eq. (2). The comment “Happens only trivially!” is explained +in detail in Sec. 2.3. +local operations on the system performed by an external agent. Since the two operations D and +I do not change the OQS state, the future dynamics is insensitive to the dephasing operations, +that is: the pointer states are stable and the dynamics is classical. Formal definitions and a +formal proof are given in Appendix A.1. +This important message together with various other notions (some of which are only +introduced in Sec. 2.3) is summarized in Fig. 2. Notably, the role of multi-time statistics +to probe the stability of pointer states and its connection to Markovianity has not yet been +made in the literature on OQS decoherence [1–4], and it is also not the focus of quantum +Darwinism [36,37]. It has been realized, however, in a numerical study of quantum Brownian +motion that non-Markovianity hinders quantum Darwinism [51]. While it is clear that the +definition of Markovianity is mathematically not in one-to-one correspondence to the concept +of quantum Darwinism, it seems worthwhile in the future to look for a closer connection of +these two notions in physical relevant situations. +Finally, we make two more important observations. First of all, the question “what is the +pointer basis?” is non-trivial and has been only answered in certain limiting cases (e.g., very +strong or very weak system-bath coupling) [1–4]. In general, for a complex open many-body +system coupled to a complex many-body environment the pointer basis is not known. It is +an advantage of the approach presented in Secs. 2.4 and 3 of this paper that no pointer basis +needs to be identified. Second, all what we said above was restricted to the OQS paradigm, +i.e., local observables defined on a system-bath tensor product structure, whose identification +can be non-trivial [52]. This restriction is also lifted in the present approach, which makes it +appealing for questions usually studied within the formalism reviewed next. +8 + +Happens only +trivially!SciPost Physics +Submission +2.3 +Consistent and decoherent histories +The consistent or decoherent histories formalism is an attempt to explain how standard rea- +soning based on classical logic can be applied in an isolated quantum system in general, and +in the cosmological Universe in particular [5,6,53–57]. As we will see, it is closely related to +the Kolmogorov consistency criterion. +The approach starts by introducing a decoherence functional D for two histories x ≡ +(xn, . . . , x2, x1) and y ≡ (yn, . . . , y2, y1) “happening” at times tn > · · · > t1: +D(x; y) ≡ tr{ΠxnUn . . . Πx2U2Πx1U1ρ0U † +1Πy1U † +2Πy2 . . . U† +nΠyn}. +(6) +Here, the projectors and unitary time evolution operators have the same meaning as in Eq. (1) +and we immediately confirm that the diagonal elements of the decoherence functional corre- +spond to our previously introduced joint probabilities: p(xn, . . . , x1) = D(x; x). +Depending on the precise reference, different notions of “consistency”, “decoherence” or +“(quasi)classicality” have been introduced based on the decoherence functional. It is beyond +the scope of this article to review them all here, so we restrict the discussion to the two +most commonly employed definitions. First, Griffith originally proposed what we here call +the consistent histories condition [53]: +consistent histories: +ℜ[D(x; y)] = 0 +for all +x ̸= y, +(7) +i.e., the vanishing of the real part of the decoherence functional for different histories. Gell- +Mann and Hartle, among others (see, e.g., Refs. [56,57] and references therein) prefer to use +the following condition that we call the decoherent histories condition: +decoherent histories: +D(x; y) = 0 +for all +x ̸= y. +(8) +Three immediately obvious remarks follow. +First, condition (8) implies Eq. (7). +Second, +D(x; y) = 0 always if xn ̸= yn, i.e., the final “measurement results” cannot be different. +Third, Eq. (7) implies the Kolmogorov consistency condition (2) (and hence so does Eq. (8)). +Confirming the last result requires a few lines of algebra, but it was already shown by Grif- +fiths [53] and many others and will thus not be repeated here. +A not so obvious conclusion is that the decoherent histories condition is strictly stronger +than the consistent histories condition. This is explained with a result of Di´osi [58], who con- +sidered two decoupled quantum systems A and B prepared in a decorrelated state ρA(t0) ⊗ +ρB(t0), unitarily evolving without interaction according to UA⊗UB and measured with decor- +related projectors ΠA +x ⊗ ΠB +x′. In this situation, one immediately confirms that the joint de- +coherence functional factorizes as DAB(x, x′; y, y′) = DA(x; y)DB(x′; y′), where unprimed +(primed) histories refer to subsystem A (B). Now, suppose that A and B separately satisfy +the decoherent histories condition. Then, this is also the case for the non-interacting com- +posite AB, as one would intuitively expect. However, this conclusion does not hold for the +consistent histories condition, thus the latter cannot imply the former. Thus, Di´osi’s argument +is typically invoked to say that Eq. (8) is a more meaningful condition than Eq. (7). +Now, if we consider the probabillities pAB(x, x′) = pA(x)pB(x′) for decoupled system, they +factorize as expected. Interestingly, if A and B separately satisfy the Kolmogorov consistency +condition, then this is also true for the composite AB. Thus, Di´osi’s argument cannot be +invoked to refute our definition of classicality based on Eq. (2).2 +Two further statements +2Di´osi also gives a second argument to argue in favour of decoherent instead of consistent histories. Again, +also this second argument does not disfavour our definition. +9 + +SciPost Physics +Submission +are noteworthy: First, what we said above implies that the decoherent histories condition +is strictly stronger than the Kolmogorov consistency condition. Second, confirming the de- +coherent histories condition experimentally is obviously much harder than confirming the +Kolmogorov consistency condition. +Next, we turn to the relation between decoherent histories and OQS decoherence, which +obviously has been already the topic of previous works, see, e.g., the above references and +in particular also Refs. [59, 60]. Quite intuitively, one would expect that histories defined +by measurements in the pointer basis naturally satisfy the decoherent histories condition, +i.e., that OQS decoherence generates decoherent histories, but it has been recognized that +this relation is not that easy [59, 60]. Indeed, since OQS decoherence alone is not sufficient +to imply Kolmogorov consistency, it also cannot be sufficient to imply decoherent histories. +Interestingly, and what seems to have not been recognized yet, is that the extra condition of +Markovianity is again sufficient to show that OQS decoherence implies decoherent histories. +The proof of this statement is given in Appendix A.1. +Finally, we turn to the question whether decoherence in a stronger, global sense can ex- +plain the emergence of decoherent histories. Here, global decoherence means that the unitarily +evolving state is block diagonal with respect to the projectors {Πx}, i.e., ρ(t) = � +x Πxρ(t)Πx +(note that this implies zero quantum discord, Eq. (5), for local system projectors). If that +is the case for all times tk appearing in definition (6) of the decoherence functional, one +immediately confirms that the histories satisfy the decoherent histories condition. Unfortu- +nately, however, if ρ(t) is unitarily evolving this can in general only be the case for trivial +situations. To see this, we restrict the discussion to pure states |ψ(t)⟩, which is sufficient +for isolated systems. Now, from � +x Πx = I we infer that every pure state can be written +as |ψ(t)⟩ = � +x +� +px(t)eiϕx(t)|ψx(t)⟩ with px(t) the probability to measure outcome x and +Πy|ψx(t)⟩ = δx,y|ψx(t)⟩. +Next, notice that the only states |ψ(t)⟩ that are block diagonal +or globally decohered are states with px(t) = δx,x∗ for some x∗, i.e., these states are fully +localized in one subspace or, with respect to the measurement outcomes, we can say that +they are deterministic. Now, this can certainly happen in some cases, for instance, if the +dimension of the subspace x∗ dominates by far all other subspaces (which corresponds to the +usual criterion of x∗ describing an equilibrium state in statistical mechanics), or if the times +tk are carefully chosen such that only on these times |ψx(t)⟩ is approximately localized in one +subspace. Moreover, if Πx commutes with the Hamiltonian, its probability remains constant +and always generates classical statistics. +However, excluding globally conserved quantities, considering interesting nonequilibrium +dynamics and rejecting the unrealistic idea that we are able to carefully choose the times tk, +the state |ψ(t)⟩ cannot remain block diagonal. For instance, if the state has a high initial +probability p(x0) ≲ 1 for some x0 and a high probablity for some final state p(xn) ≲ 1 with +xn ̸= x0, then there must be some intermediate time t where the state passed from x0 to xn +such that px0(t) = 1/2. Thus, global decoherence can only happen under trivial or unrealistic +circumstances. +A summary of this and the last section can be found in Fig. 2. +2.4 +The new approach: General picture +We now discuss the general picture behind the new approach from Refs. [7, 8, 15]. +It is +claimed to be “new” for two reasons. +First, as discussed above, it defines classicality in +terms of the Kolmogorov consistency condition. This differs from the basic question in OQS +10 + +SciPost Physics +Submission +decoherence (“What is the measurement/pointer basis?”) and it is close to but still different +from the histories approach. In particular, Kolmogorov consistency can be independently well +motivated by asking the question “When can a quantum process be simulated by a classical +process?”, as also done recently in Refs. [9–12]. Second, emphasis is put on the following +two physical aspects. First, the focus is not on OQS: even global observables of an isolated +system can behave classical. Second, non-integrability is regarded as essential, or at least +very helpful, to derive classicality. While the relation to chaos has been also studied in OQS +decoherence (see, e.g., Refs. [61–69]), it has not been regarded as essential: the traditional +workhorse model of OQS theory uses an integrable bath of harmonic oscillators (“Caldeira- +Leggett model”) prepared in a canonical Gibbs ensemble. This is avoided in the following by +considering pure states. +To the best of the author’s knowledge, the basic physical picture behind this emergence of +classicality has been already explained by van Kampen in 1954 [70]. Three basic ingredients, +which can hardly count as assumptions, plus one major assumption are necessary to see the +emergence of classicality. The three ingredients are: (i) the system has a well-defined overall +energy, i.e., the energy spread ∆E of the initial wavefunction is sufficiently narrow3; (ii) the +system has many particles N ≫ 1, i.e., the Hilbert space dimension of the aforementioned +energy shell is exponentially large: D ≡ dim H ∼ exp(N); (iii) the system is non-integrable +or, more precisely, it should obey the eigenstate thermalization hypothesis (ETH). Given the +success of the ETH this is considered a mild assumption for realistic many-body systems +found in nature [23,24]. +The major assumption concerns the observable X that one is probing: +according to +Ref. [15, 70] it should be coarse and slow. +Coarseness means that the number of poten- +tial measurement outcomes is much smaller than the Hilbert space dimension: M ≪ D. +Again, this can hardly count as an assumption. In particular, observe that an observable XS +defined for an OQS is a coarse observable in the full system-bath space. Slowness instead is +the crucial assumption and it has been discussed in detail (together with various subtleties) +in Ref. [15]. Intuitively, it means that the time scale τX on which ⟨X⟩(t) = tr{XUtρ0U † +t } +evolves is much longer than the microscopic evolution time scale ℏ/∆E. This is equivalent +to saying that the matrix with elements Xkm = ⟨k|X|m⟩ with respect to an ordered energy +eigenbasis {|k⟩} is narrowly banded. +It is interesting to contrast this approach to previous work done within the consistent +or decoherent histories formalism, where slow (or quasi-conserved) observables also played an +essential role [54,56,57,71]. Without noting the work of von Kampen, the focus in these works +was to derive the consistent or decoherent histories condition by arguing that the wave packet +remains strongly localized along some trajectory, which is described by a classical determin- +istic equation, i.e., it was argued that the wave function |ψ(t)⟩ should remain approximately +localized in one (time-dependent) subspace Πx(t) throughout the dynamics. We questioned +the adequacy of this idea already at the end of the previous section, but here we just point +out that the assumption of remaining localized around some classical trajectory is also not +necessary. As it will come clear below, the pure state |ψ(t)⟩ is allowed to have an abundance +of coherences (even maximal coherences) and can still behave classical. This marks another +and perhaps the most important novel aspect. +3Recall that energy is conserved in an isolated system. So if the initial state is a superposition of macro- +scopically different energies, the analysis should be carried out separately for each component. Moreover, if +there are further conserved quantities, the same argument has to be also applied to them. +11 + +SciPost Physics +Submission +It is also interesting to connect the assumptions above to the OQS decoherence approach. +To this end, consider an OQS and let X = HS be the system Hamiltonian. Since HS is +locally conserved ([HS, HS] = 0), HS is a slow observable provided that the coupling VSB is +weak enough. Furthermore, it is also coarse if dim HS ≪ dim HB, which is usually the case +in practice. Thus, in the weak coupling regime local measurements of the energy should give +rise to classical statistics obeying the Kolmogorov consistency condition (2). This is unison +with the predictions of the pointer basis in the decoherence approach, but we repeat that the +identification of a pointer basis is not necessary in the present approach. However, it should +be emphasized that the notion of slowness is subtle and it does not seem to be a sufficient +criterion for classicality: by precisely tuning the “fine-structure” of Xkm it appears that one +can generate arbitrary exceptions to the “rule” [72], albeit those might not be generic. In any +case, it provides a different perspective on the problem and it gives an immediate intuitive +explanation why the world around us appears classical: human senses are simply to slow and +coarse to resolve the evolution of fast observables that could potentially show quantum effects. +So how can it be that decoherence is not necessary to generate classical measurement +statistics? The following picture lacks rigour, but gives some intuition. +To start with consider a two-level system with Hamiltonian ∆ +2 σz = +∆ +2 (|1⟩⟨1| − |0⟩⟨0|) +and as the observable choose X = σx. Moreover, let the initial state ρ0 with respect to the +eigenbasis |±⟩ = (|0⟩ ± |1⟩)/ +√ +2 of σx be parametrized as +ρ0 = +�(1 + δ)/2 +reiφ +re−iφ +(1 − δ)/2 +� +, +δ ∈ [−1, +1], +0 ≤ r ≤ +√ +1 − δ2 +2 +, +φ ∈ [0, 2π). +(9) +Here, the parameter r quantifies the “strength” of the coherences in the σx basis, which is +always upper bounded by +√ +1 − δ2/2 due to the positivity requirement ρ0 ≥ 0. Now, at an +arbitrary time t the system state in the same basis reads +ρt = +� (1 + δ cos ∆t)/2 + r sin φ sin ∆t +r(cos φ + i sin φ cos ∆t) − iδ +2 sin ∆t +r(cos φ − i sin φ cos ∆t) + iδ +2 sin ∆t +(1 − δ cos ∆t)/2 − r sin φ sin ∆t +� +. +(10) +The diagonal elements equal the probabilities to measure spin up |+⟩ or spin down |−⟩ with +respect to the x-direction. Their time evolution is strongly influenced by the coherences and +therefore the dynamics is not classical. Of course, this is not a counterexample since the +system is neither a non-integrable many-body system nor is the observable coarse and slow. +So what happens for a coarse observable in a non-integrable many-body system? The +single elements of Eq. (10) now become blocks of many elements and the probability px(t) to +find the system in some state x becomes the trace over block x. It will typically contain a +sum of contributions from many coherences ⟨i|ρ0|j⟩ = rijeiφij of the initial state, schematically +written as: +px(t) ∼ +� +i,j +rij sin φij sin ∆ijt. +(11) +Now, observe the following facts. First, for a coarse observable of a many-body system the +number of terms contributing to the sum is huge (of the order eN with N the particle number). +Second, for a non-integrable system the energy differences ∆ij are incommensurable (apart +from rare accidential degeneracies) and effectively random. Thus, unless the φij are precisely +tuned or rij = 0 for most but a few pairs (i, j), Eq. (11) is a sum of many terms of random +12 + +SciPost Physics +Submission +sign and small magnitude.4 Thus, the enormous amount of coherences cannot add up to a +significant contribution and therefore it effectively does not matter whether coherences are +present or not, i.e., as long as one only asks questions about the measurement statistics in +Eq. (1) we can set rij = 0 for all (i, j). +This explanation for classical behaviour is essentially statistical and similar in spirit to +the explanation of the second law. Yes, it is possible that the positions and momenta of all +the molecules in the air surrounding you conspire such that they can be all found in one +corner of the room in the next second, yet this possibility is extremely unlikely. Similarly, it is +possible that all microscopic coherences of a coarse observable align in phase to give rise to a +strong contribution and, consequently, a strong violation of Kolmogorov consistency, yet this +is again extremely unlikely. In essence, this also underlies the decoherence approach. Yes, it +is possible that a qubit in contact with a bath suddenly “recoheres”, yet this would require +again a very unlikely because precisely tuned cooperation of many phases of the system-bath +state. Thus, the emergence of classical behaviour is related to the general phenomenon of +irreversibility, which is extremely hard to avoid given our coarse human senses. +Finally, one might wonder where above the assumption of slowness enters. This is in- +deed not directly visible here. However, our argument why Eq. (11) is small was based on +assumptions about the number of coherences rij and the correlations of the phases φij and +frequencies ∆ij. Unfortunately and in particular for states prepared out of equilibrium, these +assumptions become questionable. Slowness now helps because the microscopic state evolves +on a much shorter time scale than the observable, effectively randomizing many phases before +any noticeable change in px(t) occurs. Thus, from the perspective of the slow observable X, +the systems looks locally equilibrated and the precise microstate no longer matters [15,70]. +3 +Classicality: Derivation and numerical verification +3.1 +Derivation using random matrix theory +To show the approximate validity of the Kolmogorov consistency condition (2) one needs a +model that, ideally, is as general as possible to cover a wide range of scenarios while being at +the same time also specific enough to permit explicit calculations. Unfortunately, these two +desiderata are often mutually exclusive. In Ref. [15] the model was assumed to obey the ETH, +which is currently considered to be a mild assumption for most realistic many-body systems +found in nature [23,24]. The drawback of this generality was that some plausible but at the +end unproven assumptions entered the derivation. +Here, a random matrix theory approach is used, which has been successful in modelling a +variety of generic properties of complex systems [19–24]. Indeed, our current understanding +of the ETH is much based on random matrix theory although the ETH has been shown +to be valid for a much larger class of models. The model considered here is therefore more +restrictive than the model of Ref. [15], but it comes with the benefit that we need less unproven +assumptions (albeit we still need some). +To capture the relevant physics of a non-integrable many-body system we follow Deutsch [73] +4Recall that r2 +ij ≤ pipj (by Cauchy-Schwarz) where pi is the probability to find the system in a certain +microscopic contribution. Generically, a system has overlap with an enormous amount of microscopic states +and, since � +i pi = 1, this implies that rij must be very small. +13 + +SciPost Physics +Submission +and others [74–82] and consider a Hamiltonian of the form +H = H0 + ϵH1. +(12) +Here, H0 is some “baseline” Hamiltonian, ϵ a small parameter and H1 a banded random +Hermitian matrix chosen, e.g., from the Gaussian orthogonal or unitary ensemble. For in- +stance, H0 = HS + HB could be the bare system and bath Hamiltonian and ϵH1 = VSB their +(weak) interaction, but many more examples are imaginable. Note that H0 does not need to +be integrable, it is only assumed to describe a many-body system with an extremely dense +spectrum in the considered energy interval (recall ingredient (i) in Sec. 2.4). Moreover, the +model does not literally assume that the perturbation is random. Instead, the basic idea of +random matrix theory is that some property holds for the overwhelming majority of random +perturbations and that the real physical (and non-random) Hamiltonian then also belongs to +this overwhelming majority. Finally, the smallness of ϵ implies that the range of the spectrum +and the mean level spacing δe of H0 and H are comparable, but their eigenvectors are still +strongly perturbed as long as ϵ is larger than the extremely small level spacing δe. +Let |µ⟩ and |m⟩ be the eigenvectors of H0 and H, respectively. +A central role in the +following is played by the unitary matrix +V m +µ ≡ ⟨m|µ⟩, +(13) +which transforms between the eigenbasis of H0 and H and quantifies their overlap. Denoting +by E[. . . ] averages over the random matrix ensemble, we use that +u(m, µ) ≡ E +���V m +µ +��2� += u(m − µ), +� +n +u(n) = 1, +max +n +u(n) = O(e−N), +(14) +which holds for both the Gaussian orthogonal or unitary ensemble (and even beyond strict +Gaussianity) and whose detailed justification is left to the literature [73–85]. The important +point is that the overlap between eigenvectors of H and H0 is exponentially small in the +particle number N and for our estimate below we will set for notational simplicity maxn u(n) = +D−1 with D the Hilbert space dimension of the energy shell (which is exponentially large in +N). +Next, as an observable we allow any coarse Hermitian operator X = �M +x=1 λxΠx that +commutes with the unperturbed Hamiltonian, [H0, X] = 0, but not with the perturbation H1 +(the case [H1, X] = 0 trivially gives rise to classical dynamics for X). Coarseness means that +M ≪ D and the smallness of ϵ implies that X evolves on a slow time scale. Note that also +observables X with [H0, X] ̸= 0 can behave classical [7,8,15], but the above assumption turns +out to be very convenient for the calculation below. +Then, we consider the joint probability +p(x2, x1) ≡ tr{Πx2U2Πx1U1ρ0U † +1Πx1U † +2} +(15) +to measure x1 at time t1 and x2 at time t2 given an arbitrary initial state (perhaps far from +equilibrium) ρ0. We further introduce the single time probability +p(x2, �� +x1) ≡ tr{Πx2U2U1ρ0U † +1U † +2} +(16) +and consider the difference +Q ≡ p(x2, �� +x1) − +� +x1 +p(x2, x1) = +� +x1̸=y1 +tr{Πx2U2Πx1U1ρ0U † +1Πy1U † +2} ∈ [−1, 1]. +(17) +14 + +SciPost Physics +Submission +The goal in the following is to show that Q is extremely small. This then implies that the +Kolmogorov consistency condition (2) is satisfied at the level of arbitrary two-time probabili- +ties given any initial state ρ0. An extension of the derivation to arbitrary n-time probabilities +with n > 2 is certainly desirable, but it is a very complicated problem, likely requiring novel +techniques. However, abstracting from Refs. [86–89], where theorems about n-time correla- +tions functions for large n were proven under different circumstances, it appears likely that +approximate consistency continues to hold also for n > 2. +To make analytical progress, we first need some assumption about the initial state ρ0 +and at this stage we follow Ref. [15]. In summary, Ref. [15] has described the initial state +by a preparation procedure using some completely positive map M such that ρ0 = Mψ0 = +� +α Kαψ0K† +α, where ψ0 is the state prior to the preparation. +So far, this is completely +general [13,14,90], but now two assumptions are introduced. First, by polar decomposition we +write Kα = √PαVα, where Vα is a unitary and Pα a positive operator. It was now assumed that +the Pα = Pα(X) are functionally dependent on X. Translated into an experimental context, +this means that the experimentalist has control over X (for instance, by measuring it), but +they are not in control over the precise microstate within the subspaces of Πx. Second, it +was assumed that the prior state ψ0 is at equilibrium or, technically speaking, Haar randomly +distributed in the energy shell. Both assumptions thus express the idea that the initial state +preparation can bring a system, which is at equilibrium from a macroscopic point of view +(note that ψ0 is a pure state), arbitrarily far from equilibrium with respect to X. Then, using +a measure concentration inequality in form of Levy’s lemma [91], it was shown that smallness +of Eq. (17) is (in almost all cases) equivalent to showing smallness of +q(x2, x0) ≡ +1 +Dx0 +� +x1̸=y1 +tr{Πx2U2Πx1U1Πx0U † +1Πy1U † +2} +for +x2 ̸= x0. +(18) +Here, Dx0 ≡ tr{Πx0} is the dimension of the subspace associated to measurement outcome +x0. Thus, in essence the term Q, which is a three-point correlation function for the projectors +Πx with unknown correlations with respect to the initial state ρ0, got transformed into the +term q(x2, x0), which is a four-point correlation function for the projectors Πx without any +initial state dependence. +We evaluate Eq. (18) in the energy eigenbasis of H and introduce the following convention, +which is perhaps unconventional but useful for later considerations. Since there will be many +terms indexed by many quantities m1, m2, . . . and µ1, µ2, . . . , where mi (µi) labels energy +eigenvalues of H (H0), we decide to write labels mi (µi) as superscripts (subscripts) as in +Eq. (13) and take the freedom to simply replace them by the number i whenever appropriate. +Thus, Eq. (18) then becomes +q(x2, x0) = +1 +Dx0 +� +x1̸=y1 +1,2,3,4 +� +eiω12(t2−t1)eiω43t1Π12 +x2Π23 +x1Π34 +x0Π41 +y1, +(19) +where ω12 = E1 − E2 denotes the difference between two eigenenergies of H. Next, we recall +that X is a narrowly banded operator (due to its slowness) and this also implies that Πx is +narrowly banded (due to the coarseness of X) [15]. Thus, in an ordered energy eigenbasis +we can safely assume Πmn +x += 0 if m − n ≥ d for a sufficiently large number d. Importantly, +while d ≫ 1 can be enormous in realistic applications, a central feature of slowness and +coarseness is that still d ≪ D. +Thus, d/D serves as a small parameter in the following +15 + +SciPost Physics +Submission +and the corresponding restricted summation is denoted as �1≈2≈3≈4. Finally, notice that +q(x2, x0) = 0 if t2 = t1 or t1 = 0, which allows to turn Eq. (19) into +q(x2, x0) = +1 +Dx0 +� +x1̸=y1 +1≈2≈3≈4 +� +(eiω12(t2−t1) − 1)(eiω43t1 − 1)Π12 +x2Π23 +x1Π34 +x0Π41 +y1. +(20) +We continue by using Eq. (13) and [H0, X] = 0 to obtain +Πmn +x += ⟨m|Πx|n⟩ = +� +µ,ν +⟨m|µ⟩⟨µ|Πx|ν⟩⟨ν|n⟩ = +� +µ +χµ(x)V m +µ ¯V n +µ . +(21) +Here, χµ(x) is the indicator function which is one if and only if Πx|µ⟩ = |µ⟩ and zero otherwise. +Furthermore, note that we use an overbar to denote the complex conjugate. Inserting Eq. (21) +into Eq. (20), we arrive at +q(x2, x0) = +1 +Dx0 +� +x1̸=y1 +1≈2≈3≈4 +� +1,2,3,4 +(eiω12(t2−t1) − 1)(eiω43t1 − 1) +× χ1(x1)χ2(y1)χ3(x0)χ4(x2)V 1 +4 ¯V 2 +4 V 2 +1 ¯V 3 +1 V 3 +3 ¯V 4 +3 V 4 +2 ¯V 1 +2 . +(22) +We have now reached a point, where we can try to evaluate q(x2, x0) using random matrix +theory. However, since q(x2, x0) ∈ R can be positive or negative, showing smallness of q(x2, x0) +on average is only an indicator, but not a gurantee that q(x2, x0) is small in general (because +it could also strongly fluctuate for different realizations of the random Hamiltonian). Thus, +we will actually show that [q(x2, x0)]2 is small, which establishes smallness of q(x2, x0) and +its variance, and which is one point where we go beyond the treatment of Ref. [15]. Thus, we +aim to evaluate +E +� +[q(x2, x0)]2� +≈ +(23) +1 +D2x0 +� +x1̸=y1 +� +x′ +1̸=y′ +1 +1≈2≈3≈4 +� +1,2,3,4 +5≈6≈7≈8 +� +5,6,7,8 +(eiω12(t2−t1) − 1)(eiω43t1 − 1)(eiω56(t2−t1) − 1)(eiω87t1 − 1) +× χ1(x1)χ2(y1)χ3(x0)χ4(x2)χ5(x′ +1)χ6(y′ +1)χ7(x0)χ8(x2) +× E +� +V 1 +4 ¯V 2 +4 V 2 +1 ¯V 3 +1 V 3 +3 ¯V 4 +3 V 4 +2 ¯V 1 +2 V 5 +8 ¯V 6 +8 V 6 +5 ¯V 7 +5 V 7 +7 ¯V 8 +7 V 8 +6 ¯V 5 +6 +� +. +Note that the ensemble average is only performed over the matrix elements V m +µ , but excludes +the frequencies ωmn. In principle, these should be included in the ensemble average as well, +but the smallness of the random perturbation and the extremely small mean energy level +spacing δe suggest that the behaviour of q(x2, x0) is insensitive to small perturbations of ωmn +for times much smaller than the extremely long Heisenberg time ℏ/δe (for further justification +see Refs. [79–81,92]). +Evaluation of Eq. (23) is fascillitated by the fact that ten constraints apply. First, due +to the factors eiωijt − 1 we infer the four constraints m1 ̸= m2, m3 ̸= m4, m5 ̸= m6 and +m7 ̸= m8. Second, due to the fact that x1 ̸= y1, x′ +1 ̸= y′ +1 and x2 ̸= x0 (see Eq. (18)) we find +the six constraints µ1 ̸= µ2, µ3 ̸= µ4, µ5 ̸= µ6, µ7 ̸= µ8, µ3 ̸= µ8 and µ4 ̸= µ7. Nevertheless, +evaluation of Eq. (23) remains challenging even under the simplest approximation that we +will employ here (though we discuss corrections later on). This approximation assumes that +the V m +µ +are independent zero-mean Gaussian random numbers obeying +E +� +V m +µ +� += 0, +E +� +V m +µ V n +ν +� += E +� ¯V m +µ ¯V n +ν +� += 0, +E +� +V m +µ ¯V n +ν +� += δmnδµνu(m − µ), +(24) +16 + +SciPost Physics +Submission +with δmn and δµν denoting the standard Kronecker symbol with super- or subscripts, re- +spectively. The ensemble average in Eq. (23) can then be evaluated using Isserlis’ theorem, +which turns an expectation value of 2n random variables into sums over “pairings” where +each pairing is a product of n pairs. As an example, consider +E +� +V 1 +3 ¯V 2 +4 V 2 +4 ¯V 1 +3 +� += E +� +V 1 +3 ¯V 2 +4 +� +E +� +V 2 +4 ¯V 1 +3 +� ++ E +� +V 1 +3 ¯V 1 +3 +� +E +� +V 2 +4 ¯V 2 +4 +� += δ12δ34u2(m1 − µ3) + u(m1 − µ3)u(m2 − µ4). +(25) +Quite discomfortingly, the ensemble average in Eq. (23) involves sixteen random numbers. +In Appendix A.2 a numerical code is detailed that generates all pairings using Isserlis theorem +while respecting the ten constraints mentioned above. Then, from the total amount of 40,320 +pairings 347 distinct pairings (no multiplicity) survive. It has been found too demanding +to write a programme that automatically estimates Eq. (23) and since it is very tiring to +investigate 347 cases manually, we look for the most dominant contributions. This is justified +because we are only interested in an order-of-magnitude estimate of E{[q(x2, x0)]2}, not its +exact value. +To find the leading order contribution, we observe that each pairing gives rise to a different +number of distinct Kronecker deltas. In general, the fewer the Kronecker deltas, the larger the +contribution because each Kronecker delta “kills” a high dimensional sum. Some care, how- +ever, is required because the sums run over spaces with potentially very different dimension. +Specifically, every lower subscript runs over a subspace with dimension equal to the rank of +some projector Πx, which is always smaller than D but could still be comparable to it. In +contrast, six out of the eight superscripts run over subspaces with dimension d ≪ D. There- +fore, the leading order contributions are given by the terms that have the fewest Kronecker +deltas in total or the fewest Kronecker deltas with respect to the subscripts. +Starting with the latter, the programme from Appendix A.2 shows that the pairing with +the fewest amount of subscript-Kronecker deltas is +E +� +V 1 +2 ¯V 4 +2 +� +E +� +V 1 +4 ¯V 6 +8 +� +E +� +V 2 +1 ¯V 3 +1 +� +E +� +V 2 +4 ¯V 5 +8 +� +E +� +V 3 +3 ¯V 8 +7 +� +E +� +V 4 +3 ¯V 7 +7 +� +E +� +V 5 +6 ¯V 8 +6 +� +E +� +V 6 +5 ¯V 7 +5 +� +≈ D−8δ14δ16δ23δ25δ38δ47δ48δ37. +(26) +Here, all u(m−µ) ≥ 0 were replaced for notational simplicity with their maximum value D−1 +in agreement with the comment below Eq. (14). Then, inserting this term into Eq. (23) and +killing all the sums, one obtains the contribution +1 +D2x0D8 +1≈2 +� +(eiω12(t2−t1) − 1)(eiω12t1 − 1)(eiω21(t2−t1) − 1)(eiω21t1 − 1) +× +� +x1̸=y1 +� +x′ +1̸=y′ +1 +� +1,2,3,4,5,6 +χ1(x1)χ2(y1)χ3(x0)χ4(x2)χ5(x′ +1)χ6(y′ +1). +(27) +Now, since |eiω − 1| ≤ 2 for all ω ∈ R the first line is estimated by setting eiω − 1 = O(1) +such that �1≈2 O(1) ≈ Dd. The second line can be exactly evaluated by introducing the +Hilbert subspace dimension Dx ≡ tr{Πx} = � +1 χ1(x) associated to the projector Πx. Thus, +one obtains +1 +D2x0D8 Dd +� +x1̸=y1 +� +x′ +1̸=y′ +1 +DxDyDx0Dx2Dx′Dy′ = dDx2 +Dx0D3 +� +1 − +� +x +�Dx +D +�2�2 +. +(28) +17 + +SciPost Physics +Submission +This term is certainly negligible small. +We continue by considering the terms with the fewest Kronecker deltas in total. +The +programme from Appendix A.2 shows that these terms have six Kronecker deltas in total and +there are the following four of them (neglecting the universal prefactor D−8): +δ13δ57δ14δ67δ23δ58, +δ13δ68δ14δ68δ23δ57, +δ24δ57δ24δ67δ13δ58, +δ24δ68δ24δ68δ13δ57. +(29) +Let us look at the first one. Inserting it into Eq. (23) gives rise to the contribution +1 +D2x0D8 +1≈2≈4 +� +5≈6≈8 +� +(eiω12(t2−t1) − 1)(eiω41t1 − 1)(eiω56(t2−t1) − 1)(eiω85t1 − 1) +× +� +x1̸=y1 +� +x′ +1̸=y′ +1 +� +1,2,5,6 +χ1(x1)χ2(y1)χ2(x0)χ1(x2)χ5(x′ +1)χ6(y′ +1)χ6(x0)χ5(x2). +(30) +Using the same reasoning as above, the summation over the superscripts is approximated as +D2d4 and the summation over the subscripts gives δx1x2δy1x0δx′ +1x2δy′ +1x0D2 +x0D2 +x2. Consequently, +1 +D2x0D8 D2d4D2 +x0D2 +x2 = D2 +x2 +D2 +d4 +D4 , +(31) +which is still negligible small, though considerably larger than Eq. (28). +Using the same +strategy, it turns out that also the remaining contributions in Eq. (29) have the same scaling. +Thus, to summarize, it was shown that the dominant contributions to E{[q(x2, x0)]2} scale +like (d/D)4, which is negligible small due to the slowness and coarseness of the considered +observable X. +Remarkably, this scaling holds for all times t1 and t2 (and is thus clearly +applicable out of equilibrium). Several assumptions concerning the ETH ansatz as detailed +in Ref. [15] could be overcome due to the fact that we employed a more transparent but also +more restrictive random matrix theory approach from the beginning. Clearly, also the random +matrix theory approach is not without assumptions, but recalling its enormous success to deal +with non-integrable or chaotic many-body systems [19–24] most assumptions should turn out +to be mild in practice. +Nevertheless, a critical point concerns the approximation that the V m +µ +are Gaussian and +uncorrelated. Both cannot be strictly true. First, unitarity implies |V m +µ |2 ≤ 1, which is not +satisfied by a Gaussian distribution. Second, unitarity also implies that � +µ V m +µ ¯V n +µ = δmn +and �m V m +µ ¯V m +ν += δµν, which is satified on average, see Eqs. (14) and (24), but not for a +single realization if the V m +µ are taken uncorrelated. While one might expect corrections to be +negligible in many cases due to the hugh Hilbert space dimension and the smallness of V m +µ , +Dabelow and Reimann have shown that they can be important [79, 81]. Yet, their goal was +to determine the exact time-dependent behaviour of expectation values, instead of the rough +order-of-magnitude estimate that we were interested in here. Nevertheless, Appendix A.3 +confirms that at least the leading order correction does not give rise to a different scaling. +Unfortunately, the author found the calculation of higher order corrections to be intractable, +so the present derivation might be best interpreted as strong evidence, but no proof, that slow +and coarse observables imply classical measurement statistics in random matrix models and, +likely, also beyond. +3.2 +Numerical verification +To illustrate the main features of the present approach to classicality, we use a simple toy +model. This model cannot compete with the simulations of more realistic, non-random models +18 + +SciPost Physics +Submission +in Refs. [7,8,15]. Yet, the results clearly support our general findings and they are also used +to point out interesting features that were not yet investigated. +The toy model describes energy exchanges between two energy bands and has been studied +in detail in Ref. [93]. Each band is described by N equidistant energy levels such that the +baseline (unperturbed) Hamiltonian is +H0 = δE +N−1 +� +i=0 +i +N − 1(|i⟩⟨i| + |i + N⟩⟨i + N|). +(32) +Here, δE sets an overall energy scale, which we choose in our numerics equal to δE = 0.5. +Moreover, |i⟩ describes an energy eigenstate of the first (second) band if i ∈ {0, . . . , N − 1} +(i ∈ {N, . . . , 2N − 1}). The random coupling between the bands is mediated by +ϵH1 = ϵ +N−1 +� +i,j=0 +vij|i⟩⟨j + N| + H.c., +(33) +where the vij are independent zero mean unit variance Gaussian random numbers. +The +observable X we consider quantifies the energy imbalance between the two bands and is +defined as +X = +1 +√ +2N +N−1 +� +i=0 +(|i + N⟩⟨i + N| − |i⟩⟨i|) = +1 +√ +2N +(Π2 − Π1), +(34) +where Π1 (Π2) is the projector on the first (second) energy band. Clearly, this is a coarse +observable with two eigenspaces of dimension N. The prefactor (2N)−1/2 is pure convention, +but bears the advantage that the observable X has the same “size” (meaning that tr{X} = 0 +and tr{X2} = 1 always) for different N. The condition for X to be slow was worked out in +Ref. [93] and reads +16π2Nϵ2 +δE2 +≪ 1, +(35) +i.e., weak coupling gives rise to a slow observable as usual (note that [H0, X] = 0). In the +numerical simulations we choose ϵ = ϵ(N) such that the left hand side of Eq. (35) becomes +0.01 unless otherwise stated. +Moreover, the relaxation time-scale of X is given by τX = +(4πϵ2N)−1δE [93]. +We first consider the structure of the observable X in more detail as it played an important +role in our study. For this purpose Fig. 3 shows matrix elements of the observable X and +the projector Π1. Specifically, Fig. 3(a) shows a “matrix plot” of |Xkm| in an ordered energy +eigenbasis of H = H0 + ϵH1 for N = 600 (note that the Hilbert space dimension is D = 2N) +and we can immediately confirm that X is narrowly banded. Moreover, the coarseness of X +implies that also the projectors Π1 and Π2 are narrowly banded. This is shown in Fig. 3(b) +by plotting the (absolute value of) the matrix elements of Π1 along the “counter-diagonal” +(from the lower left to the upper right corner). This is done for the three sizes N = 60 (larger +black circles), N = 600 (medium sized pink circles) and N = 6000 (tiny blue circles). Besides +the observation of narrow bandedness, we also note that the off-diagonal elements appear +essentially random and vary erratically. Furthermore, they decay with system size. More +specifically, the black circles are roughly one order of magnitude larger than the blue circles, +which suggests a scaling 1/ +√ +D for the off-diagonal elements. These points are in complete +agreement with the general predictions of the ETH [23,24]. +19 + +SciPost Physics +Submission +Figure 3: Matrix elements explaining the structure of the considered observable. +Let us now consider the dynamics and test whether the time evolution of X is sensitive +to a dephasing operation. To this end, we plot and compare in Fig. 4 the two quantities +1 +� +xs=0 +p(1t, xs) = +1 +� +x=0 +tr +� +Π1e−iHtΠxe−iHs|ψ0⟩⟨ψ0|eiHsΠxe−iHt� +, +(36) +p(1t, �� +xs) = tr +� +Π1e−iH(t+s)|ψ0⟩⟨ψ0|eiH(t+s)� +, +(37) +i.e., the probability to find the system in the first energy band at time t with and without +dephasing operation at time s < t corresponding to the pink dashed and black solid line in +Fig. 4, respectively. This is done for a single realization of the random matrix Hamiltonian +and for a single Haar-randomly chosen initial state confined to the first energy band (Fig. 4 +was found to be representative as different realizations of the Hamiltonian or initial state give +rise to a similar picture). The dephasing operation is done at s = τX (defined below Eq. (35) +and indicated by a vertical line) and the final measurement happens at t + s = 3τX. Note +that the dephasing clearly happens before the system equilibrates. Again, we consider the +three system sizes N = 60 (a), N = 600 (b) and N = 6000 (c) and it becomes immediately +evident that the process becomes classical with increasing N. This confirms our main result, +but Fig. 4 contains two more important pieces of information. +First, the circles and crosses in Fig. 4 are generated for the same Hamiltonian and initial +state, but by using truncated projectors, which are obtained from Πx by setting all off-diagonal +elements with a distance to the diagonal greater than d/2 to zero. +The choice for d was +d = D0.7 and the reason for choosing this precise value of the exponent becomes clear later. +For now it only matters that we confirm that d/D = D−0.3 is very small for large D and that +we see in Fig. 4 that even for small D the dynamics is unchanged. This provides numerical +evidence that truncating the sums, which was a crucial step to arrive at Eq. (20) in our general +derivation, is justified. +Second, another important piece of information is revealed by the number ∆ in the top +right corner of each plot. +It equals the trace norm between the states with and without +dephasing defined as +∆ = ∆(ρ, Dρ) ≡ 1 +2tr +� +(ρ − Dρ)2 ∈ [0, 1], +(38) +20 + +SciPost Physics +Submission +0 +1000 +2000 +3000 +tδE +0.4 +0.6 +0.8 +1.0 +p(1t) +∆ = 0.5 +(a) N = 60 +0 +1000 +2000 +3000 +tδE +∆ = 0.45 +(b) N = 600 +0 +1000 +2000 +3000 +tδE +∆ = 0.46 +(c) N = 6000 +Figure 4: Exemplary check of the Kolmogorov consistency condition. +102 +103 +N +10−2 +10−1 +distance ⟨Q⟩ +α ≈ 0.9 +α ≈ 0.6 +Figure 5: Scaling behaviour of (a suitable time average of) Q. +where Dρ = � +x ΠxρΠx denotes the dephased state. The trace norm is a distance measure +characterizing the distinguishability of two quantum states and it has a wide range of ap- +plications and favorable properties [90], including that (1 + ∆)/2 is the maximum success +probability to distinguish between ρ and Dρ in an unbiased mixture given unlimited mea- +surement power. Thus, ∆ seems to be well suited to measure the amount of coherences in ρ. +Interestingly, we show in Appendix A.4 that maxρ ∆(ρ, Dρ) = 1/2. Now, observing the values +for the trace norm in Fig. 4 we see that they are very close to the maximum possible value. +In that sense, the dephasing operation is (almost) maximally invasive and a lot of coherences +is destroyed. This demonstrates not only that decoherence is not needed to explain classical +behaviour, but much more: even maximally coherent states can show classical behaviour! +Next, Fig. 5 shows the scaling behaviour of Q, Eq. (17), for s = τX and t + s = 3τX +as a function of the Hilbert space dimension. This is done by integrating the absolute value +of the difference between the black solid and pink dashed curves, shown in Fig. 4, divided +by the length 2τX of the interval to give rise to average ⟨Q⟩. Moreover, to minimize the +risk of statistical outliers, this is done for three different realizations of the random matrix +21 + +SciPost Physics +Submission +0 +1000 +2000 +3000 +tδE +0.4 +0.6 +0.8 +1.0 +(a) p(1t) +103 +N +10−2 +10−1 +(b) ⟨Q⟩ +0 +1000 +2000 +3000 +tδE +0.4 +0.6 +0.8 +1.0 +(c) p(1t) (two dephasings) +Figure 6: Violation of classicality for a highly atypical states (a), but approximate +validity for moderately atypical states (b) and for two dephasing operations and +typical states (c). +Hamiltonian and three Haar-randomly chosen initial states, thus giving nine realizations for +each N as indicated by black circles in Fig. 5. To extract the scaling, we average these nine +points for each N and fit a curve of the form +⟨Q⟩ ∼ +1 +Dα , +(39) +which is inspired by Ref. [15]. By looking at Fig. 5 (note the logarithmic scale), one might +wonder whether it is a good idea to fit all the data by a straight line (pink dashed line with +exponent α = 0.9) because the behaviour for N ≲ 400 clearly deviates from the straight line +fit obtained for N ≥ 600 (blue solid line with exponent α = 0.6). It is not completely clear +to the author what causes the discrepancy, but in Ref. [93] it was observed that the weak +coupling approximation requires the side constraint 8π2N2ϵ2/δE2 > 1, which for our choice +of ϵ implies N > 200. This might explain the “anomalous” behaviour for small N. Also note +that the exponent α = 0.6 roughly fits the scaling behaviour observed in Ref. [15] (where α +was observed to be in the range [0.25, 0.6]). +In any case, we use the fit to determine the number d at which we truncated the projectors +to generate the pink crosses and black circles in Fig. 4. Namely, our main result predicted +a scaling of the form (d/D)4 for [qt,s(x0)]2. This suggests that ⟨Q⟩ should scale as (d/D)2. +Comparing with D−α for α = 0.6 gives d = D0.7 as used in Fig. 4. +Finally, we challenge the present approach by relaxing certain assumptions. +First, we +ask what happens if the initial state is not randomly chosen within the first energy band. +Figure 6(a) shows the breakdown of classicality for a highly atypical initial state |ψ0⟩ = |i⟩ for +some randomly selected i ∈ {0, . . . , N − 1} for N = 6000 (we use the same convention as in +Fig. 4 where the black solid line refers to free evolution and the pink dashed line to an evolution +interrupted by a dephasing operation happening at the vertical black line). Experimentally, +preparing such an initial state requires precise microscopic control over the eigenstates of +each energy band, which clearly violates the agreement made above Eq. (18). Nevertheless, +classicality is quickly restored for more realistic states as demonstrated in Fig. 6(b). It shows +⟨Q⟩ for N = 600, N = 2000 and N = 6000 for five different realizations of |ψ0⟩ = |i⟩ (black +circles) and for five different initial states |ψ0⟩ ∼ � +i∈K ci|f(i)⟩, where K ⊂ {0, . . . , N − 1} is +a randomly chosen subset with 0.005N many elements (i.e., 0.5% of the energy levels in the +22 + +SciPost Physics +Submission +0 +1000 +2000 +3000 +tδE +0.6 +0.8 +1.0 +p(1t) +∆ = 0.47 +(a) weak coupling +0 +100 +200 +300 +tδE +∆ = 0.46 +(b) medium coupling +0 +10 +20 +30 +tδE +∆ = 0.45 +(c) strong coupling +Figure 7: Exemplary violation of classicality at strong coupling/for fast X. +first band are initially populated) and the ci are zero mean unit variance Gaussian random +numbers (pink triangles). Despite quite large fluctuations, Fig. 6(b) indicates a scaling law +and the pink triangles are (on average) clearly below the black circles, showing the emergence +of classicality even for moderately atypical states with a small fraction of populated levels. +Finally, Figure 6(c) shows what happens for two dephasing operations for N = 6000 and an +initial random state as used also in Fig. 4. While this is certainly not conclusive, it indicates +that for sufficiently large dimensions the here introduced concept of classicality is robust also +for n ≥ 3 measurements. +Last but not least, Fig. 7 investigates the impact of the coupling strength on classicality, +which is directly related to the slowness of X. Here, weak, medium or strong coupling means +that the right hand side of Eq. (35) was fixed to 0.01, 0.1 or 1, which is inversely proportional +to the relaxation time scale. The plots are done for N = 6000 using again an initial state +as in Fig. 4. +One sees that classicality is well satisfied up to medium coupling strength, +but fails in the strong coupling regime. This is not a deficit of the present theory because +clearly not all observables can behave classical. For strong coupling the eigenenergies of the +total Hamiltonian can no longer be approximated by the local eigenenergies of the two bands, +but are strongly hybridized, and is questionable how far X describes any meaningful energy +difference in this case. +4 +Conclusion +The first half of this paper compared and contrasted well established and important ap- +proaches to classicality, namely decoherence in OQS and consistent/decoherent histories, with +recent abstract research [9–12] as well as numerical evidence [7, 8, 15] and general deriva- +tions [15] of classicality based on the Kolmogorov consistency condition. Arguably, the differ- +ence between the consistent/decoherent histories condition and the Kolmogorov consistency +condition is small. However, Kolmogorov consistency is easier to verify experimentally than +consistent/decoherent histories and it can be independently well motivated from an opera- +tional perspective. Moreover, we established that quantum Markovianity is a key concept to +relate the decoherence approach in OQS to both the consistent/decoherent histories and the +Kolmogorov consistency condition. Figure 2 summarizes the first part. +23 + +SciPost Physics +Submission +The second half of the paper has given an independent derivation of the Kolmogorov +consistency condition based on a random matrix theory model and numerically verified the +correctness of the involved approximations. In fact, by looking at Eq. (18) it even becomes +clear that we have derived the stronger decoherent histories condition for an experimentally +relevant class of initial states. +Remarkably, it was explicitly shown that even maximally +coherent states can give rise to classical dynamics for global observables. +Several interesting research avenues open up for the future. For instance, classicality could +here be only established for “mini-histories” with two measurement results and extending +the derivation to longer histories, as done in a different context in Refs. [86–89], is highly +desirable. Moreover, various fundamental question might appear in a new light, for instance, +the relationship between quantum Darwinism [36, 37] and quantum Markovianity, or the +implications of the present findings for (quantum) cosmology [35]. +Acknowledgements +This manuscript has significantly benefited from discussions with and feedback from Lennart +Dabelow about random matrix theory, Wojciech Zurek about the quantum-to-classical tran- +sition, and John Calsamiglia and Andreas Winter about trace distance bounds. For further +discussions and feedback I thank Victor Bastidas, Giulio Gasbarri, Jochen Gemmer, Joseph +Schindler and Jiaozi Wang. +Funding information +The author is finanically supported by “la Caixa” Foundation (ID +100010434) under the fellowship code LCF/BQ/PR21/11840014 and co-funded by the Spanish +Agencia Estatal de Investigaci´on (project no. PID2019-107609GB-I00), the Spanish MINECO +(FIS2016-80681-P, AEI/FEDER, UE), the Generalitat de Catalunya (CIRIT 2017-SGR-1127), +and the European Commission QuantERA grant ExTRaQT (Spanish MICINN project PCI2022- +132965). +A +Appendix +A.1 +Decoherence, Markovianity and consistency +This appendix assumes the reader to be familiar with superoperators, in particular instru- +ments, completely positive maps and completely positive and trace preserving maps. +In- +troductions are provided, e.g., in Refs. [13, 14, 41, 90]. All superoperators are denoted with +calligraphic symbols A, E, P, . . . . +We start by defining a quantum Markov process following Ref. [16], for introductory +treatments see Refs. [13,14]. This definition recognizes the crucial role played by an external +agent (or experimenter or observer), who interrogates or intervenes the dynamics of an OQS +at a set of discrete times {tn, . . . , t2, t1}. Each intervention at each time tk is described by +an instrument {Ak(rk)}, which is a set of completely positive maps Ak(rk) adding up to a +completely positive and trace preserving map Ak ≡ � +rk Ak(rk). Importantly, these maps +only act on the system Hilbert space, encapsulating the idea that the external agent has +no precise control over the bath degrees of freedom. +Moreover, rk denotes some abstract +measurement outcome, not necessarily related to the xk appearing in the main text. Now, +24 + +SciPost Physics +Submission +a quantum process is Markovian if the response of the OQS to any sequence (or history) of +interventions {An(rn), . . . , A2(r2), A1(r1)} can be written as +˜ρS(tn|rn, . . . , r2, r1) = An(rn)En,n−1 · · · A2(r2)E2,1A1(r1)E1,0ρS(t0), +(40) +where {Ek,k−1}n +k=1 is a set of completely positive and trace preserving maps, which—importantly— +do not depend on the interventions or initial system state. Moreover, ˜ρS(tn|rn, . . . , r2, r1) is +the subnormalized OQS state conditioned on the sequence of interventions, which happens +with probability p(rn, . . . , r2, r1) = trS{˜ρS(tn|rn, . . . , r2, r1)}. Finally, notice that the Ek,k−1 +are also known as dynamical maps as they progragate the system state forward in time from +tk−1 to tk. These maps encode the influence of the bath or environment and, according to +Eq. (40), a Markov process is precisely characterized by the fact that the influence of the bath +can be neatly separated from the interventions Ak(rk) of the external agent. +A few more words of clarification might be helpful. First, one can show that Eq. (40) +reduces to the classical Markov condition in an appropriate limit. Second, for classical causal +models it reduces to the causal Markov condition of Ref. [40]. Third, the validity of Eq. (40) +can be checked by local interventions on the system only. Fourth, the existence of a Markovian +quantum master equation for ρS(t), as often studied in OQS theory, does not imply the validity +of Eq. (40), although the converse is true (see in particular Ref. [18]). Note that the identity +operator I (“do nothing!”) is an instrument too such that it is meaningful to define the +dynamical map Eℓ,k ≡ Eℓ,ℓ−1 · · · Ek+1,k for any ℓ − k > 1. Fifth, Eq. (40) really is a statement +about the multi-time behaviour of an OQS, and the validity of Eq. (40) can be shown to be +equivalent to an appropriate formulation of the quantum regression theorem [94]. +Next, we give a rigorous mathematical definition of OQS decoherence and a derivation +that it implies the decoherent histories condition for quantum Markov processes (which in +turn implies Kolmogorov consistency). To the best of the author’s knowledge, no definition of +OQS decoherence exists and the notion is used rather conceptually (what follows is, however, +closely related but not identical to the treatment of Refs. [9,10,12]). However, if one wants to +prove things mathematically, one has to start with a definition. For this purpose, we use the +dephasing operation D in the pointer basis as introduced in Sec. 2.2. Then, for a quantum +Markov process we define OQS decoherence by requiring that +[Eℓ,k, D] = 0 +for all +n ≥ ℓ > k ≥ 1, +(41) +where [A, B] = AB − BA is the commutator in superoperator space. +Is this a good definition of OQS decoherence? +At least it implies that the dynamics +induced by the environment is not able to create coherences in the pointer basis. To see this, +we introduce the superoperator Px,yρ ≡ ΠS +xρΠS +y . Next, suppose that ρS = DρS is some system +state without coherences. Then, Eq. (41) implies that Px,yEρS = 0 for all x ̸= y, i.e., it is +not possible to create coherences in the pointer basis when starting from a decohered state. +Clearly, this captures a key aspect of the OQS decoherence concept, but one could naturally +impose further constraints. For instance, Eq. (41) makes no statement about the decoherence +time tdec and, since the short time dynamics of OQS is complex, one might additionally require +that Eq. (41) is only valid on a coarse time scale, i.e., for tℓ − tk not too small. In any case, +the minimal definition given here turns out to be sufficient to prove that histories in the sense +of Eq. (8) are decoherent. +To see this, we conveniently write the decoherence functional for a quantum Markov +process using superoperators: +D(x; y) = trS{Pxn,ynEn,n−1Pxn−1,yn−1En−1,n−2 · · · Px1,y1E1,0ρS(t0)}. +(42) +25 + +SciPost Physics +Submission +Next, note that the decoherence functional does not change when subjecting it to a final +dephasing operation D in the pointer basis (in fact, the decoherence functional does not +change under any final dephasing): +D(x; y) = trS{DPxn,ynEn,n−1Pxn−1,yn−1En−1,n−2 · · · Px1,y1E1,0ρS(t0)}. +(43) +Now, let k be the first index for which xk ̸= yk, i.e., xℓ = yℓ for all ℓ > k. By the definition +of OQS decoherence, we can then permute D through until we hit the time tk: +D(x; y) = trS{Pxn,ynEn,n−1 · · · Ek+1,kDPxk,ykEk,k−1 · · · Px1,y1E1,0ρS(t0)}. +(44) +Finally, elementary algebra shows that DPxk,ykρ = 0 whatever the input state ρ is. QED. +It is interesting to note that strictly weaker conditions suffice to show Kolmogorov con- +sistency for quantum Markov processes [9, 10, 12], but they seem insufficient to show the +decoherent histories condition. +A.2 +Numerical implementation +This appendix includes some details about how to numerically fascilitate the evaluation of +the expectation values appearing in Eq. (23) using Mathematica [95]. +We are interested in expectation values of the form E[V m1 +µ4 ¯V m2 +µ4 V m2 +µ1 ¯V m3 +µ1 . . . ] with an equal +amount of V - and complex conjugate ¯V -terms. Since each pair in Isserlis theorem requires +one V - and one ¯V -term, not all permutations of (V m1 +µ4 , ¯V m2 +µ4 , V m2 +µ1 , ¯V m3 +µ1 , . . . ) contribute to the +expectation value. One way to create all contributing permutations consists in generating +two lists A = {{m1, µ4}, {m2, µ1}, . . . } and B = {{m2, µ4}, {m3, µ1}, . . . } associated to the +V - and ¯V -terms, respectively, followed by +PermA = Permutations[A]; +Pairings = Map[Sort, Table[Flatten[PermA[[α, k]], B[[k]]], {α, 1, LA!}, {k, 1, LA}], 2]; +Here, LA = Length[A] denotes the lengths of the list A (which equals LB) and consequently +LA! is the length of PermA. The output Pairings now contains all possible pairings of the +form +Pairings = {{{m1, m2, µ4, µ4}, {m2, m3, µ1, µ1}, . . . }, +{{m2, m2, µ1, µ4}, {m1, m3, µ4, µ1}, . . . }, +. . . } +(45) +The lowest level angular bracket {. . . } contains one specific pair with always two Latin and +two Greek indices. The middle level angular bracket contains the product of all pairs, which +form one specific “pairing”. +In general, Pairings will contain many forbidden pairings due to constraints such as +m1 ̸= m2, µ1 ̸= µ2, etc. To filter them out, we map each pairing to a graph with vertices +(m1, m2, . . . , µ1, µ2, . . . ) and edges created by the Kronecker symbols of each pair, e.g., the +pair {m1, m2, µ4, µ4} creates an edge between m1 and m2 and a (redundant) edge between +µ4 and µ4. Then, e.g., to respect the constraint m1 ̸= m2, a pairing is only accepted if there +exists no path in the graph from m1 to m2, see Fig. 8 for a sketch. As an example, the +26 + +SciPost Physics +Submission +Figure 8: +Three example pairings (here, each pairing has two pairs with Latin +indices) and the associated graphs. The constraint m1 ̸= m2 is violated in example +(b) and (c). +following code creates a list accepted that stores the numbers i for which the ith element of +Pairings satisfies the constraints m1 ̸= m2 (further constraints can be easily included): +For[i = 1, i ≤ LA!, i + +, +network = {}; +For[j = 1, j ≤ LA, j + +, +network = Append[network, Pairings[[i, j]][[1]] •−• Pairings[[i, j]][[2]]]; +]; +G = Graph[network]; +test = Length[Flatten[FindPath[G, m1, m2]]]; +If[test == 0, accepted = Append[accepted, i]]; +] +Here, the graph G for each pairing i is created using a set of edges stored in network, where +each edge is symbolized by •−• (typeset as “Esc ue Esc” in Mathematica). +Next, we create a list Rem that simply contains all remaining pairings that satisfy the +constraints above. This can be done by +Rem = Map[Sort, Table[Pairings[[accepted[[k]]]], {k, 1, Laccepted], 2]; +(46) +Note that Sort brings the elements of the pairing in a standard form again. +As hinted at already above, the structure of the pairings can be nicely illustrated with +a graph with edges indicating Kronecker deltas. For further manipulation, we now like to +27 + +SciPost Physics +Submission +convert Rem into a list of graphs: +graphs = {}; +vert = {m1, m2, . . . , µ1, µ2, . . . }; +For[i = 1, i ≤ Laccepted, i + +, +medges = Table[Rem[[i, j]][[1]] •−• Rem[[i, j]][[2]], {j, 1, Length[Rem[[1]]]}]; +µedges = Table[Rem[[i, j]][[3]] •−• Rem[[i, j]][[4]], {j, 1, Length[Rem[[1]]]}]; +edges = Join[medges, µedges]; +graphs = Append[graphs, Graph[vert, edges]]; +] +con = Map[Sort]@ ∗ ConnectedComponents/@graphs; +The final output con contains the connectivity of each graph associated to each accepted +pairing. For instance, if {{m1, m1, µ1, µ3}, {m2, m3, µ2, µ4}, {m2, m4, µ4, µ4}} is one accepted +pairing, then its connectivity is {{m2, m3, m4}, {µ1, µ3}, {µ2, µ4}, {m1}}. +This format has +nice properties as it directly reveals the “structure” of each pairing in terms of (multi-valued) +Kronecker deltas. To find out whether there are multiple pairings with the same structure, +one can run Tally[con]. +Given the connectivity list con, it is also straightforward to count the total number of +sums that get killed due to Kronecker deltas. In the following, the function kills computes +this number for a given pairing and final stores these numbers for each element of con: +kills[list−] := Total[Map[Length, list]] − Length[list]; +final = Table[kills[con[[m]]], {m, 1, Laccepted}]; +Clearly, the above procedure can be also applied to the graphs formed by Greek indices +only—as we needed to do to find the terms with the fewest amount of subscript-Kronecker +deltas around Eq. (26). +A.3 +Higher order corrections +In Ref. [79] (see also Sec. 3.4.4 of Ref. [92]) Dabelow and Reimann developed a systematic +way to take into account correlations among matrix elements, which we briefly summarize +here. Unfortunately, it will turn out that this procedure becomes quickly untractable due to +the fact that we already start with an expectation value over a product of sixteen random +numbers. Moreover, the method does not take into account correlations with respect to both +the perturbed and unperturbed bases |m⟩ and |µ⟩, but only with respect to one of them. +Therefore, while that method was found to work well in Refs. [79, 81, 92], it still does not +provide an exact treatment of the problem. +To start with, notice that the matrix element V m +µ +can be seen as the m’th component +of a D-dimensional vector Vµ. To each Vµ we associate two vectors vµ and wµ, where the +components of vµ are assumed to be independent Gaussian random variables with statistical +properties equal to those of Eq. (24). Then, what was effectively done in the main text was +to approximate +E +� +V 1 +4 ¯V 2 +4 V 2 +1 ¯V 3 +1 V 3 +3 ¯V 4 +3 V 4 +2 ¯V 1 +2 V 5 +8 ¯V 6 +8 V 6 +5 ¯V 7 +5 V 7 +7 ¯V 8 +7 V 8 +6 ¯V 5 +6 +� +≈ E +� +v1 +4¯v2 +4v2 +1¯v3 +1v3 +3¯v4 +3v4 +2¯v1 +2v5 +8¯v6 +8v6 +5¯v7 +5v7 +7¯v8 +7v8 +6¯v5 +6 +� +, +(47) +28 + +SciPost Physics +Submission +which disregards all correlations. Instead, we now replace +E +� +V 1 +4 ¯V 2 +4 V 2 +1 ¯V 3 +1 V 3 +3 ¯V 4 +3 V 4 +2 ¯V 1 +2 V 5 +8 ¯V 6 +8 V 6 +5 ¯V 7 +5 V 7 +7 ¯V 8 +7 V 8 +6 ¯V 5 +6 +� +≈ E +� +w1 +4 ¯w2 +4w2 +1 ¯w3 +1w3 +3 ¯w4 +3w4 +2 ¯w1 +2w5 +8 ¯w6 +8w6 +5 ¯w7 +5w7 +7 ¯w8 +7w8 +6 ¯w5 +6 +� +(48) +and obtain the vectors {wµ} from {vµ} using a Gram-Schmidt procedure, which orthonor- +malizes the set {vµ}, thereby taking into account constraints imposed by the unitarity of V m +µ . +Thus, starting from w1 = v1, we set5 +wµ = vµ − +µ−1 +� +ν=1 +⟨wν|vµ⟩wν +(49) +for all µ ≥ 2 and where ⟨w|v⟩ denotes the standard complex scalar product. Inserting the wµ +in Eq. (48) gives an explicit expression in terms of the independent Gaussian variables vm +µ +that can be calculated using Isserlis’ theorem and takes into account correlations. +Unfortunately, we would need to do this for eight vectors in Eq. (48) (and their complex +conjugates) and the total number of vm +µ -terms, and consequently the number of pairings in +Isserlis’ theorem, quickly grows to astronomically large numbers, even when respecting the +constraints identified in the main text (m1 ̸= m2, µ1 ̸= µ2, etc.). To see this, it might be +helpful to explicity write down the components obtained via the Gram-Schmidt procedure. +Clearly, everything is simple for the first vector: wm +1 = vm +1 . The second vector is also still +managable: wm +2 = vm +2 − � +n ¯vn +1 vn +2 vm +1 . The third vector, however, contains already six terms +with up to seven v-components: +wm +3 = vm +3 − +� +n +¯vn +1 vn +3 vm +1 − +� +n +¯vn +2 vn +3 vm +2 + +� +no +vo +1¯vo +2¯vn +1 vn +3 vm +2 + +� +np +¯vn +2 vn +3 ¯vp +1vp +2vm +1 +− +� +nop +vo +1¯vo +2¯vn +1 vn +3 ¯vp +1vp +2vm +1 , +(50) +and it does not get simpler for the remaining vectors. +However, recall that we are only interested in an order-of-magnitude estimate. Each added +pair of v-terms comes with an extra D-dimensional summation, but also contributes a factor +of the order D−1 due to Eq. (24). In general, one therefore expects that these contributions +roughly cancel each other in an order-of-magnitude estimate provided that the minimum +number of Kronecker deltas as identified in the main text remains the same (if one term in +Isserlis’ theorem gives rise to fewer Kronecker deltas than before, then an additional sum +appears potentially contributing a huge factor). +Whether this is the case has been explicitly checked to lowest order in the Gram-Schmidt +procedure. This means one first sets +wµ ≈ vµ − +µ−1 +� +ν=1 +⟨vν|vµ⟩vν +(51) +for µ ∈ {2, 3, . . . , 8}, which follows from Eq. (49) by replacing wµ by vµ on the right hand side. +This approximation is then inserted into Eq. (48) and only terms with a single additional sum +5To be precise, the here presented procedure only ensures orthogonality, but not normalization. However, +since the real and imaginary coefficients of vµ are each drawn from a zero mean (approximate) Gaussian +distribution with variance 1/2D, it follows that vµ is not only normalized on average, but each single realization +of vµ is strongly concentrated around vectors with unit norm for large D [96]. +29 + +SciPost Physics +Submission +are kept. There are 2·(1+2+· · ·+7) = 56 such single sum contributions, where the factor two +arises because one has to take into account wµ and ¯wµ for µ ∈ {2, 3, . . . , 8}. However, from +the structure of the problem it does not appear that the complex conjugate entries contribute +differently (since we are interested in a real-valued object), so these additional terms can be +neglected. On the other hand, which of the 28 terms gives the worst contribution to the +scaling is not clear. +Therefore, this has been tested with the programme from Appendix A.2 and it was found +that each correction term contains enough (multi-valued) Kronecker deltas to kill at least 6 +sums, i.e., the same number as identified in Eq. (29). The following table explicitly list the 28 +contributions together with their “Kronecker order” L(δ), which equals the number of sums +killed or, equivalently, the minimum of the list final defined at the end of Sec. A.2. +replace... +by... +L(δ) +replace... +by... +L(δ) +w1 +4 +− �9 ¯v9 +1v9 +4v1 +1 +6 +w6 +5 +− �9 ¯v9 +2v9 +5v6 +2 +7 +− �9 ¯v9 +2v9 +4v1 +2 +6 +− �9 ¯v9 +3v9 +5v6 +3 +7 +− �9 ¯v9 +3v9 +4v1 +3 +6 +− �9 ¯v9 +4v9 +5v6 +4 +7 +w3 +3 +− �9 ¯v9 +1v9 +3v3 +1 +6 +w7 +7 +− �9 ¯v9 +1v9 +7v7 +1 +7 +− �9 ¯v9 +2v9 +3v3 +2 +6 +− �9 ¯v9 +2v9 +7v7 +2 +7 +w4 +2 +− �9 ¯v9 +1v9 +2v4 +1 +6 +− �9 ¯v9 +3v9 +7v7 +3 +7 +w5 +8 +− �9 ¯v9 +1v9 +8v5 +1 +7 +− �9 ¯v9 +4v9 +7v7 +4 +7 +− �9 ¯v9 +2v9 +8v5 +2 +7 +− �9 ¯v9 +5v9 +7v7 +5 +6 +− �9 ¯v9 +3v9 +8v5 +3 +7 +− �9 ¯v9 +6v9 +7v7 +6 +6 +− �9 ¯v9 +4v9 +8v5 +4 +7 +w8 +6 +− �9 ¯v9 +1v9 +6v8 +1 +7 +− �9 ¯v9 +5v9 +8v5 +5 +6 +− �9 ¯v9 +2v9 +6v8 +2 +7 +− �9 ¯v9 +6v9 +8v5 +6 +6 +− �9 ¯v9 +3v9 +6v8 +3 +7 +− �9 ¯v9 +7v9 +8v5 +7 +6 +− �9 ¯v9 +4v9 +6v8 +4 +7 +w6 +5 +− �9 ¯v9 +1v9 +5v6 +1 +7 +− �9 ¯v9 +5v9 +6v8 +5 +6 +Finally, one might worry that, even when each single contribution to the expectation +value is very small, the sum of the enormous amount of terms involved gives rise to a giant +prefactor. However, this is unlikely a problem because, first, this prefactor does not scale +with the particle number N and, second, recall that the contributions have different signs, +see, e.g., Eq. (50). Additonal cancellations are therefore likely and were indeed an important +observation in Ref. [79,81]. +A.4 +Trace distance bound under dephasing +The author owes the details of the following proof to Ref. [97]. +We start by noting that strong convexity of the trace norm [90] implies +max +ρ +∆(ρ, Dρ) = max +ψ +∆(ψ, Dψ), +(52) +where ψ = |ψ⟩⟨ψ| is here used to denote pure states. Next, any such |ψ⟩ can be written as +|ψ⟩ = �M +x=1 αx|ψx⟩ with Πy|ψx⟩ = δx,y|ψx⟩ and |αx|2 = ⟨ψ|Πx|ψ⟩ such that � +x |αx|2 = 1. +One then finds +∆(ψ, Dψ) = 1 +2tr +� +� +� +� +� +� +� +M +� +x,y=1 +αxα∗y|ψx⟩⟨ψy| − +M +� +x=1 +|αx|2|ψx⟩⟨ψx| +� +�. +(53) +30 + +SciPost Physics +Submission +Since the {|ψx⟩} are orthonormal for different x and because the trace norm is invariant +under unitary rotations [90] such that we can map |ψx⟩ to some fixed standard vector for each +subspace x, Eq. (53) makes it evident that we can restrict the problem to an M-dimensional +Hilbert space, i.e., +max +ρ +∆(ρ, Dρ) = max +˜ψ +∆ +� +˜ψ, ˜D ˜ψ +� +, +(54) +where | ˜ψ⟩ ∈ CM and the dephasing operation becomes ˜D˜ρ = �M +x=1 |x⟩⟨x|˜ρ|x⟩⟨x| for some set +of one dimensional projectors {|x⟩⟨x|} spanning CM. +Next, we note that we can write the dephasing map as ˜D˜ρ = +1 +M +�M−1 +k=0 Zk ˜ρZ−k with +the M-dimensional diagonal phase unitary Z = �M +x=1 e2πi(x−1)/M|x⟩⟨x|. 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Winter, private communication (2022). +37 + diff --git a/rdE0T4oBgHgl3EQfrQFV/content/tmp_files/load_file.txt b/rdE0T4oBgHgl3EQfrQFV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea491054b7bd0dc80d7a95db1617952b6f696823 --- /dev/null +++ b/rdE0T4oBgHgl3EQfrQFV/content/tmp_files/load_file.txt @@ -0,0 +1,2031 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf,len=2030 +page_content='SciPost Physics Submission Classicality with(out) decoherence: Concepts, relation to Markovianity, and a random matrix theory approach Philipp Strasberg F´ısica Te`orica: Informaci´o i Fen`omens Qu`antics, Departament de F´ısica, Universitat Aut`onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain philipp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='strasberg@uab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='cat January 9, 2023 Abstract Answers to the question how a classical world emerges from underlying quan- tum physics are revisited, connected and extended as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, three dis- tinct concepts are compared: decoherence in open quantum systems, consis- tent/decoherent histories and Kolmogorov consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, the crucial role of quantum Markovianity (defined rigorously) to connect these concepts is es- tablished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Third, using a random matrix theory model, quantum effects are shown to be exponentially suppressed in the measurement statistics of slow and coarse observables despite the presence of large amount of coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is also numerically exemplified, and it highlights the potential and importance of non-integrability and chaos for the emergence of classicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Classicality: Definitions and approaches 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1 Definition used in this work 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2 The decoherence approach for open quantum systems 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='3 Consistent and decoherent histories 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 The new approach: General picture 10 3 Classicality: Derivation and numerical verification 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1 Derivation using random matrix theory 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2 Numerical verification 18 4 Conclusion 23 A Appendix 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1 Decoherence, Markovianity and consistency 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2 Numerical implementation 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='3 Higher order corrections 28 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 Trace distance bound under dephasing 30 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='02563v1 [quant-ph] 6 Jan 2023 SciPost Physics Submission References 31 1 Introduction It is an obvious everyday fact that the world around us does not show direct quantum effects: we can safely disregard the wave-like behaviour of matter and do not need to worry about the effects of measurement backaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' But this causes a conundrum because our everyday world is built out of particles that are fundamentally quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Studying the emergence of classicality from underlying quantum physics is thus of foundational importance, but also has great practical relevance for the realization of quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Yet, the questions how to prove the emergence of classicality and also the prior question what needs to be proved are not fully settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The present paper aims to clarify and extend the discussion and it is divided into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The first part (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2) is about clarifications and makes two contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The first contribution is to give a focused overview over three different approaches to the question what needs to be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' These are the decoherence approach in open quantum systems (OQS) [1–4], the consistent/decoherent histories formalism studied often in the cosmological context [5, 6], and a recent approach based on the notion of Kolmogorov consistency [7–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The second contribution emphasizes the crucial role played by the rigorous definition of (multi- time) quantum Markovianity [16–18] (for introductions see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [13,14]), which connects the decoherence approach in OQS to the other two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To the the best of the author’s knowledge, this connection has not yet been made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The second part (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 3) is about extensions inspired by recent research results [7,8,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Therein it has been observed that isolated many-body systems can behave classical even in presence of large amount of coherences and that non-integrability and chaos might be the key to understand this behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In particular, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [15] argued that this is a generic effect for a large class of observables (specified in greater detail later on) and estimated that deviations from classical behaviour are exponentially small in the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Here, we confirm this estimate and provide an alternative derivation of it in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1, which is inspired by the idea to model complex isolated quantum systems by random matrix theory [19–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, a simple model is used to illustrate important features of this new approach to classicality in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2, before concluding in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For completeness, we remark that there are alternative explanations of classicality, which we will not review here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For instance, classicality is sometimes explained by gravity as the fun- damental cause for decoherence [25] or by collapse theories that directly modify Schr¨odinger’s equation [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, we are here interested in explanations within the conventional frame- work of non-relativistic quantum mechanics, where gravity or collapse theories cannot play any role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, also the semiclassical limit of large action S/ℏ ≫ 1—formalized by taking ℏ → 0 [27] among other limiting procedures related to temperature, mass, angular momentum, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='—is often invoked to explain classicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' While this provides an important consistency check, we here avoid such limiting procedures (after all, decoherence is a major obstacle to build a quantum computer and one can certainly not claim that a quantum computer oper- ates in the high temperature limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Finally, for fundamental criticism about decoherence we 2 SciPost Physics Submission refer the reader to Leggett [28], the importance of decoherence for macroscopic objects was discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [29–34], and for a general criticism of prevailing notions of classicality in the cosmological context see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Furthermore, a topic which we will only briefly touch is quantum Darwinism [36,37], which further refines the notion of decoherence in OQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2 Classicality: Definitions and approaches 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1 Definition used in this work For any discussion of the quantum-to-classical transition, it is important to precisely define what “classical” behaviour means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' One here hits the first obstacle as the boundary between the quantum and classical world is not one-dimensional: depending on the problem different boundaries can be drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For instance, one legitimate way to define classicality could be to ask whether a state of a bipartite quantum system obeys all Bell inequalities or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This definition, however, only reveals static quantum features of a state, and does not allow to draw any conclusion about how classicality emerges from an underlying quantum description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Here, we are precisely interested in this emergence and ask whether a process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', an ex- perimentally well-defined procedure to access the time evolution of a quantum system [13,14], can look classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To be precise, consider an isolated quantum system with (time indepen- dent) Hamiltonian H prepared in some state (density matrix) ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Let X = �M x=1 λxΠx be some observable with eigenvalues λx, eigenprojectors Πx and M denoting the total number of distinct eigenvalues (measurement outcomes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The probability to measure xn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , x1 at times tn > · · · > t1 is p(xn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , x1) ≡ tr{ΠxnUn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Πx1U1ρ0U † 1Πx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' U† n} (1) with Uk ≡ e−iH(tk−tk−1) the unitary time evolution operator between two times (ℏ ≡ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (1) can be experimentally reconstructed by performing n repeated measurements on a quantum system and by repeating this procedure many times to create sufficient statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, pick some k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , n−1} and define p(xn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , �� xk , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , x1) to be the same probability as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (1) except that no measurement is performed at time tk (and thus no outcome xk is recorded), which is indicated with the notation �� xk and obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (1) by dropping the two projectors Πxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Then, the process is classical if the following “probability sum rule” is satisfied for all k < n and all n > 1 (up to some error much smaller than the considered probabilities): � xk p(xn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , x1) = p(xn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , �� xk , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2) In words, a process is classical if marginalizing over measurement results is identical to not measuring at any given time tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Since measurements can be disturbing in quantum mechan- ics, even on average, the validity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2) signifies the absence of quantum effects from the perspective of measuring X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' An example violating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2) is the famous double slit experi- ment, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The following facts further support the idea that this is a good definition of classicality (though, as emphasized above, not the only one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, observe that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2) defines a classical stochastic process [38], where it is also known as the Kolmogorov consistency condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Classicality as considered here therefore has a clear operational meaning, which was also used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [7–15]: a process is classical if (at least 3 SciPost Physics Submission Figure 1: The double slit experiment where a coherent source of particles ρ0 hits a detection screen at position x2 after passing a wall with two holes (the double slit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (a) The particle’s location x1 is also measured at the double slit, allowing to decide through which slit it passed (corresponding trajectories indicated by dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' No interference pattern is seen on the detection screen, also not after averaging over x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (b) There is no measurement of the particle’s location at the double slit, it thus retains its coherent wave-like properties and an interference pattern emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Clearly, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2) is violated: the dynamics is non-classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' in principle) a classical stochastic process can be used to generate the same measurement statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The idea of defining classicality in this way is rooted in a “black-box-mentality”: there might be some very expensive quantum computer in front of you, but if the available measurement statistics can be simulated, or emulated, with a classical stochastic processes, then the measurement statistics alone do not allow you to draw the conclusion that there is anything quantum going on in the computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Furthermore, this definition of classicality also has a clear practical motivation because classical stochastic processes are much easier to analyse and simulate than quantum stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2) implies the validity of Leggett-Garg inequalities [39] and it is closely related but not equivalent to the conditions imposed in the consistent or decoherent histories formalism [5, 6], which we review below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Importantly, however, the definition of classicality used here does not hinge on any specific interpretation of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Confirming Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2) experimentally only requires measurements of X, no further hidden as- sumption is contained in its definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Clearly, classicality is defined with respect to some observable X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', a system that behaves classical with respect to X can behave non-classical with respect to a different observable Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Finally, notice that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2) could be also vio- lated in a classical context, for instance, whenever an external agent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', some observer or experimenter) actively intervenes in the process, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', by performing feedback control opera- tions [13,14,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' We exclude these scenarios here by definition of the probabilities in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (1), which in the classical limit (replacing projectors on Hilbert space by characteristic functions on phase space) clearly obey Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In the remainder of this section, we first review the well known decoherence approach and ask whether it explains classicality according to the definition used here (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Afterwards, we comment on the relation to the perhaps less well known consistent or decoherent histories approach (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Finally, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 concludes with an intuitive explanation why Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2) can be satisfied for an isolated quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4 SciPost Physics Submission 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2 The decoherence approach for open quantum systems We consider an open quantum system (OQS) S coupled to some environment or bath B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The total Hilbert space is thus a tensor product HS⊗HB of the system and bath Hilbert spaces HS and HB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The dynamics in the full system-bath space is unitary and generated by a Hamiltonian HSB = HS +HB +VSB with HS (HB) the system (bath) Hamiltonian alone (suppressing tensor products with the identity in the notation) and VSB their interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The reduced system state ρS(t) = trB{ρSB(t)} is obtained from a partial trace of the full system-bath state over the bath degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In contrast to ρSB(t), ρS(t) does not evolve in a unitary way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Decoherence happens whenever it is possible to identify a fixed special basis {|ψx⟩}, which is called the pointer basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The special role of this basis is to ensure that any initial OQS state ρS(0) becomes after a characteristic (and typically very short) decoherence time tdec diagonal in that basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In equations, ρS(0) = � x,y cx,y(0)|ψx⟩⟨ψy| −→ t≥tdec ρS(t) ≈ � x px(t)|ψx⟩⟨ψx|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (3) Here, the cx,y(0) = ⟨ψx|ρS(0)|ψy⟩ are complex numbers, which ensure positivity and normal- ization of ρS(0) but are otherwise arbitrary, and the px(t) ≈ cx,x(0) are close to the initial probabilities to be in state |ψx⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Equation (3) is indeed a remarkable robust prediction of OQS theory [1–4,41,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In particular, we repeat that the pointer basis is fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', it does not depend on the initial system state, but it is determined by the system-bath Hamiltonian and the initial bath state (though the dependence on the latter should be mild in realistic situations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The pointer basis is also often described as “stable”, “robust” or “objective” [1–4] and we come back to these properties below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' We further add some clarifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, for the sake of generality one should stress that the pointer basis might not be a basis of pure states |ψx⟩, but rather a complete set of orthonormal projectors {ΠS x} acting on the system Hilbert space, where certain projectors can have a rank greater than one [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In that case, there exist “decoherence-free subspaces” (caused, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', by additional conservation laws), but they do not change the fundamental point of our discussion and we continue to call {ΠS x} the pointer basis for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, we here assume pointer states to be orthogonal, which is typically the case for finite dimen- sional OQS, but pointer states can be non-orthogonal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', coherent states of a harmonic oscillator [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (3) we allowed the probabilities px(t) to be time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Their change, however, typically happens on a time scale much slower than the decoherence time scale (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [45]) and is called “dissipation”—a phenomenon already known from classical open systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Third, we remark that a more nuanced presentation of decoherence is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For instance, in order to determine the measurement basis, Zurek in his seminal paper was actually interested in the decoherence of the measurement apparatus, which was in turn coupled to the system to be measured and an environment [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, also the measurement apparatus is an OQS, and for the remainder of this paper it is not necessary to explicitly distinguish between system and measurement apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In the following we call the phenenology explained above OQS decoherence to distinguish it from the decoherent histories mentioned later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, we ask whether decoherence explains the emergence of classicality according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2) if applied to a system observable XS = �M x=1 λxΠS x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', an observable commuting with the pointer basis and acting trivially on the bath space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To this end, we first confirm 5 SciPost Physics Submission that for all times larger than the decoherence time tdec we have DρS ≡ � x ΠS xρS(t)ΠS x = ρS(t), (4) where D is a dephasing operation in the pointer basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1 Thus, measuring and averaging is identical to not measuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, let us additionally assume that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (4) holds for the full system-bath state: DρSB(tk) = � x ΠS xρSB(tk)ΠS x = ρSB(tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (5) If that is the case, one can confirm our definition of classicality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', the validity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (4) does not imply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (5), even though the converse is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, OQS decoherence does not imply classical measurement statistics according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It is thus worth thinking about which condition on top of decoherence could imply classi- cality, and it seems that two fundamentally different strategies are conceivable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The first strategy takes a more detailed look at the environmental degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Indeed, the validity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (5) is equivalent to having vanishing quantum discord [47] in the pointer basis, and testing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (4) has been suggested as a tool to probe non-classical system- bath correlations [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, deciding whether the system-bath state has zero quantum discord or not requires knowledge of the full system-bath state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This knowledge is unavailable experimentally and therefore the condition of zero quantum discord is inaccessible from an operational perspective of OQS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, the idea of having zero quantum discord is problematic from the perspective of having a unitarily evolving “universe” consisting of the system and the bath as explained later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, a refinement of this idea is possible and has lead to the recently much studied approach of quantum Darwinism, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [36, 37] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In a nutshell, quantum Darwinism starts by dividing the bath into many different “fragments” F ⊂ B and asserts that most fragments, even those of small size, have (close) to zero quantum discord with respect to the pointer basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (5) applies to most states ρSF (tk) = trB\\F {ρSB(tk)}, where B \\ F denotes all bath degrees of freedom except those of the fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The resulting classical correlations between the system S and most fragments F allow external observers to learn about the system state even by only looking at a small fragment F of the bath, and different observers looking at different small fragments will agree about the state of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, an objective world emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For the important mechanism of photons scattered off some material object the idea of quantum Darwinism is indeed intuitively appealing because the scattered photons allow different observers, by looking at different narrow angles at the object, to infer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', the same colour or position of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, since photons are non-interacting and scatter off to infinity, it becomes clear that their detection does not change the future evolution of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' As a consequence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Quantum Darwinism thus provides a sufficient criterium for objectivity and classical mea- surement statistics, but it is questionable whether it is always necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In condensed matter and other situations, the bath does not split into non-interacting fragments and perturbations might not be able to escape to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In this case, quantum Darwinism will generically 1In principle, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (3) implies only an approximate equality (≈) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, since classical behaviour should be always understood as some approximation, we replace ≈ by = whenever we mean “equal up to some irrelevant measurement error”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 6 SciPost Physics Submission hold at most for transient times [49], yet objectivity and Kolmogorov consistency might nev- ertheless arise—as this paper will indeed confirm now within and later also without OQS decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Within the paradigm of OQS decoherence, this brings us to the second strategy imply- ing classical behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This strategy is different from the first by rejecting the idea that information about (fragments of) the bath is directly accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Instead, solely the degrees of freedom of the OQS are deemed operationally accessible, and it then becomes necessary to think about how could one locally decide whether the pointer states are stable, robust or objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Historically, Zurek introduced for this purpose the “predictability sieve” [50], which requires to compute the change in von Neumann entropy of the OQS state ρS(t) as a function of a pure initial state ρS(0) = |ψ(0)⟩⟨ψ(0)|S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' If it changes very slowly, the dynamics are predictable as the state remains approximately pure, but if it changes very rapidly, the dynamics are unpredictable as the state becomes very mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Now, from what we said above, we see that the initial pointer states are characterized by a slow change in von Neumann entropy, whereas superpositions of pointer states quickly decohere into a mixture on a time scale tdec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The predictability sieve thus selects out the pointer states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Although the predictability sieve has appealing properties, it is ultimatively not satisfac- tory for the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' If we want to find out whether something is predictable (or stable, robust or objective), it is best to really “take a look at it”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For instance, the memories in our computers are stable because we can repeatedly read them out without changing their state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' How can this idea be formalized mathematically?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Clearly, one way to test this prop- erty is to measure the OQS in the pointer basis, say at some time t1 ≥ tdec, and then to look whether this measurement influences the future evolution of the OQS at some time t2 > t1, for instance, by checking whether the future probabilities of the pointer states depend on the measurement at time t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' We now notice that this idea to check for predictability is exactly equal to testing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', our definition of classicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Clearly, other ways are possible, but within this second strategy they should always be related to watching the response to some form of external perturbation or intervention on the OQS: do the pointer states remain stable if we shake them a bit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' After having spelled out the basic idea, it remains mostly a technical problem to realize that the condition of Markovianity as defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [16] (for introductions see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [13,14]) is sufficient to imply classical measurement statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In short, this definition of Markovianity is based on the idea that local operations on the system performed by an external agent do not influence the OQS dynamics generated by the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Importantly, the property of Markovianity can be checked by local system operations only (no knowledge of the bath state is required) [13, 14, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, since the definition of Markovianity requires to check multi-time correlations (in complete analogy to the classical definition), knowledge of the time evolution of ρS(t) alone is insufficient to check for Markovianity (a discussion focused on this point can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The connection to Markovianity now becomes transparent by realizing that “shaking a bit the system” is an external intervention that will sooner or later also influence the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Can this influence of the environment cause a different behaviour of the system?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' If the answer is no, then this precisely means that the dynamics is Markovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In that case, we can conclude the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, we found above that OQS decoherence implies that DρS(t1) = ρS(t1) for t1 ≥ tdec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Obviously, one also has IρS(t1) = ρS(t1) where I is the identity operation which, operationally speaking, literally means “do nothing!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Now, according to the definition of Markovianity explained above, the dynamics induced by the environment is insensitive to 7 SciPost Physics Submission Figure 2: Overview over the relations between different concepts defined in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The arrows mean strict mathematical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The equivalence between global decoherence and zero discord has to be understood for local observables (discord is undefined without a system-bath tensor product structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, global decoherence and zero discord are here understood as applying repeatedly for all times tk considered in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The comment “Happens only trivially!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' is explained in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' local operations on the system performed by an external agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Since the two operations D and I do not change the OQS state, the future dynamics is insensitive to the dephasing operations, that is: the pointer states are stable and the dynamics is classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Formal definitions and a formal proof are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This important message together with various other notions (some of which are only introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='3) is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Notably, the role of multi-time statistics to probe the stability of pointer states and its connection to Markovianity has not yet been made in the literature on OQS decoherence [1–4], and it is also not the focus of quantum Darwinism [36,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It has been realized, however, in a numerical study of quantum Brownian motion that non-Markovianity hinders quantum Darwinism [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' While it is clear that the definition of Markovianity is mathematically not in one-to-one correspondence to the concept of quantum Darwinism, it seems worthwhile in the future to look for a closer connection of these two notions in physical relevant situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Finally, we make two more important observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First of all, the question “what is the pointer basis?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' is non-trivial and has been only answered in certain limiting cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', very strong or very weak system-bath coupling) [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In general, for a complex open many-body system coupled to a complex many-body environment the pointer basis is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It is an advantage of the approach presented in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 and 3 of this paper that no pointer basis needs to be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, all what we said above was restricted to the OQS paradigm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', local observables defined on a system-bath tensor product structure, whose identification can be non-trivial [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This restriction is also lifted in the present approach, which makes it appealing for questions usually studied within the formalism reviewed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 8 Happens only trivially!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='SciPost Physics Submission 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='3 Consistent and decoherent histories The consistent or decoherent histories formalism is an attempt to explain how standard rea- soning based on classical logic can be applied in an isolated quantum system in general, and in the cosmological Universe in particular [5,6,53–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' As we will see, it is closely related to the Kolmogorov consistency criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The approach starts by introducing a decoherence functional D for two histories x ≡ (xn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , x2, x1) and y ≡ (yn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , y2, y1) “happening” at times tn > · · · > t1: D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' y) ≡ tr{ΠxnUn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Πx2U2Πx1U1ρ0U † 1Πy1U † 2Πy2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' U† nΠyn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (6) Here, the projectors and unitary time evolution operators have the same meaning as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (1) and we immediately confirm that the diagonal elements of the decoherence functional corre- spond to our previously introduced joint probabilities: p(xn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , x1) = D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Depending on the precise reference, different notions of “consistency”, “decoherence” or “(quasi)classicality” have been introduced based on the decoherence functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It is beyond the scope of this article to review them all here, so we restrict the discussion to the two most commonly employed definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, Griffith originally proposed what we here call the consistent histories condition [53]: consistent histories: ℜ[D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' y)] = 0 for all x ̸= y, (7) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', the vanishing of the real part of the decoherence functional for different histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Gell- Mann and Hartle, among others (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [56,57] and references therein) prefer to use the following condition that we call the decoherent histories condition: decoherent histories: D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' y) = 0 for all x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (8) Three immediately obvious remarks follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, condition (8) implies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' y) = 0 always if xn ̸= yn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', the final “measurement results” cannot be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Third, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (7) implies the Kolmogorov consistency condition (2) (and hence so does Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Confirming the last result requires a few lines of algebra, but it was already shown by Grif- fiths [53] and many others and will thus not be repeated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' A not so obvious conclusion is that the decoherent histories condition is strictly stronger than the consistent histories condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is explained with a result of Di´osi [58], who con- sidered two decoupled quantum systems A and B prepared in a decorrelated state ρA(t0) ⊗ ρB(t0), unitarily evolving without interaction according to UA⊗UB and measured with decor- related projectors ΠA x ⊗ ΠB x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In this situation, one immediately confirms that the joint de- coherence functional factorizes as DAB(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' y, y′) = DA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' y)DB(x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' y′), where unprimed (primed) histories refer to subsystem A (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Now, suppose that A and B separately satisfy the decoherent histories condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Then, this is also the case for the non-interacting com- posite AB, as one would intuitively expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, this conclusion does not hold for the consistent histories condition, thus the latter cannot imply the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, Di´osi’s argument is typically invoked to say that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (8) is a more meaningful condition than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Now, if we consider the probabillities pAB(x, x′) = pA(x)pB(x′) for decoupled system, they factorize as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Interestingly, if A and B separately satisfy the Kolmogorov consistency condition, then this is also true for the composite AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, Di´osi’s argument cannot be invoked to refute our definition of classicality based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2 Two further statements 2Di´osi also gives a second argument to argue in favour of decoherent instead of consistent histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Again, also this second argument does not disfavour our definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 9 SciPost Physics Submission are noteworthy: First, what we said above implies that the decoherent histories condition is strictly stronger than the Kolmogorov consistency condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, confirming the de- coherent histories condition experimentally is obviously much harder than confirming the Kolmogorov consistency condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, we turn to the relation between decoherent histories and OQS decoherence, which obviously has been already the topic of previous works, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', the above references and in particular also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Quite intuitively, one would expect that histories defined by measurements in the pointer basis naturally satisfy the decoherent histories condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', that OQS decoherence generates decoherent histories, but it has been recognized that this relation is not that easy [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Indeed, since OQS decoherence alone is not sufficient to imply Kolmogorov consistency, it also cannot be sufficient to imply decoherent histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Interestingly, and what seems to have not been recognized yet, is that the extra condition of Markovianity is again sufficient to show that OQS decoherence implies decoherent histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The proof of this statement is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Finally, we turn to the question whether decoherence in a stronger, global sense can ex- plain the emergence of decoherent histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Here, global decoherence means that the unitarily evolving state is block diagonal with respect to the projectors {Πx}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', ρ(t) = � x Πxρ(t)Πx (note that this implies zero quantum discord, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (5), for local system projectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' If that is the case for all times tk appearing in definition (6) of the decoherence functional, one immediately confirms that the histories satisfy the decoherent histories condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Unfortu- nately, however, if ρ(t) is unitarily evolving this can in general only be the case for trivial situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To see this, we restrict the discussion to pure states |ψ(t)⟩, which is sufficient for isolated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Now, from � x Πx = I we infer that every pure state can be written as |ψ(t)⟩ = � x � px(t)eiϕx(t)|ψx(t)⟩ with px(t) the probability to measure outcome x and Πy|ψx(t)⟩ = δx,y|ψx(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, notice that the only states |ψ(t)⟩ that are block diagonal or globally decohered are states with px(t) = δx,x∗ for some x∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', these states are fully localized in one subspace or, with respect to the measurement outcomes, we can say that they are deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Now, this can certainly happen in some cases, for instance, if the dimension of the subspace x∗ dominates by far all other subspaces (which corresponds to the usual criterion of x∗ describing an equilibrium state in statistical mechanics), or if the times tk are carefully chosen such that only on these times |ψx(t)⟩ is approximately localized in one subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, if Πx commutes with the Hamiltonian, its probability remains constant and always generates classical statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, excluding globally conserved quantities, considering interesting nonequilibrium dynamics and rejecting the unrealistic idea that we are able to carefully choose the times tk, the state |ψ(t)⟩ cannot remain block diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For instance, if the state has a high initial probability p(x0) ≲ 1 for some x0 and a high probablity for some final state p(xn) ≲ 1 with xn ̸= x0, then there must be some intermediate time t where the state passed from x0 to xn such that px0(t) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, global decoherence can only happen under trivial or unrealistic circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' A summary of this and the last section can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 The new approach: General picture We now discuss the general picture behind the new approach from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [7, 8, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It is claimed to be “new” for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, as discussed above, it defines classicality in terms of the Kolmogorov consistency condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This differs from the basic question in OQS 10 SciPost Physics Submission decoherence (“What is the measurement/pointer basis?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=') and it is close to but still different from the histories approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In particular, Kolmogorov consistency can be independently well motivated by asking the question “When can a quantum process be simulated by a classical process?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', as also done recently in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [9–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, emphasis is put on the following two physical aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, the focus is not on OQS: even global observables of an isolated system can behave classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, non-integrability is regarded as essential, or at least very helpful, to derive classicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' While the relation to chaos has been also studied in OQS decoherence (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [61–69]), it has not been regarded as essential: the traditional workhorse model of OQS theory uses an integrable bath of harmonic oscillators (“Caldeira- Leggett model”) prepared in a canonical Gibbs ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is avoided in the following by considering pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To the best of the author’s knowledge, the basic physical picture behind this emergence of classicality has been already explained by van Kampen in 1954 [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Three basic ingredients, which can hardly count as assumptions, plus one major assumption are necessary to see the emergence of classicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The three ingredients are: (i) the system has a well-defined overall energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', the energy spread ∆E of the initial wavefunction is sufficiently narrow3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (ii) the system has many particles N ≫ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', the Hilbert space dimension of the aforementioned energy shell is exponentially large: D ≡ dim H ∼ exp(N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (iii) the system is non-integrable or, more precisely, it should obey the eigenstate thermalization hypothesis (ETH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Given the success of the ETH this is considered a mild assumption for realistic many-body systems found in nature [23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The major assumption concerns the observable X that one is probing: according to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [15, 70] it should be coarse and slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Coarseness means that the number of poten- tial measurement outcomes is much smaller than the Hilbert space dimension: M ≪ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Again, this can hardly count as an assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In particular, observe that an observable XS defined for an OQS is a coarse observable in the full system-bath space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Slowness instead is the crucial assumption and it has been discussed in detail (together with various subtleties) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Intuitively, it means that the time scale τX on which ⟨X⟩(t) = tr{XUtρ0U † t } evolves is much longer than the microscopic evolution time scale ℏ/∆E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is equivalent to saying that the matrix with elements Xkm = ⟨k|X|m⟩ with respect to an ordered energy eigenbasis {|k⟩} is narrowly banded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It is interesting to contrast this approach to previous work done within the consistent or decoherent histories formalism, where slow (or quasi-conserved) observables also played an essential role [54,56,57,71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Without noting the work of von Kampen, the focus in these works was to derive the consistent or decoherent histories condition by arguing that the wave packet remains strongly localized along some trajectory, which is described by a classical determin- istic equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', it was argued that the wave function |ψ(t)⟩ should remain approximately localized in one (time-dependent) subspace Πx(t) throughout the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' We questioned the adequacy of this idea already at the end of the previous section, but here we just point out that the assumption of remaining localized around some classical trajectory is also not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' As it will come clear below, the pure state |ψ(t)⟩ is allowed to have an abundance of coherences (even maximal coherences) and can still behave classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This marks another and perhaps the most important novel aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 3Recall that energy is conserved in an isolated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' So if the initial state is a superposition of macro- scopically different energies, the analysis should be carried out separately for each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, if there are further conserved quantities, the same argument has to be also applied to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 11 SciPost Physics Submission It is also interesting to connect the assumptions above to the OQS decoherence approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To this end, consider an OQS and let X = HS be the system Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Since HS is locally conserved ([HS, HS] = 0), HS is a slow observable provided that the coupling VSB is weak enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Furthermore, it is also coarse if dim HS ≪ dim HB, which is usually the case in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, in the weak coupling regime local measurements of the energy should give rise to classical statistics obeying the Kolmogorov consistency condition (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is unison with the predictions of the pointer basis in the decoherence approach, but we repeat that the identification of a pointer basis is not necessary in the present approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, it should be emphasized that the notion of slowness is subtle and it does not seem to be a sufficient criterion for classicality: by precisely tuning the “fine-structure” of Xkm it appears that one can generate arbitrary exceptions to the “rule” [72], albeit those might not be generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In any case, it provides a different perspective on the problem and it gives an immediate intuitive explanation why the world around us appears classical: human senses are simply to slow and coarse to resolve the evolution of fast observables that could potentially show quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' So how can it be that decoherence is not necessary to generate classical measurement statistics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The following picture lacks rigour, but gives some intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To start with consider a two-level system with Hamiltonian ∆ 2 σz = ∆ 2 (|1⟩⟨1| − |0⟩⟨0|) and as the observable choose X = σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, let the initial state ρ0 with respect to the eigenbasis |±⟩ = (|0⟩ ± |1⟩)/ √ 2 of σx be parametrized as ρ0 = �(1 + δ)/2 reiφ re−iφ (1 − δ)/2 � , δ ∈ [−1, +1], 0 ≤ r ≤ √ 1 − δ2 2 , φ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (9) Here, the parameter r quantifies the “strength” of the coherences in the σx basis, which is always upper bounded by √ 1 − δ2/2 due to the positivity requirement ρ0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Now, at an arbitrary time t the system state in the same basis reads ρt = � (1 + δ cos ∆t)/2 + r sin φ sin ∆t r(cos φ + i sin φ cos ∆t) − iδ 2 sin ∆t r(cos φ − i sin φ cos ∆t) + iδ 2 sin ∆t (1 − δ cos ∆t)/2 − r sin φ sin ∆t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (10) The diagonal elements equal the probabilities to measure spin up |+⟩ or spin down |−⟩ with respect to the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Their time evolution is strongly influenced by the coherences and therefore the dynamics is not classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Of course, this is not a counterexample since the system is neither a non-integrable many-body system nor is the observable coarse and slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' So what happens for a coarse observable in a non-integrable many-body system?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The single elements of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (10) now become blocks of many elements and the probability px(t) to find the system in some state x becomes the trace over block x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It will typically contain a sum of contributions from many coherences ⟨i|ρ0|j⟩ = rijeiφij of the initial state, schematically written as: px(t) ∼ � i,j rij sin φij sin ∆ijt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (11) Now, observe the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, for a coarse observable of a many-body system the number of terms contributing to the sum is huge (of the order eN with N the particle number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, for a non-integrable system the energy differences ∆ij are incommensurable (apart from rare accidential degeneracies) and effectively random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, unless the φij are precisely tuned or rij = 0 for most but a few pairs (i, j), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (11) is a sum of many terms of random 12 SciPost Physics Submission sign and small magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 Thus, the enormous amount of coherences cannot add up to a significant contribution and therefore it effectively does not matter whether coherences are present or not, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', as long as one only asks questions about the measurement statistics in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (1) we can set rij = 0 for all (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This explanation for classical behaviour is essentially statistical and similar in spirit to the explanation of the second law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Yes, it is possible that the positions and momenta of all the molecules in the air surrounding you conspire such that they can be all found in one corner of the room in the next second, yet this possibility is extremely unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Similarly, it is possible that all microscopic coherences of a coarse observable align in phase to give rise to a strong contribution and, consequently, a strong violation of Kolmogorov consistency, yet this is again extremely unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In essence, this also underlies the decoherence approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Yes, it is possible that a qubit in contact with a bath suddenly “recoheres”, yet this would require again a very unlikely because precisely tuned cooperation of many phases of the system-bath state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, the emergence of classical behaviour is related to the general phenomenon of irreversibility, which is extremely hard to avoid given our coarse human senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Finally, one might wonder where above the assumption of slowness enters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is in- deed not directly visible here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, our argument why Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (11) is small was based on assumptions about the number of coherences rij and the correlations of the phases φij and frequencies ∆ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Unfortunately and in particular for states prepared out of equilibrium, these assumptions become questionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Slowness now helps because the microscopic state evolves on a much shorter time scale than the observable, effectively randomizing many phases before any noticeable change in px(t) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, from the perspective of the slow observable X, the systems looks locally equilibrated and the precise microstate no longer matters [15,70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 3 Classicality: Derivation and numerical verification 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1 Derivation using random matrix theory To show the approximate validity of the Kolmogorov consistency condition (2) one needs a model that, ideally, is as general as possible to cover a wide range of scenarios while being at the same time also specific enough to permit explicit calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Unfortunately, these two desiderata are often mutually exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [15] the model was assumed to obey the ETH, which is currently considered to be a mild assumption for most realistic many-body systems found in nature [23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The drawback of this generality was that some plausible but at the end unproven assumptions entered the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Here, a random matrix theory approach is used, which has been successful in modelling a variety of generic properties of complex systems [19–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Indeed, our current understanding of the ETH is much based on random matrix theory although the ETH has been shown to be valid for a much larger class of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The model considered here is therefore more restrictive than the model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [15], but it comes with the benefit that we need less unproven assumptions (albeit we still need some).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To capture the relevant physics of a non-integrable many-body system we follow Deutsch [73] 4Recall that r2 ij ≤ pipj (by Cauchy-Schwarz) where pi is the probability to find the system in a certain microscopic contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Generically, a system has overlap with an enormous amount of microscopic states and, since � i pi = 1, this implies that rij must be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 13 SciPost Physics Submission and others [74–82] and consider a Hamiltonian of the form H = H0 + ϵH1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (12) Here, H0 is some “baseline” Hamiltonian, ϵ a small parameter and H1 a banded random Hermitian matrix chosen, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', from the Gaussian orthogonal or unitary ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For in- stance, H0 = HS + HB could be the bare system and bath Hamiltonian and ϵH1 = VSB their (weak) interaction, but many more examples are imaginable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Note that H0 does not need to be integrable, it is only assumed to describe a many-body system with an extremely dense spectrum in the considered energy interval (recall ingredient (i) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, the model does not literally assume that the perturbation is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Instead, the basic idea of random matrix theory is that some property holds for the overwhelming majority of random perturbations and that the real physical (and non-random) Hamiltonian then also belongs to this overwhelming majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Finally, the smallness of ϵ implies that the range of the spectrum and the mean level spacing δe of H0 and H are comparable, but their eigenvectors are still strongly perturbed as long as ϵ is larger than the extremely small level spacing δe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Let |µ⟩ and |m⟩ be the eigenvectors of H0 and H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' A central role in the following is played by the unitary matrix V m µ ≡ ⟨m|µ⟩, (13) which transforms between the eigenbasis of H0 and H and quantifies their overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Denoting by E[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' ] averages over the random matrix ensemble, we use that u(m, µ) ≡ E ���V m µ ��2� = u(m − µ), � n u(n) = 1, max n u(n) = O(e−N), (14) which holds for both the Gaussian orthogonal or unitary ensemble (and even beyond strict Gaussianity) and whose detailed justification is left to the literature [73–85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The important point is that the overlap between eigenvectors of H and H0 is exponentially small in the particle number N and for our estimate below we will set for notational simplicity maxn u(n) = D−1 with D the Hilbert space dimension of the energy shell (which is exponentially large in N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, as an observable we allow any coarse Hermitian operator X = �M x=1 λxΠx that commutes with the unperturbed Hamiltonian, [H0, X] = 0, but not with the perturbation H1 (the case [H1, X] = 0 trivially gives rise to classical dynamics for X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Coarseness means that M ≪ D and the smallness of ϵ implies that X evolves on a slow time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Note that also observables X with [H0, X] ̸= 0 can behave classical [7,8,15], but the above assumption turns out to be very convenient for the calculation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Then, we consider the joint probability p(x2, x1) ≡ tr{Πx2U2Πx1U1ρ0U † 1Πx1U † 2} (15) to measure x1 at time t1 and x2 at time t2 given an arbitrary initial state (perhaps far from equilibrium) ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' We further introduce the single time probability p(x2, �� x1) ≡ tr{Πx2U2U1ρ0U † 1U † 2} (16) and consider the difference Q ≡ p(x2, �� x1) − � x1 p(x2, x1) = � x1̸=y1 tr{Πx2U2Πx1U1ρ0U † 1Πy1U † 2} ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (17) 14 SciPost Physics Submission The goal in the following is to show that Q is extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This then implies that the Kolmogorov consistency condition (2) is satisfied at the level of arbitrary two-time probabili- ties given any initial state ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' An extension of the derivation to arbitrary n-time probabilities with n > 2 is certainly desirable, but it is a very complicated problem, likely requiring novel techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, abstracting from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [86–89], where theorems about n-time correla- tions functions for large n were proven under different circumstances, it appears likely that approximate consistency continues to hold also for n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To make analytical progress, we first need some assumption about the initial state ρ0 and at this stage we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In summary, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [15] has described the initial state by a preparation procedure using some completely positive map M such that ρ0 = Mψ0 = � α Kαψ0K† α, where ψ0 is the state prior to the preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' So far, this is completely general [13,14,90], but now two assumptions are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, by polar decomposition we write Kα = √PαVα, where Vα is a unitary and Pα a positive operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It was now assumed that the Pα = Pα(X) are functionally dependent on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Translated into an experimental context, this means that the experimentalist has control over X (for instance, by measuring it), but they are not in control over the precise microstate within the subspaces of Πx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, it was assumed that the prior state ψ0 is at equilibrium or, technically speaking, Haar randomly distributed in the energy shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Both assumptions thus express the idea that the initial state preparation can bring a system, which is at equilibrium from a macroscopic point of view (note that ψ0 is a pure state), arbitrarily far from equilibrium with respect to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Then, using a measure concentration inequality in form of Levy’s lemma [91], it was shown that smallness of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (17) is (in almost all cases) equivalent to showing smallness of q(x2, x0) ≡ 1 Dx0 � x1̸=y1 tr{Πx2U2Πx1U1Πx0U † 1Πy1U † 2} for x2 ̸= x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (18) Here, Dx0 ≡ tr{Πx0} is the dimension of the subspace associated to measurement outcome x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, in essence the term Q, which is a three-point correlation function for the projectors Πx with unknown correlations with respect to the initial state ρ0, got transformed into the term q(x2, x0), which is a four-point correlation function for the projectors Πx without any initial state dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' We evaluate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (18) in the energy eigenbasis of H and introduce the following convention, which is perhaps unconventional but useful for later considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Since there will be many terms indexed by many quantities m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' and µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , where mi (µi) labels energy eigenvalues of H (H0), we decide to write labels mi (µi) as superscripts (subscripts) as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (13) and take the freedom to simply replace them by the number i whenever appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (18) then becomes q(x2, x0) = 1 Dx0 � x1̸=y1 1,2,3,4 � eiω12(t2−t1)eiω43t1Π12 x2Π23 x1Π34 x0Π41 y1, (19) where ω12 = E1 − E2 denotes the difference between two eigenenergies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, we recall that X is a narrowly banded operator (due to its slowness) and this also implies that Πx is narrowly banded (due to the coarseness of X) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, in an ordered energy eigenbasis we can safely assume Πmn x = 0 if m − n ≥ d for a sufficiently large number d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Importantly, while d ≫ 1 can be enormous in realistic applications, a central feature of slowness and coarseness is that still d ≪ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, d/D serves as a small parameter in the following 15 SciPost Physics Submission and the corresponding restricted summation is denoted as �1≈2≈3≈4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Finally, notice that q(x2, x0) = 0 if t2 = t1 or t1 = 0, which allows to turn Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (19) into q(x2, x0) = 1 Dx0 � x1̸=y1 1≈2≈3≈4 � (eiω12(t2−t1) − 1)(eiω43t1 − 1)Π12 x2Π23 x1Π34 x0Π41 y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (20) We continue by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (13) and [H0, X] = 0 to obtain Πmn x = ⟨m|Πx|n⟩ = � µ,ν ⟨m|µ⟩⟨µ|Πx|ν⟩⟨ν|n⟩ = � µ χµ(x)V m µ ¯V n µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (21) Here, χµ(x) is the indicator function which is one if and only if Πx|µ⟩ = |µ⟩ and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Furthermore, note that we use an overbar to denote the complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (21) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (20), we arrive at q(x2, x0) = 1 Dx0 � x1̸=y1 1≈2≈3≈4 � 1,2,3,4 (eiω12(t2−t1) − 1)(eiω43t1 − 1) × χ1(x1)χ2(y1)χ3(x0)χ4(x2)V 1 4 ¯V 2 4 V 2 1 ¯V 3 1 V 3 3 ¯V 4 3 V 4 2 ¯V 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (22) We have now reached a point, where we can try to evaluate q(x2, x0) using random matrix theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, since q(x2, x0) ∈ R can be positive or negative, showing smallness of q(x2, x0) on average is only an indicator, but not a gurantee that q(x2, x0) is small in general (because it could also strongly fluctuate for different realizations of the random Hamiltonian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, we will actually show that [q(x2, x0)]2 is small, which establishes smallness of q(x2, x0) and its variance, and which is one point where we go beyond the treatment of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, we aim to evaluate E � [q(x2, x0)]2� ≈ (23) 1 D2x0 � x1̸=y1 � x′ 1̸=y′ 1 1≈2≈3≈4 � 1,2,3,4 5≈6≈7≈8 � 5,6,7,8 (eiω12(t2−t1) − 1)(eiω43t1 − 1)(eiω56(t2−t1) − 1)(eiω87t1 − 1) × χ1(x1)χ2(y1)χ3(x0)χ4(x2)χ5(x′ 1)χ6(y′ 1)χ7(x0)χ8(x2) × E � V 1 4 ¯V 2 4 V 2 1 ¯V 3 1 V 3 3 ¯V 4 3 V 4 2 ¯V 1 2 V 5 8 ¯V 6 8 V 6 5 ¯V 7 5 V 7 7 ¯V 8 7 V 8 6 ¯V 5 6 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Note that the ensemble average is only performed over the matrix elements V m µ , but excludes the frequencies ωmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In principle, these should be included in the ensemble average as well, but the smallness of the random perturbation and the extremely small mean energy level spacing δe suggest that the behaviour of q(x2, x0) is insensitive to small perturbations of ωmn for times much smaller than the extremely long Heisenberg time ℏ/δe (for further justification see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [79–81,92]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Evaluation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (23) is fascillitated by the fact that ten constraints apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, due to the factors eiωijt − 1 we infer the four constraints m1 ̸= m2, m3 ̸= m4, m5 ̸= m6 and m7 ̸= m8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, due to the fact that x1 ̸= y1, x′ 1 ̸= y′ 1 and x2 ̸= x0 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (18)) we find the six constraints µ1 ̸= µ2, µ3 ̸= µ4, µ5 ̸= µ6, µ7 ̸= µ8, µ3 ̸= µ8 and µ4 ̸= µ7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Nevertheless, evaluation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (23) remains challenging even under the simplest approximation that we will employ here (though we discuss corrections later on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This approximation assumes that the V m µ are independent zero-mean Gaussian random numbers obeying E � V m µ � = 0, E � V m µ V n ν � = E � ¯V m µ ¯V n ν � = 0, E � V m µ ¯V n ν � = δmnδµνu(m − µ), (24) 16 SciPost Physics Submission with δmn and δµν denoting the standard Kronecker symbol with super- or subscripts, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The ensemble average in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (23) can then be evaluated using Isserlis’ theorem, which turns an expectation value of 2n random variables into sums over “pairings” where each pairing is a product of n pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' As an example, consider E � V 1 3 ¯V 2 4 V 2 4 ¯V 1 3 � = E � V 1 3 ¯V 2 4 � E � V 2 4 ¯V 1 3 � + E � V 1 3 ¯V 1 3 � E � V 2 4 ¯V 2 4 � = δ12δ34u2(m1 − µ3) + u(m1 − µ3)u(m2 − µ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (25) Quite discomfortingly, the ensemble average in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (23) involves sixteen random numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2 a numerical code is detailed that generates all pairings using Isserlis theorem while respecting the ten constraints mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Then, from the total amount of 40,320 pairings 347 distinct pairings (no multiplicity) survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It has been found too demanding to write a programme that automatically estimates Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (23) and since it is very tiring to investigate 347 cases manually, we look for the most dominant contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is justified because we are only interested in an order-of-magnitude estimate of E{[q(x2, x0)]2}, not its exact value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To find the leading order contribution, we observe that each pairing gives rise to a different number of distinct Kronecker deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In general, the fewer the Kronecker deltas, the larger the contribution because each Kronecker delta “kills” a high dimensional sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Some care, how- ever, is required because the sums run over spaces with potentially very different dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Specifically, every lower subscript runs over a subspace with dimension equal to the rank of some projector Πx, which is always smaller than D but could still be comparable to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In contrast, six out of the eight superscripts run over subspaces with dimension d ≪ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' There- fore, the leading order contributions are given by the terms that have the fewest Kronecker deltas in total or the fewest Kronecker deltas with respect to the subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Starting with the latter, the programme from Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2 shows that the pairing with the fewest amount of subscript-Kronecker deltas is E � V 1 2 ¯V 4 2 � E � V 1 4 ¯V 6 8 � E � V 2 1 ¯V 3 1 � E � V 2 4 ¯V 5 8 � E � V 3 3 ¯V 8 7 � E � V 4 3 ¯V 7 7 � E � V 5 6 ¯V 8 6 � E � V 6 5 ¯V 7 5 � ≈ D−8δ14δ16δ23δ25δ38δ47δ48δ37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (26) Here, all u(m−µ) ≥ 0 were replaced for notational simplicity with their maximum value D−1 in agreement with the comment below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Then, inserting this term into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (23) and killing all the sums, one obtains the contribution 1 D2x0D8 1≈2 � (eiω12(t2−t1) − 1)(eiω12t1 − 1)(eiω21(t2−t1) − 1)(eiω21t1 − 1) × � x1̸=y1 � x′ 1̸=y′ 1 � 1,2,3,4,5,6 χ1(x1)χ2(y1)χ3(x0)χ4(x2)χ5(x′ 1)χ6(y′ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (27) Now, since |eiω − 1| ≤ 2 for all ω ∈ R the first line is estimated by setting eiω − 1 = O(1) such that �1≈2 O(1) ≈ Dd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The second line can be exactly evaluated by introducing the Hilbert subspace dimension Dx ≡ tr{Πx} = � 1 χ1(x) associated to the projector Πx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, one obtains 1 D2x0D8 Dd � x1̸=y1 � x′ 1̸=y′ 1 DxDyDx0Dx2Dx′Dy′ = dDx2 Dx0D3 � 1 − � x �Dx D �2�2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (28) 17 SciPost Physics Submission This term is certainly negligible small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' We continue by considering the terms with the fewest Kronecker deltas in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The programme from Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2 shows that these terms have six Kronecker deltas in total and there are the following four of them (neglecting the universal prefactor D−8): δ13δ57δ14δ67δ23δ58, δ13δ68δ14δ68δ23δ57, δ24δ57δ24δ67δ13δ58, δ24δ68δ24δ68δ13δ57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (29) Let us look at the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Inserting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (23) gives rise to the contribution 1 D2x0D8 1≈2≈4 � 5≈6≈8 � (eiω12(t2−t1) − 1)(eiω41t1 − 1)(eiω56(t2−t1) − 1)(eiω85t1 − 1) × � x1̸=y1 � x′ 1̸=y′ 1 � 1,2,5,6 χ1(x1)χ2(y1)χ2(x0)χ1(x2)χ5(x′ 1)χ6(y′ 1)χ6(x0)χ5(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (30) Using the same reasoning as above, the summation over the superscripts is approximated as D2d4 and the summation over the subscripts gives δx1x2δy1x0δx′ 1x2δy′ 1x0D2 x0D2 x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Consequently, 1 D2x0D8 D2d4D2 x0D2 x2 = D2 x2 D2 d4 D4 , (31) which is still negligible small, though considerably larger than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Using the same strategy, it turns out that also the remaining contributions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (29) have the same scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, to summarize, it was shown that the dominant contributions to E{[q(x2, x0)]2} scale like (d/D)4, which is negligible small due to the slowness and coarseness of the considered observable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Remarkably, this scaling holds for all times t1 and t2 (and is thus clearly applicable out of equilibrium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Several assumptions concerning the ETH ansatz as detailed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [15] could be overcome due to the fact that we employed a more transparent but also more restrictive random matrix theory approach from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Clearly, also the random matrix theory approach is not without assumptions, but recalling its enormous success to deal with non-integrable or chaotic many-body systems [19–24] most assumptions should turn out to be mild in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Nevertheless, a critical point concerns the approximation that the V m µ are Gaussian and uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Both cannot be strictly true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, unitarity implies |V m µ |2 ≤ 1, which is not satisfied by a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, unitarity also implies that � µ V m µ ¯V n µ = δmn and �m V m µ ¯V m ν = δµν, which is satified on average, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (14) and (24), but not for a single realization if the V m µ are taken uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' While one might expect corrections to be negligible in many cases due to the hugh Hilbert space dimension and the smallness of V m µ , Dabelow and Reimann have shown that they can be important [79, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Yet, their goal was to determine the exact time-dependent behaviour of expectation values, instead of the rough order-of-magnitude estimate that we were interested in here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Nevertheless, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='3 confirms that at least the leading order correction does not give rise to a different scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Unfortunately, the author found the calculation of higher order corrections to be intractable, so the present derivation might be best interpreted as strong evidence, but no proof, that slow and coarse observables imply classical measurement statistics in random matrix models and, likely, also beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2 Numerical verification To illustrate the main features of the present approach to classicality, we use a simple toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This model cannot compete with the simulations of more realistic, non-random models 18 SciPost Physics Submission in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [7,8,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Yet, the results clearly support our general findings and they are also used to point out interesting features that were not yet investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The toy model describes energy exchanges between two energy bands and has been studied in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Each band is described by N equidistant energy levels such that the baseline (unperturbed) Hamiltonian is H0 = δE N−1 � i=0 i N − 1(|i⟩⟨i| + |i + N⟩⟨i + N|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (32) Here, δE sets an overall energy scale, which we choose in our numerics equal to δE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, |i⟩ describes an energy eigenstate of the first (second) band if i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , N − 1} (i ∈ {N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , 2N − 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The random coupling between the bands is mediated by ϵH1 = ϵ N−1 � i,j=0 vij|i⟩⟨j + N| + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', (33) where the vij are independent zero mean unit variance Gaussian random numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The observable X we consider quantifies the energy imbalance between the two bands and is defined as X = 1 √ 2N N−1 � i=0 (|i + N⟩⟨i + N| − |i⟩⟨i|) = 1 √ 2N (Π2 − Π1), (34) where Π1 (Π2) is the projector on the first (second) energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Clearly, this is a coarse observable with two eigenspaces of dimension N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The prefactor (2N)−1/2 is pure convention, but bears the advantage that the observable X has the same “size” (meaning that tr{X} = 0 and tr{X2} = 1 always) for different N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The condition for X to be slow was worked out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [93] and reads 16π2Nϵ2 δE2 ≪ 1, (35) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', weak coupling gives rise to a slow observable as usual (note that [H0, X] = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In the numerical simulations we choose ϵ = ϵ(N) such that the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (35) becomes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='01 unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, the relaxation time-scale of X is given by τX = (4πϵ2N)−1δE [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' We first consider the structure of the observable X in more detail as it played an important role in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For this purpose Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 3 shows matrix elements of the observable X and the projector Π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Specifically, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 3(a) shows a “matrix plot” of |Xkm| in an ordered energy eigenbasis of H = H0 + ϵH1 for N = 600 (note that the Hilbert space dimension is D = 2N) and we can immediately confirm that X is narrowly banded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, the coarseness of X implies that also the projectors Π1 and Π2 are narrowly banded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 3(b) by plotting the (absolute value of) the matrix elements of Π1 along the “counter-diagonal” (from the lower left to the upper right corner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is done for the three sizes N = 60 (larger black circles), N = 600 (medium sized pink circles) and N = 6000 (tiny blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Besides the observation of narrow bandedness, we also note that the off-diagonal elements appear essentially random and vary erratically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Furthermore, they decay with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' More specifically, the black circles are roughly one order of magnitude larger than the blue circles, which suggests a scaling 1/ √ D for the off-diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' These points are in complete agreement with the general predictions of the ETH [23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 19 SciPost Physics Submission Figure 3: Matrix elements explaining the structure of the considered observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Let us now consider the dynamics and test whether the time evolution of X is sensitive to a dephasing operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To this end, we plot and compare in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4 the two quantities 1 � xs=0 p(1t, xs) = 1 � x=0 tr � Π1e−iHtΠxe−iHs|ψ0⟩⟨ψ0|eiHsΠxe−iHt� , (36) p(1t, �� xs) = tr � Π1e−iH(t+s)|ψ0⟩⟨ψ0|eiH(t+s)� , (37) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', the probability to find the system in the first energy band at time t with and without dephasing operation at time s < t corresponding to the pink dashed and black solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is done for a single realization of the random matrix Hamiltonian and for a single Haar-randomly chosen initial state confined to the first energy band (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4 was found to be representative as different realizations of the Hamiltonian or initial state give rise to a similar picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The dephasing operation is done at s = τX (defined below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (35) and indicated by a vertical line) and the final measurement happens at t + s = 3τX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Note that the dephasing clearly happens before the system equilibrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Again, we consider the three system sizes N = 60 (a), N = 600 (b) and N = 6000 (c) and it becomes immediately evident that the process becomes classical with increasing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This confirms our main result, but Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4 contains two more important pieces of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, the circles and crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4 are generated for the same Hamiltonian and initial state, but by using truncated projectors, which are obtained from Πx by setting all off-diagonal elements with a distance to the diagonal greater than d/2 to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The choice for d was d = D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='7 and the reason for choosing this precise value of the exponent becomes clear later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For now it only matters that we confirm that d/D = D−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='3 is very small for large D and that we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4 that even for small D the dynamics is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This provides numerical evidence that truncating the sums, which was a crucial step to arrive at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (20) in our general derivation, is justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, another important piece of information is revealed by the number ∆ in the top right corner of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It equals the trace norm between the states with and without dephasing defined as ∆ = ∆(ρ, Dρ) ≡ 1 2tr � (ρ − Dρ)2 ∈ [0, 1], (38) 20 SciPost Physics Submission 0 1000 2000 3000 tδE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='0 p(1t) ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='5 (a) N = 60 0 1000 2000 3000 tδE ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='45 (b) N = 600 0 1000 2000 3000 tδE ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='46 (c) N = 6000 Figure 4: Exemplary check of the Kolmogorov consistency condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 102 103 N 10−2 10−1 distance ⟨Q⟩ α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='9 α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6 Figure 5: Scaling behaviour of (a suitable time average of) Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' where Dρ = � x ΠxρΠx denotes the dephased state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The trace norm is a distance measure characterizing the distinguishability of two quantum states and it has a wide range of ap- plications and favorable properties [90], including that (1 + ∆)/2 is the maximum success probability to distinguish between ρ and Dρ in an unbiased mixture given unlimited mea- surement power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, ∆ seems to be well suited to measure the amount of coherences in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Interestingly, we show in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 that maxρ ∆(ρ, Dρ) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Now, observing the values for the trace norm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4 we see that they are very close to the maximum possible value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In that sense, the dephasing operation is (almost) maximally invasive and a lot of coherences is destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This demonstrates not only that decoherence is not needed to explain classical behaviour, but much more: even maximally coherent states can show classical behaviour!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 5 shows the scaling behaviour of Q, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (17), for s = τX and t + s = 3τX as a function of the Hilbert space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is done by integrating the absolute value of the difference between the black solid and pink dashed curves, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4, divided by the length 2τX of the interval to give rise to average ⟨Q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, to minimize the risk of statistical outliers, this is done for three different realizations of the random matrix 21 SciPost Physics Submission 0 1000 2000 3000 tδE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='0 (a) p(1t) 103 N 10−2 10−1 (b) ⟨Q⟩ 0 1000 2000 3000 tδE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='0 (c) p(1t) (two dephasings) Figure 6: Violation of classicality for a highly atypical states (a), but approximate validity for moderately atypical states (b) and for two dephasing operations and typical states (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Hamiltonian and three Haar-randomly chosen initial states, thus giving nine realizations for each N as indicated by black circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To extract the scaling, we average these nine points for each N and fit a curve of the form ⟨Q⟩ ∼ 1 Dα , (39) which is inspired by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' By looking at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 5 (note the logarithmic scale), one might wonder whether it is a good idea to fit all the data by a straight line (pink dashed line with exponent α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='9) because the behaviour for N ≲ 400 clearly deviates from the straight line fit obtained for N ≥ 600 (blue solid line with exponent α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It is not completely clear to the author what causes the discrepancy, but in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [93] it was observed that the weak coupling approximation requires the side constraint 8π2N2ϵ2/δE2 > 1, which for our choice of ϵ implies N > 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This might explain the “anomalous” behaviour for small N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Also note that the exponent α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6 roughly fits the scaling behaviour observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [15] (where α was observed to be in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In any case, we use the fit to determine the number d at which we truncated the projectors to generate the pink crosses and black circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Namely, our main result predicted a scaling of the form (d/D)4 for [qt,s(x0)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This suggests that ⟨Q⟩ should scale as (d/D)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Comparing with D−α for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6 gives d = D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='7 as used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Finally, we challenge the present approach by relaxing certain assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, we ask what happens if the initial state is not randomly chosen within the first energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Figure 6(a) shows the breakdown of classicality for a highly atypical initial state |ψ0⟩ = |i⟩ for some randomly selected i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , N − 1} for N = 6000 (we use the same convention as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4 where the black solid line refers to free evolution and the pink dashed line to an evolution interrupted by a dephasing operation happening at the vertical black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Experimentally, preparing such an initial state requires precise microscopic control over the eigenstates of each energy band, which clearly violates the agreement made above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Nevertheless, classicality is quickly restored for more realistic states as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It shows ⟨Q⟩ for N = 600, N = 2000 and N = 6000 for five different realizations of |ψ0⟩ = |i⟩ (black circles) and for five different initial states |ψ0⟩ ∼ � i∈K ci|f(i)⟩, where K ⊂ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , N − 1} is a randomly chosen subset with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='005N many elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='5% of the energy levels in the 22 SciPost Physics Submission 0 1000 2000 3000 tδE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='0 p(1t) ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='47 (a) weak coupling 0 100 200 300 tδE ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='46 (b) medium coupling 0 10 20 30 tδE ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='45 (c) strong coupling Figure 7: Exemplary violation of classicality at strong coupling/for fast X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' first band are initially populated) and the ci are zero mean unit variance Gaussian random numbers (pink triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Despite quite large fluctuations, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 6(b) indicates a scaling law and the pink triangles are (on average) clearly below the black circles, showing the emergence of classicality even for moderately atypical states with a small fraction of populated levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Finally, Figure 6(c) shows what happens for two dephasing operations for N = 6000 and an initial random state as used also in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' While this is certainly not conclusive, it indicates that for sufficiently large dimensions the here introduced concept of classicality is robust also for n ≥ 3 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Last but not least, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 7 investigates the impact of the coupling strength on classicality, which is directly related to the slowness of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Here, weak, medium or strong coupling means that the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (35) was fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1 or 1, which is inversely proportional to the relaxation time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The plots are done for N = 6000 using again an initial state as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' One sees that classicality is well satisfied up to medium coupling strength, but fails in the strong coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This is not a deficit of the present theory because clearly not all observables can behave classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For strong coupling the eigenenergies of the total Hamiltonian can no longer be approximated by the local eigenenergies of the two bands, but are strongly hybridized, and is questionable how far X describes any meaningful energy difference in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 4 Conclusion The first half of this paper compared and contrasted well established and important ap- proaches to classicality, namely decoherence in OQS and consistent/decoherent histories, with recent abstract research [9–12] as well as numerical evidence [7, 8, 15] and general deriva- tions [15] of classicality based on the Kolmogorov consistency condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Arguably, the differ- ence between the consistent/decoherent histories condition and the Kolmogorov consistency condition is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, Kolmogorov consistency is easier to verify experimentally than consistent/decoherent histories and it can be independently well motivated from an opera- tional perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, we established that quantum Markovianity is a key concept to relate the decoherence approach in OQS to both the consistent/decoherent histories and the Kolmogorov consistency condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Figure 2 summarizes the first part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 23 SciPost Physics Submission The second half of the paper has given an independent derivation of the Kolmogorov consistency condition based on a random matrix theory model and numerically verified the correctness of the involved approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In fact, by looking at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (18) it even becomes clear that we have derived the stronger decoherent histories condition for an experimentally relevant class of initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Remarkably, it was explicitly shown that even maximally coherent states can give rise to classical dynamics for global observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Several interesting research avenues open up for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For instance, classicality could here be only established for “mini-histories” with two measurement results and extending the derivation to longer histories, as done in a different context in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [86–89], is highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, various fundamental question might appear in a new light, for instance, the relationship between quantum Darwinism [36, 37] and quantum Markovianity, or the implications of the present findings for (quantum) cosmology [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Acknowledgements This manuscript has significantly benefited from discussions with and feedback from Lennart Dabelow about random matrix theory, Wojciech Zurek about the quantum-to-classical tran- sition, and John Calsamiglia and Andreas Winter about trace distance bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For further discussions and feedback I thank Victor Bastidas, Giulio Gasbarri, Jochen Gemmer, Joseph Schindler and Jiaozi Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Funding information The author is finanically supported by “la Caixa” Foundation (ID 100010434) under the fellowship code LCF/BQ/PR21/11840014 and co-funded by the Spanish Agencia Estatal de Investigaci´on (project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' PID2019-107609GB-I00), the Spanish MINECO (FIS2016-80681-P, AEI/FEDER, UE), the Generalitat de Catalunya (CIRIT 2017-SGR-1127), and the European Commission QuantERA grant ExTRaQT (Spanish MICINN project PCI2022- 132965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1 Decoherence, Markovianity and consistency This appendix assumes the reader to be familiar with superoperators, in particular instru- ments, completely positive maps and completely positive and trace preserving maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In- troductions are provided, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [13, 14, 41, 90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' All superoperators are denoted with calligraphic symbols A, E, P, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' We start by defining a quantum Markov process following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [16], for introductory treatments see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This definition recognizes the crucial role played by an external agent (or experimenter or observer), who interrogates or intervenes the dynamics of an OQS at a set of discrete times {tn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , t2, t1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Each intervention at each time tk is described by an instrument {Ak(rk)}, which is a set of completely positive maps Ak(rk) adding up to a completely positive and trace preserving map Ak ≡ � rk Ak(rk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Importantly, these maps only act on the system Hilbert space, encapsulating the idea that the external agent has no precise control over the bath degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, rk denotes some abstract measurement outcome, not necessarily related to the xk appearing in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Now, 24 SciPost Physics Submission a quantum process is Markovian if the response of the OQS to any sequence (or history) of interventions {An(rn), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , A2(r2), A1(r1)} can be written as ˜ρS(tn|rn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , r2, r1) = An(rn)En,n−1 · · · A2(r2)E2,1A1(r1)E1,0ρS(t0), (40) where {Ek,k−1}n k=1 is a set of completely positive and trace preserving maps, which—importantly— do not depend on the interventions or initial system state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, ˜ρS(tn|rn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , r2, r1) is the subnormalized OQS state conditioned on the sequence of interventions, which happens with probability p(rn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , r2, r1) = trS{˜ρS(tn|rn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , r2, r1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Finally, notice that the Ek,k−1 are also known as dynamical maps as they progragate the system state forward in time from tk−1 to tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' These maps encode the influence of the bath or environment and, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (40), a Markov process is precisely characterized by the fact that the influence of the bath can be neatly separated from the interventions Ak(rk) of the external agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' A few more words of clarification might be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' First, one can show that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (40) reduces to the classical Markov condition in an appropriate limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Second, for classical causal models it reduces to the causal Markov condition of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Third, the validity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (40) can be checked by local interventions on the system only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Fourth, the existence of a Markovian quantum master equation for ρS(t), as often studied in OQS theory, does not imply the validity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (40), although the converse is true (see in particular Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Note that the identity operator I (“do nothing!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=') is an instrument too such that it is meaningful to define the dynamical map Eℓ,k ≡ Eℓ,ℓ−1 · · · Ek+1,k for any ℓ − k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Fifth, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (40) really is a statement about the multi-time behaviour of an OQS, and the validity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (40) can be shown to be equivalent to an appropriate formulation of the quantum regression theorem [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, we give a rigorous mathematical definition of OQS decoherence and a derivation that it implies the decoherent histories condition for quantum Markov processes (which in turn implies Kolmogorov consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To the best of the author’s knowledge, no definition of OQS decoherence exists and the notion is used rather conceptually (what follows is, however, closely related but not identical to the treatment of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [9,10,12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, if one wants to prove things mathematically, one has to start with a definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For this purpose, we use the dephasing operation D in the pointer basis as introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Then, for a quantum Markov process we define OQS decoherence by requiring that [Eℓ,k, D] = 0 for all n ≥ ℓ > k ≥ 1, (41) where [A, B] = AB − BA is the commutator in superoperator space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Is this a good definition of OQS decoherence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' At least it implies that the dynamics induced by the environment is not able to create coherences in the pointer basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To see this, we introduce the superoperator Px,yρ ≡ ΠS xρΠS y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, suppose that ρS = DρS is some system state without coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (41) implies that Px,yEρS = 0 for all x ̸= y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', it is not possible to create coherences in the pointer basis when starting from a decohered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Clearly, this captures a key aspect of the OQS decoherence concept, but one could naturally impose further constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For instance, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (41) makes no statement about the decoherence time tdec and, since the short time dynamics of OQS is complex, one might additionally require that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (41) is only valid on a coarse time scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', for tℓ − tk not too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In any case, the minimal definition given here turns out to be sufficient to prove that histories in the sense of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (8) are decoherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To see this, we conveniently write the decoherence functional for a quantum Markov process using superoperators: D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' y) = trS{Pxn,ynEn,n−1Pxn−1,yn−1En−1,n−2 · · · Px1,y1E1,0ρS(t0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (42) 25 SciPost Physics Submission Next, note that the decoherence functional does not change when subjecting it to a final dephasing operation D in the pointer basis (in fact, the decoherence functional does not change under any final dephasing): D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' y) = trS{DPxn,ynEn,n−1Pxn−1,yn−1En−1,n−2 · · · Px1,y1E1,0ρS(t0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (43) Now, let k be the first index for which xk ̸= yk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', xℓ = yℓ for all ℓ > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' By the definition of OQS decoherence, we can then permute D through until we hit the time tk: D(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' y) = trS{Pxn,ynEn,n−1 · · · Ek+1,kDPxk,ykEk,k−1 · · · Px1,y1E1,0ρS(t0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (44) Finally, elementary algebra shows that DPxk,ykρ = 0 whatever the input state ρ is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' It is interesting to note that strictly weaker conditions suffice to show Kolmogorov con- sistency for quantum Markov processes [9, 10, 12], but they seem insufficient to show the decoherent histories condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2 Numerical implementation This appendix includes some details about how to numerically fascilitate the evaluation of the expectation values appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (23) using Mathematica [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' We are interested in expectation values of the form E[V m1 µ4 ¯V m2 µ4 V m2 µ1 ¯V m3 µ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' ] with an equal amount of V - and complex conjugate ¯V -terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Since each pair in Isserlis theorem requires one V - and one ¯V -term, not all permutations of (V m1 µ4 , ¯V m2 µ4 , V m2 µ1 , ¯V m3 µ1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' ) contribute to the expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' One way to create all contributing permutations consists in generating two lists A = {{m1, µ4}, {m2, µ1}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' } and B = {{m2, µ4}, {m3, µ1}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' } associated to the V - and ¯V -terms, respectively, followed by PermA = Permutations[A];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Pairings = Map[Sort, Table[Flatten[PermA[[α, k]], B[[k]]], {α, 1, LA!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' }, {k, 1, LA}], 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Here, LA = Length[A] denotes the lengths of the list A (which equals LB) and consequently LA!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' is the length of PermA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The output Pairings now contains all possible pairings of the form Pairings = {{{m1, m2, µ4, µ4}, {m2, m3, µ1, µ1}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' }, {{m2, m2, µ1, µ4}, {m1, m3, µ4, µ1}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' }, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' } (45) The lowest level angular bracket {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' } contains one specific pair with always two Latin and two Greek indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The middle level angular bracket contains the product of all pairs, which form one specific “pairing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In general, Pairings will contain many forbidden pairings due to constraints such as m1 ̸= m2, µ1 ̸= µ2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To filter them out, we map each pairing to a graph with vertices (m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' ) and edges created by the Kronecker symbols of each pair, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', the pair {m1, m2, µ4, µ4} creates an edge between m1 and m2 and a (redundant) edge between µ4 and µ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Then, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', to respect the constraint m1 ̸= m2, a pairing is only accepted if there exists no path in the graph from m1 to m2, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 8 for a sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' As an example, the 26 SciPost Physics Submission Figure 8: Three example pairings (here, each pairing has two pairs with Latin indices) and the associated graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The constraint m1 ̸= m2 is violated in example (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' following code creates a list accepted that stores the numbers i for which the ith element of Pairings satisfies the constraints m1 ̸= m2 (further constraints can be easily included): For[i = 1, i ≤ LA!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', i + +, network = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For[j = 1, j ≤ LA, j + +, network = Append[network, Pairings[[i, j]][[1]] •−• Pairings[[i, j]][[2]]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' G = Graph[network];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' test = Length[Flatten[FindPath[G, m1, m2]]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' If[test == 0, accepted = Append[accepted, i]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' ] Here, the graph G for each pairing i is created using a set of edges stored in network, where each edge is symbolized by •−• (typeset as “Esc ue Esc” in Mathematica).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, we create a list Rem that simply contains all remaining pairings that satisfy the constraints above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This can be done by Rem = Map[Sort, Table[Pairings[[accepted[[k]]]], {k, 1, Laccepted], 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (46) Note that Sort brings the elements of the pairing in a standard form again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' As hinted at already above, the structure of the pairings can be nicely illustrated with a graph with edges indicating Kronecker deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For further manipulation, we now like to 27 SciPost Physics Submission convert Rem into a list of graphs: graphs = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' vert = {m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For[i = 1, i ≤ Laccepted, i + +, medges = Table[Rem[[i, j]][[1]] •−• Rem[[i, j]][[2]], {j, 1, Length[Rem[[1]]]}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' µedges = Table[Rem[[i, j]][[3]] •−• Rem[[i, j]][[4]], {j, 1, Length[Rem[[1]]]}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' edges = Join[medges, µedges];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' graphs = Append[graphs, Graph[vert, edges]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' ] con = Map[Sort]@ ∗ ConnectedComponents/@graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The final output con contains the connectivity of each graph associated to each accepted pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' For instance, if {{m1, m1, µ1, µ3}, {m2, m3, µ2, µ4}, {m2, m4, µ4, µ4}} is one accepted pairing, then its connectivity is {{m2, m3, m4}, {µ1, µ3}, {µ2, µ4}, {m1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This format has nice properties as it directly reveals the “structure” of each pairing in terms of (multi-valued) Kronecker deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To find out whether there are multiple pairings with the same structure, one can run Tally[con].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Given the connectivity list con, it is also straightforward to count the total number of sums that get killed due to Kronecker deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In the following, the function kills computes this number for a given pairing and final stores these numbers for each element of con: kills[list−] := Total[Map[Length, list]] − Length[list];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' final = Table[kills[con[[m]]], {m, 1, Laccepted}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Clearly, the above procedure can be also applied to the graphs formed by Greek indices only—as we needed to do to find the terms with the fewest amount of subscript-Kronecker deltas around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='3 Higher order corrections In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [79] (see also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [92]) Dabelow and Reimann developed a systematic way to take into account correlations among matrix elements, which we briefly summarize here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Unfortunately, it will turn out that this procedure becomes quickly untractable due to the fact that we already start with an expectation value over a product of sixteen random numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Moreover, the method does not take into account correlations with respect to both the perturbed and unperturbed bases |m⟩ and |µ⟩, but only with respect to one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Therefore, while that method was found to work well in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [79, 81, 92], it still does not provide an exact treatment of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To start with, notice that the matrix element V m µ can be seen as the m’th component of a D-dimensional vector Vµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To each Vµ we associate two vectors vµ and wµ, where the components of vµ are assumed to be independent Gaussian random variables with statistical properties equal to those of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Then, what was effectively done in the main text was to approximate E � V 1 4 ¯V 2 4 V 2 1 ¯V 3 1 V 3 3 ¯V 4 3 V 4 2 ¯V 1 2 V 5 8 ¯V 6 8 V 6 5 ¯V 7 5 V 7 7 ¯V 8 7 V 8 6 ¯V 5 6 � ≈ E � v1 4¯v2 4v2 1¯v3 1v3 3¯v4 3v4 2¯v1 2v5 8¯v6 8v6 5¯v7 5v7 7¯v8 7v8 6¯v5 6 � , (47) 28 SciPost Physics Submission which disregards all correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Instead, we now replace E � V 1 4 ¯V 2 4 V 2 1 ¯V 3 1 V 3 3 ¯V 4 3 V 4 2 ¯V 1 2 V 5 8 ¯V 6 8 V 6 5 ¯V 7 5 V 7 7 ¯V 8 7 V 8 6 ¯V 5 6 � ≈ E � w1 4 ¯w2 4w2 1 ¯w3 1w3 3 ¯w4 3w4 2 ¯w1 2w5 8 ¯w6 8w6 5 ¯w7 5w7 7 ¯w8 7w8 6 ¯w5 6 � (48) and obtain the vectors {wµ} from {vµ} using a Gram-Schmidt procedure, which orthonor- malizes the set {vµ}, thereby taking into account constraints imposed by the unitarity of V m µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, starting from w1 = v1, we set5 wµ = vµ − µ−1 � ν=1 ⟨wν|vµ⟩wν (49) for all µ ≥ 2 and where ⟨w|v⟩ denotes the standard complex scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Inserting the wµ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (48) gives an explicit expression in terms of the independent Gaussian variables vm µ that can be calculated using Isserlis’ theorem and takes into account correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Unfortunately, we would need to do this for eight vectors in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (48) (and their complex conjugates) and the total number of vm µ -terms, and consequently the number of pairings in Isserlis’ theorem, quickly grows to astronomically large numbers, even when respecting the constraints identified in the main text (m1 ̸= m2, µ1 ̸= µ2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' To see this, it might be helpful to explicity write down the components obtained via the Gram-Schmidt procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Clearly, everything is simple for the first vector: wm 1 = vm 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The second vector is also still managable: wm 2 = vm 2 − � n ¯vn 1 vn 2 vm 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The third vector, however, contains already six terms with up to seven v-components: wm 3 = vm 3 − � n ¯vn 1 vn 3 vm 1 − � n ¯vn 2 vn 3 vm 2 + � no vo 1¯vo 2¯vn 1 vn 3 vm 2 + � np ¯vn 2 vn 3 ¯vp 1vp 2vm 1 − � nop vo 1¯vo 2¯vn 1 vn 3 ¯vp 1vp 2vm 1 , (50) and it does not get simpler for the remaining vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, recall that we are only interested in an order-of-magnitude estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Each added pair of v-terms comes with an extra D-dimensional summation, but also contributes a factor of the order D−1 due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' In general, one therefore expects that these contributions roughly cancel each other in an order-of-magnitude estimate provided that the minimum number of Kronecker deltas as identified in the main text remains the same (if one term in Isserlis’ theorem gives rise to fewer Kronecker deltas than before, then an additional sum appears potentially contributing a huge factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Whether this is the case has been explicitly checked to lowest order in the Gram-Schmidt procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This means one first sets wµ ≈ vµ − µ−1 � ν=1 ⟨vν|vµ⟩vν (51) for µ ∈ {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , 8}, which follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (49) by replacing wµ by vµ on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' This approximation is then inserted into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (48) and only terms with a single additional sum 5To be precise, the here presented procedure only ensures orthogonality, but not normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, since the real and imaginary coefficients of vµ are each drawn from a zero mean (approximate) Gaussian distribution with variance 1/2D, it follows that vµ is not only normalized on average, but each single realization of vµ is strongly concentrated around vectors with unit norm for large D [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' 29 SciPost Physics Submission are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' There are 2·(1+2+· · ·+7) = 56 such single sum contributions, where the factor two arises because one has to take into account wµ and ¯wµ for µ ∈ {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' , 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, from the structure of the problem it does not appear that the complex conjugate entries contribute differently (since we are interested in a real-valued object), so these additional terms can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' On the other hand, which of the 28 terms gives the worst contribution to the scaling is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Therefore, this has been tested with the programme from Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2 and it was found that each correction term contains enough (multi-valued) Kronecker deltas to kill at least 6 sums, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', the same number as identified in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' The following table explicitly list the 28 contributions together with their “Kronecker order” L(δ), which equals the number of sums killed or, equivalently, the minimum of the list final defined at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' replace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' L(δ) replace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='L(δ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='w1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='− �9 ¯v9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1v9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='w6 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6v8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='w6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='− �9 ¯v9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1v9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='5v6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='− �9 ¯v9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='5v9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6v8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' one might worry that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' even when each single contribution to the expectation value is very small,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' the sum of the enormous amount of terms involved gives rise to a giant prefactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' However, this is unlikely a problem because, first, this prefactor does not scale with the particle number N and, second, recall that the contributions have different signs, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Additonal cancellations are therefore likely and were indeed an important observation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [79,81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='4 Trace distance bound under dephasing The author owes the details of the following proof to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' We start by noting that strong convexity of the trace norm [90] implies max ρ ∆(ρ, Dρ) = max ψ ∆(ψ, Dψ), (52) where ψ = |ψ⟩⟨ψ| is here used to denote pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, any such |ψ⟩ can be written as |ψ⟩ = �M x=1 αx|ψx⟩ with Πy|ψx⟩ = δx,y|ψx⟩ and |αx|2 = ⟨ψ|Πx|ψ⟩ such that � x |αx|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' One then finds ∆(ψ, Dψ) = 1 2tr � � � � � � � M � x,y=1 αxα∗y|ψx⟩⟨ψy| − M � x=1 |αx|2|ψx⟩⟨ψx| � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (53) 30 SciPost Physics Submission Since the {|ψx⟩} are orthonormal for different x and because the trace norm is invariant under unitary rotations [90] such that we can map |ψx⟩ to some fixed standard vector for each subspace x, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (53) makes it evident that we can restrict the problem to an M-dimensional Hilbert space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=', max ρ ∆(ρ, Dρ) = max ˜ψ ∆ � ˜ψ, ˜D ˜ψ � , (54) where | ˜ψ⟩ ∈ CM and the dephasing operation becomes ˜D˜ρ = �M x=1 |x⟩⟨x|˜ρ|x⟩⟨x| for some set of one dimensional projectors {|x⟩⟨x|} spanning CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Next, we note that we can write the dephasing map as ˜D˜ρ = 1 M �M−1 k=0 Zk ˜ρZ−k with the M-dimensional diagonal phase unitary Z = �M x=1 e2πi(x−1)/M|x⟩⟨x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, ˜ρ − ˜D˜ρ = 1 M �M−1 k=1 (˜ρ − Zk ˜ρZ−k) and it then follows from the triangle inequality that ∆( ˜ψ, ˜D ˜ψ) ≤ 1 M M−1 � k=1 ∆ � ˜ψ, Zk ˜ψZ−k� ≤ M − 1 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' (55) Finally, one straightforwardly shows that the upper bound is satisfied by the maximally coherent state | ˜ψ⟩ = � x |x⟩/ √ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Thus, for M = 2 we obtain the result used in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' References [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfrQFV/content/2301.02563v1.pdf'} +page_content=' Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev.' metadata={'source': 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Bolatto,1 Deanne B. Fisher,3, 4 Roberto Abraham,5 Karl Glazebrook,3, 4 +Rodrigo Herrera-Camus,6 Rebecca C. Levy,7, ∗ Danail Obreschkow,8 and Carrie Volpert1 +1Department of Astronomy, University of Maryland, College Park, MD 20742, USA +2SOFIA Science Center, USRA, NASA Ames Research Center, M.S. N232-12, Moffett Field, CA 94035, USA +3Centre for Astrophysics and Supercomputing, Swinburne University of Technology, PO Box 218, Hawthorn, VIC 3122, Australia +4ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D) +5Department of Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada +6Departamento de Astronom´ıa, Universidad de Concepci´on, Barrio Universitario, Concepci´on, Chile +7Steward Observatory, University of Arizona, Tucson, AZ 85721, USA +8International Centre for Radio Astronomy Research (ICRAR), University of Western Australia, Crawley, WA 6009, Australia +(Received; Revised; Accepted) +Submitted to +ABSTRACT +The spectral line energy distribution of carbon monoxide contains information about the physical +conditions of the star forming molecular hydrogen gas; however, the relation to local radiation field +properties is poorly constrained. Using ∼ 1−2 kpc scale ALMA observations of CO(3−2) and CO(4−3), +we characterize the CO(4−3)/CO(3−2) line ratios of local analogues of main sequence galaxies at +z ∼ 1 − 2, drawn from the DYNAMO sample. We measure CO(4−3)/CO(3−2) across the disk of +each galaxy and find a median line ratio of R43 = 0.54 ++0.16 +−0.15 for the sample. +This is higher than +literature estimates of local star-forming galaxies and is consistent with multiple lines of evidence that +indicate DYNAMO galaxies, despite residing in the local Universe, resemble main-sequence galaxies at +z ∼ 1 − 2. Comparing to existing lower resolution CO(1−0) observations, we find R41 and R31 values +in the range ∼ 0.2 − 0.3 and ∼ 0.4 − 0.8 respectively. We combine our kpc-scale resolved line ratio +measurements with HST observations of Hα to investigate the relation to star formation rate surface +density and compare this relation to expectations from models. We find increasing CO(4−3)/CO(3−2) +with increasing star formation rate surface density; however, models over-predict the line ratios across +the range of star formation rate surface densities we probe, particularly at the lower range. Finally, +SOFIA observations with HAWC+ and FIFI-LS reveal low dust temperatures and no deficit of [CII] +emission with respect to the total infrared luminosity. +Keywords: interstellar line emission – CO line emission, galactic and extragalactic astronomy – extra- +galactic galaxies, galaxies – disk galaxies +1. INTRODUCTION +Molecular hydrogen gas (H2) is routinely mapped in +high-redshift (high−z) galaxies with instruments such +as the Atacama Large Millimeter/sub-millimeter Ar- +ray (ALMA) and NOrthern Extended Millimeter Ar- +Corresponding author: Laura Lenki´c +laura.lenkic@nasa.gov, llenkic@usra.edu +∗ NSF Astronomy and Astrophysics Postdoctoral Fellow +ray (NOEMA) through the use of high rotational lines +(high−J) of carbon monoxide (CO; see e.g., Genzel et al. +2010; Tacconi et al. 2010; Decarli et al. 2014; Walter +et al. 2016; Freundlich et al. 2019). +High−J lines of +CO can be used to address important topics such as the +evolution of molecular gas reservoirs in galaxies across +cosmic time (see e.g., Walter et al. 2014; Decarli et al. +2016; Riechers et al. 2019; Decarli et al. 2019; Lenki´c +et al. 2020). +Mapping H2 through high−J CO emis- +sion in high−z galaxies provides certain advantages over +arXiv:2301.05251v1 [astro-ph.GA] 12 Jan 2023 + +2 +Lenki´c et al. +CO(1−0), because these lines are bright and allow for +higher resolution observations. +However, limited con- +straints of the CO excitation ladder, or the CO spectral +line energy distribution (SLED), render the conversion +to the ground state transition, CO(1−0), uncertain. The +CO excitation ladder contains information about the +temperature and density of the H2 material (see e.g., +Carilli & Walter 2013, for a review), and understand- +ing how those properties relate to local star formation +activity will improve our understanding of how high−J +CO lines map to CO(1−0) and H2 mass. +Several studies have characterized the CO excitation +ladder in various local galaxy populations. In the Milky +Way galactic center, observations from the COBE Far +Infrared Absolute Spectrophotometer, which constrain +the CO SLED from J = 1 − 0 to J = 8 − 7, show +that the line ratios can be modeled with an excitation +temperature of 40 K and that the CO SLED peaks at +J = 3 (Fixsen et al. 1999). A recent systematic study of +CO(1−0) to CO(3−2) in nearby galaxies finds that the +Rayleigh-Jeans brightness temperature ratios are gen- +erally higher in galaxy centers, decreasing with radius +and diminishing star formation rate (SFR) surface den- +sity (Leroy et al. 2022). +Kamenetzky et al. (2016) study CO emission up to +J = 13−12 in ultraluminous infrared galaxies (ULIRGS; +see also Greve et al. 2014), active galactic nuclei (AGN), +and non-ULIRGs to find that CO SLEDs peak at in- +creasingly higher−J with increasing far-infrared (FIR) +luminosity, indicating higher kinetic temperatures or +densities are required (see also Figure 1 of Obreschkow +et al. 2009, for how CO excitation depends on galaxy +type and excitation temperature). Observations of sub- +millimeter galaxies (SMGs) show that they also have +CO SLEDs with an “excess” of CO excitation with re- +spect to the Milky Way; these generally rise up to J = 5 +and then turn over for higher rotational states (Bothwell +et al. 2013; Spilker et al. 2014). These high excitation +CO SLEDs suggest that alternative heating sources are +required in these extreme galaxies such as mechanical +heating via shocks, turbulence, or cosmic rays. +Although several large studies probe the CO SLEDs +of extreme systems like SMGs and U/LIRGS, the CO +SLEDs of normal z ∼ 1−2 star-forming galaxies are not +as well characterized. Valentino et al. (2020) conduct a +large survey of mid− and high−J CO lines with ALMA +in main sequence galaxies at z ∼ 1.1−1.7, and find that +they have higher excitation than the Milky Way but +are not quite as highly excited as ULIRGs, SMGs, or +QSOs. Daddi et al. (2015) find similar results in a sam- +ple of four main-sequence near-IR selected galaxies at +z ∼ 1.5, using CO(2−1), CO(3−2), and CO(5−4) obser- +vations. Bolatto et al. (2015) also present the Rayleigh- +Jeans brightness temperature CO(3−2)/CO(1−0) line +ratio of four main sequence galaxies observed with the +Plateau de Bure Interferometer (PdBI), and find a ratio +of about unity denoting high excitation. Finally, multi- +ple case studies of the CO ladder in specific galaxies exist +(see e.g., Barvainis et al. 1997; van der Werf et al. 2010; +Kamenetzky et al. 2012; Aravena et al. 2014; Dessauges- +Zavadsky et al. 2017; Brisbin et al. 2019; Sharon et al. +2019; Henr´ıquez-Brocal et al. 2022; Klitsch et al. 2022), +and show that while general trends exist in different +galaxy populations, the CO ladder of every galaxy is +unique. +This indicates that it is necessary to understand how +CO emission and excitation vary within galaxies and +how they relate to other physical properties in order +to correctly interpret H2 masses derived from high−J +transitions. This requires resolved studies of CO line ra- +tios; however, observational limitations at high−z make +this challenging. +To address this, we present a sam- +ple of nine galaxies drawn from the DYnamics of Newly +Assembled Massive Objects (DYNAMO; Green et al. +2014) observed by ALMA in CO(3−2) and CO(4−3) +on ∼ 1 − 2 kpc scales. DYNAMO galaxies are nearby +(z ∼ 0.1) objects with high gas fractions, high star +formation rates, and widespread turbulence, consistent +with known properties of high−z main-sequence galax- +ies, and many are indeed found to lie on the main- +sequence of star formation at z ∼ 1 − 2 (Fisher et al. +2019). Their resemblance to high−z systems and prox- +imity allows us to probe the CO excitation in gas-rich, +turbulent galaxies at scales that are not yet achievable at +z ∼ 2 in unlensed systems. Furthermore, theories seek +to explain CO line ratios by their local radiation field +properties (Lagos et al. 2012; Narayanan & Krumholz +2014; Popping et al. 2014; Bournaud et al. 2015), and +our ALMA observations allow us to compare to model +expectations. +This paper is structured as follows: §2 describes our +observations, data reduction, and methods; §3 and §4 +describe and discuss our results, and finally we con- +clude in §5. Throughout this work, we assume ΛCDM +cosmology with H0 = 69.6 km s−1, Ωm = 0.286, and +ΩΛ = 0.714 (Bennett et al. 2014), and a Kroupa initial +mass function (IMF; Kroupa 2001). +2. OBSERVATIONS AND DATA REDUCTION +The DYNAMO sample of galaxies was first defined by +Green et al. (2014), who selected galaxies from the MPA- +JHU Value Added Catalog of the Sloan Digital Sky Sur- +vey DR4 (SDSS; Adelman-McCarthy et al. 2006) based +on their redshift and Hα emission. The sample consists + +Resolved CO Excitation in DYNAMO +3 +of 67 galaxies, half of which have LHα > 1042 erg s−1 in +the 3′′ diameter SDSS fiber, lying in two redshift win- +dows centered at z ∼ 0.075 and z ∼ 0.13. Their stel- +lar masses range from 109 − 1011 M⊙ and their SFRs +from ∼ 0.1 − 100 M⊙ yr−1, while their metallicities are +about solar (Tremonti et al. 2004). Employing integral- +field spectroscopy of Hα, Green et al. (2014) derive Hα +rotation curves and find high ionized gas velocity dis- +persions with a mean of ∼ 50 km s−1, and gas frac- +tions as high as fgas ∼ 0.8 (fgas ≡ Mgas/(Mgas + M∗); +Mgas = MHI + MH2). +Furthermore, they find that +DYNAMO galaxies are more “turbulent” than local +disks, as parameterized by their ratio of rotation ve- +locity to velocity dispersion (V/σ). +These properties +make DYNAMO galaxies promising candidates for local +analogues of high-redshift, star-forming galaxies. +Here, we consider a sub-set of nine DYNAMO galax- +ies that have robustly been identified as consistently +more similar to z ∼ 1 − 2 star-forming systems: DY- +NAMO C13-1, C22-2, D13-5, D15-3, G04-1, G08-5, +G14-1, G20-2, and SDSS J013527.10-103938.6 (hereafter +SDSS 013527-1039). This builds on a multi-wavelength +campaign to investigate the nature of star formation at +high redshift (Bassett et al. 2014; Fisher et al. 2014; +Obreschkow et al. 2015; Bassett et al. 2017; Fisher et al. +2017a,b; White et al. 2017; Girard et al. 2021; Lenki´c +et al. 2021; Ambachew et al. 2022; White et al. 2022). +All galaxies in our sample are classified as rotating disks +based on their Hα kinematics. Galaxies C22-2, G04-1, +G14-1, and G20-2 are furthermore classified as “com- +pact” rotating disks, because their SDSS r−band expo- +nential scale lengths are smaller than 3 kpc. For these, +the poorer resolution results in less reliable kinematic +classifications (Green et al. 2014). +All galaxies in our sample have CO(1−0) observations +from either the PdBI or NOEMA, from which molecular +gas fractions (MH2/(MH2+M∗)) of fgas ∼ 20−30 % and +molecular gas depletion times of tdep ∼ 0.5 Gyr are in- +ferred (Fisher et al. 2014; White et al. 2017; Fisher et al. +2019). These high molecular gas fractions are consis- +tent with those of z ∼ 1 − 2 main-sequence star-forming +galaxies (Daddi et al. 2010; Tacconi et al. 2010, 2013; +Genzel et al. 2015; Tacconi et al. 2020). Similarly, sub- +sequent studies of the gas kinematics in these galaxies +consistently show that they do indeed have high ion- +ized gas velocity dispersions (Bassett et al. 2014; Oliva- +Altamirano et al. 2018; Girard et al. 2021), similar to +main-sequence galaxies at z ∼ 1 − 2 (F¨orster Schreiber +et al. 2006; ¨Ubler et al. 2019). +In addition, Fisher +et al. (2017b); White et al. (2017) both show that these +DYNAMO galaxies are consistent with marginally sta- +ble disks (Toomre Q ∼ 1). +DYNAMO galaxies also +conform to established definitions of clumpy galaxies +(Fisher et al. 2017b) at high-redshift (e.g., CANDELS; +Guo et al. 2015). +Finally, their clumps are arranged +within their host disks such that the redder clumps are +preferentially more centrally located than the bluer ones +(Lenki´c et al. 2021; White et al. 2022), which has also +been observed in z ∼ 1 − 2 clumpy galaxies (F¨orster +Schreiber et al. 2011; Soto et al. 2017; Guo et al. 2018). +2.1. ALMA CO Observations +We make use of the CO(3−2) and CO(4−3) obser- +vations of nine DYNAMO galaxies with the Atacama +Large Millimeter/Submillimeter Array (ALMA), associ- +ated with project code 2017.1.00239.S (PI: D. B. Fisher). +Observations were taken in Band 7 (275−373 GHz) and +Band 8 (385 − 500 GHz) between 2018-06-01 and 2018- +07-10. The spectral windows were configured with band- +widths of 2.00 GHz and channel widths of 15.625 MHz +(128 channels). In addition, we also make use of higher +resolution CO(3−2) ALMA observations of three DY- +NAMO galaxies (G04-1, G08-5, and G14-1) associated +with the project code 2019.1.00447.S (PI: R. Herrera- +Camus). These observations were taken in Band 7 be- +tween 2019-10-09 and 2019-10-10. +The spectral win- +dows were configured with bandwidths of 1.875 GHz and +channel widths of 7.8125 MHz (240 channels). The data +associated with both projects were presented in Girard +et al. (2021). +The visibilities were calibrated and flagged by the +observatory with the Common Astronomy Software +Application (casa, McMullin et al. 2007) pipeline ver- +sions listed in the fifth column of Table 1. After cal- +ibrating the visibilities, we imaged each observation +using tclean in casa version 6.1.0.188 with param- +eters deconvolver=‘hogbom’, +weighting=‘briggs’, +robust=0.5, +usemask=‘auto-multithresh’, +and +restfreq set to the redshifted frequency of the ob- +served CO line. We cleaned the data until the residuals +were consistent with the root-mean-square (rms) noise +levels that are listed in the fourth column of Table 1. +To derive these thresholds, we consider data cubes with +just a shallow clean, mask the emission (see below), and +calculate the standard deviation of the masked cubes; +i.e., non-line channels. These values are listed in col- +umn four of Table 1, and we re-clean the data cubes to +that rms level. For visualization purposes and ease of +comparison to the Hα maps, we convolve the final cubes +to a circular beam, listed in the second (angular size) +and third (physical size) columns of Table 1, with the +casa imsmooth function. At the redshifts of DYNAMO +galaxies in our sample, the beam sizes correspond to +physical scales of ∼ 1 − 2 kpc. Finally, we export all + +4 +Lenki´c et al. +Table 1. CO Data Cube Parameters +CO Trans. +Beam FWHM +rms Noise +casa Cal. +(arcsec) +(kpc) +(mK) +DYNAMO C13-1 +3 − 2 +1.07 +1.60 +11.5 +v5.1.1-5 +DYNAMO C22-2 +3 − 2 +1.07 +1.46 +16.0 +v5.1.1-5 +4 − 3 +0.81 +1.11 +12.5 +v5.1.1-5 +DYNAMO D13-5 +3 − 2 +1.10 +1.58 +8.2 +v5.1.1-5 +4 − 3 +0.79 +1.14 +12.1 +v5.1.1-5 +DYNAMO D15-3 +4 − 3 +0.96 +1.24 +10.8 +v5.1.1-5 +DYNAMO G04-1 +3 − 2 +0.42 +0.98 +29.1 +v5.6.1-8 +4 − 3 +0.84 +1.96 +21.5 +v5.1.1-5 +DYNAMO G08-5 +3 − 2 +0.40 +0.95 +32.4 +v5.6.1-8 +DYNAMO G14-1 +3 − 2 +0.43 +1.02 +3.3 +v5.6.1-8 +4 − 3 +0.85 +2.01 +6.3 +v5.1.1-5 +DYNAMO G20-2 +3 − 2 +1.23 +3.08 +3.7 +v5.1.1-5 +4 − 3 +0.86 +2.15 +4.8 +v5.1.1-5 +SDSS 013527-1039 +3 − 2 +1.23 +2.81 +5.2 +v5.1.1-5 +4 − 3 +0.86 +1.97 +4.5 +v5.1.1-5 +data cubes with the spectral axis in units of velocity, +in the local standard of rest frame, adopting the radio +convention. We present channel maps of CO(3−2) for +DYNAMO G04-1 in Figure 1 to show an example of +the final data, with the circularized beam shown in the +bottom left corner of each panel. The complete figure +set (14 images) is available in the online journal. +We produce moment zero maps (integrated intensity) +by first masking each cleaned data cube along both the +spatial and spectral axes. +To produce our masks, we +first smooth each cleaned data cube to twice the cir- +cularized beam full width at half maximum (FWHM). +We then compute the rms of the data cube, and mask +all pixels that are below 3× the cube rms. For the re- +maining pixels, we compute the integrated intensity over +the channels that are not masked out. We do this for +both the CO(3−2) and CO(4−3) observations. Figure 2 +presents these moment zero maps in the two right-most +panels. +Finally, for the goal of calculating CO(4−3)/CO(3−2) +line ratios, we match the pixel scale and resolution of all +2017 CO(4−3) observations to the pixel scale and res- +olution of the 2017 CO(3−2) observations, where data +for both transitions are available. Similarly, we match +the pixel scale and resolution of the 2019 CO(3−2) ob- +servations, where available, to the 2017 CO(4−3) obser- +vations. We match the pixel scales using the casa func- +tion imregrid, while to match the resolution, we use the +casa imsmooth tool to convolve the higher resolution +data with a Gaussian kernel to produce the lower reso- +lution Gaussian beam. We note that these transforma- +tions are done on the cleaned data cubes with the origi- +nal, non-circular beams, to ensure we are not introduc- +ing errors or artifacts in the data. Finally, we apply the +masking of the CO(3−2) observations to the CO(4−3) +to produce matching integrated intensity maps. +This +ensures that the intensities we derive for both lines are +integrated over the same velocity ranges and regions. +2.2. HST Hα Observations +In addition to the ALMA observations of CO, we make +use of Hubble Space Telescope (HST) observations of +Hα (PID 12977; P.I.: I. Damjanov) as a tracer of the +star formation rate (left-most panel of Figure 2). Ob- +servations were taken with the Wide Field Camera on +the Advanced Camera for Surveys (WFC/ACS) using +the FR716N and FR728N narrow-band filters, and were +processed with the standard HST pipeline. Continuum +observations with the FR647M filter were also taken and +used to create continuum-subtracted Hα maps (for de- +tails, see §3.2 of Fisher et al. 2017a). The final Hα maps +have a pixel scale of ∼ 0.05′′ and a resolution corre- +sponding to physical scales of ∼ 50 − 200 pc (Fisher +et al. 2017a). +Our ability to make resolved measurements in these +DYNAMO galaxies is limited by the resolution of the +ALMA data; therefore, we match the pixel scale and res- +olution of the Hα observations to that of the CO(3−2), +where available, and CO(4−3) otherwise. +To achieve +this, we convolve each Hα observation with a two- +dimensional Gaussian function whose FWHM is equal to +the circularized beam of the corresponding ALMA ob- +servation. Then, we re-project and re-grid the Hα obser- +vations to match the WCS information and pixel scale of +the CO observations using the Python astropy pack- + +Resolved CO Excitation in DYNAMO +5 +Figure 1. Channel maps of CO(3−2) in brightness units of Jy beam−1 for the galaxy DYNAMO G04-1. Each panel is centered +at 04h12m19.713s, -05d54m48.62s, and is 10.8×10.8′′ in size. The velocity range is −172 to 96 km s−1 in steps of ∼8 km s−1, +as indicated in the top right corners. The circularized beam is shown in white in the bottom left corner of each panel. The +complete figure set (14 images) is available in the online journal. +age reproject1, noting that the reproject functions as- +sume that input images have surface brightness units. +2.3. SOFIA FIFI-LS and HAWC+ Observations +Finally, we make use of observations from the Strato- +spheric Observatory for Infrared Astronomy (SOFIA) +of DYNAMO galaxies taken by the FIFI-LS (Colditz +et al. 2018; Fischer et al. 2018) and HAWC+ (Harper +et al. 2018) instruments (PLAN ID 08 0238 and 09 158; +1 https://reproject.readthedocs.io/en/stable/index.html +P.I.: L. Lenki´c) as part of Cycles 8 and 9. The Field- +Imaging Far-Infrared Line Spectrometer (FIFI-LS) is an +integral field spectrometer with two channels observ- +ing simultaneously from 50 − 125 µm (blue channel) +and 105 − 200 µm (red channel). The FIFI-LS obser- +vations targeted the [CII] 158 µm fine-structure emis- +sion line in the red channel and the [OIII] 88 µm fine- +structure line (or [OI] at 63 µm depending on atmo- +spheric transmission) in the blue channel for six galaxies +(DYNAMO B08-3, D10-4, D14-1, D15-3, F08-2, F09- +1, and F12-4) at 15.6′′ resolution. These data cover a + +96.0 +88.0 +80.0 +73.0 +65.0 +57.0 +25 +50.0 +42.0 +34.0 +27.0 +19.0 +11.0 +20 +-12.0 +4.0 +-4.0 +-19.0 +-27.0 +-35.0 +15 +-42.0 +-50.0 +-57.0 +65.0 +-73.0 +-80.0 +10 +-88.0 +-96.0 +-103.0 +-111.0 +-119.0 +-126.0 +5 +-134.0 +-149.0 +-172.0 +-142.0 +-157.0 +-165.0 +06 +Lenki´c et al. + +30 +25 +C13- +CO(3-2) +1 kpc +Ho +20 +15 +1°30'04" +10 +Dec (ICRS) +5 +00" +29'56" +39.4s +39.1s +39.4s +39.1s +13h26m39.6s +13h26m39.6s +13h26m39.6s +39.4s +39.1s50 +CO(4-3) +50 +C22-2 +CO(3-2) +1 kpc +40 +40 +-8°04'16" +30 +30 +20 +20 +(ICRS) +Dec( +10 +10 +19" +0 +22h39m49.4s +49.2s +49.2s +22h39m49.4s +49.2s +22h39m49.4s80 +CO(4-3) +80 +D13-5 +CO(3-2) +1 kpc +OH +60 +60 +40 +40 +0°31'55" +Dec (ICRS) +20 +20 +52" +48" +0 +13h30m07.2s +07.0s +06.7s +13h30m07.2s +07.0s +06.7s +13h30m07.2s +07.0s +06.7s50 +-0°28'41" +D15- +CO(4-3) +Ha +1 kpc +40 +30 +20 +Dec (ICRS) +44" +10 +48" +0 +15h34m35.5s +35.3s +15h34m35.5s +35.3s +15h34m35.5s +35.3s70 +CO(3-2) +200 +CO(4-3) +60 +G04-1 +1 kpc +Haα +50 +150 +40 +5°54'47" +100 +30 +(ICRS) +20 +50 +Dec +10 +50" +0 +0 +4h12m19.9s +19.7s +4h12m19.9s +19.7s +19.7s +4h12m19.9sResolved CO Excitation in DYNAMO +7 +Figure 2. Summary of data sets analyzed in this work for each galaxy in our sample, as indicated in the top right corners of +the left-most panels: HST Hα (left), CO(3−2) integrated intensity (middle), CO(4−3) integrated intensity (right; both in units +of K km s−1). We show all images using an arcsinh stretch. The ALMA CO beam sizes are in the bottom left corners of the +middle and right-most panels, while 1 kpc scalebars are shown in the top right corner of the right-most panels. Empty panels +indicate that data is absent for the given galaxy. + +20.0 +12 +SDSS013527-1039 +CO(3-2) +17.5 +CO(4-3) +1 kpc +10 +15.0 +8 +12.5 +-10°39'36". +10.0 +7.5 +4 + (ICRS) +5.0 +Dec +2 +40" +2.5 +0 +0.0 +43" +1h35m27.4s +27.1s +26.9s +1h35m27.4s +26.9s +1h35m27.4s +27.1s +26.9s +27.1s +RA (ICRS) +RA (ICRS) +RA (ICRS)300 +Ha +G08-5 ++CO(3-2) +1 kpc +250 +200 +6°46'23" +150 +100 +(ICRS) +Dec +50 +19" +0 +18.7s +18.7s +18.7s +8h54m19.0s +8h54m19.0s +8h54m19.0s80 +35 +70 +Ha +G14-1 +CO(3-2 +CO(4-3) +1 kpc +30 +60 +25 +50 +20 +40 +15 +30 +Dec (ICRS) +0°44'35" +10 +20 +5 +10 +0 +31" +14h54m28.3s +14h54m28.3s +14h54m28.3s-25 +20 +G20-2 +CO(3-2) +CO(4-3) +1 kpc +Ho +20 +15 +-6°46'55" +15 +10 +10 + (ICRS) +5 +Dec ( +5 +59" +0 +0 +02.9s +02.9s +20h44m03.1s +02.6s +20h44m03.1s +20h44m03.1s +02.9s8 +Lenki´c et al. +1 × 1 arcmin2 field-of-view (FOV) in the red channel +and a 30 × 30 arcsec2 FOV in the blue channel. FIFI- +LS observations were taken on six nights in April 2021 +in the nod-match-chop mode, and were reduced using +the FIFI-LS pipeline2 (Vacca et al. 2020). The data re- +duction steps include ramp fitting and flagging bad pix- +els, subtracting the chops, wavelength and spatial cal- +ibration, flat-field correction, atmospheric transmission +correction using the ATRAN models (Lord 1992), flux +calibration, and finally resampling to a regular grid to +produce the final data cubes. The observations resulted +in [CII] detections for all galaxies in the sample, and +an [OIII] detection in DYNAMO F08-2 (see Figure 9 in +Appendix A). +The High-resolution Airborne Wideband Camera Plus +(HAWC+) instrument is a FIR camera and imaging po- +larimeter with a wavelength coverage of 50 − 240 µm. +The HAWC+ observations targeted four galaxies (DY- +NAMO D14-1, D15-3, F08-2, and F12-4) in bands C, D, +and E. These data provide measurements of the 89, 155, +and 216 µm fluxes at a resolution of 7.8′′, 14′′, and 19′′ +respectively. Observations were taken on three nights +in May 2021 and one night in November 2021 in the +on-the-fly mapping mode with a Lissajous scan pattern, +and were reduced using the HAWC+ pipeline3. The ob- +servations resulted in detections for all galaxies in the +sample (see Figure 8 in Appendix A). +The typical sizes of galaxies in this sample are ∼ 4′′ +and our sources are thus point sources for both the FIFI- +LS and HAWC+ observations. +Appendix A presents +the HAWC+ observations in Figure 8 and the FIFI- +LS integrated intensity maps in Figure 9. While DY- +NAMO D13-5 is the only galaxy that overlaps with our +ALMA sample, we make use of all SOFIA observations +described here to measure the SEDs of DYNAMO galax- +ies, and place the measured dust temperatures within +the global context of the line ratio measurements we will +present. We also make use of these observations to mea- +sure the [CII] luminosity and measure the [CII]-to-total +far-infrared luminosity ratios. +2.4. Resolved Measurements +This work aims to investigate the CO(4−3)/CO(3−2) +properties of DYNAMO galaxies resolved on a 1−2 kpc +scale, and to relate this line ratio to the star formation +rate surface density (ΣSFR) on the same scale. Thus, +we describe here our method for extracting these mea- +surements from the data. For each of our resolution and +WCS matched ALMA and HST data sets (excluding +2 FIFI-LS Redux User’s Manual +3 HAWC+ DRP User’s Manual +SOFIA observations because they are unresolved), we +define two sets of “grids” of circular, beam-sized aper- +tures: one that is centered on the galaxy, and a second +that is offset from the center by 0.5× the beam FWHM +in both the x and y directions. This is to ensure that we +cover the gaps of the first grid and results in measure- +ments that are not entirely independent. Within each +aperture, we measure the median brightness tempera- +ture of both the CO(3−2) and CO(4−3) lines from the +integrated intensity maps and take their ratio. +We measure the SFR surface density from our CO- +matched Hα observations. We perform aperture pho- +tometry within each ALMA beam-sized aperture in our +two grids, described above, to obtain the Hα flux (in +electrons per second). We convert these fluxes to units +of erg s−1 cm−2 ˚A−1, apply a correction for extinction by +relating AV to AHα assuming the Cardelli et al. (1989) +extinction law and the AV measurements from Lenki´c +et al. (2021). Bassett et al. (2017) use Paα observations +from the OSIRIS instrument at Keck to make resolved +extinction measurements in four DYNAMO galaxies. +Their results show up to a magnitude difference in AHα; +however, strong variation in the adaptive optics point +spread function introduces significant systematic uncer- +tainties in measuring the Paα flux. Furthermore, Bas- +sett et al. (2017) also find that the average AHα in +clump and non-clump regions are, within the uncer- +tainties, consistent with one another (see their Table +3). This is consistent with the results of Lenki´c et al. +(2021), who find that within a given DYNAMO galaxy, +the extinction-sensitive color they measure shows little +variation between clumps, and the clump colors are con- +sistent with their host disks (see their Figures 5 and 8). +For these reasons, we choose to adopt a single AV value +for each galaxy. Finally, we calculate Hα luminosities +and convert them to SFRs using the Hao et al. (2011) +calibration for a Kroupa IMF, constant star formation +history, and age of 100 Myr (see their Table 2): +SFR [M⊙ yr−1] = 5.53 × 10−42 × LHα [erg s−1] +(1) +3. RESULTS +3.1. CO(4−3)/CO(3−2) Line Ratios +In §2.1, we describe our process for matching our +CO(4−3) and CO(3−2) observations and deriving in- +tegrated intensity maps. We adopt brightness tempera- +ture units, thus our integrated intensity maps have units +of K km s−1. +To visually determine how this line ra- +tio varies across each galaxy disk, if at all, we simply +divide our CO(4−3) integrated intensity map by that +of the CO(3−2). This is what we present in Figure 3, +where the color scale indicates the ratio variations across + +Resolved CO Excitation in DYNAMO +9 +Figure 3. +CO(4−3)/CO(3−2) line ratios (in brightness temperature units) measured from the pixel scale and resolution +matched integrated intensity maps, integrated over the same velocity ranges. These maps show CO(4−3)/CO(3−2) only in +regions where the line ratio S/N ≥ 3. The black contours correspond to Hα emission, where available, ranging from 1 − 10σ in +increments of 1σ. The galaxy name is indicated in the top left corner of each panel, the median line ratios and their associated +uncertainties are in the top right corners, and the black hatched circles in the bottom left corners indicate the circularized +beam sizes. Finally, we show a 1 kpc scale bar in the bottom right corners. We see that galaxies generally have mildly varying +line ratios within the regions where the uncertainties do not dominate, and that they lie typically around 0.4 − 0.7, with the +exception of DYNAMO G14-1 which has a stronger varying line ratio. +each galaxy disk for which both line transitions were ob- +served, and where the line ratio S/N ≥ 3, and the black +contours correspond to Hα emission in the pixel scale +and resolution matched HST observations. +The con- +tours span 1−10σ in increments of 1σ, where we take σ +to correspond to the rms of each HST observation cal- +culated in galaxy emission-free regions. We note that +there are no HST Hα observations for DYNAMO C22- +2 and SDSS 013527-1039. We derive uncertainties for +the integrated intensity maps (σJ→J−1) by summing in +quadrature the rms of every channel over which we inte- +grate, excluding line emission from the rms calculation, +and multiplying by the channel width: +σJ→J−1 = ∆v +� +� +� +� +N +� +i +(rmsi)2 +(2) +where ∆v is the channel width, rmsi is the rms of the ith +channel, and N is the number of channels over which the +emission is integrated. To obtain the final uncertainty +on the line ratio per pixel, we propagate the integrated +intensity uncertainties by taking: +σlr = CO(4 − 3) +CO(3 − 2) +�� +σ43 +CO(4 − 3) +�2 ++ +� +σ32 +CO(3 − 2) +�2 +(3) +which results in line ratio uncertainty maps. +From Figure 3, we see that the line ratio for galaxies +in our sample vary mildly across the disks, with typical +values ranging from R43 ∼ 0.4 − 0.7. However, galaxy +DYNAMO G14-1 shows a strong gradient in the line +ratio, with values approaching unity. The Hα image of +G14-1 in Figure 2 shows two bright clumps with a fainter +“stream” connecting the two. The Hα contours we over- +plot in Figure 3 show that these two bright features with +the connecting filament coincide with the elevated line +ratio values and the strong gradient. This may be in- +dicative of an interaction taking place; however, the Hα +kinematics of G14-1 show a rotating disk and no com- +plex kinematics (Green et al. 2014). Overall, the line +ratio maps we show in Figure 3 suggest a potential cen- +tral enhancement in R43. Such a central enhancement +has been observed in the Milky Way and other nearby + +1.0 +-8°04'14" +C22-2 +0.62±0.13 +0.9 +0.8 +16" +Dec (ICRS) +0.7 +18" +0.6 +0.5 +20" +0.4 +0.3 +22" +1 kpc +0.2 +22h39m49.6s +49.4s +49.2s1.0 +0°31'58" +D13-5 +0.57±0.08 +0.9 +56" +0.8 +0.7 +54" +0.6 +52" +0.5 +0.4 +50" +0.3 +1 kpc +0.2 +13h30m07.2s +07.0s +06.8s1.0 +G04-1 +0.50±0.08 +-5°54'46" +0.9 +0.8 +0.7 +48" +0.6 +0.5 +50" +0.4 +0.3 +1 kpc +0.2 +4h12m19.9s19.8s +19.7s +19.6s +19.5s1.0 +0°44'37" +G14-1 +0.71±0.11 +0.9 +36" +0.8 +35" +0.7 +Dec (ICRS) +0.6 +34" +0.5 +33" +0.4 +0.3 +32" +1 kpc +0.2 +14h54m28.5s28.4s +28.3s +28.2s +RA (ICRS)1.0 +G20-2 +0.61±0.07 +-6°46'54" +0.9 +0.8 +56" +0.7 +58" +0.6 +0.5 +47'00" +0.4 +0.3 +kpc +02" +0.2 +20h44m03.2s +03.0s +02.8s +02.6s +RA (ICRS)1.0 +10°39'34" +SDSS013527-1039 +0.44±0.05 +0.9 +36" +0.8 +0.7 +38" +0.6 +40" +0.5 +0.4 +42" +0.3 +l kpc +0.2 +1h35m27.4s +27.2s +27.0s +26.8s +RA (ICRS)10 +Lenki´c et al. +star-forming galaxies for CO(2−1)/CO(1−0) (see e.g., +Sakamoto et al. 1997; Sawada et al. 2001; Leroy et al. +2009, 2013; den Brok et al. 2021). To verify this, we +separate pixels that are located within the central kpc +of each galaxy from pixels that lie outside this region, +and compare the median line ratios. +Indeed, we find +enhanced CO(4−3)/CO(3−2) values in the central kpc +of all galaxies in Figure 3, except for G04-1 and G14-1, +on the order of ∼ 10% (see Table 2). Finally, Figure 3 +shows in particular for galaxy G04-1, variations in R43 +between the spiral arm and inter-arm region, a trend also +observed for CO(2−1)/CO(1−0) in M51 (Koda et al. +2012). +Next, we perform ∼ 1−2 kpc sized sightline measure- +ments of the line ratio across the disk of each galaxy, +as described in §2.4, to characterize the typical line ra- +tio we measure across the sample and the magnitude of +the spread. To this end, we construct a global proba- +bility density function (PDF) by modeling each beam- +averaged line ratio measurement with a kernel density +estimate (KDE). We construct the individual KDEs by +modeling each beam-averaged line ratio measurement as +a one-dimensional Gaussian with centroid correspond- +ing to the measured line ratio and with width equal to +the line ratio uncertainty. The area of each Gaussian is +normalized to unity, then we sum all Gaussians to pro- +duce a final global PDF (see for example §4 and Figure +5 of Levy et al. 2018). This is what we show in Fig- +ure 4. +From this, we find that the median line ratio +and 68% confidence interval for DYNAMO galaxies are: +R43 = 0.54 ++0.16 +−0.15. +These values are taken at the 15.9, +50, and 84.1 percentiles of the cumulative distribution +function of the PDF. +For comparison, we compile estimates of the CO(4−3) +to CO(3−2) line ratio from the literature and include +these in Figure 4. We describe our derivation of all line +ratios we compile from the literature in Appendix B, and +we summarize them along with the median line ratios +we measure for each DYNAMO galaxy individually, and +the median line ratio for the entire DYNAMO sample +studied here in Table 2. +This comparison reveals that the CO(4−3)/CO(3−2) +line ratio of non-ULIRGs from Kamenetzky et al. (2016) +(local galaxies with LFIR ≤ 6 × 1010 L⊙) is much lower +and incompatible with what we find in our DYNAMO +sample. +In contrast, the U/LIRG line ratio estimate +from Kamenetzky et al. (2016) for LFIR = 1011 L⊙ is in +much better agreement with what we find across the DY- +NAMO sample. Likewise, the CO(4−3)/CO(3−2) line +ratios measured in main-sequence galaxies at z ∼ 1 − 2 +(Daddi et al. 2015; Boogaard et al. 2020; Henr´ıquez- +Brocal et al. 2022) are, within the uncertainties, con- +Table 2. +CO(4−3)/CO(3−2) Line Brightness Temperature +Ratios Compiled from the Literature Compared to DYNAMO +Object(s) +Line Ratio +Reference +C22-2 +0.62 ± 0.13 +This work +C22-2 (≤ 1 kpc) +0.60 ± 0.10 +This work +C22-2 (> 1 kpc) +0.52 ± 0.08 +This work +D13-5 +0.57 ± 0.08 +This work +D13-5 (≤ 1 kpc) +0.60 ± 0.08 +This work +D13-5 (> 1 kpc) +0.56 ± 0.09 +This work +G04-1 +0.50 ± 0.08 +This work +G04-1 (≤ 1 kpc) +0.48 ± 0.20 +This work +G04-1 (.1 kpc) +0.52 ± 0.16 +This work +G14-1 +0.71 ± 0.11 +This work +G14-1 (≤ 1 kpc) +0.70 ± 0.13 +This work +G14-1 (> 1 kpc) +0.71 ± 0.16 +This work +G20-2 +0.61 ± 0.07 +This work +G20-2 (≤ 1 kpc) +0.62 ± 0.10 +This work +G20-2 (> 1 kpc) +0.57 ± 0.10 +This work +SDSS J013527.10-103938.6 +0.44 ± 0.05 +This work +SDSS J013527.10-103938.6 (≤ 1 kpc) +0.46 ± 0.06 +This work +SDSS J013527.10-103938.6 (> 1 kpc) +0.42 ± 0.06 +This work +DYNAMO all +0.54 ++0.16 +−0.15 +This work +z = 1.5 MS Galaxies +0.74 ± 0.26 +D15 +ASPECS z = 1.0 − 1.6 SFGs +0.52 ± 0.16 +B20 +G1700-MD94 one component +0.92 ± 0.18 +HB22 +G1700-MD94 two component +0.77 ± 0.15 +HB22 +non-U/LIRGs (LFIR = 1010L⊙) +0.25 ± 0.05 +K16 +LIRGs (LFIR = 1011L⊙) +0.51 ± 0.10 +K16 +LIRGs +1.23 ± 0.38 +P12 +ULIRGs low CO excitation +1.08 +R15 +ULIRGs mid CO excitation +0.70 +R15 +ULIRGs high CO excitation +1.02 +R15 +sistent with DYNAMO. In particular, the eight star- +forming galaxies at z = 1.0 − 1.6 from the ALMA Spec- +troscopic Survey (ASPECS; Boogaard et al. 2020), are +an especially good match to the R43 we measure across +our sample. DYNAMO galaxies lie on the star forma- +tion main-sequence at z ∼ 2 (Fisher et al. 2019) and +have gas fractions and velocity dispersions that are more +similar to main-sequence galaxies of that epoch than lo- +cal ones. Therefore, this result is consistent with lines +of evidence that indicate DYNAMO galaxies are local +analogues of high−z main-sequence systems. +ULIRG +samples (Rosenberg et al. 2015) have much larger line +ratios than we observe in DYNAMO, and this too is +consistent with previous observations. Using Herschel +PACS+SPIRE observations of five DYNAMO galaxies, +White et al. (2017) found that despite their large FIR +luminosities, LFIR > 1011 L⊙, these galaxies have much +lower dust temperatures (∼ 30 K) than ULIRGs. There- + +Resolved CO Excitation in DYNAMO +11 +Figure 4. Global PDF for the resolved CO(4−3)/CO(3−2) +line ratio measurements. We construct the PDF by model- +ing each line-of sight R43 measurement (where S/N ≥ 3) as +a Gaussian whose width is the line ratio uncertainty. +We +show these individual Gaussians as light grey lines (not to +scale); summing them and normalizing the area of the result- +ing Gaussian to unity results in the solid black line shown +here. From the cumulative distribution function, we infer a +median line ratio of R43 = 0.54. For comparison, we include +estimates from the literature: R43 = 0.74 ± 0.26 for three +z ∼ 1.5 main sequence star forming galaxies (black circle; +Daddi et al. 2015), R43 = 0.52 ± 0.16 in eight star-forming +galaxies at z = 1.0 − 1.6 (black square; Boogaard et al. +2020), R43 = 0.25 ± 0.05, 0.51 ± 0.10 for non-U/LIRGs with +LFIR = 1010 L⊙ and U/LIRGs with LFIR = 1011 L⊙ respec- +tively (black stars; Kamenetzky et al. 2016), R43 = 0.70, 1.02 +for mid- and high-excitation ULIRGs respectively (black di- +amonds; Rosenberg et al. 2015), and R43 = 0.96 ± 0.12 for +LIRGs (black pentagon; Papadopoulos et al. 2012). +fore, unlike ULIRGs, the star formation in DYNAMO +galaxies is more distributed throughout the disks; thus, +colder dust temperatures would be expected and like- +wise lower CO(4−3)/CO(3−2) line ratios. +3.2. Relating High−J CO to CO(1−0) +We make use of existing CO(1−0) measurements +from the PdBI and NOEMA (angular resolution ∼ +5 − 10′′; Fisher et al. 2014; White et al. 2017; Fisher +et al. 2019) to derive the CO(4−3)/CO(1−0) and +CO(3−2)/CO(1−0) line ratios across our sample. We +measure the total CO(4−3) and CO(3−2) fluxes by sum- +ming all pixels with S/N ≥ 3 in our integrated intensity +Figure 5. CO ladders normalized to CO(1−0) in integrated +brightness temperature units, for DYNAMO galaxies (red +small diamond), z ∼ 1−2 main-sequence BzK galaxies (black +circles; Daddi et al. 2015), z = 1.0−1.6 star-forming galaxies +from ASPECS (black squares Boogaard et al. 2020) nearby +star forming galaxies (blue pentagons; Leroy et al. 2022), and +the Milky Way inner disk (blue large diamonds; Fixsen et al. +1999). DYNAMO line ratios are consistent with z ∼ 1 − 2 +star-forming galaxies, while the nearby star-forming galaxies +and the Milky Way show overall lower CO excitation. One +galaxy from the sample of Daddi et al. (2015) (BzK-16000) +is more consistent with nearby galaxies and the Milky Way +than with DYNAMO. BzK-16000 is more evolved and has +no massive clumps. +moment maps, then scaling by the number of pixels per +beam. We then convert the total fluxes to luminosities +(L′; K km s−1 pc2) using equation 3 in Solomon et al. +(1997). We present these results in Table 3. +We find line ratio values across our sample that range +from R31 ∼ 0.4 − 0.8, with a mean (median) R31 = +0.56 (0.55) and R41 ∼ 0.2 − 0.4, with a mean (me- +dian) R41 = 0.27 (0.28). +Our R31 result is consistent +with multiple studies of CO excitation in z ∼ 1 − 3 +star-forming galaxies: Daddi et al. (2015) found that +the brightness temperature line ratio of CO(3−2) to +CO(1−0) ranges from R31 ∼ 0.4 − 0.6, with an aver- +age R31 = 0.42 ± 0.07, for their three star-forming BzK +z ∼ 1.5 galaxies, Dessauges-Zavadsky et al. (2015) find +R31 = 0.57 ± 0.15 for five lensed star-forming galaxies +(SFR < 40 M⊙ yr−1) at z ∼ 1.5 galaxies, Riechers et al. +(2020) find R31 = 0.84±0.26 for six galaxies at z ∼ 2−3, +Birkin et al. (2021) find R31 = 0.63 ± 0.12 for a large +sample of SMGs at z ∼ 1.2 − 4.8, and Harrington et al. + +DYNAMO Median = 0.54±0-19 +★ +LIRGs (K16) +-0.15 +z～ 1 - 2 MS Galaxies (D15) +ULIRGs, Mid Ex. (R15) +ASPECS z = 1.0 - 1.6 SFGs (B20) +ULiRGs, High Ex (R15) +Non-U/LIRGs (K16) +U/LIRGS (P12) +0.025 +0.020 +Density +0.015 +0.010 +0.005 +0.000 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +CO(4-3)/CO(3-2)DYNAMO +1.0 +z ～ 1 -2MS Galaxies (D15) +ASPECS z = 1.0 - 1.6 SFGs (B20) +Nearby Galaxies (L22) +Milky Way (F99) +0.8 +CO(J →J-1)/CO(1-0) +0.6 +0.4 +0.2 +0.0 +2 +3 +4 +5 +6 +Upper Rotational Quantum Number, Jupper12 +Lenki´c et al. +Table 3. Galaxy Integrated CO(3−2)/CO(1−0) and CO(4−3)/CO(1−0) Line Ratios and Model Predictions +Galaxy +L′ +CO(1−0) +L′ +CO(3−2) +L′ +CO(4−3) +R31 +R41 +R31 +R41 +(109 K km s−1 pc2) +Observed +Predicted +C13-1 +1.91 ± 0.05 +0.47 ± 0.05 +· · · +0.40 ± 0.05 +· · · +· · · +· · · +C22-2 +0.66 ± 0.05 +0.47 ± 0.04 +0.23 ± 0.02 +0.71 ± 0.08 +0.35 ± 0.04 +· · · +· · · +D13-5 +2.69 ± 0.08 +1.48 ± 0.03 +0.87 ± 0.03 +0.55 ± 0.02 +0.32 ± 0.01 +0.63 +0.39 +D15-3 +3.02 ± 0.06 +· · · +0.49 ± 0.02 +· · · +0.16 ± 0.01 +0.57 +0.32 +G04-1 +5.41 ± 0.39 +2.90 ± 0.10 +1.54 ± 0.08 +0.54 ± 0.04 +0.28 ± 0.02 +0.56 +0.31 +G08-5 +2.29 ± 0.26 +1.83 ± 0.08 +· · · +0.80 ± 0.10 +· · · +0.57 +0.32 +G14-1 +1.59 ± 0.19 +0.77 ± 0.08 +0.427 ± 0.001 +0.48 ± 0.08 +0.27 ± 0.03 +0.56 +0.31 +G20-2 +1.68 ± 0.19 +0.97 ± 0.03 +0.56 ± 0.03 +0.58 ± 0.07 +0.33 ± 0.04 +0.61 +0.36 +SDSS 013527-1039 +3.45 ± 0.16 +1.48 ± 0.05 +0.65 ± 0.02 +0.43 ± 0.02 +0.19 ± 0.01 +· · · +· · · +(2021) find R31 = 0.69 ± 0.12 for 24 dusty star forming +galaxies at 1 < z < 3. +However, we note that Bo- +latto et al. (2015) find R31 ∼ 1 for two main-sequence +galaxies at z ∼ 2; however, one of the two galaxies is +classified as an AGN, and the other may host an weak +AGN. In contrast, Leroy et al. (2022) analyze the global +R31 for nearby normal star-forming galaxies and find a +mean (median) of R31 = 0.30 (0.29), which are lower +and inconsistent with the DYNAMO results, but simi- +lar to the Milky Way (Fixsen et al. 1999). We illustrate +this comparison in Figure 5 where we plot the brightness +temperature ratios of DYNAMO galaxies as a function +of upper-J number, along with the ratios of z ∼ 1 − 2 +star-forming galaxies (Daddi et al. 2015; Boogaard et al. +2020), nearby star-forming galaxies (Leroy et al. 2022), +and the Milky Way inner disk (Fixsen et al. 1999). We +can see that the ratios of nearby galaxies and the Milky +Way are incompatible with those of DYNAMO. The CO +SLED of the ASPECS galaxies and two of the three +BzK galaxies are in agreement with DYNAMO, while +the third galaxy (referred to as BzK-16000 in Daddi +et al. 2015) shows overall lower line ratios. +Interest- +ingly, Daddi et al. (2015) describe this galaxy as the +most evolved in their sample, with no massive clumps. +3.3. DYNAMO SEDs and [CII] Emission +We extract background subtracted 89, +155, +and +216 µm fluxes for four DYNAMO galaxies, including +D15-3 which overlaps with our ALMA sample, from +our HAWC+ SOFIA observations using the Photutils +software (Bradley et al. 2021). We define the flux ex- +traction apertures to correspond to the FWHM beam +size of each corresponding HAWC+ band, while we de- +fine the background annuli to have an inner radius equal +to 5× beam FWHM and an outer radius of 7× beam +FWHM (see Figure 8 in Appendix A). We record these +flux measurements in Table 4. To fit the SED, we com- +bine the HAWC+ fluxes with WISE measurements at +22 µm (which is not contaminated by line emission and +traces the warm dust continuum; Cluver et al. 2017), +and use the modified blackbody (MBB) SED fitting tool +mbb emcee4, described in Riechers et al. (2013) and +Dowell et al. (2014). The MBB is joined to a power law +of the form να at short wavelengths. mbb emcee fits +the dust temperature, Td, the extinction curve power +law slope, β, the power law slope of the blue side, α, the +wavelength where the optical depth reaches one, λ0, and +the normalization. We impose a prior on β to constrain +it between 1.5 − 2, and leave all other parameters un- +constrained. We record the resulting fit parameters and +total infrared luminosity (8−1000 µm; TIR) in Table 4. +We present the resulting SEDs in the left panel of +Figure 6, where the filled colored data points represent +fluxes from the HAWC+ (three longest wavelength data +points), open colored data points represent WISE bands +(shortest wavelength point) for each of the four galaxies, +and the matching colored line represents the SED fit for +that galaxy. The dust temperatures derived from these +SEDs are shown in the upper left corner. For DYNAMO +D14-1 and D15-3, the resulting dust temperatures are +Td = 25.94 ++5.10 +−4.88 K and Td = 29.56 ++3.06 +−2.89 K respectively, +consistent with the dust temperature measurements of +Td = 28.09 ± 0.86 K and Td = 25.64 ± 0.52 K from SED +fitting of Herschel PACS and SPIRE observations by +White et al. (2017). +4 https://github.com/aconley/mbb emcee + +Resolved CO Excitation in DYNAMO +13 +Table 4. SOFIA [CII] and IR Measurements +Galaxy +22.2 µm +89 µm +155 µm +216 µm +Td +TIR +log10 L[CII] +(mJy) +(mJy) +(mJy) +(mJy) +(K) +(1010 L⊙) +(erg s−1) +B08-3 +· · · +· · · +· · · +· · · +· · · +· · · +41.82 ± 0.46 +D10-4 +· · · +· · · +· · · +· · · +· · · +· · · +41.91 ± 0.46 +D14-1 +12.3 ± 3.5 +148 ± 19 +423 ± 47 +209 ± 25 +25.94 ++5.10 +−4.88 +3.48 ++0.69 +−0.78 +· · · +D15-3 +8.1 ± 2.8 +282 ± 33 +393 ± 44 +685 ± 73 +29.56 ++3.06 +−2.89 +4.26 ++0.52 +−0.49 +41.74 ± 0.46 +F08-2 +· · · +522 ± 57 +326 ± 37 +563 ± 61 +46.15 ++7.66 +−6.72 +6.27 ++1.30 +−1.34 +41.95 ± 0.46 +F09-1 +· · · +· · · +· · · +· · · +· · · +· · · +42.20 ± 0.46 +F12-4 +15.3 ± 3.9 +224 ± 27 +279 ± 32 +594 ± 64 +23.43 ++7.65 +−7.05 +4.11 ++0.74 +−1.02 +42.21 ± 0.46 +Figure 6. Left: Spectral energy distribution of galaxies DYNAMO D14-1, D15-3, F08-2, and F12-4 based on fluxes from SOFIA +HAWC+ (colored symbols) and WISE observations (open colored symbols). The solid colored lines are the resulting SED fits +using mbb emcee, with the corresponding dust temperatures appearing in the top left corner. For DYNAMO D14-1 and D15-3, +the dust temperatures derived from the HAWC+ measurements are consistent with those derived by White et al. (2017) using +Herschel PACS and SPIRE photometry. Right: The [CII]-to-TIR ratio as a function of TIR. The colored symbols with black +outlines correspond to DYNAMO measurements; the magenta symbols have TIR measurements derived from SED fitting, while +the teal symbols have TIR estimated from SFRs. For galaxies where we have both SED measurements and SFRs, we link the +data points via a black dashed line. Grey error bars are the assumed 40% calibration uncertainty for the [CII] observations and +TIR uncertainties propagated through the ratio. We compare our DYNAMO measurements to those of Herrera-Camus et al. +(2018a) and find that DYNAMO galaxies do not show a deficit of [CII] emission, consistent with their cooler dust temperatures. + +103 +Td = 25.94+5.1 +K +.88 +Td = 29.56±3.06 +Observed Flux (mJy) +102 +101 +DYNAMO D14-1 +DYNAMO D15-3 +DYNAMO F08-2 +DYNAMO F12-4 +102 +Observed Wavelength (μm)10-1 +High-z (z > 1) +LINER +Hll Galaxy +Seyfert & QSO +10-2 +[CII]/ LTIR +10-3 +10-4 +DYNAMO TIR from SFR +DYNAMO TIR from SED +109 +1010 +1011 +1012 +1013 +LTIR (Lo)14 +Lenki´c et al. +The SEDs provide us with estimates of TIR for four +out of the seven galaxies in the SOFIA sample. +We +combine these measurements with the SOFIA FIFI-LS +observations to explore the “[CII]-deficit”: the observed +decreasing fraction of [CII] emission with respect to TIR +in increasingly more infrared luminous objects (see e.g., +Malhotra et al. 2001; Brauher et al. 2008; Smith et al. +2017; Herrera-Camus et al. 2018a). For the remaining +four galaxies in the SOFIA sample where no HAWC+ +observations are available, we instead use the SFRs re- +ported in Green et al. (2014) to estimate TIR and the +calibration in equation (3) of Cluver et al. (2017): +SFR [M⊙ yr−1] = 2.8 × 10−44 LT IR [erg s−1] +(4) +which is derived from Starburst99 for solar metallic- +ity, continuous star formation over 100 Myr, a Kroupa +IMF, and assumes that the ultraviolet (UV) compo- +nent of stellar emission is completely absorbed and re- +radiated in the infrared (see also Calzetti 2013). +To determine the [CII] luminosities, we produce in- +tegrated intensity maps from the FIFI-LS observations +(see Figure 9 in Appendix A) and take the peak value +within a beam located at the position of each galaxy. In +the right panel of Figure 6, we present the [CII]/TIR as a +function of TIR measured in this sample of DYNAMO +galaxies. +The magenta squares represent galaxies for +which SEDs were used to derive TIR, while the teal di- +amonds represent the galaxies for which the SFRs were +used instead. In both cases, we show error bars where +the errors on the [CII] and TIR luminosities have been +propagated into the ratio. The two approaches to es- +timating the TIR luminosities yield consistent results. +To illustrate this, we join with black dashed lines the +data points for which we have SEDs and SFRs. When +we compare DYNAMO to existing measurements in dif- +ferent types of galaxies (Herrera-Camus et al. 2018a), +we find that DYNAMO galaxies do not exhibit a [CII]- +deficit. Herrera-Camus et al. (2018a) shows that at a +fixed IR luminosity, the [CII]/FIR ratio decreases as +galaxies become more compact, and Lutz et al. (2016) +shows that the line-to-FIR ratios form a much tighter +relation with FIR surface brightness than luminosity. +To investigate this, Herrera-Camus et al. (2018b) con- +struct two toy models with the PDR toolbox (Kaufman +et al. 2006): one where OB stars are closely associated +with molecular gas clouds, and another where OB stars +and clouds are randomly distributed. +They find that +as galaxies become more compact, a combination of ef- +fects give rise to the [CII]-deficit. These include a reduc- +tion of the photo-electric heating efficiency, an increase +in the ionization parameter, and as the interstellar ra- +diation field increases, the [CII] line saturates and be- +comes nearly independent of the far-UV flux. Although +DYNAMO galaxies generally lie above the star-forming +main sequence at z ∼ 0.1, their star formation is dis- +tributed throughout their disks within numerous star- +forming clumps, rather than being confined to a com- +pact region. Their low dust temperatures and lack of a +[CII]-deficit is consistent with this morphology. +4. DISCUSSION +The ∼ 1−2 kpc-scale ALMA observations allow us to +investigate how the line ratios we measure are affected +by the surface density of star formation. +We expect +that the CO(4−3) transition will be more highly excited +in regions of higher ΣSFR, because these regions will +have larger UV radiation fields and thus warmer dust +temperatures (Narayanan & Krumholz 2014). To test +this, we compare our resolved line ratio measurements +to the ΣSFR measurements we make in the same beam- +sized apertures. Figure 7 shows the CO(4−3)/CO(3−2) +line ratio as a function of ΣSFR for four galaxies for +which all necessary observations are available, as indi- +cated by the legend. For each galaxy, we plot the set +of resolved beam-sized measurements as previously de- +scribed. Though the line ratio uncertainties are large, +there is a moderate positive correlation between the line +ratios and ΣSFR measurements, indicating that in this +sample of DYNAMO galaxies, higher ΣSFR regions are +indeed correlated with higher line ratios. We perform a +Spearman Rank Order correlation and find a coefficient +of ρ = 0.6. The correlation between resolved measure- +ments within a single galaxy is stronger for DYNAMO +D13-5 and G20-2 (ρ = 0.8, 0.7 respectively), and weak- +est for DYNAMO G04-1 (ρ = 0.4), while for G14-1 it is +ρ = 0.6. In addition, we perform a linear fit to our ob- +served line ratio−ΣSFR relation using scipy.curve fit, +which performs a non-linear least squares analysis with +errors on the y−data as a parameter, and show the re- +sults with the black solid line. The black dashed line +corresponds to the parameterization of CO line emission +intensity as a function of ΣSFR, derived by Narayanan +& Krumholz (2014) (their equation 19): +Iij +I1−0 += A × [log10(ΣSFR) − χ]B + C +(5) +where Iij is the intensity of the CO(i − j) transition, A, +B, and C are fit parameters, and χ = −1.85 (an off- +set introduced to produce only real values of Iij/I1−0). +Narayanan & Krumholz (2014) calculate the CO SLED +of high−z star-forming galaxies from CO intensities that +are modeled at ∼ 70 pc resolution. For real observations +with coarser beams, such as in our case, the resolved line + +Resolved CO Excitation in DYNAMO +15 +ratio−ΣSFR parameterization is not an appropriate com- +parison. Therefore, Narayanan & Krumholz (2014) de- +termine the luminosity-weighted emitting area for each +CO transition and scale the resolved line intensities, and +then refit the line ratio−ΣSFR relation. Because our ob- +servations probe ∼ 1 − 2 kpc scales, this is primarily +what we compare to here. However, we show compar- +isons to the resolved parameterization for completeness +We adopt values for A, B, and C for unresolved obser- +vations from their Table 3 for CO(3−2) and CO(4−3), +and substitute in our measured values of ΣSFR. Finally, +we take the ratio of the two equations and divide by +J2 +u/J2 +l = 42/32 to convert from Jy to K and produce +the dashed black line in Figure 7. We repeat the same +procedure for the resolved galaxy observations parame- +terization from their Table 2 and plot this as the black +dashed-dotted line in Figure 7. +Overall, both model parameterizations under-predict +the steepness of the CO(4−3)/CO(3−2)−ΣSFR rela- +tion that our observations suggest, and over-predict +the line ratio across the entire range of ΣSFR val- +ues that our observations probe. +Similarly, Boogaard +et al. (2020) find that the unresolved models also over- +predict their CO(4−3)/CO(2−1) measurements (see +their Figure 13), while providing a better match to their +CO(5−4)/CO(2−1) values. Sharon et al. (2019), who +present ∼ 2 kpc resolution CO(1−0) and CO(3−2) ob- +servations of a lensed galaxy at z = 2.26, also find that +the Narayanan & Krumholz (2014) models do not repro- +duce their observations; however, they do not attribute +much meaning to this difference due to the limited ΣSFR +values probed by a single galaxy. In contrast, Valentino +et al. (2020) find qualitative agreement between the un- +resolved Narayanan & Krumholz (2014) model and their +CO(5−4)/CO(2−1) observations in z = 1.1 − 1.7 IR- +selected galaxies on and above the main sequence of star +formation. +It is possible that because these models do not ex- +plicitly model gas-rich clumpy disks like DYNAMO and +high-redshift star-forming galaxies, that their properties +are not completely captured in the early-phase snap- +shots of the model disks and model mergers (Narayanan +& Krumholz 2014). +It is also possible that a model +which characterizes global CO excitation properties for +an average ΣSFR may not be well-suited to investigate +the internal variations within a single galaxy. To test +this, we convolve our Hα maps to the CO(1−0) beam +sizes (∼ 5 − 10′′) of Fisher et al. (2019) and measure +the global ΣSFR of each galaxy for which data are avail- +able. We then use 5 and the unresolved parameters of +Narayanan & Krumholz (2014) to predict R31 and R41. +We list these predictions in the last two columns of Ta- +Figure 7. +CO(4−3)/CO(3−2) line ratio as a function of +SFR surface density, measured in beam-sized regions across +the disk of each galaxy, indicated by the color and symbol +coding in the legend. We present this data for galaxies where +observations of both CO transitions and Hα exist. Despite +the large uncertainties, there is an indication of an increas- +ing line ratio with increasing SFR surface density trend. This +is parameterized by the Spearman’s Rank Order correlation +coefficient of ρ = 0.6, suggesting a moderate positive cor- +relation between these two quantities. We present a linear +fit to our measurements (black solid line) and for compari- +son, include the predicted trend using the unresolved relation +between CO intensity and ΣSFR of Narayanan & Krumholz +(2014) (black dashed line), and their 70 pc resolved relations +(black dash-dotted line) +. +ble 3. We find that overall, the Narayanan & Krumholz +(2014) models give better predictions of our global R31 +and R41 measurements than our kpc-scale R43 measure- +ments, which may indicate that the unresolved mod- +els do not capture the kpc-scale variation in CO ex- +citation. Using hydrodynamical simulations, Bournaud +et al. (2015) study the CO SLEDs of high-redshift galax- +ies (as well as spirals and mergers), and investigate the +contribution of giant clumps to the global CO SLED. +They derive CO SLEDs for clumps and the inter-clump +gas and show that there is a considerable difference in +the CO excitation (see their Figures 3 and 4). This may +indicate a need for models that specifically relate CO ex- +citation, measured at various physical scales, in gas-rich +clumpy disks to observable quantities such as ΣSFR. + +DYNAMO D13-5 +DYNAMO G14-1 +DYNAMO G04-1 +DYNAMO G20-2 +r43 = (0.2 ± 0.04) × log ZsFR + (0.67 ± 0.02) +1.0 +Unresolved Narayanan+2014 +Resolved (～ 70 pc) Narayanan+2014 +0.9 +0.8 +CO(4-3)/CO(3-2) +0.7 +0.6 +0.5 +0.4 +0.3 +p = 0.6 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +log ZsFR [Mo yr-1 kpc-2]16 +Lenki´c et al. +5. CONCLUSIONS +In this work, +we have combined ∼ +1 − 2 kpc +scale ALMA observations of CO(3−2) and CO(4−3) +with +HST +observations +of +Hα, +to +study +the +CO(4−3)/CO(3−2) line ratio and its dependence on +ΣSFR. +We have combined this with SOFIA HAWC+ +and FIFI-LS observations of [CII] which provide addi- +tional measurements of the ISM gas physical conditions. +We summarize our findings here: +1. DYNAMO galaxies have typical CO(4−3)/CO(3−2) +line ratios of R43 = 0.54 ++0.16 +−0.15, which is most consistent +with samples of star forming ∼ 1 − 2 main-sequence +galaxies (e.g., Daddi et al. 2015; Boogaard et al. 2020; +Henr´ıquez-Brocal et al. 2022). +2. +Likewise, +the +global +CO(3−2)/CO(1−0) +and +CO(4−3)/CO(1−0) measurements in DYNAMO are +higher than global measurements of nearby star-forming +galaxies (Leroy et al. 2022) and are more consistent with +the measurements of z ∼ 1−2 star-forming galaxies (see +e.g., Daddi et al. 2015; Dessauges-Zavadsky et al. 2015; +Birkin et al. 2021; Harrington et al. 2021). +3. The DYNAMO SEDs derived from SOFIA HAWC+ +suggest cooler dust temperatures than those observed +in local starburst galaxies and U/LIRGs. +This sug- +gests that while DYNAMO galaxies are strongly star +forming, their star formation must be distributed rather +than very compact. This is consistent with the picture +we obtain from the CO(4−3)/CO(3−2) line ratio mea- +surements and the clumpy morphology of these systems. +4. +The DYNAMO CO(4−3)/CO(3−2) line ratios +are positively correlated with the ΣSFR measurements, +with a Spearman Rank Order correlation coefficient +of ρ += +0.6. +Our best fit relation between the +CO(4−3)/CO(3−2) line ratio and ΣSFR is R43 = (0.2 ± +0.04) × log ΣSFR + (0.67 ± 0.02). +This relation sug- +gests a steeper relation than predicted by the parame- +terization of Narayanan & Krumholz (2014), which also +over-predicts the line ratio over the whole range of ΣSFR +values probed by observations. It is possible that this is +consistent with the low dust temperatures of DYNAMO +galaxies. However, Sharon et al. (2019), who also study +∼ kpc scale line ratios in a high-redshift lensed galaxy, +also find a discrepancy between the models and obser- +vations. This may indicate that models that investigate +CO emission variations with internal galaxy properties +for gas-rich clumpy disks, are required. +ACKNOWLEDGMENTS +We thank the anonymous referee for comments +and +suggestions +that +have +greatly +improved +this +work. +This +paper +makes +use +of +the +following +ALMA data: +ADS/JAO.ALMA#2017.1.00239.S. and +ADS/JAO/ALMA#2019.1.00447.S. ALMA is a part- +nership of ESO (representing its member states), +NSF (USA) and NINS (Japan), together with NRC +(Canada), MOST and ASIAA (Taiwan), and KASI (Re- +public of Korea), in cooperation with the Republic of +Chile. +The Joint ALMA Observatory is operated by +ESO, AUI/NRAO and NAOJ. The National Radio As- +tronomy Observatory is a facility of the National Sci- +ence Foundation operated under cooperative agreement +by Associated Universities, Inc. Some of the data pre- +sented in this paper were obtained from the Mikulski +Archive for Space Telescopes (MAST) at the Space Tele- +scope Science Institute. The specific observations ana- +lyzed can be accessed via 10.17909/faa7-sw34. Based in +part on observations made with the NASA/DLR Strato- +spheric Observatory for Infrared Astronomy (SOFIA). +SOFIA is jointly operated by the Universities Space Re- +search Association, Inc. +(USRA), under NASA con- +tract NNA17BF53C, and the Deutsches SOFIA Institut +(DSI) under DLR contract 50 OK 0901 to the University +of Stuttgart. Financial support for this work was pro- +vided by NASA through award #SOFIA-080238 issued +by USRA. L.L. and A.D.B. acknowledges support from +USRA SOFIA-080238 and NASA HSTGO15069002A, +and NSF-AST2108140. +R.C.L. acknowledges support +from a NSF Astronomy and Astrophysics Postdoctoral +Fellowship under award AST-2102625. D.O. is a recip- +ient of an Australian Research Council Future Fellow- +ship (FT190100083) funded by the Australian Govern- +ment. R.H.-C. thanks the Max Planck Society for sup- +port under the Partner Group project ”The Baryon Cy- +cle in Galaxies” between the Max Planck for Extrater- +restrial Physics and the Universidad de Concepci´on. +R.H-C also acknowledge financial support from Mil- +lenium Nucleus NCN19058 (TITANs) and support by +the ANID BASAL projects ACE210002 and FB210003. +This research made use of Photutils, an Astropy package +for detection and photometry of astronomical sources +(Bradley et al. 2021). +Facilities: +ALMA, HST(WFC), SOFIA(FIFI-LS, +HAWC+) +Software: aplpy (Robitaille 2019), astropy (Astropy +Collaboration et al. 2013, 2018), casa (McMullin et al. +2007), numpy (Harris et al. 2020), Photutils (Bradley + +Resolved CO Excitation in DYNAMO +17 +et al. 2021), reproject (Robitaille 2018), spectral-cube +(Ginsburg et al. 2019) +APPENDIX +A. SOFIA OBSERVATIONS +B. LINE RATIO LITERATURE COMPILATION +Daddi et al. (2015) use IRAM PdBI observations of CO(2−1), CO(3−2), and CO(5−4), and Very Large Array +observation of CO(1−0) in three main-sequence star forming disk galaxies at z ∼ 1.5 to study their CO excitations. +We use their average R31 and interpolate their models from their Figure 10 to extract R41, then take the ratio R41/R31 +to obtain R43 = 0.74 ± 0.26, which we include in Figure 4 as a black circle. +Kamenetzky et al. (2016) find a linear relation between LFIR and L′ +CO for low- to mid-J CO lines and a slightly +sub-linear relation for high-J CO lines. We adopt the slope and intercepts of the relations for CO(4−3) and CO(3−2) +from their Tables 6 and 7 (for U/LIRGs and non-U/LIRGs (LFIR ≤ 6 × 1010 L⊙) respectively), and assume a FIR +luminosity of 1011 for the U/LIRG case and 1010 for the non-U/LIRG case to derive the LF IR −L′ +CO relations. Taking +the ratio of these we find R43 = 0.51 ± 0.10 and 0.25 ± 0.05 for U/LIRGs and non-U/LIRGs respectively, assuming +20% uncertainty. We plot these as a black stars in Figure 4. +Rosenberg et al. (2015) study the CO SLEDs of 29 (Ultra) Luminous Infrared Galaxies (U/LIRGs) from CO(1−0) +through CO(13−12). They classify their objects into three classes based on their excitation level. Where available, +we compiled CO(4−3) and CO(3−2) fluxes from their Tables 2 and 3, and divided the resulting ratios by (J3 +u/J3 +l ) +to convert from units of W m−2 to K. Finally, we separate the galaxies according to their classification, and plot the +median line ratio for each class as black diamonds in Figure 4. The error bars represent the standard deviation of +line ratios in each class to illustrate the spread. We note that most of the Rosenberg et al. (2015) sample is contained +within the Kamenetzky et al. (2016) sample. +Papadopoulos et al. (2012) study the CO SLEDs of 70 U/LIRGs; we average the R43 values from their Table 7 (eight +galaxies in total) and calculate the standard error on the mean. This results in R43 = 0.96 ± 0.12 and plot this as a +black pentagon in Figure 4. We note that 11/70 galaxies from the Papadopoulos et al. (2012) sample overlap with the +sample of Rosenberg et al. (2015). +Finally, Henr´ıquez-Brocal et al. (2022) combine NOEMA observations of [CI](1−0), [CI](2−1), and CO(7−6) with +ancillary CO(1−0) and CO(3−2) observations to model the CO SLED of Q1700-MD94, a massive main-sequence +galaxy at z ∼ 2, with a one- and two-temperature component model using RADEX (van der Tak et al. 2007). We +interpolate the model curves in their Figure 3 to extract R43 = 0.92 ± 0.18 and 0.77 ± 0.15 for the one- and two- +component models respectively (taking a 20% uncertainty). We do not plot these values in Figure 4, but include them +in Table 2. +REFERENCES +Adelman-McCarthy, J. K., Ag¨ueros, M. A., Allam, S. 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B., Murray, N., et al. 2017, ApJ, +846, 35, doi: 10.3847/1538-4357/aa7fbf + diff --git a/sNE4T4oBgHgl3EQfwg2l/content/tmp_files/load_file.txt b/sNE4T4oBgHgl3EQfwg2l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e55a3dc30b9bef585b7ae15eaa96a9dd4a8e3bee --- /dev/null +++ b/sNE4T4oBgHgl3EQfwg2l/content/tmp_files/load_file.txt @@ -0,0 +1,2182 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf,len=2181 +page_content='Draft version January 16, 2023 Typeset using LATEX twocolumn style in AASTeX63 CO Excitation in High−z Main Sequence Analogues: Resolved CO(4−3)/CO(3−2) Line Ratios in DYNAMO Galaxies Laura Lenki´c,1, 2 Alberto D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Bolatto,1 Deanne B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Fisher,3, 4 Roberto Abraham,5 Karl Glazebrook,3, 4 Rodrigo Herrera-Camus,6 Rebecca C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Levy,7, ∗ Danail Obreschkow,8 and Carrie Volpert1 1Department of Astronomy, University of Maryland, College Park, MD 20742, USA 2SOFIA Science Center, USRA, NASA Ames Research Center, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' N232-12, Moffett Field, CA 94035, USA 3Centre for Astrophysics and Supercomputing, Swinburne University of Technology, PO Box 218, Hawthorn, VIC 3122, Australia 4ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D) 5Department of Astronomy & Astrophysics, University of Toronto, 50 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' George Street, Toronto, ON M5S 3H4, Canada 6Departamento de Astronom´ıa, Universidad de Concepci´on, Barrio Universitario, Concepci´on, Chile 7Steward Observatory, University of Arizona, Tucson, AZ 85721, USA 8International Centre for Radio Astronomy Research (ICRAR), University of Western Australia, Crawley, WA 6009, Australia (Received;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Revised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Accepted) Submitted to ABSTRACT The spectral line energy distribution of carbon monoxide contains information about the physical conditions of the star forming molecular hydrogen gas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' however, the relation to local radiation field properties is poorly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Using ∼ 1−2 kpc scale ALMA observations of CO(3−2) and CO(4−3), we characterize the CO(4−3)/CO(3−2) line ratios of local analogues of main sequence galaxies at z ∼ 1 − 2, drawn from the DYNAMO sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We measure CO(4−3)/CO(3−2) across the disk of each galaxy and find a median line ratio of R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='54 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='15 for the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This is higher than literature estimates of local star-forming galaxies and is consistent with multiple lines of evidence that indicate DYNAMO galaxies, despite residing in the local Universe, resemble main-sequence galaxies at z ∼ 1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Comparing to existing lower resolution CO(1−0) observations, we find R41 and R31 values in the range ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3 and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We combine our kpc-scale resolved line ratio measurements with HST observations of Hα to investigate the relation to star formation rate surface density and compare this relation to expectations from models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We find increasing CO(4−3)/CO(3−2) with increasing star formation rate surface density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' however, models over-predict the line ratios across the range of star formation rate surface densities we probe, particularly at the lower range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, SOFIA observations with HAWC+ and FIFI-LS reveal low dust temperatures and no deficit of [CII] emission with respect to the total infrared luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Keywords: interstellar line emission – CO line emission, galactic and extragalactic astronomy – extra- galactic galaxies, galaxies – disk galaxies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' INTRODUCTION Molecular hydrogen gas (H2) is routinely mapped in high-redshift (high−z) galaxies with instruments such as the Atacama Large Millimeter/sub-millimeter Ar- ray (ALMA) and NOrthern Extended Millimeter Ar- Corresponding author: Laura Lenki´c laura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='lenkic@nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='gov, llenkic@usra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='edu ∗ NSF Astronomy and Astrophysics Postdoctoral Fellow ray (NOEMA) through the use of high rotational lines (high−J) of carbon monoxide (CO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Tacconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Decarli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Walter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Freundlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' High−J lines of CO can be used to address important topics such as the evolution of molecular gas reservoirs in galaxies across cosmic time (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Walter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Decarli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Riechers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Decarli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Mapping H2 through high−J CO emis- sion in high−z galaxies provides certain advantages over arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05251v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='GA] 12 Jan 2023 2 Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' CO(1−0), because these lines are bright and allow for higher resolution observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' However, limited con- straints of the CO excitation ladder, or the CO spectral line energy distribution (SLED), render the conversion to the ground state transition, CO(1−0), uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The CO excitation ladder contains information about the temperature and density of the H2 material (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Carilli & Walter 2013, for a review), and understand- ing how those properties relate to local star formation activity will improve our understanding of how high−J CO lines map to CO(1−0) and H2 mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Several studies have characterized the CO excitation ladder in various local galaxy populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' In the Milky Way galactic center, observations from the COBE Far Infrared Absolute Spectrophotometer, which constrain the CO SLED from J = 1 − 0 to J = 8 − 7, show that the line ratios can be modeled with an excitation temperature of 40 K and that the CO SLED peaks at J = 3 (Fixsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' A recent systematic study of CO(1−0) to CO(3−2) in nearby galaxies finds that the Rayleigh-Jeans brightness temperature ratios are gen- erally higher in galaxy centers, decreasing with radius and diminishing star formation rate (SFR) surface den- sity (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Kamenetzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2016) study CO emission up to J = 13−12 in ultraluminous infrared galaxies (ULIRGS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' see also Greve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014), active galactic nuclei (AGN), and non-ULIRGs to find that CO SLEDs peak at in- creasingly higher−J with increasing far-infrared (FIR) luminosity, indicating higher kinetic temperatures or densities are required (see also Figure 1 of Obreschkow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2009, for how CO excitation depends on galaxy type and excitation temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Observations of sub- millimeter galaxies (SMGs) show that they also have CO SLEDs with an “excess” of CO excitation with re- spect to the Milky Way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' these generally rise up to J = 5 and then turn over for higher rotational states (Bothwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Spilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' These high excitation CO SLEDs suggest that alternative heating sources are required in these extreme galaxies such as mechanical heating via shocks, turbulence, or cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Although several large studies probe the CO SLEDs of extreme systems like SMGs and U/LIRGS, the CO SLEDs of normal z ∼ 1−2 star-forming galaxies are not as well characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Valentino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2020) conduct a large survey of mid− and high−J CO lines with ALMA in main sequence galaxies at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7, and find that they have higher excitation than the Milky Way but are not quite as highly excited as ULIRGs, SMGs, or QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015) find similar results in a sam- ple of four main-sequence near-IR selected galaxies at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5, using CO(2−1), CO(3−2), and CO(5−4) obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Bolatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015) also present the Rayleigh- Jeans brightness temperature CO(3−2)/CO(1−0) line ratio of four main sequence galaxies observed with the Plateau de Bure Interferometer (PdBI), and find a ratio of about unity denoting high excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, multi- ple case studies of the CO ladder in specific galaxies exist (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Barvainis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' van der Werf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Kamenetzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Aravena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Dessauges- Zavadsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Brisbin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Sharon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Henr´ıquez-Brocal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Klitsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2022), and show that while general trends exist in different galaxy populations, the CO ladder of every galaxy is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This indicates that it is necessary to understand how CO emission and excitation vary within galaxies and how they relate to other physical properties in order to correctly interpret H2 masses derived from high−J transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This requires resolved studies of CO line ra- tios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' however, observational limitations at high−z make this challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To address this, we present a sam- ple of nine galaxies drawn from the DYnamics of Newly Assembled Massive Objects (DYNAMO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014) observed by ALMA in CO(3−2) and CO(4−3) on ∼ 1 − 2 kpc scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' DYNAMO galaxies are nearby (z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1) objects with high gas fractions, high star formation rates, and widespread turbulence, consistent with known properties of high−z main-sequence galax- ies, and many are indeed found to lie on the main- sequence of star formation at z ∼ 1 − 2 (Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Their resemblance to high−z systems and prox- imity allows us to probe the CO excitation in gas-rich, turbulent galaxies at scales that are not yet achievable at z ∼ 2 in unlensed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Furthermore, theories seek to explain CO line ratios by their local radiation field properties (Lagos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Narayanan & Krumholz 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Popping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Bournaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015), and our ALMA observations allow us to compare to model expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This paper is structured as follows: §2 describes our observations, data reduction, and methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' §3 and §4 describe and discuss our results, and finally we con- clude in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Throughout this work, we assume ΛCDM cosmology with H0 = 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 km s−1, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='286, and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='714 (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014), and a Kroupa initial mass function (IMF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Kroupa 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' OBSERVATIONS AND DATA REDUCTION The DYNAMO sample of galaxies was first defined by Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2014), who selected galaxies from the MPA- JHU Value Added Catalog of the Sloan Digital Sky Sur- vey DR4 (SDSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Adelman-McCarthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2006) based on their redshift and Hα emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The sample consists Resolved CO Excitation in DYNAMO 3 of 67 galaxies, half of which have LHα > 1042 erg s−1 in the 3′′ diameter SDSS fiber, lying in two redshift win- dows centered at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='075 and z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Their stel- lar masses range from 109 − 1011 M⊙ and their SFRs from ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1 − 100 M⊙ yr−1, while their metallicities are about solar (Tremonti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Employing integral- field spectroscopy of Hα, Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2014) derive Hα rotation curves and find high ionized gas velocity dis- persions with a mean of ∼ 50 km s−1, and gas frac- tions as high as fgas ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8 (fgas ≡ Mgas/(Mgas + M∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Mgas = MHI + MH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Furthermore, they find that DYNAMO galaxies are more “turbulent” than local disks, as parameterized by their ratio of rotation ve- locity to velocity dispersion (V/σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' These properties make DYNAMO galaxies promising candidates for local analogues of high-redshift, star-forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Here, we consider a sub-set of nine DYNAMO galax- ies that have robustly been identified as consistently more similar to z ∼ 1 − 2 star-forming systems: DY- NAMO C13-1, C22-2, D13-5, D15-3, G04-1, G08-5, G14-1, G20-2, and SDSS J013527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10-103938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 (hereafter SDSS 013527-1039).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This builds on a multi-wavelength campaign to investigate the nature of star formation at high redshift (Bassett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Obreschkow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Bassett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Girard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Ambachew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' All galaxies in our sample are classified as rotating disks based on their Hα kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Galaxies C22-2, G04-1, G14-1, and G20-2 are furthermore classified as “com- pact” rotating disks, because their SDSS r−band expo- nential scale lengths are smaller than 3 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For these, the poorer resolution results in less reliable kinematic classifications (Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' All galaxies in our sample have CO(1−0) observations from either the PdBI or NOEMA, from which molecular gas fractions (MH2/(MH2+M∗)) of fgas ∼ 20−30 % and molecular gas depletion times of tdep ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 Gyr are in- ferred (Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' These high molecular gas fractions are consis- tent with those of z ∼ 1 − 2 main-sequence star-forming galaxies (Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Tacconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2010, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Tacconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Similarly, sub- sequent studies of the gas kinematics in these galaxies consistently show that they do indeed have high ion- ized gas velocity dispersions (Bassett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Oliva- Altamirano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Girard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2021), similar to main-sequence galaxies at z ∼ 1 − 2 (F¨orster Schreiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' ¨Ubler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' In addition, Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2017b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2017) both show that these DYNAMO galaxies are consistent with marginally sta- ble disks (Toomre Q ∼ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' DYNAMO galaxies also conform to established definitions of clumpy galaxies (Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017b) at high-redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', CANDELS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, their clumps are arranged within their host disks such that the redder clumps are preferentially more centrally located than the bluer ones (Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2022), which has also been observed in z ∼ 1 − 2 clumpy galaxies (F¨orster Schreiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Soto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' ALMA CO Observations We make use of the CO(3−2) and CO(4−3) obser- vations of nine DYNAMO galaxies with the Atacama Large Millimeter/Submillimeter Array (ALMA), associ- ated with project code 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='00239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='S (PI: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Fisher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Observations were taken in Band 7 (275−373 GHz) and Band 8 (385 − 500 GHz) between 2018-06-01 and 2018- 07-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The spectral windows were configured with band- widths of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='00 GHz and channel widths of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='625 MHz (128 channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' In addition, we also make use of higher resolution CO(3−2) ALMA observations of three DY- NAMO galaxies (G04-1, G08-5, and G14-1) associated with the project code 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='00447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='S (PI: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Herrera- Camus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' These observations were taken in Band 7 be- tween 2019-10-09 and 2019-10-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The spectral win- dows were configured with bandwidths of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='875 GHz and channel widths of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8125 MHz (240 channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The data associated with both projects were presented in Girard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The visibilities were calibrated and flagged by the observatory with the Common Astronomy Software Application (casa, McMullin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2007) pipeline ver- sions listed in the fifth column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' After cal- ibrating the visibilities, we imaged each observation using tclean in casa version 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='188 with param- eters deconvolver=‘hogbom’, weighting=‘briggs’, robust=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5, usemask=‘auto-multithresh’, and restfreq set to the redshifted frequency of the ob- served CO line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We cleaned the data until the residuals were consistent with the root-mean-square (rms) noise levels that are listed in the fourth column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To derive these thresholds, we consider data cubes with just a shallow clean, mask the emission (see below), and calculate the standard deviation of the masked cubes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', non-line channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' These values are listed in col- umn four of Table 1, and we re-clean the data cubes to that rms level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For visualization purposes and ease of comparison to the Hα maps, we convolve the final cubes to a circular beam, listed in the second (angular size) and third (physical size) columns of Table 1, with the casa imsmooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' At the redshifts of DYNAMO galaxies in our sample, the beam sizes correspond to physical scales of ∼ 1 − 2 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, we export all 4 Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' CO Data Cube Parameters CO Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Beam FWHM rms Noise casa Cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (arcsec) (kpc) (mK) DYNAMO C13-1 3 − 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='60 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 DYNAMO C22-2 3 − 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='46 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 4 − 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='11 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 DYNAMO D13-5 3 − 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='58 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 4 − 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='14 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 DYNAMO D15-3 4 − 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='24 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 DYNAMO G04-1 3 − 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='98 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-8 4 − 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='96 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 DYNAMO G08-5 3 − 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='95 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-8 DYNAMO G14-1 3 − 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-8 4 − 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 DYNAMO G20-2 3 − 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 4 − 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 SDSS 013527-1039 3 − 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='81 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 4 − 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1-5 data cubes with the spectral axis in units of velocity, in the local standard of rest frame, adopting the radio convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We present channel maps of CO(3−2) for DYNAMO G04-1 in Figure 1 to show an example of the final data, with the circularized beam shown in the bottom left corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The complete figure set (14 images) is available in the online journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We produce moment zero maps (integrated intensity) by first masking each cleaned data cube along both the spatial and spectral axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To produce our masks, we first smooth each cleaned data cube to twice the cir- cularized beam full width at half maximum (FWHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We then compute the rms of the data cube, and mask all pixels that are below 3× the cube rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For the re- maining pixels, we compute the integrated intensity over the channels that are not masked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We do this for both the CO(3−2) and CO(4−3) observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Figure 2 presents these moment zero maps in the two right-most panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, for the goal of calculating CO(4−3)/CO(3−2) line ratios, we match the pixel scale and resolution of all 2017 CO(4−3) observations to the pixel scale and res- olution of the 2017 CO(3−2) observations, where data for both transitions are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Similarly, we match the pixel scale and resolution of the 2019 CO(3−2) ob- servations, where available, to the 2017 CO(4−3) obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We match the pixel scales using the casa func- tion imregrid, while to match the resolution, we use the casa imsmooth tool to convolve the higher resolution data with a Gaussian kernel to produce the lower reso- lution Gaussian beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We note that these transforma- tions are done on the cleaned data cubes with the origi- nal, non-circular beams, to ensure we are not introduc- ing errors or artifacts in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, we apply the masking of the CO(3−2) observations to the CO(4−3) to produce matching integrated intensity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This ensures that the intensities we derive for both lines are integrated over the same velocity ranges and regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' HST Hα Observations In addition to the ALMA observations of CO, we make use of Hubble Space Telescope (HST) observations of Hα (PID 12977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' : I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Damjanov) as a tracer of the star formation rate (left-most panel of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Ob- servations were taken with the Wide Field Camera on the Advanced Camera for Surveys (WFC/ACS) using the FR716N and FR728N narrow-band filters, and were processed with the standard HST pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Continuum observations with the FR647M filter were also taken and used to create continuum-subtracted Hα maps (for de- tails, see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 of Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The final Hα maps have a pixel scale of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05′′ and a resolution corre- sponding to physical scales of ∼ 50 − 200 pc (Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Our ability to make resolved measurements in these DYNAMO galaxies is limited by the resolution of the ALMA data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' therefore, we match the pixel scale and res- olution of the Hα observations to that of the CO(3−2), where available, and CO(4−3) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To achieve this, we convolve each Hα observation with a two- dimensional Gaussian function whose FWHM is equal to the circularized beam of the corresponding ALMA ob- servation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Then, we re-project and re-grid the Hα obser- vations to match the WCS information and pixel scale of the CO observations using the Python astropy pack- Resolved CO Excitation in DYNAMO 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Channel maps of CO(3−2) in brightness units of Jy beam−1 for the galaxy DYNAMO G04-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Each panel is centered at 04h12m19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='713s, -05d54m48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='62s, and is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8×10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8′′ in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The velocity range is −172 to 96 km s−1 in steps of ∼8 km s−1, as indicated in the top right corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The circularized beam is shown in white in the bottom left corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The complete figure set (14 images) is available in the online journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' age reproject1, noting that the reproject functions as- sume that input images have surface brightness units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' SOFIA FIFI-LS and HAWC+ Observations Finally, we make use of observations from the Strato- spheric Observatory for Infrared Astronomy (SOFIA) of DYNAMO galaxies taken by the FIFI-LS (Colditz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2018) and HAWC+ (Harper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2018) instruments (PLAN ID 08 0238 and 09 158;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 1 https://reproject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='readthedocs.' metadata={'source': 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125 µm (blue channel) and 105 − 200 µm (red channel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The FIFI-LS obser- vations targeted the [CII] 158 µm fine-structure emis- sion line in the red channel and the [OIII] 88 µm fine- structure line (or [OI] at 63 µm depending on atmo- spheric transmission) in the blue channel for six galaxies (DYNAMO B08-3, D10-4, D14-1, D15-3, F08-2, F09- 1, and F12-4) at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6′′ resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' These data cover a 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 88.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 30 25 C13- CO(3-2) 1 kpc Ho 20 15 1°30\'04" 10 Dec (ICRS) 5 00" 29\'56" 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1s 13h26m39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6s 13h26m39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6s 13h26m39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4s 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1s50 CO(4-3) 50 C22-2 CO(3-2) 1 kpc 40 40 8°04\'16" 30 30 20 20 (ICRS) Dec( 10 10 19" 0 22h39m49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4s 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2s 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2s 22h39m49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4s 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2s 22h39m49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4s80 CO(4-3) 80 D13-5 CO(3-2) 1 kpc OH 60 60 40 40 0°31\'55" Dec (ICRS) 20 20 52" 48" 0 13h30m07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2s 07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0s 06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7s 13h30m07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2s 07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0s 06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7s 13h30m07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2s 07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0s 06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7s50 0°28\'41" D15- CO(4-3) Ha 1 kpc 40 30 20 Dec (ICRS) 44" 10 48" 0 15h34m35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5s 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3s 15h34m35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5s 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3s 15h34m35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5s 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3s70 CO(3-2) 200 CO(4-3) 60 G04-1 1 kpc Haα 50 150 40 5°54\'47" 100 30 (ICRS) 20 50 Dec 10 50" 0 0 4h12m19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9s 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7s 4h12m19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9s 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7s 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7s 4h12m19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9sResolved CO Excitation in DYNAMO 7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Summary of data sets analyzed in this work for each galaxy in our sample, as indicated in the top right corners of the left-most panels: HST Hα (left), CO(3−2) integrated intensity (middle), CO(4−3) integrated intensity (right;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' both in units of K km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We show all images using an arcsinh stretch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The ALMA CO beam sizes are in the bottom left corners of the middle and right-most panels, while 1 kpc scalebars are shown in the top right corner of the right-most panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Empty panels indicate that data is absent for the given galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 12 SDSS013527-1039 CO(3-2) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 CO(4-3) 1 kpc 10 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 10°39\'36".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 4 (ICRS) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 Dec 2 40" 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 43" 1h35m27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4s 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1s 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9s 1h35m27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4s 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9s 1h35m27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4s 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1s 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9s 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1s RA (ICRS) RA (ICRS) RA (ICRS)300 Ha G08-5 +CO(3-2) 1 kpc 250 200 6°46\'23" 150 100 (ICRS) Dec 50 19" 0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7s 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7s 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7s 8h54m19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0s 8h54m19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0s 8h54m19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0s80 35 70 Ha G14-1 CO(3-2 CO(4-3) 1 kpc 30 60 25 50 20 40 15 30 Dec (ICRS) 0°44\'35" 10 20 5 10 0 31" 14h54m28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3s 14h54m28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3s 14h54m28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3s-25 20 G20-2 CO(3-2) CO(4-3) 1 kpc Ho 20 15 6°46\'55" 15 10 10 (ICRS) 5 Dec ( 5 59" 0 0 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9s 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9s 20h44m03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1s 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6s 20h44m03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1s 20h44m03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1s 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9s8 Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 1 × 1 arcmin2 field-of-view (FOV) in the red channel and a 30 × 30 arcsec2 FOV in the blue channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' FIFI- LS observations were taken on six nights in April 2021 in the nod-match-chop mode, and were reduced using the FIFI-LS pipeline2 (Vacca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The data re- duction steps include ramp fitting and flagging bad pix- els, subtracting the chops, wavelength and spatial cal- ibration, flat-field correction, atmospheric transmission correction using the ATRAN models (Lord 1992), flux calibration, and finally resampling to a regular grid to produce the final data cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The observations resulted in [CII] detections for all galaxies in the sample, and an [OIII] detection in DYNAMO F08-2 (see Figure 9 in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The High-resolution Airborne Wideband Camera Plus (HAWC+) instrument is a FIR camera and imaging po- larimeter with a wavelength coverage of 50 − 240 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The HAWC+ observations targeted four galaxies (DY- NAMO D14-1, D15-3, F08-2, and F12-4) in bands C, D, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' These data provide measurements of the 89, 155, and 216 µm fluxes at a resolution of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8′′, 14′′, and 19′′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Observations were taken on three nights in May 2021 and one night in November 2021 in the on-the-fly mapping mode with a Lissajous scan pattern, and were reduced using the HAWC+ pipeline3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The ob- servations resulted in detections for all galaxies in the sample (see Figure 8 in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The typical sizes of galaxies in this sample are ∼ 4′′ and our sources are thus point sources for both the FIFI- LS and HAWC+ observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Appendix A presents the HAWC+ observations in Figure 8 and the FIFI- LS integrated intensity maps in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' While DY- NAMO D13-5 is the only galaxy that overlaps with our ALMA sample, we make use of all SOFIA observations described here to measure the SEDs of DYNAMO galax- ies, and place the measured dust temperatures within the global context of the line ratio measurements we will present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We also make use of these observations to mea- sure the [CII] luminosity and measure the [CII]-to-total far-infrared luminosity ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Resolved Measurements This work aims to investigate the CO(4−3)/CO(3−2) properties of DYNAMO galaxies resolved on a 1−2 kpc scale, and to relate this line ratio to the star formation rate surface density (ΣSFR) on the same scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Thus, we describe here our method for extracting these mea- surements from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For each of our resolution and WCS matched ALMA and HST data sets (excluding 2 FIFI-LS Redux User’s Manual 3 HAWC+ DRP User’s Manual SOFIA observations because they are unresolved), we define two sets of “grids” of circular, beam-sized aper- tures: one that is centered on the galaxy, and a second that is offset from the center by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5× the beam FWHM in both the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This is to ensure that we cover the gaps of the first grid and results in measure- ments that are not entirely independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Within each aperture, we measure the median brightness tempera- ture of both the CO(3−2) and CO(4−3) lines from the integrated intensity maps and take their ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We measure the SFR surface density from our CO- matched Hα observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We perform aperture pho- tometry within each ALMA beam-sized aperture in our two grids, described above, to obtain the Hα flux (in electrons per second).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We convert these fluxes to units of erg s−1 cm−2 ˚A−1, apply a correction for extinction by relating AV to AHα assuming the Cardelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (1989) extinction law and the AV measurements from Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Bassett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2017) use Paα observations from the OSIRIS instrument at Keck to make resolved extinction measurements in four DYNAMO galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Their results show up to a magnitude difference in AHα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' however, strong variation in the adaptive optics point spread function introduces significant systematic uncer- tainties in measuring the Paα flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Furthermore, Bas- sett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2017) also find that the average AHα in clump and non-clump regions are, within the uncer- tainties, consistent with one another (see their Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This is consistent with the results of Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2021), who find that within a given DYNAMO galaxy, the extinction-sensitive color they measure shows little variation between clumps, and the clump colors are con- sistent with their host disks (see their Figures 5 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For these reasons, we choose to adopt a single AV value for each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, we calculate Hα luminosities and convert them to SFRs using the Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2011) calibration for a Kroupa IMF, constant star formation history, and age of 100 Myr (see their Table 2): SFR [M⊙ yr−1] = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='53 × 10−42 × LHα [erg s−1] (1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' CO(4−3)/CO(3−2) Line Ratios In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1, we describe our process for matching our CO(4−3) and CO(3−2) observations and deriving in- tegrated intensity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We adopt brightness tempera- ture units, thus our integrated intensity maps have units of K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To visually determine how this line ra- tio varies across each galaxy disk, if at all, we simply divide our CO(4−3) integrated intensity map by that of the CO(3−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This is what we present in Figure 3, where the color scale indicates the ratio variations across Resolved CO Excitation in DYNAMO 9 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' CO(4−3)/CO(3−2) line ratios (in brightness temperature units) measured from the pixel scale and resolution matched integrated intensity maps, integrated over the same velocity ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' These maps show CO(4−3)/CO(3−2) only in regions where the line ratio S/N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The black contours correspond to Hα emission, where available, ranging from 1 − 10σ in increments of 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The galaxy name is indicated in the top left corner of each panel, the median line ratios and their associated uncertainties are in the top right corners, and the black hatched circles in the bottom left corners indicate the circularized beam sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, we show a 1 kpc scale bar in the bottom right corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We see that galaxies generally have mildly varying line ratios within the regions where the uncertainties do not dominate, and that they lie typically around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7, with the exception of DYNAMO G14-1 which has a stronger varying line ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' each galaxy disk for which both line transitions were ob- served, and where the line ratio S/N ≥ 3, and the black contours correspond to Hα emission in the pixel scale and resolution matched HST observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The con- tours span 1−10σ in increments of 1σ, where we take σ to correspond to the rms of each HST observation cal- culated in galaxy emission-free regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We note that there are no HST Hα observations for DYNAMO C22- 2 and SDSS 013527-1039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We derive uncertainties for the integrated intensity maps (σJ→J−1) by summing in quadrature the rms of every channel over which we inte- grate, excluding line emission from the rms calculation, and multiplying by the channel width: σJ→J−1 = ∆v � � � � N � i (rmsi)2 (2) where ∆v is the channel width, rmsi is the rms of the ith channel, and N is the number of channels over which the emission is integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To obtain the final uncertainty on the line ratio per pixel, we propagate the integrated intensity uncertainties by taking: σlr = CO(4 − 3) CO(3 − 2) �� σ43 CO(4 − 3) �2 + � σ32 CO(3 − 2) �2 (3) which results in line ratio uncertainty maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' From Figure 3, we see that the line ratio for galaxies in our sample vary mildly across the disks, with typical values ranging from R43 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' However, galaxy DYNAMO G14-1 shows a strong gradient in the line ratio, with values approaching unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The Hα image of G14-1 in Figure 2 shows two bright clumps with a fainter “stream” connecting the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The Hα contours we over- plot in Figure 3 show that these two bright features with the connecting filament coincide with the elevated line ratio values and the strong gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This may be in- dicative of an interaction taking place;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' however, the Hα kinematics of G14-1 show a rotating disk and no com- plex kinematics (Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Overall, the line ratio maps we show in Figure 3 suggest a potential cen- tral enhancement in R43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Such a central enhancement has been observed in the Milky Way and other nearby 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 8°04\'14" C22-2 0.' metadata={'source': 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+page_content='2 1h35m27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4s 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2s 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0s 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8s RA (ICRS)10 Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' star-forming galaxies for CO(2−1)/CO(1−0) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Sakamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Sawada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2009, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' den Brok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To verify this, we separate pixels that are located within the central kpc of each galaxy from pixels that lie outside this region, and compare the median line ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Indeed, we find enhanced CO(4−3)/CO(3−2) values in the central kpc of all galaxies in Figure 3, except for G04-1 and G14-1, on the order of ∼ 10% (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, Figure 3 shows in particular for galaxy G04-1, variations in R43 between the spiral arm and inter-arm region, a trend also observed for CO(2−1)/CO(1−0) in M51 (Koda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Next, we perform ∼ 1−2 kpc sized sightline measure- ments of the line ratio across the disk of each galaxy, as described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4, to characterize the typical line ra- tio we measure across the sample and the magnitude of the spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To this end, we construct a global proba- bility density function (PDF) by modeling each beam- averaged line ratio measurement with a kernel density estimate (KDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We construct the individual KDEs by modeling each beam-averaged line ratio measurement as a one-dimensional Gaussian with centroid correspond- ing to the measured line ratio and with width equal to the line ratio uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The area of each Gaussian is normalized to unity, then we sum all Gaussians to pro- duce a final global PDF (see for example §4 and Figure 5 of Levy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This is what we show in Fig- ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' From this, we find that the median line ratio and 68% confidence interval for DYNAMO galaxies are: R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='54 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' These values are taken at the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9, 50, and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1 percentiles of the cumulative distribution function of the PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For comparison, we compile estimates of the CO(4−3) to CO(3−2) line ratio from the literature and include these in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We describe our derivation of all line ratios we compile from the literature in Appendix B, and we summarize them along with the median line ratios we measure for each DYNAMO galaxy individually, and the median line ratio for the entire DYNAMO sample studied here in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This comparison reveals that the CO(4−3)/CO(3−2) line ratio of non-ULIRGs from Kamenetzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2016) (local galaxies with LFIR ≤ 6 × 1010 L⊙) is much lower and incompatible with what we find in our DYNAMO sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' In contrast, the U/LIRG line ratio estimate from Kamenetzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2016) for LFIR = 1011 L⊙ is in much better agreement with what we find across the DY- NAMO sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Likewise, the CO(4−3)/CO(3−2) line ratios measured in main-sequence galaxies at z ∼ 1 − 2 (Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Boogaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Henr´ıquez- Brocal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2022) are, within the uncertainties, con- Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' CO(4−3)/CO(3−2) Line Brightness Temperature Ratios Compiled from the Literature Compared to DYNAMO Object(s) Line Ratio Reference C22-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='13 This work C22-2 (≤ 1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 This work C22-2 (> 1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 This work D13-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 This work D13-5 (≤ 1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 This work D13-5 (> 1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='09 This work G04-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 This work G04-1 (≤ 1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='20 This work G04-1 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='16 This work G14-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='11 This work G14-1 (≤ 1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='13 This work G14-1 (> 1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='16 This work G20-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='07 This work G20-2 (≤ 1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 This work G20-2 (> 1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 This work SDSS J013527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10-103938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05 This work SDSS J013527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10-103938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 (≤ 1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='06 This work SDSS J013527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10-103938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 (> 1 kpc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='06 This work DYNAMO all 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='54 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='15 This work z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 MS Galaxies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='26 D15 ASPECS z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 SFGs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='16 B20 G1700-MD94 one component 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='18 HB22 G1700-MD94 two component 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='15 HB22 non-U/LIRGs (LFIR = 1010L⊙) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05 K16 LIRGs (LFIR = 1011L⊙) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 K16 LIRGs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='38 P12 ULIRGs low CO excitation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 R15 ULIRGs mid CO excitation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='70 R15 ULIRGs high CO excitation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 R15 sistent with DYNAMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' In particular, the eight star- forming galaxies at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 from the ALMA Spec- troscopic Survey (ASPECS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Boogaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2020), are an especially good match to the R43 we measure across our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' DYNAMO galaxies lie on the star forma- tion main-sequence at z ∼ 2 (Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2019) and have gas fractions and velocity dispersions that are more similar to main-sequence galaxies of that epoch than lo- cal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Therefore, this result is consistent with lines of evidence that indicate DYNAMO galaxies are local analogues of high−z main-sequence systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' ULIRG samples (Rosenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015) have much larger line ratios than we observe in DYNAMO, and this too is consistent with previous observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Using Herschel PACS+SPIRE observations of five DYNAMO galaxies, White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2017) found that despite their large FIR luminosities, LFIR > 1011 L⊙, these galaxies have much lower dust temperatures (∼ 30 K) than ULIRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' There- Resolved CO Excitation in DYNAMO 11 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Global PDF for the resolved CO(4−3)/CO(3−2) line ratio measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We construct the PDF by model- ing each line-of sight R43 measurement (where S/N ≥ 3) as a Gaussian whose width is the line ratio uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We show these individual Gaussians as light grey lines (not to scale);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' summing them and normalizing the area of the result- ing Gaussian to unity results in the solid black line shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' From the cumulative distribution function, we infer a median line ratio of R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For comparison, we include estimates from the literature: R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='26 for three z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 main sequence star forming galaxies (black circle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015), R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='16 in eight star-forming galaxies at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 (black square;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Boogaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2020), R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 for non-U/LIRGs with LFIR = 1010 L⊙ and U/LIRGs with LFIR = 1011 L⊙ respec- tively (black stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Kamenetzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2016), R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='70, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 for mid- and high-excitation ULIRGs respectively (black di- amonds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Rosenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015), and R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='12 for LIRGs (black pentagon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Papadopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' fore, unlike ULIRGs, the star formation in DYNAMO galaxies is more distributed throughout the disks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' thus, colder dust temperatures would be expected and like- wise lower CO(4−3)/CO(3−2) line ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Relating High−J CO to CO(1−0) We make use of existing CO(1−0) measurements from the PdBI and NOEMA (angular resolution ∼ 5 − 10′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2019) to derive the CO(4−3)/CO(1−0) and CO(3−2)/CO(1−0) line ratios across our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We measure the total CO(4−3) and CO(3−2) fluxes by sum- ming all pixels with S/N ≥ 3 in our integrated intensity Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' CO ladders normalized to CO(1−0) in integrated brightness temperature units, for DYNAMO galaxies (red small diamond), z ∼ 1−2 main-sequence BzK galaxies (black circles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015), z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 star-forming galaxies from ASPECS (black squares Boogaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2020) nearby star forming galaxies (blue pentagons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2022), and the Milky Way inner disk (blue large diamonds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Fixsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' DYNAMO line ratios are consistent with z ∼ 1 − 2 star-forming galaxies, while the nearby star-forming galaxies and the Milky Way show overall lower CO excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' One galaxy from the sample of Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015) (BzK-16000) is more consistent with nearby galaxies and the Milky Way than with DYNAMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' BzK-16000 is more evolved and has no massive clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' moment maps, then scaling by the number of pixels per beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We then convert the total fluxes to luminosities (L′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' K km s−1 pc2) using equation 3 in Solomon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We present these results in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We find line ratio values across our sample that range from R31 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8, with a mean (median) R31 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='56 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='55) and R41 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4, with a mean (me- dian) R41 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='27 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Our R31 result is consistent with multiple studies of CO excitation in z ∼ 1 − 3 star-forming galaxies: Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015) found that the brightness temperature line ratio of CO(3−2) to CO(1−0) ranges from R31 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6, with an aver- age R31 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='07, for their three star-forming BzK z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 galaxies, Dessauges-Zavadsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015) find R31 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='15 for five lensed star-forming galaxies (SFR < 40 M⊙ yr−1) at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 galaxies, Riechers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2020) find R31 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='26 for six galaxies at z ∼ 2−3, Birkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2021) find R31 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='12 for a large sample of SMGs at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8, and Harrington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' DYNAMO Median = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='54±0-19 ★ LIRGs (K16) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='15 z～ 1 - 2 MS Galaxies (D15) ULIRGs, Mid Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (R15) ASPECS z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 SFGs (B20) ULiRGs, High Ex (R15) Non-U/LIRGs (K16) U/LIRGS (P12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='020 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 CO(4-3)/CO(3-2)DYNAMO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 z ～ 1 -2MS Galaxies (D15) ASPECS z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 SFGs (B20) Nearby Galaxies (L22) Milky Way (F99) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8 CO(J →J-1)/CO(1-0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 2 3 4 5 6 Upper Rotational Quantum Number, Jupper12 Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Galaxy Integrated CO(3−2)/CO(1−0) and CO(4−3)/CO(1−0) Line Ratios and Model Predictions Galaxy L′ CO(1−0) L′ CO(3−2) L′ CO(4−3) R31 R41 R31 R41 (109 K km s−1 pc2) Observed Predicted C13-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05 · · · · · · C22-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='04 · · · · D13-5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='39 D15-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='06 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='32 G04-1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='31 G08-5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='32 G14-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='427 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='31 G20-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='36 SDSS 013527-1039 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='01 · · · · (2021) find R31 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='12 for 24 dusty star forming galaxies at 1 < z < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' However, we note that Bo- latto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015) find R31 ∼ 1 for two main-sequence galaxies at z ∼ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' however, one of the two galaxies is classified as an AGN, and the other may host an weak AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' In contrast, Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2022) analyze the global R31 for nearby normal star-forming galaxies and find a mean (median) of R31 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='30 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='29), which are lower and inconsistent with the DYNAMO results, but simi- lar to the Milky Way (Fixsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We illustrate this comparison in Figure 5 where we plot the brightness temperature ratios of DYNAMO galaxies as a function of upper-J number, along with the ratios of z ∼ 1 − 2 star-forming galaxies (Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Boogaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2020), nearby star-forming galaxies (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2022), and the Milky Way inner disk (Fixsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We can see that the ratios of nearby galaxies and the Milky Way are incompatible with those of DYNAMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The CO SLED of the ASPECS galaxies and two of the three BzK galaxies are in agreement with DYNAMO, while the third galaxy (referred to as BzK-16000 in Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015) shows overall lower line ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Interest- ingly, Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015) describe this galaxy as the most evolved in their sample, with no massive clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' DYNAMO SEDs and [CII] Emission We extract background subtracted 89, 155, and 216 µm fluxes for four DYNAMO galaxies, including D15-3 which overlaps with our ALMA sample, from our HAWC+ SOFIA observations using the Photutils software (Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We define the flux ex- traction apertures to correspond to the FWHM beam size of each corresponding HAWC+ band, while we de- fine the background annuli to have an inner radius equal to 5× beam FWHM and an outer radius of 7× beam FWHM (see Figure 8 in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We record these flux measurements in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To fit the SED, we com- bine the HAWC+ fluxes with WISE measurements at 22 µm (which is not contaminated by line emission and traces the warm dust continuum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017), and use the modified blackbody (MBB) SED fitting tool mbb emcee4, described in Riechers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2013) and Dowell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The MBB is joined to a power law of the form να at short wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' mbb emcee fits the dust temperature, Td, the extinction curve power law slope, β, the power law slope of the blue side, α, the wavelength where the optical depth reaches one, λ0, and the normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We impose a prior on β to constrain it between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 − 2, and leave all other parameters un- constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We record the resulting fit parameters and total infrared luminosity (8−1000 µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' TIR) in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We present the resulting SEDs in the left panel of Figure 6, where the filled colored data points represent fluxes from the HAWC+ (three longest wavelength data points), open colored data points represent WISE bands (shortest wavelength point) for each of the four galaxies, and the matching colored line represents the SED fit for that galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The dust temperatures derived from these SEDs are shown in the upper left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For DYNAMO D14-1 and D15-3, the resulting dust temperatures are Td = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='94 +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='88 K and Td = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='56 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='06 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='89 K respectively, consistent with the dust temperature measurements of Td = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='86 K and Td = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='52 K from SED fitting of Herschel PACS and SPIRE observations by White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='com/aconley/mbb emcee Resolved CO Excitation in DYNAMO 13 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' SOFIA [CII] and IR Measurements Galaxy 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 µm 89 µm 155 µm 216 µm Td TIR log10 L[CII] (mJy) (mJy) (mJy) (mJy) (K) (1010 L⊙) (erg s−1) B08-3 · · · · · · · · · · · · 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='46 D10-4 · · · · · · · · · · · · 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='46 D14-1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 148 ± 19 423 ± 47 209 ± 25 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='94 +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='48 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='78 · · D15-3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8 282 ± 33 393 ± 44 685 ± 73 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='56 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='06 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='89 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='26 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='49 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='46 F08-2 · · 522 ± 57 326 ± 37 563 ± 61 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='15 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='66 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='72 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='27 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='30 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='34 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='46 F09-1 · · · · · · · · · · · · 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='46 F12-4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9 224 ± 27 279 ± 32 594 ± 64 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='43 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='65 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='11 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='74 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='46 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Left: Spectral energy distribution of galaxies DYNAMO D14-1, D15-3, F08-2, and F12-4 based on fluxes from SOFIA HAWC+ (colored symbols) and WISE observations (open colored symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The solid colored lines are the resulting SED fits using mbb emcee, with the corresponding dust temperatures appearing in the top left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For DYNAMO D14-1 and D15-3, the dust temperatures derived from the HAWC+ measurements are consistent with those derived by White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2017) using Herschel PACS and SPIRE photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Right: The [CII]-to-TIR ratio as a function of TIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The colored symbols with black outlines correspond to DYNAMO measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' the magenta symbols have TIR measurements derived from SED fitting, while the teal symbols have TIR estimated from SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For galaxies where we have both SED measurements and SFRs, we link the data points via a black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Grey error bars are the assumed 40% calibration uncertainty for the [CII] observations and TIR uncertainties propagated through the ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We compare our DYNAMO measurements to those of Herrera-Camus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2018a) and find that DYNAMO galaxies do not show a deficit of [CII] emission, consistent with their cooler dust temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 103 Td = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='94+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1 K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='88 Td = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='56±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='06 Observed Flux (mJy) 102 101 DYNAMO D14-1 DYNAMO D15-3 DYNAMO F08-2 DYNAMO F12-4 102 Observed Wavelength (μm)10-1 High-z (z > 1) LINER Hll Galaxy Seyfert & QSO 10-2 [CII]/ LTIR 10-3 10-4 DYNAMO TIR from SFR DYNAMO TIR from SED 109 1010 1011 1012 1013 LTIR (Lo)14 Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The SEDs provide us with estimates of TIR for four out of the seven galaxies in the SOFIA sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We combine these measurements with the SOFIA FIFI-LS observations to explore the “[CII]-deficit”: the observed decreasing fraction of [CII] emission with respect to TIR in increasingly more infrared luminous objects (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Malhotra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Brauher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Herrera-Camus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For the remaining four galaxies in the SOFIA sample where no HAWC+ observations are available, we instead use the SFRs re- ported in Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2014) to estimate TIR and the calibration in equation (3) of Cluver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2017): SFR [M⊙ yr−1] = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8 × 10−44 LT IR [erg s−1] (4) which is derived from Starburst99 for solar metallic- ity, continuous star formation over 100 Myr, a Kroupa IMF, and assumes that the ultraviolet (UV) compo- nent of stellar emission is completely absorbed and re- radiated in the infrared (see also Calzetti 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To determine the [CII] luminosities, we produce in- tegrated intensity maps from the FIFI-LS observations (see Figure 9 in Appendix A) and take the peak value within a beam located at the position of each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' In the right panel of Figure 6, we present the [CII]/TIR as a function of TIR measured in this sample of DYNAMO galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The magenta squares represent galaxies for which SEDs were used to derive TIR, while the teal di- amonds represent the galaxies for which the SFRs were used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' In both cases, we show error bars where the errors on the [CII] and TIR luminosities have been propagated into the ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The two approaches to es- timating the TIR luminosities yield consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To illustrate this, we join with black dashed lines the data points for which we have SEDs and SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' When we compare DYNAMO to existing measurements in dif- ferent types of galaxies (Herrera-Camus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2018a), we find that DYNAMO galaxies do not exhibit a [CII]- deficit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Herrera-Camus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2018a) shows that at a fixed IR luminosity, the [CII]/FIR ratio decreases as galaxies become more compact, and Lutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2016) shows that the line-to-FIR ratios form a much tighter relation with FIR surface brightness than luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To investigate this, Herrera-Camus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2018b) con- struct two toy models with the PDR toolbox (Kaufman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2006): one where OB stars are closely associated with molecular gas clouds, and another where OB stars and clouds are randomly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' They find that as galaxies become more compact, a combination of ef- fects give rise to the [CII]-deficit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' These include a reduc- tion of the photo-electric heating efficiency, an increase in the ionization parameter, and as the interstellar ra- diation field increases, the [CII] line saturates and be- comes nearly independent of the far-UV flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Although DYNAMO galaxies generally lie above the star-forming main sequence at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1, their star formation is dis- tributed throughout their disks within numerous star- forming clumps, rather than being confined to a com- pact region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Their low dust temperatures and lack of a [CII]-deficit is consistent with this morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' DISCUSSION The ∼ 1−2 kpc-scale ALMA observations allow us to investigate how the line ratios we measure are affected by the surface density of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We expect that the CO(4−3) transition will be more highly excited in regions of higher ΣSFR, because these regions will have larger UV radiation fields and thus warmer dust temperatures (Narayanan & Krumholz 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To test this, we compare our resolved line ratio measurements to the ΣSFR measurements we make in the same beam- sized apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Figure 7 shows the CO(4−3)/CO(3−2) line ratio as a function of ΣSFR for four galaxies for which all necessary observations are available, as indi- cated by the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For each galaxy, we plot the set of resolved beam-sized measurements as previously de- scribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Though the line ratio uncertainties are large, there is a moderate positive correlation between the line ratios and ΣSFR measurements, indicating that in this sample of DYNAMO galaxies, higher ΣSFR regions are indeed correlated with higher line ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We perform a Spearman Rank Order correlation and find a coefficient of ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The correlation between resolved measure- ments within a single galaxy is stronger for DYNAMO D13-5 and G20-2 (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7 respectively), and weak- est for DYNAMO G04-1 (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4), while for G14-1 it is ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' In addition, we perform a linear fit to our ob- served line ratio−ΣSFR relation using scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='curve fit, which performs a non-linear least squares analysis with errors on the y−data as a parameter, and show the re- sults with the black solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The black dashed line corresponds to the parameterization of CO line emission intensity as a function of ΣSFR, derived by Narayanan & Krumholz (2014) (their equation 19): Iij I1−0 = A × [log10(ΣSFR) − χ]B + C (5) where Iij is the intensity of the CO(i − j) transition, A, B, and C are fit parameters, and χ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='85 (an off- set introduced to produce only real values of Iij/I1−0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Narayanan & Krumholz (2014) calculate the CO SLED of high−z star-forming galaxies from CO intensities that are modeled at ∼ 70 pc resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' For real observations with coarser beams, such as in our case, the resolved line Resolved CO Excitation in DYNAMO 15 ratio−ΣSFR parameterization is not an appropriate com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Therefore, Narayanan & Krumholz (2014) de- termine the luminosity-weighted emitting area for each CO transition and scale the resolved line intensities, and then refit the line ratio−ΣSFR relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Because our ob- servations probe ∼ 1 − 2 kpc scales, this is primarily what we compare to here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' However, we show compar- isons to the resolved parameterization for completeness We adopt values for A, B, and C for unresolved obser- vations from their Table 3 for CO(3−2) and CO(4−3), and substitute in our measured values of ΣSFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, we take the ratio of the two equations and divide by J2 u/J2 l = 42/32 to convert from Jy to K and produce the dashed black line in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We repeat the same procedure for the resolved galaxy observations parame- terization from their Table 2 and plot this as the black dashed-dotted line in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Overall, both model parameterizations under-predict the steepness of the CO(4−3)/CO(3−2)−ΣSFR rela- tion that our observations suggest, and over-predict the line ratio across the entire range of ΣSFR val- ues that our observations probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Similarly, Boogaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2020) find that the unresolved models also over- predict their CO(4−3)/CO(2−1) measurements (see their Figure 13), while providing a better match to their CO(5−4)/CO(2−1) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Sharon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2019), who present ∼ 2 kpc resolution CO(1−0) and CO(3−2) ob- servations of a lensed galaxy at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='26, also find that the Narayanan & Krumholz (2014) models do not repro- duce their observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' however, they do not attribute much meaning to this difference due to the limited ΣSFR values probed by a single galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' In contrast, Valentino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2020) find qualitative agreement between the un- resolved Narayanan & Krumholz (2014) model and their CO(5−4)/CO(2−1) observations in z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7 IR- selected galaxies on and above the main sequence of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' It is possible that because these models do not ex- plicitly model gas-rich clumpy disks like DYNAMO and high-redshift star-forming galaxies, that their properties are not completely captured in the early-phase snap- shots of the model disks and model mergers (Narayanan & Krumholz 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' It is also possible that a model which characterizes global CO excitation properties for an average ΣSFR may not be well-suited to investigate the internal variations within a single galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' To test this, we convolve our Hα maps to the CO(1−0) beam sizes (∼ 5 − 10′′) of Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2019) and measure the global ΣSFR of each galaxy for which data are avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We then use 5 and the unresolved parameters of Narayanan & Krumholz (2014) to predict R31 and R41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We list these predictions in the last two columns of Ta- Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' CO(4−3)/CO(3−2) line ratio as a function of SFR surface density, measured in beam-sized regions across the disk of each galaxy, indicated by the color and symbol coding in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We present this data for galaxies where observations of both CO transitions and Hα exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Despite the large uncertainties, there is an indication of an increas- ing line ratio with increasing SFR surface density trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This is parameterized by the Spearman’s Rank Order correlation coefficient of ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6, suggesting a moderate positive cor- relation between these two quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We present a linear fit to our measurements (black solid line) and for compari- son, include the predicted trend using the unresolved relation between CO intensity and ΣSFR of Narayanan & Krumholz (2014) (black dashed line), and their 70 pc resolved relations (black dash-dotted line) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We find that overall, the Narayanan & Krumholz (2014) models give better predictions of our global R31 and R41 measurements than our kpc-scale R43 measure- ments, which may indicate that the unresolved mod- els do not capture the kpc-scale variation in CO ex- citation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Using hydrodynamical simulations, Bournaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015) study the CO SLEDs of high-redshift galax- ies (as well as spirals and mergers), and investigate the contribution of giant clumps to the global CO SLED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' They derive CO SLEDs for clumps and the inter-clump gas and show that there is a considerable difference in the CO excitation (see their Figures 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This may indicate a need for models that specifically relate CO ex- citation, measured at various physical scales, in gas-rich clumpy disks to observable quantities such as ΣSFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' DYNAMO D13-5 DYNAMO G14-1 DYNAMO G04-1 DYNAMO G20-2 r43 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='04) × log ZsFR + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 Unresolved Narayanan+2014 Resolved (～ 70 pc) Narayanan+2014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='8 CO(4-3)/CO(3-2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0 log ZsFR [Mo yr-1 kpc-2]16 Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' CONCLUSIONS In this work, we have combined ∼ 1 − 2 kpc scale ALMA observations of CO(3−2) and CO(4−3) with HST observations of Hα, to study the CO(4−3)/CO(3−2) line ratio and its dependence on ΣSFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We have combined this with SOFIA HAWC+ and FIFI-LS observations of [CII] which provide addi- tional measurements of the ISM gas physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We summarize our findings here: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' DYNAMO galaxies have typical CO(4−3)/CO(3−2) line ratios of R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='54 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='15, which is most consistent with samples of star forming ∼ 1 − 2 main-sequence galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Boogaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Henr´ıquez-Brocal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Likewise, the global CO(3−2)/CO(1−0) and CO(4−3)/CO(1−0) measurements in DYNAMO are higher than global measurements of nearby star-forming galaxies (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2022) and are more consistent with the measurements of z ∼ 1−2 star-forming galaxies (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Dessauges-Zavadsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Birkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Harrington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The DYNAMO SEDs derived from SOFIA HAWC+ suggest cooler dust temperatures than those observed in local starburst galaxies and U/LIRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This sug- gests that while DYNAMO galaxies are strongly star forming, their star formation must be distributed rather than very compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This is consistent with the picture we obtain from the CO(4−3)/CO(3−2) line ratio mea- surements and the clumpy morphology of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The DYNAMO CO(4−3)/CO(3−2) line ratios are positively correlated with the ΣSFR measurements, with a Spearman Rank Order correlation coefficient of ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Our best fit relation between the CO(4−3)/CO(3−2) line ratio and ΣSFR is R43 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='04) × log ΣSFR + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This relation sug- gests a steeper relation than predicted by the parame- terization of Narayanan & Krumholz (2014), which also over-predicts the line ratio over the whole range of ΣSFR values probed by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' It is possible that this is consistent with the low dust temperatures of DYNAMO galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' However, Sharon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2019), who also study ∼ kpc scale line ratios in a high-redshift lensed galaxy, also find a discrepancy between the models and obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This may indicate that models that investigate CO emission variations with internal galaxy properties for gas-rich clumpy disks, are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank the anonymous referee for comments and suggestions that have greatly improved this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This paper makes use of the following ALMA data: ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='ALMA#2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='00239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' and ADS/JAO/ALMA#2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='00447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' ALMA is a part- nership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Re- public of Korea), in cooperation with the Republic of Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The National Radio As- tronomy Observatory is a facility of the National Sci- ence Foundation operated under cooperative agreement by Associated Universities, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Some of the data pre- sented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Tele- scope Science Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The specific observations ana- lyzed can be accessed via 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='17909/faa7-sw34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Based in part on observations made with the NASA/DLR Strato- spheric Observatory for Infrared Astronomy (SOFIA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' SOFIA is jointly operated by the Universities Space Re- search Association, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (USRA), under NASA con- tract NNA17BF53C, and the Deutsches SOFIA Institut (DSI) under DLR contract 50 OK 0901 to the University of Stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Financial support for this work was pro- vided by NASA through award #SOFIA-080238 issued by USRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' acknowledges support from USRA SOFIA-080238 and NASA HSTGO15069002A, and NSF-AST2108140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' acknowledges support from a NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-2102625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' is a recip- ient of an Australian Research Council Future Fellow- ship (FT190100083) funded by the Australian Govern- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' thanks the Max Planck Society for sup- port under the Partner Group project ”The Baryon Cy- cle in Galaxies” between the Max Planck for Extrater- restrial Physics and the Universidad de Concepci´on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='H-C also acknowledge financial support from Mil- lenium Nucleus NCN19058 (TITANs) and support by the ANID BASAL projects ACE210002 and FB210003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This research made use of Photutils, an Astropy package for detection and photometry of astronomical sources (Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Facilities: ALMA, HST(WFC), SOFIA(FIFI-LS, HAWC+) Software: aplpy (Robitaille 2019), astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2013, 2018), casa (McMullin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2007), numpy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2020), Photutils (Bradley Resolved CO Excitation in DYNAMO 17 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2021), reproject (Robitaille 2018), spectral-cube (Ginsburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2019) APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' SOFIA OBSERVATIONS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' LINE RATIO LITERATURE COMPILATION Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015) use IRAM PdBI observations of CO(2−1), CO(3−2), and CO(5−4), and Very Large Array observation of CO(1−0) in three main-sequence star forming disk galaxies at z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='5 to study their CO excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We use their average R31 and interpolate their models from their Figure 10 to extract R41, then take the ratio R41/R31 to obtain R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='26, which we include in Figure 4 as a black circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Kamenetzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2016) find a linear relation between LFIR and L′ CO for low- to mid-J CO lines and a slightly sub-linear relation for high-J CO lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We adopt the slope and intercepts of the relations for CO(4−3) and CO(3−2) from their Tables 6 and 7 (for U/LIRGs and non-U/LIRGs (LFIR ≤ 6 × 1010 L⊙) respectively), and assume a FIR luminosity of 1011 for the U/LIRG case and 1010 for the non-U/LIRG case to derive the LF IR −L′ CO relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Taking the ratio of these we find R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='05 for U/LIRGs and non-U/LIRGs respectively, assuming 20% uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We plot these as a black stars in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Rosenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015) study the CO SLEDs of 29 (Ultra) Luminous Infrared Galaxies (U/LIRGs) from CO(1−0) through CO(13−12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' They classify their objects into three classes based on their excitation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Where available, we compiled CO(4−3) and CO(3−2) fluxes from their Tables 2 and 3, and divided the resulting ratios by (J3 u/J3 l ) to convert from units of W m−2 to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, we separate the galaxies according to their classification, and plot the median line ratio for each class as black diamonds in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The error bars represent the standard deviation of line ratios in each class to illustrate the spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We note that most of the Rosenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015) sample is contained within the Kamenetzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2016) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Papadopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2012) study the CO SLEDs of 70 U/LIRGs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' we average the R43 values from their Table 7 (eight galaxies in total) and calculate the standard error on the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' This results in R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='12 and plot this as a black pentagon in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We note that 11/70 galaxies from the Papadopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2012) sample overlap with the sample of Rosenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Finally, Henr´ıquez-Brocal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' (2022) combine NOEMA observations of [CI](1−0), [CI](2−1), and CO(7−6) with ancillary CO(1−0) and CO(3−2) observations to model the CO SLED of Q1700-MD94, a massive main-sequence galaxy at z ∼ 2, with a one- and two-temperature component model using RADEX (van der Tak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We interpolate the model curves in their Figure 3 to extract R43 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='18 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='15 for the one- and two- component models respectively (taking a 20% uncertainty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' We do not plot these values in Figure 4, but include them in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' REFERENCES Adelman-McCarthy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' K.' 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+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2013, MNRAS, 429, 3047, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1093/mnras/sts562 18 Lenki´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' SOFIA HAWC+ observations of DYNAMO galaxies: 89 µm (left), 155 µm (middle), and 216 µm (right) in units of Jy pixel−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The black circles are centered on the position of the DYNAMO galaxy observed (indicated in the top right corner of the left-most panels), and their size corresponds to the angular resolution of each band (wavelength is indicated in the top left corner of each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' The crimson dashed circles define the background annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' DYNAMO D15-3 (second row) is the only galaxy that overlaps with the ALMA sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 0°53\'00" 89μm D14-1 52\'30" Dec (ICRS) 00" 51\'30" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='.00 14h46m12s 10s 08s 06s 04s RA (ICRS)155 μm 14h46m12s 10s 08s 06s 04s RA (ICRS)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='025 216 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='015 14h46m12s 10s 08s 06s 04s RA (ICRS)89μum D15-3 0°28\'00" Dec (ICRS) 30" O 29\'00" 30" 15h34m40s 38s 36s 34s 32s RA (ICRS)155μm 15h34m40s 38s 36s 34s 32s RA (ICRS)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='10 216μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='08 15h34m40s38s 36s 34s 32s RA (ICRS)4°04\'30" 89 μm F08-2 00" Dec (ICRS) 03\'30" 00" 02\'30" 8h31m06s 04s 02s s00 30m58s RA (ICRS)155μm 8h31m06s 04s 02s 00s 30m58s RA (ICRS)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='04 216 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='04 8h31m06s 04s 02s s00 30m58s RA (ICRS)89 μm F12-4 2°49\'30" 50\'00" Dec (ICRS) 30" 51\'00" 30" 12h25m38s 36s 34s 32s 30s RA (ICRS)155 μm 12h25m38s 36s 34s 32s 30s RA (ICRS)216 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} 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(first row, right-most panel) is the only galaxy that overlaps with the ALMA sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='B08-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='9°31\'30" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='15" ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' Bournaud, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Daddi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Weiß, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015, A&A, 575, A56, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1051/0004-6361/201425078 Bradley, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Sip˝ocz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Robitaille, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2021, astropy/photutils: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='2.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1142/S2251171718400032 Fisher, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Bolatto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', White, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Tacconi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', Lutz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2015, ApJ, 800, 20, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='1088/0004-637X/800/1/20 Ginsburg, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content=' 2017, ApJ, 846, 35, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} +page_content='3847/1538-4357/aa7fbf' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfwg2l/content/2301.05251v1.pdf'} diff --git a/ttAzT4oBgHgl3EQfr_2o/content/tmp_files/2301.01653v1.pdf.txt b/ttAzT4oBgHgl3EQfr_2o/content/tmp_files/2301.01653v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..19a97391a386343714085ee99abe2de809ba6ff4 --- /dev/null +++ b/ttAzT4oBgHgl3EQfr_2o/content/tmp_files/2301.01653v1.pdf.txt @@ -0,0 +1,2235 @@ +arXiv:2301.01653v1 [stat.ME] 4 Jan 2023 +Simultaneous directional inference +Ruth Heller∗ +Department of Statistics and Operations Research, Tel-Aviv University +ruheller@gmail.com +Aldo Solari +Department of Economics, Management and Statistics, University of Milano-Bicocca +solari.aldo@gmail.com +November 2022 +Abstract +We consider the problem of inference on the signs of n > 1 parameters. Within +a simultaneous inference framework, we aim to: identify as many of the signs of +the individual parameters as possible; provide confidence bounds on the number +of positive (or negative) parameters on subsets of interest. +Our suggestion is as +follows: start by using the data to select the direction of the hypothesis test for +each parameter; then, adjust the one-sided p-values for the selection, and use them +for simultaneous inference on the selected n one-sided hypotheses. The adjustment +is straightforward assuming that the one-sided p-values are conditionally valid and +mutually independent. Such assumptions are commonly satisfied in a meta-analysis, +and we can apply our approach following a test of the global null hypothesis that all +parameters are zero, or of the hypothesis of no qualitative interaction. We consider +the use of two multiple testing principles: closed testing and partitioning. The novel +procedure based on partitioning is more powerful, but slightly less informative: it only +infers on positive and non-positive signs. The procedure takes at most a polynomial +time, and we show its usefulness on a subgroup analysis of a medical intervention, +and on a meta-analysis of an educational intervention. +∗The authors gratefully acknowledge the Rita Levi-Montalcini prize for Scientific Cooperation between +Italy and Israel. +1 + +Keywords: conditional inference; directional inference; meta-analysis; multiple testing; +partitioning principle; simultaneous confidence bounds +1 +Introduction +Let θ = (θ1, . . . , θn) be a vector of n unknown real-valued parameters. A conventional +analysis is two-sided multiple testing, e.g., testing the family of point null hypotheses +{θ1 = 0, . . . , θn = 0} with a procedure that guarantees familywise error rate (FWER) +control at a pre-specified level α. If the ith null hypothesis is rejected, the conclusion is +that θi ̸= 0. Tukey (1991) argued that such a conclusion is unsatisfactory, and the analysis +should aim instead at concluding on the sign of the parameter, i.e., that θi > 0 or θi < 0. +Following rejection, it is tempting to conclude for the ith parameter its direction based on +the data. However, a directional error, referred to often as a type III error (see Finner 1999 +and references within) may occur if we decide after rejection of the point null hypothesis +θi = 0 that θi > 0 when in fact θi < 0. Therefore, the probability of making at least one +type I or type III error, henceforth referred to as the directional FWER (dFWER), may +be larger than α even though FWER ≤ α. +Existing multiple testing procedures on the family of point null hypotheses have been +shown to control the dFWER for independent two-sided p-values satisfying additional dis- +tributional assumptions: Holm’s procedure (Shaffer, 1980), Hochberg’s procedure (Finner, +1994; Liu, 1997) and, in general, the closed testing procedure (Finner, 1999), which includes +the previous two as special cases. +We are interested in simultaneous inference on the signs of the coordinates of θ. As in +previous works, we would like to control both type I and type III errors. However, we aim +not only to identify positive and negative parameters, but also to provide simultaneous +confidence bounds on the number of positive and on the number of negative parameters +for any subsets of interest. For this purpose, rather than adopting existing procedures for +two-sided tests for the purpose of directional identification, we start by considering the +problem of testing n pairs of one-sided hypotheses given by +H− +i +: +θi ≤ 0 +H+ +i +: +θi ≥ 0. +(1) +Lehmann (1950, 1957); Duncan (1955); Kaiser (1960); Shaffer (1974) considered (1) refer- +ring to it as a directional hypothesis-pair or a three-decision problem. Multiple directional +hypothesis-pairs were considered in Holm (1979); Shaffer (1980); Bauer et al. (1986); Finner +2 + +(1999); Shaffer (2002, 2006); Guo and Romano (2015), among others. We suggest the fol- +lowing two step approach, referred to henceforth as directional closed testing: first, select +from each pair the hypothesis to test (based on the data); second, on the selected n one- +sided hypotheses, apply an α level closed testing procedure. Of course, the second step has +to take care to adjust for the first step of selection from the same data. The adjustment +is straightforward for independent p-values that are conditionally valid (i.e., for which we +can compute valid p-values conditional on the first selection step, see Zhao et al. 2019; +Ellis et al. 2020). With this approach, we can provide simultaneous confidence bounds +on the number of positive and on the number of negative parameters for any subsets of +interest, building upon the work of Goeman and Solari (2011) that showed how to obtain +simultaneous bounds for a closed testing procedure. +This approach can be viewed as a direct generalization of dFWER controlling proce- +dures suggested previously, since it provides bounds of interest in addition to identification +of positive and negative parameters. For some local tests used in the α level closed testing +procedure in the second step, the bounds can be much tighter than the number of param- +eters identified with positive or negative signs. In some applications, tight bounds may +be more important than identification, so the local test should be selected with care. For +example, when evaluating a treatment effect on multiple subgroups (or cohorts), so θi is +the average treatment effect for subgroup i, a positive lower bound on both the number of +subgroups for which θi > 0, and on the number of subgroups for which θi < 0, conveys the +heterogeneity of the treatment effect. This can be of major clinical importance, since it +provides the researcher with the knowledge that at least for some subgroups the treatment +is harmful rather than effective (i.e., that there is a qualitative interaction, Gail and Simon +1985; Zhao et al. 2019). With our approach, we can provide tight lower bounds on the +number of positive and on the number of negative parameters, as well as identify positive +and negative parameters, with the guarantee that the probability of any wrong inference +is at most α. +For some applications, it is enough to infer on positive and non-positive findings, i.e., +we modify H+ +i in (1) by removing the equality sign, and thus exactly one of the hypothesis +pair is true: +H− +i +: +θi ≤ 0 +Ki +: +θi > 0. +(2) +In such a case, more powerful procedures than typically used with two-sided testing can be +devised (Bauer et al., 1986; Guo and Romano, 2015). For example, the Bonferroni proce- +dure compares each one sided p-value to α/n instead of to the two-sided threshold α/(2n). +3 + +This can be understood from the work of Shaffer (1986), who showed that when performing +a Bonferroni procedure, the proper correction factor for FWER control is not the number +of hypotheses tested, but the maximum number of hypotheses that can simultaneously +be true (Goeman et al., 2010). We suggest the following two step approach, referred to +henceforth as adaptive partitioning: first, select (based on the data) from each pair the +one-sided hypothesis; second, using the selected n one-sided hypotheses, test each parti- +tioning hypothesis at level α. The first step is identical to the first step in the directional +closed testing procedure, and the requirement that the p-values be independent and condi- +tionally valid is the same as well. The second step applies an α level partitioning procedure +(Stefansson et al., 1988; Finner and Strassburger, 2002; Finner et al., 2021) based on the +selected p-values. With this novel approach, we can provide tight lower and upper bounds +on the number of positive parameters (where the upper bound is n minus the lower bound +on the number of nonpositive parameters), as well as identify positive and nonpositive pa- +rameters, with the guarantee that the probability of any wrong inference is at most α. In +systematic reviews in clinical trials, for example, it is important to evaluate the bounds on +the number of studies with an effective intervention effect, and a non-trivial upper bound +is of concern since it suggests that the intervention effect is null or harmful (IntHout et al., +2016). We show that the bounds are uniformly tighter than with directional closed testing. +So unless it is important to infer on the positive as well as the negative parameters, this +approach should always be preferred. Importantly, it should always be preferred in settings +where the point null hypotheses are never true (which is always the case according to Tukey +1991; Jones and Tukey 2000). +In the next subsection, we illustrate our suggested approach to a small problem of a +subgroup analysis. The results highlight the difference between directional closed testing +and adaptive partitioning, and the potential usefulness of our approach. Then, we explain +in detail directional closed testing in § 2, and adaptive partitioning in § 3. We provide +the conditional and unconditional inferential guarantees for each procedure. +In § 4 we +compare and contrast the directional closed testing and adaptive partitioning procedures +in simulations, using various local tests. In § 5 we provide two usages of our suggested +methods: for enhancing meta-analysis, and for providing inference following the test of +qualitative interactions. Finally, in § 6 we discuss extensions and some final remarks. All +proofs are in Appendix A. +4 + +i +1 +2 +3 +4 +di +0.163 +-0.114 +-0.047 +-0.151 +sei +0.0788 +0.0689 +0.0614 +0.0547 +pi +0.0193 +0.9510 +0.7780 +0.9971 +Table 1: Analysis of the proportions free of breast cancer at 3 years in 1260 patients divided +in four subgroups defined by age and progesterone receptor levels (i = 1: Age < 50, PR +< 10; i = 2: Age ≥ 50, PR < 10; i = 3: Age < 50, PR ≥ 10; i = 4: Age ≥ 50, PR ≥ 10). +Reproduced from Table 2 in Gail and Simon (1985). The differences, di, in proportions +disease-free at 3 years and their relative standard errors, sei, are used to compute the right +tailed p-values pi by using the large-sample normal approximation. +1.1 +An example: PFT therapy for breast cancer +The data of Fisher et al. (1983), re-analysed in Gail and Simon (1985), consists of the +difference in the proportion of patients that are disease free with the PFT treatment versus +with the FT treatment, in n = 4 subgroups defined by age and progesterone receptor levels. +Table 1 provides for each subgroup the difference, standard error, and right-tailed p-value. +Applying the first step of hypothesis selection based on the direction favored by the data, +the directional closed testing procedure selects H− +1 , H+ +2 , H+ +3 , H+ +4 . Similarly, the adaptive +partitioning procedure selects H− +1 , K2, K3, K4. For both procedures, the p-values used for +testing in the second step, are therefore: 0.0193, 1−0.9510, 1−0.7780, and 1−0.99714, for +groups 1, 2, 3, and 4, respectively. The procedures are illustrated in Figure 1. Nodes in the +graph represent closed testing intersection hypotheses, labelled by their corresponding index +set and selected direction, as well as the partition of the parameter space, for which the +same local test is used. The local test p-value, obtained by the adaptive Simes combination +test (described in § 4), is shown for each node. +Applying directional closed testing at level α = 0.05, we have 0 ≤ n+ and 1 ≤ n− with +95% confidence. Furthermore, we conclude that θ4 < 0, since H+ +4 is rejected by closed +testing. +We obtain a trivial lower bound for n+ because H− +1 is not rejected by closed +testing, and a non-trivial lower bound for n− because H+ +2 ∩ H+ +3 ∩ H+ +4 is rejected by closed +testing. +The non-rejected partitioning hypotheses at the 5% level are denoted by − + +−, − − ++−, + + +−, + + −−, + − +−, + − −−. For example, − + +− corresponds to testing +the intersection hypothesis H− +1 ∩ K2 ∩ K3 ∩ H− +4 , and this is carried out in the adaptive +partitioning procedure by a local test of H− +1 ∩ K2 ∩ K3. This is valid, since the actual +5 + +test carried out is for a larger null (H− +1 ∩ K2 ∩ K3 ⊇ H− +1 ∩ K2 ∩ K3 ∩ H− +4 ). The reason +for testing H− +1 ∩ K2 ∩ K3 is that the fourth p-value is greater than half (so K4 is selected +for testing, not H− +4 ). The local test p-value is 0.0772 in this case. We can extract the +confidence set for n+ from these non-rejected partitioning null hypotheses as follows. Since +the number of positive parameters in each non-rejected partition null hypothesis is at least +one and at most three, the adaptive partitioning procedure provides bounds 1 ≤ n+ ≤ 3 +with 95% confidence. This is an improvement over the directional closed testing procedure. +Alternatively, we can see that the lower bound is one because H− +1 ∩H− +2 ∩H− +3 ∩H− +4 (denoted +by −−−− in Figure 1) is rejected by the partitioning procedure (by a local test of H− +1 , since +for the remaining coordinates K2, K3, K4 were selected for testing). Moreover, since the +fourth individual hypothesis in each non-rejected partition null hypothesis is non-positive, +we further conclude that θ4 ≤ 0. This is a weaker conclusion compared to θ4 < 0 obtained +by closed testing. +We revisit this example in § 5.2. +6 + +− + ++ +1−2+3+4+ +0.0115 +− + +− +1−2+3+ +0.0772 +− + −+ +1−2+4+ +0.0115 +− − ++ +1−3+4+ +0.0115 ++ + ++ +2+3+4+ +0.0115 +− + −− +1−2+ +0.0115 +− − +− +1−3+ +0.0772 +− − −+ +1−4+ +0.0115 ++ + +− +2+3+ +0.1960 ++ + −+ +2+4+ +0.0115 ++ − ++ +3+4+ +0.0115 +− − −− +1− +0.0386 ++ + −− +2+ +0.0980 ++ − +− +3+ +0.4440 ++ − −+ +4+ +0.0058 ++ − −− +∅ +1 +Figure 1: The example from Table 1. Nodes in the graph represent closed testing intersec- +tion hypothesis, labeled by their corresponding index set and selected direction (superscript ++ or −); e.g. 1−2+3+ corresponds to H− +1 ∩ H+ +2 ∩ H+ +3 . Below each intersection hypothesis, +the corresponding p-value obtained by the adaptive Simes combination test (described in +§ 4). Above each intersection hypothesis, the corresponding partitioning hypothesis (rep- +resented by a sequence of signs); e.g. − + +− corresponds to H− +1 ∩ K2 ∩ K3 ∩ H− +4 . For the +closed testing hypotheses only, arrows represent subset relationships. Hypotheses rejected +by the adaptive Simes local test at the 5% level are marked in bold; hypotheses rejected +by the closed testing procedure at 5% level are filled in gray. +7 + +2 +Directional closed testing +Suppose that we have right-tailed p-values p1, . . . , pn for the hypotheses H− +1 , . . . , H− +n , and +left-tailed p-values q1, . . . , qn for the hypotheses H+ +1 , . . . , H+ +n . Let p = (p1, . . . , pn) denote +the p-value vector, and [n] the index set {1, . . . , n}. Our assumptions regarding the p-values +are as follows: +(A0) Each p-value is valid, in the sense +∀x ∈ [0, 1], +sup +θ∈H− +i +Pθ(pi ≤ x) ≤ x, +sup +θ∈H+ +i +Pθ(qi ≤ x) ≤ x. +where the inequality is an equality for θi = 0. +(A1) Each p-value is conditionally valid: +sup +θ∈H− +i +Pθ(2pi ≤ x | pi ≤ 1/2) ≤ x +∀x ∈ [0, 1], +sup +θ∈H+ +i +Pθ(2qi ≤ x | pi > 1/2) ≤ x +∀x ∈ [0, 1]. +(A2) p1, . . . , pn are mutually independent. +Remark 2.1. For continuous exponential families (where the p-value is a monotone trans- +formation of the sufficient statistic), the p-values satisfy (A0) as well as uniform validity, +i.e., ∀θi ∈ H− +i , Pθi(pi/τ ≤ x | pi ≤ τ) ≤ x ∀ 0 ≤ x, τ ≤ 1, cf. Karlin and Rubin (1956); +Zhao et al. (2019). We only concentrate on the most natural selection value τ = 1/2, hence +assumption (A1) which is slightly more general than uniform validity. We address other +values of τ in the § 6. Moreover, for simplicity, we further assume in (A0) that the p-value +is uniform when θi = 0, but this assumption can be relaxed for discrete distributions, where +the adjustment for selection may be 1/P0(pi ≤ 1/2) instead of 2. +We are interested in simultaneous inference on the selected family of n directional +hypotheses {H+ +i +: pi ≤ 1/2} ∪ {H− +i +: pi > 1/2}. A general way to derive procedures +with simultaneous error control is provided by closed testing (Goeman and Solari, 2011; +Goeman et al., 2021), introduced by Marcus et al. (1976) for FWER control. We propose +the directional closed testing procedure, which is the closed testing procedure on the selected +one-sided hypotheses: +Procedure 2.1 (Directional closed testing). +8 + +Step 1 Select the n one-sided hypotheses for testing: � +H = {H− +i : pi ≤ 1/2}∪{H+ +i : pi > 1/2}. +Let S− = {i : pi ≤ 1/2} and S+ = {i : pi > 1/2} be, respectively, the indices for +which we test H− +i and H+ +i . +Step 2 Apply a level α closed testing procedure on the family of hypotheses � +H, using the +conditional p-values +{2pi, i ∈ S−} ∪ {2qj, j ∈ S+}. +So, the procedure tests all intersection hypotheses: +HI : +� +� +i∈I∩S− +H− +i +� +∩ +� +� +i∈I∩S+ +H+ +i +� +, +I ⊆ [n], where HI is true if and only if all selected hypotheses with i ∈ I are true. +Testing HI at level α is done by the local test +φI = 1{f({2pi, i ∈ I ∩ S−} ∪ {2qj, j ∈ I ∩ S+}) ≤ α} +(3) +using a combining function f(·). +We denote by S the vector of signs that we condition on: +S = (sign(p1 − 1/2), . . . , sign(pn − 1/2)), +sign(pi − 1/2) = +� +−1 +if pi ≤ 1/2, +1 +if pi > 1/2. +Conditional on the vector of signs S ∈ {−1, 1}n, the probability of rejecting the intersection +hypotheses of the true nulls among the selected is at most α, if we use the local test in +(3), since we combine (conditional) p-values that have each a distribution that is at least +as large as uniform given S. We formalize this in the next proposition. +Proposition 2.1. Let p1, . . . , pn fulfil conditions (A1)−(A2), and let θ be the true unknown +vector of parameters. +So among the n selected for testing in Step 1 of Procedure 2.1, +T − = {i : θi ∈ H− +i } ∩ S− and T + = {i : θi ∈ H+ +i } ∩ S+ are the true left sided and right +sided null hypotheses, respectively. T = T −∪T + is the (unknown) index set of the true null +hypotheses. Then the test of the intersection of the hypotheses in T satisfies conditional +type I error control: +sup +θ′∈HT +Pθ′ +� +φT = 1 | S +� +≤ α, +(4) +9 + +Proposition 2.1 implies that all the probability statements that hold for the closed test- +ing procedure, will hold conditional on S for Procedure 2.1. So the conditional guarantees +imply the unconditional guarantees, but are stronger since they are conditional. +How does directional closed testing compare with the direct approach of using two-sided +p-values in a closed testing procedure? The direct approach with two-sided p-values infers +on intersection hypothesis of the form ∩i∈I{θi = 0}. The indices of rejected individual +hypotheses (i.e., point null hypotheses θi = 0) is identical to the set of indices identified +using the directional closed testing procedure 2.1. However, using procedure 2.1 it is clear +that we can also infer about the signs (as long as each intersection hypothesis is tested +with a valid local test), and it is clear how to provide simultaneous confidence bounds on +the number of positive and negative parameters of any subset of hypotheses. To the best +of our knowledge, these bounds have not been available up to now. More generally, all +the guarantees provided by closed testing procedures follow immediately using directional +closed testing, while many of these guarantees were unclear with the direct approach using +two-sided p-values, due to the worry about type III errors. Moreover, to the best of our +knowledge, the specific assumption (A1) − (A2) are more general than the assumptions +considered in Shaffer (1980); Bauer et al. (1986); Finner (1999); Guo and Romano (2015) +for directional FWER control on individual discoveries. Specifically, conditional validity in +(A1) is a weaker condition than assuming the monotone likelihood ratio order. +We conclude this section by showing explicitly how to obtain individual discoveries and +confidence bounds on subsets of interest with Procedure 2.1. The closed testing procedure +corrects the local tests for multiple testing by +¯φI = min{φJ : J ⊇ I}. +Marcus et al. (1976) showed that the adjusted tests ¯φI have FWER control, so with Pro- +cedure 2.1 we have: +Pθ +� +¯φI = 0 for all I ⊆ T | S +� +≥ 1 − α. +Let +Dα = D+ +α ∪ D− +α = {i ∈ S− : ¯φi = 1} ∪ {i ∈ S+ : ¯φi = 1} +be the index set of the discoveries, i.e. the elementary hypotheses rejected by the closed +testing procedure at level α. The procedure concludes θi > 0 for all i ∈ D+ +α and θi < 0 for +all i ∈ D− +α while controlling the conditional FWER at level α. +10 + +Let n+(I) and n−(I) be the number of parameters θi, i ∈ I with positive value and +negative values, respectively: +n+(I) = |{i ∈ I : θi > 0}|, +n−(I) = |{i ∈ I : θi < 0}|. +For simplicity of notation, we use n+ and n− instead of n+([n]) and n−([n]). Goeman and Solari +(2011) showed that +eα(I) = max(|J | : J ⊆ I, ¯φJ = 0) +provide false discovery control over all I, so with Procedure 2.1 we have: +Pθ +� +|I ∩ T | ≤ eα(I) for all I | S +� +≥ 1 − α. +(5) +eα(I) is an upper bound for the false discoveries in I, for all I. Let dα(I) = |I| − eα(I) +be the lower bound on true discoveries in I. From (5) it follows that +� +l+ +α (I) = dα(I ∩ S−) +u+ +α(I) = |I| − dα(I ∩ S+) , +� +l− +α (I) = dα(I ∩ S+) +u− +α(I) = |I| − dα(I ∩ S−) +(6) +are such that +Pθ +� +l+ +α (I) ≤ n+(I) ≤ u+ +α(I), l− +α (I) ≤ n−(I) ≤ u− +α(I), for all I | S +� +≥ 1 − α. +(7) +In particular: +� +l+ +α = dα(S−) +u+ +α = n − dα(S+) , +� +l− +α = dα(S+) +u− +α = n − dα(S−) , +are the lower and upper bound on n+ and n−, the number of parameters θ1, . . . , θn with +positive value and negative values, and +Dα = D+ +α ∪ D− +α = {i : l+ +α (i) = u+ +α(i) = 1} ∪ {i : l− +α (i) = u− +α(i) = 1}. +A computational problem remains: computing (6) involves the evaluation of exponen- +tially many tests, which hinders its practical application. For specific combining functions, +however, there exist shortcuts to derive the closed testing results in polynomial time. +Our assumption regarding the combining function f(·) is as follows: +11 + +(A3) The combining function f(|I|; (x)i∈I) depends on the size of I and the vector of +p-values (x)i∈I, and it satisfies monotonicity: +f(|I|; (x1, . . . , x|I|)) ≤ f(|I|; (x′ +1, . . . , x′ +|I|)) +(8) +for xi ≤ x′ +i for all i = 1, . . . , |I|; and symmetry: +f(|I|; (x1, . . . , x|I|)) = f(|I|; (xj1, . . . , xj|I|)) +(9) +for any permutation (j1, . . . , j|I|) of (1, . . . , |I|). +Quadratic time shortcuts for combining functions satisfying (A3) have been developed +for FWER control (Dobriban, 2020) and simultaneous FDP control (Goeman and Solari, +2011; Goeman et al., 2019, 2021). Another condition that could reduce computation time +is separability (Tian et al., 2021). +3 +Adaptive partitioning +If one is interested only in inference on the number of positive parameters, we can provide +bounds that are uniformly better than the bounds in (6). The individual hypotheses we +consider for providing the (lower and upper) bounds on the number of positive parameters +are the n pairs in (2). +We derive (1 − α) confidence bounds l+ +α (I) and u+ +α(I) for n+(I) that are simultaneous +for all I, i.e., +Pθ +� +l+ +α (I) ≤ n+(I) ≤ u+ +α(I) for all I | S +� +≥ 1 − α. +(10) +Remark 3.1. The conditional guarantee in (10) imply of course the unconditional guar- +antee. We shall further show that the unconditional coverage is slightly larger than (1−α), +due to the fact that there is positive probability that the realized S cannot produce an infer- +ential error. This is in contrast with the conditional guarantee in Proposition 2.1, which +can reach α when θi = 0 for at least one i ∈ {1, . . . , n}. This is due to the fact that zero is +included in both H+ +i and H− +i , so for any realized S there will be a danger of falsely rejecting +the intersection null among the selected, HT . +We start by considering the case of n = 2 parameters in § 3.1. The inference using +adaptive partitioning is easy to explain in this case, since all partitions and all the possibil- +ities for the vector of signs S are easily enumerated. Moreover, we can visualize the power +advantage of adaptive partitioning over directional closed testing. Then we proceed to the +general case of n > 2 parameters in § 3.2. +12 + +3.1 +Inference for n = 2 +Figure 2 shows directional closed testing and adaptive partitioning procedures as a function +of the selection event (Step 1 of the procedures), for the case n = 2. Possible selections of +one-sided hypotheses are {H− +1 , H− +2 }, {H− +1 , H+ +2 }, {H+ +1 , H− +2 } and {H+ +1 , H+ +2 }. +p1 ≤ 1 +2, p2 ≤ 1 +2 +ˆH = {H− +1 , H− +2 } +−− +1−2− +f(2p1, 2p2) +−+ +1+ +2p1 ++− +2+ +2p2 +++ +∅ +1 +p1 ≤ 1 +2, p2 > 1 +2 +ˆH = {H− +1 , H+ +2 } +−+ +1−2+ +f(2p1, 2q2) +−− +1− +2p1 +++ +2+ +2q2 ++− +∅ +1 +p1 > 1 +2, p2 ≤ 1 +2 +ˆH = {H+ +1 , H− +2 } ++− +1+2− +f(2q1, 2p2) +++ +1+ +2q1 +−− +2− +2p2 +−+ +∅ +1 +p1 > 1 +2, p2 > 1 +2 +ˆH = {H+ +1 , H+ +2 } +++ +1+2+ +f(2q1, 2q2) ++− +1+ +2q1 +−+ +2+ +2q2 +−− +∅ +1 +Figure 2: Directional closed testing and adaptive partitioning procedures as a function of +the selection event (displayed on top), for the case n = 2. Nodes in the graph show the par- +titioning hypothesis (first row, represented by a sequence of signs; e.g. −+ corresponds to +H− +1 ∩K2), the closed testing hypothesis (second row, represented by an index set along with +the selected direction; e.g. 1−2+ corresponds to H− +1 ∩ H+ +2 ) and the corresponding p-value +(third row). For the closed testing hypotheses only, arrows represent subset relationships. +For example, when ˆH{H− +1 , H+ +2 } is selected, the closed testing intersection hypothe- +sis H− +1 ∩ H+ +2 +(1−2+) and the partitioning hypothesis H− +1 ∩ K2 (−+) are both tested +by the p-value f(2p1, 2p2) (e.g., with Fisher’s combining function, f(2p1, 2p2) = P(χ2 +4 ≥ +13 + +−2(log(2p1) + log(2p2)), H− +1 (1−) and H− +1 ∩ H− +2 (−−) by the p-value 2p1; and H+ +2 (2+) +and K1 ∩ K2 (++) by the p-value 2q2. +The partitioning hypothesis K1 ∩ H− +2 (+−) is +not tested, i.e. +the p-value is set to 1. +Because H− +1 ∩ H− +2 implies both H− +1 and H+ +2 , +directional closed testing adjusts the p-values for H− +1 and H+ +2 by max(2p1, f(2p1, 2p2)) +and max(2q2, f(2p1, 2p2)), respectively. If e.g. f(2p1, 2p2) > α and p1 < α/2, then di- +rectional closed testing does not reject any hypothesis; conversely, adaptive partitioning +rejects H− +1 ∩H− +2 (−−), which implies n+ ≥ 1. This situation is illustrated in Figure 3 with +Fisher’s combining function, where the top-left and top-right plots show the areas leading +to inference about n+ with (1−α) confidence for the directional closed testing and adaptive +partitioning, respectively. +14 + +Directional Closed Testing +α/2 +p1 +p2 +Adaptive Partitioning +α/2 +p1 +p2 +Rejection of individual hypotheses +1−, 2− +1−, 2+ +1+, 2+ +1+, 2− +1− +1+ +2− +2+ +p1 +p2 +α/2 +Unconditional Adaptive Partitioning +2α/3 +p1 +p2 +n+ = 2 +n+ ≥ 1 +n+ = 1 +n+ ≤ 1 +n+ = 0 +Figure 3: Inference for n = 2 using Fisher’s combining function. Areas leading to inference +about n+ with (1 − α) confidence for the directional closed testing (top-left plot) and the +adaptive partitioning (top-right plot with conditional guarantee and bottom-right plot with +unconditional guarantee). The gray scale indicates the type of inference. Bottom-left plot: +areas leading to the rejection of individual hypotheses. The dot pattern indicates areas in +which one hypothesis is rejected at level α; the north east lines pattern indicates areas in +which two hypotheses are rejected at level α. Hypotheses are represented by their indices +and directions, e.g. 1−, 2+ corresponds to the hypotheses H− +1 and H+ +2 . Each plot is based +on α = 0.2. +15 + +We see that the area is larger for the adaptive partitioning procedure when {p1 ≤ +1/2, p2 > 1/2} or {p1 > 1/2, p2 ≤ 1/2}, i.e. +adaptive partitioning is uniformly more +powerful than directional closed testing for inference on n+. If we are only interested in +providing level α unconditional error control, the power gain of adaptive partitioning is +even larger, as illustrated in the bottom-right plot of Figure 3. +Indeed, we will see in +Section 4 that the adaptive partitioning procedure can be carried out at the larger level +α∗ = α/(1 − 2−n) = 4α/3 while providing level α unconditional error control. +Figure 4 shows the power gain of using adaptive partitioning over directional closed +testing, in discovering that n+ ≥ 1, for a range of parameters. The power gain is substantial, +e.g., when θ1 > 0 and θ2 < 0 there can be a power gain of 8%. In addition, we see the +additional gain we get from using α∗ instead of α = 0.05. +16 + +−3 +−2.5 +−2 +−1.5 +−1 +−0.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +0 +0.2 +0.4 +0.6 +0.8 +1 +θ2 +Power +DCT (θ1 = 0) +AP (θ1 = 0) +UAP (θ1 = 0) +DCT (θ1 = 2) +AP (θ1 = 2) +UAP (θ1 = 2) +Figure 4: Power to discover that the lower bound for n+ is at least one with 95% confidence +versus θ2 with: directional closed testing (DCT, solid line); adaptive partitioning (AP, +dashed line); the unconditional adaptive partitioning (UAP, dotted line). The test statistics +are generated independently from a Gaussian distribution with standard deviation 1 and +mean centered at θ1 = 0 (black curves), θ1 = 2 (blue curves). +Finally, the bottom-left plot of Figure 3 shows the area leading to the rejection of +individual hypotheses for the directional closed testing with Fisher’s combining function, +which is contained in the area leading to inference about n+. Adaptive partitioning has +the same rejection region but with H+ +i replaced by Ki, i = 1, 2. Appendix B discusses the +two procedures with Simes’ combining function for n = 2, comparing them to the dFWER +controlling procedures of Bauer et al. (1986) and Guo and Romano (2015). +17 + +3.2 +Simultaneous confidence bounds for n+(I) +We derive simultaneous (1 − α)-confidence bounds l+ +α (I) and u+ +α(I) for n+(I) by using the +partitioning principle (Stefansson et al., 1988; Finner and Strassburger, 2002; Finner et al., +2021). The main idea is to partition the parameter space Θ into disjoint subspaces: consider +the 2n orthants +ΘK += +{θ ∈ Θ : θi > 0 for all i ∈ K, θj ≤ 0 for all j /∈ K} +for all K, so that exactly one ΘK contains the true parameter θ. +Each orthant ΘK has a corresponding null hypothesis JK : θ ∈ ΘK that the orthant +includes the true parameter. This hypothesis is true if and only if all Ki, i ∈ K are true +and all H− +j , j /∈ K are true: +JK : { +� +i∈K +Ki} ∩ { +� +i/∈K +H− +i }. +Suppose that for every K there is a partitioning local test ψK taking values in {0, 1}, +with 1 indicating the rejection of JK at level α. The hypotheses JK are all disjoint, and +only one of them is true. Assume that ψK is a valid statistical test for the true JK, i.e. +sup +θ∈ΘK +Pθ(ψK = 1) ≤ α. +Consider the hypothesis Jv(I) : n+(I) = v, which can be equivalently expressed as +Jv(I) : � +K:|K∩I|=v JK. Then Jv(I) is rejected if and only if all JK with |K ∩ I| = v are +rejected, and denote by +ψv(I) = min{ψK : K ⊆ [n] : |K ∩ I| = v} +(11) +the statistical test for Jv(I). The collection of values v for which we failed to reject Jv(I) +at level α: +N +(I)α += +{v ∈ {0, . . . , |I|} : ψv(I) = 0}, +constitutes a (1 − α) confidence set for n+(I). +Proposition 3.1. Pθ(n+(I) ∈ N +(I)α for all I) ≥ 1−α for any θ ∈ Θ. Furthermore, the +lower and upper bounds given by +l+ +α (I) = min(N +(I)α), +u+ +α(I) = max(N +(I)α) +(12) +satisfy +Pθ +� +l+ +α (I) ≤ n+(I) ≤ u+ +α(I) for all I +� +≥ 1 − α. +18 + +However, the computational problem remains: direct application of (11) takes expo- +nential time. We propose a procedure that apply to a combining function f(·) satisfying +condition (A3) and that can be computed in polynomial time. +Procedure 3.1 (Adaptive partitioning). +Step 1 Apply Step 1 of procedure 2.1 +Step 2 Apply Algorithm 1 to obtain the bounds l+ +α (I) and u+ +α(I) for n+(I) for one or several +I ⊆ [n] of your choice. Algorithm 1 performs a level α partitioning procedure testing +JK by the test for the larger intersection hypothesis with indices in {Kc ∩ S−} ∪ {K ∩ +S+}: +ψK += +φ{Kc∩S−}∪{K∩S+} +(13) += +1{f({2pi, i ∈ Kc ∩ S−} ∪ {2qj, j ∈ K ∩ S+}) ≤ α}. +19 + +Algorithm 1: Shortcut for computing the confidence bounds l+ +α (I) and u+ +α(I) +Input +: right-tailed p-values p1, . . . , pn; combining function f(·); subset +I ⊂ [n]; level α +Output : Confidence bounds l+ +α (I) and u+ +α(I), p-values fv(I) for +Jv(I) : n+(I) = v, for v = 0, . . . , |I| +Initialize: +S− = {i ∈ [n] : pi ≤ 1/2}, |S−| = s, fs(I) = 0 ; +a = (a1, . . . , a|S−∩I|) with a1 ≥ . . . ≥ a|S∩I| sorted values of {2pi : i ∈ S− ∩ I} ; +b = (b1, . . . , b|S−∩Ic|) with b1 ≥ . . . ≥ b|S−∩Ic| sorted values of {2pi : i ∈ S− ∩ Ic} ; +c = (c1, . . . , c|S+∩I|) with c1 ≥ . . . ≥ c|S+∩I| sorted values of {2qi : i ∈ S+ ∩ I} ; +d = (d1, . . . , d|S+∩Ic|) with d1 ≥ . . . ≥ d|S+∩Ic| sorted values of {2qi : i ∈ S+ ∩ Ic} ; +1 for v ← 0 to |I| do +2 +for u ← 0 to |Ic| do +3 +for k ← max(0, v − |S− ∩ I|) to min(v, |S+ ∩ I|) do +4 +for j ← max(0, u − |S− ∩ Ic|) to min(u, |S+ ∩ Ic|) do +5 +fk,j = f({a1, . . . , ak−v+|S−∩I|} ∪ {b1, . . . , bj−u+|S−∩Ic|} ∪ {c1, . . . , ck} ∪ +{d1, . . . , dj}) ; +6 +end +7 +end +8 +fv,u(I) ← max{fk,j} ; +9 +end +10 +fv(I) = max{fv,u(I), u = 0, . . . , |Ic|} ; +11 end +12 l+ +α (I) ← min(v ∈ {0, . . . , s} : fv(I) > α) ; +13 u+ +α(I) ← max(v ∈ {s, . . . , |I|} : fv(I) > α) ; +14 return l+ +α (I), u+ +α(I), f0(I), . . . , f|I|(I) ; +The following proposition asserts that Procedure 3.1 guarantees conditional coverage +and it also improves upon the directional closed testing Procedure 2.1. +Proposition 3.2. Let p1, . . . , pn fulfil conditions (A1) − (A2), and the combining function +f(·) fulfils condition (A3). Then Procedure 3.1 returns the bounds l+ +α (I) and u+ +α(I) for +n+(I) with at most O(|I|2 · max(1, |Ic|2)) computation. The bounds satisfy the conditional +coverage guarantee in (12) and are at least as good as those obtained by the directional +closed testing Procedure 2.1. Furthermore, the unconditional coverage guarantee is +Pθ +� +l+ +α (I) ≤ n+(I) ≤ u+ +α(I) for all I +� +≥ 1 − (1 − 2−n)α > 1 − α. +(14) +20 + +Remark 3.2. For inference on n+ only, in Appendix C we derive (1−α) confidence bounds +for n+, i.e., a lower bound l+ +α and an upper bound u+ +α such that +Pθ +� +l+ +α ≤ n+ ≤ u+ +α | S +� +≥ 1 − α. +(15) +These bounds may be tighter for n+, than taking I = [n] in procedure 3.1, but there is +no follow-up identification. Interestingly, the gap between these bounds and the bounds for +I = [n] in the adaptive partitioning procedure 3.1 is minimal. This finding is based on a +wide range of data generations examined (omitted for brevity), including the ones detailed +in § 4. Since we are also concerned with follow-up identification and the cost of further +inferences is minimal, we suggest using Procedure 3.1 in order to find the bounds for n+ +and identify positive and non-positive parameters. Nevertheless, procedure C.1 should be +used if interest is only in bounds for n+, since it is computationally much simpler and the +bounds are at least as tight as with I = [n] in procedure 3.1. See Proposition D.1 in the +Appendix. +Remark 3.3. Theorem 2 in Goeman et al. (2021) shows that closed testing and partition- +ing are equivalent principles, which are also equivalent to sequential rejection (Goeman and Solari, +2010). We employed the closed testing principle in procedure 2.1 and the partitioning prin- +ciple in procedure 3.1 simply because they seemed the more natural formulations. +4 +Simulations +We carry out simulation studies in order to evaluate the performance of our suggested meth- +ods. Our main focus is on the evaluation of two procedures: the directional closed testing +(DCT) procedure 2.1 and the adaptive partitioning (AP) procedure 3.1. Proposition 3.2 +states that AP is uniformly better than DCT, but the scientific discovery is slightly weaker: +inference is only on the positive and nonpositive parameters, rather than on the positive +and negative parameters. Therefore, our primary aim is to quantify the potential gain in +terms of tighter bounds and better identification using AP versus DCT. We shall use two +representative combining methods for this purpose: Fisher, which is a sum-based method, +and Simes, which is quantile based. Sum based tests, such as Fisher, are better than Simes +for the bounds on n+ when more than a single parameter is positive or negative, but are +worse for identification. This finding is demonstrated in our simulations but is also expected +based on previous work (see, e.g., Benjamini and Heller 2008; Heard and Rubin-Delanchy +2018 and references within). +21 + +For adaptive partitioning, we shall further consider the adaptive likelihood ratio test +(ALRT, Al Mohamad et al. 2020), which is a sum based test for normal test statistics, and +the adaptive Simes (Bogomolov, 2021) combination test statistic. We pause to motivate +and introduce adaptive Simes, which is superior to Simes for the bounds as well as for +identification in our simulations. The Simes combining method is essentially based on a +single quantile, and thus does not take full advantage of the pooled evidence when the +number of positive (or negative) parameters is large. +Bogomolov (2021) suggested the +adaptive Simes combination test, which uses the following combination function: +fadaptive Simes(x(1), . . . , x(d)) = min +1≤i≤d +� +2(�d +i=1 1(xi > 0.5) + 1) +i +x(i) +� +, +where in our setting x(1) ≤ . . . ≤ x(d) are the ordered conditional p-values within the selected +set for the conditional approach. The adaptive Simes p-value may be much smaller than +Simes, except in the case where the intersection is only of two hypotheses. Therefore, it +makes sense to switch to the Simes combining method in that case. So we shall denote by +adaptive Simes the following combining function to be used in Procedure 3.1: +f(x(1), . . . , x(d)) = +� +fadaptive Simes(x(1), . . . , x(d)) +if d > 2, +min1≤i≤d +�d +i x(i) +� +otherwise. +We compare our procedures to the FWER and FDR controlling procedures in Guo +and Romano (Guo and Romano, 2015). +These procedures aim to identify the positive +and nonpositive parameters. The FWER controlling procedure is a good competitor for +identification, but we expect it to do poorly for the bounds. The FDR controlling procedure +is less relevant for identification, since it targets a more lenient error guarantee (i.e., the +FDR), whereas all other methods control the FWER. However, we included it since it +is interesting that for the bounds it is not necessarily better, even though there is no +confidence guarantee. +Our data generation is as follows. For a total of n = 50 parameters, there are n+ ∈ +{0, . . . , n/2} positive parameters and n+ negative parameters (we exclude simulations with +a different ratio of the number of positive and negative parameters, since the qualitative +conclusions remain unchanged). The remaining parameters are zero. The test statistics +are generated independently from a Gaussian distribution with standard deviation one +and mean centered at the parameter value, ˆθi ∼ N(θi, 1). The right sided p-values are +pi = 1 − Φ(ˆθi), i = 1, . . . , n. Our analysis is based on B = 2000 data generations, and +α = 0.05. +22 + +Figure 5 provides the average length of the bounds and the number of parameters +discovered for the settings considered. Our key findings are the following. First, adaptive +partitioning provides much tighter bounds than directional closest testing, with barely any +difference in the number of discoveries. +Second, the sum-based tests are better for the bounds than the quantile based tests +and than the FWER controlling procedure in Guo and Romano (2015). +They are also +much better than the FDR controlling procedure in Guo and Romano (2015) in the data +generation with fairly weak signal (in row 1 of Figure 5). +Third, the adaptive Simes method dominates Simes in the adaptive partitioning proce- +dure, and is competitive with the FWER controlling procedure in Guo and Romano (2015) +for identification. It (as well as Simes) is also much better than the sum-based tests for +identification. From the simulations, we can support the following general guidance: adap- +tive Simes is a good choice if identification is important in addition to the bounds, or if it +is expected that the fraction of parameters far enough from zero is small. Otherwise it is +best to use a sum based combining method, such as Fisher or the ALRT. +23 + +0 +10 +20 +25 +30 +35 +40 +45 +50 +n+ +Average length u+ − l+ +0 +10 +20 +25 +0 +5 +10 +15 +20 +n+ +Average number of hypotheses rejected +0 +10 +20 +25 +30 +35 +40 +45 +50 +n+ +Average length u+ − l+ +0 +10 +20 +25 +0 +5 +10 +15 +20 +n+ +Average number of hypotheses rejected +Figure 5: The average length (left column) and number of individual hypotheses rejected +(right column), versus n+, for the following methods: the partitioning algorithm with +combining functions Fisher (black circles), ALRT (black square), Simes (black triangle), +adaptive Simes (black diamond); the CT procedure 2.1 with combining functions Fisher +(blue circles),and Simes (blue triangle); the Guo-Romano procedure for FWER control (red +x’s) and for FDR control (red pluses). The rows differ by the θ configuration: θi = 1.5, i = +1, . . . , n+, θi = −1.5, i = n+ + 1, . . . , 2n+, θi = 0, i = 2n+ + 1, . . . , n in row 1; the nonzero +θi’s are generated independently from the absolute value of ∼ N(0, 2) in row 2. In the right +column, the absence of black circles and triangles is due to the fact that they coincide with +their blue counterparts. +24 + +5 +Applications +5.1 +Enhancing meta-analysis +The first step in a meta-analysis is often the computation of a global null p-value, which +communicates the strength of the evidence against the null hypothesis of no association +in any of the studies. Rejecting the global null does not rule out the possibility that this +is due to a non-zero association only in a single study (e.g., due to the particulars of the +cohort in that study, or due to bias). Therefore, it is of interest provide tight lower and +upper bounds on the number of studies with, e.g., a positive effect. A lower bound greater +than one is related to the replicability goal of establishing that the effect is present in at +least r (r ≥ 2) of n studies (see Bogomolov and Heller 2022 and references within). An +upper bound smaller than n conveys the limit on replicability or the specificity of the +association to a subset of studies. Moreover, for non-trivial bounds it is natural also to +try to identify the studies where there is an association. Directional closed testing should +be used for this purpose if inference on positive and negative associations are desired. +Adaptive partitioning should be used if it is enough to infer on n+, and to identify positive +and non-positive associations (e.g., when studying an intervention effect, identify studies +where the intervention is beneficial and studies where the intervention has no effect or may +be harmful). Since in a typical meta-analysis the studies are independent and the test +statistic in each study is approximately normal, the required conditions (A0) − (A2) are +satisfied. +Fore a concrete example, we consider the data of Cooper et al. (2003), made public by +Konstantopoulos (2011), where the effect of a modified school calendar (with more frequent +but shorter breaks) on student achievement is examined. The meta-analysis uses studies +of n = 56 schools in 11 districts. +The experimental design is a two-group comparison +(modified calendar vs traditional calendar) which involves computing the standardized +mean difference Xi ∼ N(θi, 1), where θi > 0 indicates a positive effects, i.e. a higher level +of achievement in the group following the modified school calendar. Zhao et al. (2019) +showed that there is evidence of a qualitative interaction at the 5% level, i.e., that the +effect of the modified school calendar on student achievement is positive in at least one +study, and negative in at least one study. +We re-analyze the data by providing the lower and upper bounds on the number of +studies with positive/negative effect in all schools (Table 2), and in the (data-driven) top +k schools (Figure 6). +Directional closed testing makes the same number of individual +discoveries as adaptive partitioning, but the upper bound on n+ is the trivial bound of +25 + +56. With adaptive partitioning the bounds are much more informative: using Fisher’s +combining method, the 95% confidence bound for n+ is 10 to 53, and we have four individual +discoveries; using Adaptive Simes, the 95% confidence bound for n+ is 8 to 54, and we +have eight individual discoveries. Using the procedure of Guo and Romano (2015) which +is targeted for dFWER control, the 95% confidence bound for n+ is 8 to 55, with nine +individual discoveries. +Figure 6 shows simultaneous 95% confidence intervals for the top k schools: for example, +within the top 38 schools, we have at least 9 positive effects and at least 1 non-positive +effect, or within the top 51 schools, we have at least 10 positive effects and at least 2 +non-positive effect, etc. +n+ +n− +Local test +Procedure +l+ +u+ +|D+| +l− +u− +|D−| +Fisher +DCT +9 +56 +4 +0 +47 +0 +AP +10 +53 +4 +3 +46 +0 +Simes +DCT +8 +56 +8 +0 +48 +0 +AP +8 +55 +8 +1 +48 +0 +Adaptive Simes +AP +8 +54 +8 +2 +48 +0 +ALRT +AP +11 +53 +5 +3 +45 +0 +Guo and Romano (2015) +8 +55 +8 +1 +48 +1 +Table 2: The effect of modified school calendars on student achievement: intervals for n+ +and n− and corresponding number of discoveries |D+| and |D−|, for different combining +methods and procedures, with 95% confidence. DCT and AP are directional closed testing +and adaptive partitioning procedures 2.1 and 3.1, respectively. +n− is the number of negative parameters for DCT and non-positive parameters for AP +and Guo and Romano (2015); |D−| is the number of discoveries of negative parameters for +DCT and non-positive parameters for AP and Guo and Romano (2015). +26 + +1 +56 +0 +56 +k +95% confidence interval for n+(Ik) +Figure 6: Simultaneous 95% confidence intervals for n+(Ik) with Fisher’s combining func- +tion, where Ik are the indexes of the top k = |Ik| schools with largest (in absolute value) +effects. For example, within the top 38 schools, we have at least 9 positive effects and at +least 1 non-positive effect (red interval); or within the top 51 schools, we have at least 10 +positive effects and at least 2 non-positive effect (blue interval). +5.2 +Testing for qualitative interactions with follow-up inference +If testing for mixed signs of the parameter vector θ is of primary concern, the approach +described in § 5.1 can be adjusted so that directional closed is applied only if the hypothesis +of no qualitative interaction (i.e., no positive and negative parameters in θ) is rejected. Sev- +eral tests have been developed in the literature (Gail and Simon, 1985; Zelterman, 1990; +Piantadosi and Gail, 1993; Ciminera et al., 1993; Pan and Wolfe, 1997; Silvapulle, 2001; +Li and Chan, 2006; Zhao et al., 2019; Hudson and Shojaie, 2020) for testing the null hy- +27 + +pothesis of no qualitative interaction: +H0 : +� n� +i=1 +H− +i +� +∪ +� n� +i=1 +H+ +i +� +. +Gail and Simon (1985) analyzed the data discussed in Section 1.1 to see whether they could +demonstrate a qualitative interaction at the α = 5% level. However, the likelihood ratio +test proposed by Gail and Simon (1985) results in a p-value of 0.0877. +Zhao et al. (2019), when their selection threshold τ = 1/2, uses the p-value of HS− to +test �n +i=1 H− +i and the p-value of HS+ to test �n +i=1 H+ +i , which are valid tests since HS− ⊇ +�n +i=1 H− +i and HS+ ⊇ �n +i=1 H+ +i . So the p-value for H0 is the larger of p1/n = f({2pi, i ∈ S−}) +(the p-value for testing �n +i=1 H− +i ) and q1/n = f({2qi, i ∈ S+}) (the p-value for testing +�n +i=1 H+ +i ). +We see from Figure 1 that this approach gives a p-value of max(0.0386, 0.0115) = 0.0386 +with the adaptive Simes combination test. (With Fisher, Simes and ALRT combination +tests we obtain the same result: p1/4 = 2p1 = 0.0386 and q1/4 = f(2q2, 2q3, 2q4) = 0.0110, +0.0173 and 0.0120 respectively). +We propose the following closed testing procedure tailored for qualitative interactions: +start with Zhao et al. (2019) global test for H0 and, if rejected, continue with the directional +closed testing procedure. +Procedure 5.1 (Closed testing for qualitative interactions). +Step 1 Apply Step 1 of procedure 2.1 +Step 2 Apply a level α closed testing procedure on the family of hypotheses ˆH. Testing HI +at level α is done by the local test +φI = +� +1{max{f({2pi, i ∈ S−}), f({2qi, i ∈ S+})} ≤ α} +if I ⊇ S− or I ⊇ S+ +1{f({2pi, i ∈ I ∩ S−} ∪ {2qj, j ∈ I ∩ S+}) ≤ α} +otherwise +From Figure 1 we see that at level 5% procedure 5.1 rejects all hypotheses but H+ +2 ∩H+ +3 , +H+ +2 and H+ +3 , giving 1 ≤ n+ ≤ 3, 1 ≤ n− ≤ 3 and θ1 > 0, θ4 < 0. This result improves +the one obtained by the directional closed testing procedure 2.1 (1 ≤ n− ≤ 4 and θ4 < 0). +However, with the closed testing for qualitative interactions procedure 5.1, rejecting H0 is +crucial: if H0 is not rejected, the closed testing procedure 5.1 does not reject any intersection +hypothesis. For instance, at α = 2.5%, procedure 2.1 gives the same conclusions as at +α = 5%, while procedure 5.1 is uninformative since no rejections are made. On the other +hand, if H0 is rejected, then the lower bounds for n+ and n− are both informative. +28 + +6 +Concluding remarks +In Procedure 2.1, the selection Step 1 has a cost of inflating the p-values by a factor of 2. +It is a reasonable cost, since it is no worse than the cost of 2-sided testing. It is possible +to consider more generally the selection of {H+ +i : pi ≤ τ} ∪ {H− +i : pi > τ} for a τ ∈ (0, 1). +Such a selection step will inflate the right-sided p-value by 1/τ, and the left-sided p-value +by 1/(1 − τ). It may be useful when it is more important to detect one direction over +another, and the inferential tools can easily be adjusted for independent p-values. +Each parameter may be accompanied by its own predefined selection threshold τi, i = +1, . . . , n. In particular, if it is a-priori known for a parameter that only the right-sided +alternative is of interest, set τi = 1. Similarly, set τi = 0 if it is a-priori known for the +parameter that only the left-sided alternative is of interest. +The selection step is thus +more general, and the following step of directional closed testing or adaptive procedures is +carried as described, using the conditional p-values {pi/τi : pi ≤ τi}∪{qi/(1−τi) : pi > τi}. +Importantly, the same computational shortcuts are carried out on these conditional p-values +for the desired inference. +An open problem is how to deal with dependent p-values. Adjusting for the selection +event S by conditioning may incur an inflation factor much greater than 2 (or more gen- +erally 1/τ). For example, using the polyhedral lemma and the data carving methods of +Fithian et al. (2017). Currently, it is unclear to us whether using conditional p-values is +useful in settings where the p-values are not independent. +The conditional (1 − α) confidence bounds for n+ and n+(I) have an unconditional +1−α(1−2−n) confidence guarantee. Therefore, adaptive partitioning can be carried out at +level α∗ = α/(1 − 2−n) if the conditional guarantee is not necessary. Using α∗ falls within +the framework of the holistic approach of Goeman and Solari (2022), that view selection +and conditioning as means for useful inferences with unconditional error guarantees. Inter- +estingly, our directional closed testing procedure provides a conditional error control that +can be equal to the unconditional control if θi = 0. It is interesting to consider in our +setting what can be gained from having a conditional versus an unconditional guarantee, +and what are the implications when the conditional and unconditional guarantee coincide. +As mentioned in Remark 3.2, if only n+ is of interest, (1 −α) confidence bounds can be +achieved with the simpler and (slightly) more powerful procedure than adaptive partition- +ing described in Appendix C. The procedure is in line with all other procedures suggested +in this paper, in that the local tests are conditional on the vector of signs S. In Jaljuli et al. +(2022); Solari and Goeman (2021) suggested combining α/2 one-sided bounds, that were +computed with more general local tests of intersection hypotheses. For most data gen- +29 + +erations encountered in practice, we argue in Appendix C.1, that combining α one-sided +bounds is enough (this is provably so for the conditional procedure, see Proposition C.1). +In our numerical experiments (omitted for brevity), we see that the tightest bounds are +provided with our conditional approach if there are several positive and several negative +parameters, but with the unconditional approach if all parameters are non-positive or non- +negative. Deriving a data driven approach that adapts to the choice of conditional versus +unconditional bounds for n+ is left for future work. +This work concentrated on simultaneous inference. For large scale testing, however, a +popular error for identification is the false discovery rate (FDR). Overall bounds can be +calculated from the FDR controlling procedure by counting the number of discoveries in +each direction. Of course, there is no (1 − α) confidence guarantee on the overall bounds, +but for large problems knowing that it is based on an FDR guarantee may be enough for +some purposes. However, it is interesting to note that the bounds can be smaller than +with our proposed approach, which provides the desired coverage guarantee. Specifically, +we considered the FDR controlling procedure in Guo and Romano (2015). They showed +that for independent p-values (if U(0, 1) ≤rh pi for null p-values, where ≤rh is the reverse +hazard rate order), when considering the family {H− +i , Ki : i = 1, . . . , n}, it is enough to +apply the Benjamini Hochberg procedure (Benjamini and Hochberg, 1995) on the set of +smallest one-sided p-values. 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Since the p-values are independent (assumption (A2)), it follows that +Pθ(φT = 1 | S) = P{θi:i∈T }(φT = 1 | {sign(pi − 1/2) : i ∈ T }). +Since φT = 1{f({2pi, i ∈ T −}∩{2qi, i ∈ T +}) ≤ α} is a valid local test, i.e., the probability +of rejection is at most α if the p-values in the intersection test are all valid, it follows from +the independence across p-values (assumption (A2)) that the right hand side is at most α +if each conditional p-value is (marginally) valid. The validity of each conditional p-values +is guaranteed by assumption (A1), thus concluding the proof. +A.2 +Proof of Proposition 3.1 +Proof. Suppose that the true θ lies in ΘK for some K ⊆ [n]. We have ψ|K∩I|(I) ≤ ψK for +every I. Therefore, if ψK = 0, then |K ∩ I| = n+(I) ∈ N +(I)α for all I, thus +Pθ(l+ +α (I) ≤ n+(I) ≤ u+ +α(I) ∀I) ≥ Pθ(n+(I) ∈ N +(I)α ∀I) ≥ Pθ(ψK = 0) ≥ 1 − α. +A.3 +Proof of Proposition 3.2 +Proof. Proposition D.2 shows that Algorithm 3.1 returns the bounds l+ +α (I) and u+ +α(I) +defined in (12) with at most O(|I|2 · max(1, |Ic|2)) computation. +Suppose the true θ ∈ ΘK for some K ⊆ [n], and let ˜T = T −∪ ˜T + = {Kc∩S−}∪{K∩S+}. +The partitioning Procedure 3.1 tests JK by using ψK = φ ˜T , and +sup +θ∈ΘK +Pθ(φK = 1 | S) ≤ sup +θ∈H ˜ +T +Pθ(φ ˜T = 1 | S) ≤ α +35 + +where the first inequality follows from JK ⊆ H ˜T and the second inequality follows from +Proposition 2.1. Furthermore, +Pθ( ˜T = ∅) = Pθ(S− = K) = +� +i∈K +Pθ(pi ≤ 0.5) +� +j /∈K +Pθ(pj > 0.5) ≥ P0(S− = K) ≥ 2−n, +thus +Pθ(ψK = 1) = +� +S +Pθ(φ ˜T = 1|S)Pθ(S) ≤ α(1 − Pθ( ˜T = ∅)) ≤ α(1 − 2−n). +The previous results together with Proposition 3.1 imply the conditional and unconditional +coverage (12) and (14). See also § 1.2 of the Supplementary Material to Al Mohamad et al. +(2020) for the unconditional type I error control with the adaptive likelihood ratio test. +Now we want to prove that l+ +DCT(I) ≤ l+ +AP(I) for every I. We proceed by contradiction: +assume that ∃ I such that l+ +AP(I) < l+ +DCT(I). By Lemma 1 in Goeman et al. (2021): +l+ +DCT(I) = min +V⊆[n](|(I ∩ S−) \ V| : φV = 0). +(16) +Because the null hypothesis that n+(I) is equal to l+ +AP(I) is not rejected at level α, +then ∃ K such that |K ∩ I| = l+ +AP(I) for which JK is not rejected at level α. Equivalently, +∃ U = (Kc ∩ S−) ∪ (K ∩ S+) such that φU = 0 and +|(I ∩ S−) \ U| = |I ∩ S− ∩ K| ≤ l+ +AP(I) < l+ +DCT(I). +On the other hand, from equation (16) it follows that l+ +DCT(I) ≤ |(I∩S−)\U| contradicting +the assumption that l+ +AP(I) < l+ +DCT(I). Finally, u+ +AP(I) ≤ u+ +DCT(I) for every I can be +proved in analogous way. +B +Inference for n = 2 with Simes combining function +The top-left plot of Figure 7 displays inference about n+ with (1−α) confidence for the di- +rectional closed testing procedure with Simes combining function. For n = 2, the directional +closed testing procedure with Simes’ local test is a consonant procedure (Romano et al., +2011), i.e. the rejection of the intersection hypothesis implies the rejection of at least one of +its component hypotheses. We see that each area leading to inference about n+ gives also +the rejection of one or two individual hypotheses (hypotheses are represented by indices +and directions, e.g. 1−, 2+ corresponds to the hypotheses H− +1 and H+ +2 .). +36 + +Bauer et al. (1986) proposed a modified Bonferroni procedure for testing the hypotheses +pairs (H− +i , Ki), i = 1, . . . , n, comparing each one sided p-value (pi, qi) to α/n instead of +α/2n. +Bauer et al.’s modified Bonferroni procedure requires condition (A0) only. +In a +similar fashion, if p1, . . . , pn fulfil conditions (A0) and (A2), we can consider a modified +ˇSid´ak procedure comparing (pi, qi) to 1 − (1 − α)1/n instead of 1 − (1 − α)1/(2n). +The bottom-left plot of Figure 7 shows Bauer et al.’s modified ˇSid´ak procedure for +n = 2. We see that for inference about n+, Bauer et al.’s modified ˇSid´ak procedure is +uniformly more powerful than directional closed testing. However, directional closed testing +provides also inference about n−, whereas Bauer et al.’s procedure does not. +The top-right plot of Figure 7 shows the adaptive partitioning procedure with Simes test +performed at the larger level α∗ = 4α/3 providing level α unconditional error control. Note +that the adaptive partitioning procedure performed at level α is not admissible because is +dominated by Bauer et al. modified ˇSid´ak procedure. +The bottom-right plot of Figure 7 shows Guo and Romano (2015) Holm-type proce- +dure (Procedure 3 of their paper), which requires assumptions similar to (A0) − (A2). +Guo and Romano (2015) proposed another procedure (Procedure 2 of their paper) that +is uniformly more powerful than Bauer et al.’s modified Bonferroni procedure, altough it +does not dominate Bauer et al.’s modified ˇSid´ak procedure. +37 + +Directional Closed Testing +α/2 +α/4 +p1 +p2 +1−, 2− +1−, 2+ +1+, 2+ +1+, 2− +1− +1+ +2− +2+ +Unconditional Adaptive Partitioning +2α/3 +α/3 +p1 +p2 +Bauer et al. (1986) +1 − (1 − α)1/2 +p1 +p2 +Guo and Romano (2015) +α/(1 + α) +α/(2 + α) +p1 +p2 +n+ = 2 +n+ ≥ 1 +n+ = 1 +n+ ≤ 1 +n+ = 0 +Figure 7: +Areas leading to inference about n+ with (1 − α) confidence for the di- +rectional closed testing procedure with Simes combining function (top-left plot), the +unconditional adaptive partitioning procedure with Simes combining function (top- +right plot), Bauer et al. (1986) procedure with ˇSid´ak correction (bottom-left plot), and +Guo and Romano (2015) Holm-type procedure (bottom-right plot). Each plot is based on +α = 0.2. +38 + +C +Adaptive confidence bounds for n+ only +Let Hr/n : n+ ≤ r − 1, be the partial conjunction (PC) null hypothesis that at most r − 1 +hypotheses among H− +1 , . . . , H− +n are false; and Kr/n : n+ ≥ n−r+1, the PC null hypothesis +that at most r − 1 hypotheses among K1, . . . , Kn are false. For r = 1, Hr/n is the global +null hypothesis that none of the parameters are positive. For r = 2, rejection of Hr/n +leads to establishing minimal replicability in the positive direction (Benjamini et al., 2009; +Jaljuli et al., 2022). +Since Hr/n is false if and only if every intersection hypothesis of size n − r + 1 is false +(Benjamini et al., 2009), a valid p-value pr/n for Hr/n is the largest intersection hypothesis +p-value, over all intersections of n − r + 1 null hypotheses: +pr/n = +max +{I:I⊆[n],|I|=n−r+1}pI, +where pI is the p-value for the intersection hypothesis ∩i∈IH− +i . Similarly, a valid p-value +qr/n for Kr/n is +qr/n = +max +{I:I⊆[n],|I|=n−r+1}qI, +where qI is the p-value for the intersection hypothesis ∩i∈IKi. +For example, the PC p-values using Fisher’s combining method (Fisher, 1934) are: +pr/n = P +� +χ2 +2(n−r+1) ≥ −2 +n +� +k=r +log(p(k)) +� +, +qr/n = P +� +χ2 +2(n−r+1) ≥ −2 +n +� +k=r +log(q(k)) +� +where p(1) ≤ . . . ≤ p(n) and q(1) ≤ . . . ≤ q(n) denote the sorted values of p1, . . . , pn and +q1, . . . , qn, respectively. Note that 1 − p(k) = q(n−k+1) for continuous test statistics. +For f that satisfies the monotonicity and symmetry conditions (8) and (9), pr/n = +f(p(r), . . . , p(n)) is a valid p-value, satisfying +sup +θ∈Hr/n Pθ(pr/n ≤ x) ≤ x +∀x ∈ [0, 1]. +The inequality is an equality, i.e., pr/n is uniformly distributed, for the least favorable +parameter configuration (LFC) θLF C ∈ Hr/n, for which r − 1 parameters have an infinite +value and their corresponding p-values are zero (almost surely), and n − r + 1 parameters +are zero and their corresponding p-values are uniformly distributed. By considering only +hypotheses in S, we can avoid including p-values that are stochastically much larger than +39 + +uniform when their null hypotheses are true. Therefore, as in § 2, we shall restrict ourselves +to the directions guided by the data, so we shall use for testing Hr/n +pr/n = +max +{I:I⊆[n],|I|=n−r+1}f({2pi : i ∈ I ∩ S−}) = +� +f +� +2p(r), . . . , 2p(|S−|) +� +if r ≤ |S−|, +1 +otherwise. +(17) +and for testing Kr/n +qr/n = +max +{I:I⊆[n],|I|=n−r+1}f({2qi : i ∈ I ∩ S+}) = +� +f +� +2q(r), . . . , 2q(|S+|) +� +if r ≤ |S+|, +1 +otherwise. +(18) +Procedure C.1 (Adaptive PC testing). +Step 1 Apply Step 1 of procedure 2.1. +Step 2 Test in order {Hr/n : r = 1, . . . , |S−|}, using pr/n in (17), at level α. Stop at the first +non-rejection, pr/n > α. Let l+ +α be the number of rejections, with l+ +α ∈ {0, . . . , |S−|}. +If l+ +α = n, return l+ +α = u+ +α = n, otherwise go to the next step. +Step 3 Test in order {Kr/n : r = 1, . . . , n − |S−|}, using qr/n in (18), at level α. +Stop +at the first non-rejection, qr/n > α. Let n − u+ +α be the number of rejections, with +n − u+ +α ∈ {0, . . . , n − |S−|}. Return l+ +α and u+ +α (with l+ +α ≤ u+ +α). +Note that the bounds of the procedure will be the same if the testing in order is continued +until n in each step. This is so because p(|S−|+1)/n > α and q(n−|S−|+1)/n > α. +The guaranteed coverage is formalized in the following proposition. +Proposition C.1. Let +� +pr/n : r ∈ [n] +� +be valid conditional PC p-values for {Hr/n : r ∈ +[n]}, and let +� +qr/n : r ∈ [n] +� +be valid conditional PC p-values for {Kr/n : r ∈ [n]}. Then +l+ +α and u+ +α satisfy (15). Furthermore, the unconditional coverage is +Pθ +� +l+ +α ≤ n+ ≤ u+ +α +� +≥ 1 − (1 − 2−n)α. +(19) +Proof. Suppose that θ is the true parameter value with n+(θ) = t. We have l+ +α ≤ t ≤ u+ +α +if and only if p(t+1)/n > α and q(n−t+1)/n > α. The result follows since conditional on +S−, it is only possible to make an error in one direction. Specifically, if t > |S−|, then +40 + +p(|S−|+1)/n > α, since it is not possible to reject H(|S−|+1)/n, so it is not possible to err with +regard to the lower bound. Therefore, if t > |S−|, then +Pθ(t /∈ [l+ +α (p), u+ +α(p)] | S) = Pθ(t > u+ +α(p) | S) ≤ Pθ(q(n−t+1)/n ≤ α | S) ≤ α. +Similarly, if t < |S−|, then q(n−|S−|+1)/n > α, since it is not possible to reject K(n−|S−|+1)/n, +so it is not possible to err with regard to the upper bound. Therefore, if t < |S−|, then +Pθ(t /∈ [l+ +α (p), u+ +α(p)] | S) = Pθ(t < l+ +α (p) | S) ≤ Pθ(p(t+1)/n ≤ α | S) ≤ α. +If t = |S−|, then since the lower bound is at most |S−| and the upper bound is at least +n − |S+|, it is not possible to make an error on either bound. Therefore, the unconditional +error of non-covering n+ is +Pθ(t /∈ [l+ +α (p), u+ +α(p)]) = E +� +Pθ(t /∈ [l+ +α (p), u+ +α(p)] | S) +� += Eθ +� +I(t > |S−|)Pθ(t > u+ +α(p) | S−) + I(t < |S−|)Pθ(t < l+ +α (p) | S−) +� +≤ αEθ +� +I(t > |S−|) + I(t < |S−|) +� += αPθ(t ̸= |S−|) ≤ α(1 − 2−n), +where the last inequality follows since Pθ(t = |S−|) is bounded below by +� +{i:θi>0} +Pθ(pi ≤ 0.5) +� +{j:θj≤0} +Pθ(pj > 0.5) ≥ +� +{i:θi>0} +P0(pi ≤ 0.5) +� +{j:θj≤0} +P0(pj > 0.5) = 2−n. +Remark C.1. The number of nonpositive parameters, n − n+, can be decomposed into the +number of negative parameters n− and the number of parameters with value zero n0, so +n = n+ +n− +n0. The 1-α confidence bound on the number of positive parameters, [l+ +α , u+ +α] +has also the following interpretation if n0 = 0: with (1 − α) confidence, the number of +positive parameters is at least l+ +α and the number of negative parameters is at least n − u+ +α. +However, if n0 > 0 then the probability that the lower bounds do not cover at least one of +n+, n− may exceed α. If n0 = n and all p values are uniform then P(p1/n ≤ α ∪ q1/n ≤ +α) ≈ 2α; as n0 decreases the probability that the lower bounds do not cover at least one +parameter decreases from 2α to α (for n0 = 0). +Remark C.2. For a combination function f(), the test for qualitative interactions in +Zhao et al. (2019) is rejected at level α if and only if l+ +α ≥ 1 and u+ +α ≤ n − 1 in the above +procedure. Therefore, if the assumption n0 = 0 is reasonable, then the above procedure +41 + +complements nicely a conclusion that there is qualitative interaction, by providing with +(1 − α) confidence the (interval) estimate of the parameter tested, n+. More generally, a +level α test of a generalized qualitative interaction null hypothesis that n+ < a or n+ > b, +for predefined 1 ≤ a < b ≤ n − 1, has the following rejection rule: reject if l+ +α ≥ a and +u+ +α ≤ b. To see that this is an α level test, consider the null value θ such that n+(θ) /∈ [a, b]. +Without loss of generality, suppose n+(θ) > b. Then the probability of falsely rejecting the +generalized qualitative interaction true null hypothesis is +Pθ(l+ +α ≥ a and u+ +α ≤ b) ≤ Pθ(u+ +α ≤ b) ≤ Pθ(u+ +α < n+) ≤ α. +C.1 +A note on general confidence bounds for n+ +If pr/n is a valid p-value for testing Hr/n for r = 1, . . . , n, then +lα(p) += +max{l ∈ {0, . . . , n} : pr/n ≤ α for r = 0, . . . , l} +(20) +satisfies Pθ +� +lα(p) ≤ n+� +≥ 1−α for all θ ∈ Θ, where p0/n ≡ 0 since H0/n : n+ < 0 is always +false. Analogously, if qr/n is a valid p-value for testing Kr/n for r = 1, . . . , n, then +uα(p) += +n − max{u ∈ {0, . . . , n} : qr/n ≤ α for r = 0, . . . , u} +(21) +satisfies Pθ +� +n+ ≤ uα(p) +� +≥ 1 − α for all θ ∈ Θ, where q0/n ≡ 0 since K0/n : n+ > n is +always false. For notational simplicity, we shall often write lα and uα instead of lα(p) and +uα(p), but of course these bounds are functions of the p-value vector p. +A straightforward application of the Bonferroni inequality shows that if level α/2 is +used for each bound, i.e., lα/2 in (20) and uα/2 in (21), then Pθ +� +lα/2 ≤ n+ ≤ uα/2 +� +≥ 1 − α. +These bounds where used in Jaljuli et al. (2022) in order to complement meta-analyses in +systematic reviews. +The correction of using α/2 in each direction (instead of α, as in the adaptive PC testing +procedure ) is, however, conservative. Intuitively, the correction should be less severe since +in a given configuration, the probability of erring by exceeding one bound is much larger +than the probability of erring by exceeding the other bound. To see this, note that +Pθ(n+ /∈ [lα/2, uα/2] = Pθ(n+ < lα/2) + Pθ(n+ > uα/2) +has value α/2 for the following least favorable parameter configurations (LFCs) when test- +ing PC null hypotheses: the positive parameter value is θi = ∞ and the non-positive +parameter value is θi = 0 in θ; or the positive parameter value is θi = 0+ (where 0+ is +42 + +an arbitrary fixed small positive value, e.g., the machine precision) and the non-positive +parameter value is θi = −∞ in θ. To see this, note that for the LFC with θi = ∞ for +n+ parameters, p(n++1)/n ∼ U(0, 1) and q(n−n++1)/n = 1 almost surely. For the LFC with +θi = −∞ for n − n+ parameters, q(n−n++1)/n ∼ U(0, 1) and p(n++1)/n = 1 almost surely. +Non-coverage can occur if the lower bound is violated, so p(n++1)/n ≤ α/2, or if the upper +bound is violated, so q(n−n++1)/n ≤ α/2. Therefore +PLF C(n+ /∈ [lα/2, uα/2]) = P(U ≤ α/2) = α/2. +We conjecture that the coverage guarantee is typically (1 − α) if the lower and upper +confidence bounds are lα and uα. In particular, whenever the test statistics are continuous, +from one dimensional exponential families. Without loss of generality, suppose the first +t coordinates are positive, i.e., θ1, . . . , θt > 0 and θt+1, . . . , θn ≤ 0, and n+ = t. +Our +conjecture is thus that the solution to the following optimization problem is α for a large +class of valid PC p-values: +max +θ +Pθ(p(t+1)/n ≤ α) + Pθ(q(n−t+1)/n ≤ α) +s.t. +θi > 0, i = 1, . . . , t, +θj ≤ 0, j = t + 1, . . . , n. +To see that this is not true in general for unconditional combination tests (i.e., using local +tests that do not condition on the vector of signs S), consider the following stylized example. +Let n = 2 and n+ = 1 be such that θ1 = 0 and θ2 is positive. Assume that the distribution +of a p-value from Hi, xi, has the following distribution: P(xi = α) = α, P(xi = 1) = 1 − α. +Similarly, the distribution of a p-value from Ki, yi = 1 − xi, has distribution: P(yi = α) = +α, P(yi = 1) = 1 − α. These p-values are valid since +PHi(pi ≤ a) ≤ a, PKi(qi ≤ a) ≤ a, ∀a ∈ [0, 1]. +The unconditional PC p-value (i.e., it is derived from a local test that does not condition +on the vector of signs S) for H2/2 and K2/2 is, respectively, p2/2 = max(p1, p2) which has +distribution max(1−x1, x2), and q2/2 = max(q1, q2) which has distribution max(x1, 1−x2). +Therefore: +Pθ(1 /∈ [lα, uα] = Pθ(1 < lα) + Pθ(1 > uα) += Pθ(max(p1/2, p2/2) < α) + Pθ(max(q1/2, q2/2) < α) += P((1 − x1, x2) = (0, α)) + P((x1, 1 − x2) = (α, 0)) = 2 × α × (1 − α) > α. +43 + +D +Exact shortcuts for Procedure 3.1 +We first discuss the computation of the confidence bounds l+ +α and u+ +α for n+. Suppose we +have observed S− of size |S−| = s, and we want to check whether the test in (13) rejects +all the hypotheses JK with |K| = k at level α: +fk = max +K:|K|=k f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) +≤ +α. +Vandermonde’s convolution +�n +k +� += +k +� +v=0 +� +s +k − v +��n − s +v +� +states that for each v = 0, . . . , k there are +� s +k−v +��n−s +v +� +sets K of size k such that |S+ ∩K| = v +and |S− ∩ K| = k − v (or equivalently, |S− ∩ Kc| = s − k + v). Then, the maximization +problem becomes +fk = +max +v∈{max(0,k−s),...,min(k,n−s)} +max +K:|S+∩K|=v, +|S−∩K|=k−v +f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) +For any increasing function f, the maximum has solution with the v largest p-values qi +with i ∈ S+ and the s − k + v largest p-values pi with i ∈ S−, i.e. +fk = +max +v∈{max(0,k−s),...,min(k,n−s)} f({2p(k−v+1), . . . , 2p(s)} ∪ {2q(n−s−v+1), . . . , 2q(n−s)}). +(22) +In order to compute fk in (22) for k = 0, . . . , n, we can use a nested loop, where the number +of iterations of the inner loop (i.e. the index v in (22) from max(0, k − s) to min(k, n − s)) +depends on the value of the outer loop’s index (i.e. k from 0 to n) and the size s of S−. The +total complexity for the two loops is O(n2). The maximum number of iterations happens +when s ∈ {n/2 − 1, n/2, n/2 + 1} if n is even, and s ∈ {(n − 1)/2, (n + 1)/2} if n is odd; +the mininum number of iterations happens when s ∈ {0, n}. Thus we proved the first part +of the following proposition. +Proposition D.1. Algorithm 2 returns the (1 − α) confidence bounds l+ +α and u+ +α based on +the test in (13), with at most O(n2) computation. The bounds are at most as good as those +obtained by the Procedure C.1. +44 + +Proof. For prove the second part of the proposition, it is sufficient to note that if k < s, +then the index v in (22) starts at max(0, k − s) = 0 by computing the PC conditional +p-value pk+1/n = f({2p(k+1), . . . , 2p(s)}). Then pk+1/n > α implies +fk = max +K:|K|=k f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) ≥ pk+1/n > α +for any k ∈ {0, . . . , s − 1}, i.e. +the lower bound l+ +α from Algorithm 2 is smaller than +or equal to the lower bound of Procedure C.1. Likewise, if k > s then the index v in +(22) starts at max(0, k − s) = k − s by computing the PC conditional p-value qn−k+1/n = +f({2q(n−k+1), . . . , 2q(n−s)}), thus qn−k+1/n > α implies +fk = max +K:|K|=k f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) ≥ qn−k+1/n > α +for any k ∈ {s + 1, . . . , n}, i.e. the upper bound u+ +α from Algorithm 2 is larger than or +equal to the upper bound of Procedure C.1. +45 + +Algorithm 2: Shortcut for computing the confidence bounds l+ +α and u+ +α +Input +: right-tailed p-values p1, . . . , pn; combining function f(·); level α +Output : Confidence bounds lα and uα, p-values fk for Jk : n+ = k, k = 0, . . . , n +Initialize: +S− = {i : pi ≤ 1/2}, |S−| = s, fs = 1 ; +a = (a1, . . . , as) with a1 ≥ . . . ≥ as sorted values of {2pi : i ∈ S−} ; +b = (b1, . . . , bn−s) with b1 ≥ . . . ≥ bn−s sorted values of {2qi : i ∈ S+} ; +1 if s > 0 then +2 +for k ← 0 to s − 1 do +3 +A ← {1, . . . , s − k}, B ← ∅ ; +4 +for v ← 1 to min(k, n − s) + 1 do +5 +fk,v ← f({ai, i ∈ A} ∪ {bi, i ∈ B}) ; +6 +A ← A ∪ {s − k + v}, B ← B ∪ {v} ; +7 +end +8 +fk = maxv{fk,v} +9 +end +10 else if s < n then +11 +for k ← n to s + 1 do +12 +A ← ∅, B ← {1, . . . , k − s} ; +13 +for v ← 1 to min(k, n − s) − (k − s) + 1 do +14 +fk,v ← f({ai, i ∈ A} ∪ {bi, i ∈ B}) ; +15 +A ← A ∪ {v}, B ← B ∪ {k − s + v} ; +16 +end +17 +fk = maxv{fk,v} +18 +end +19 l+ +α ← min(k ∈ {0, . . . , s} : fk > α) ; +20 u+ +α ← max(k ∈ {s, . . . , n} : fk > α) ; +21 return l+ +α , u+ +α, f0, . . . , fn ; +Algorithm 2 is just a special case of the following Algorithm for the derivation of the +confidence bounds l+ +α (I) and u+ +α(I) for a generic subset I. +Proposition D.2. For any I ⊆ [n], Algorithm 1 returns the simultaneous (1−α) confidence +bounds l+ +α (I) and u+ +α(I) in (12) based on the test in (13), with at most O(|I|2·max(1, |Ic|2)) +computation. In particular, calculation of the adjusted p-values for Hi and Ki requires +O(n2) time. +46 + +Proof. Suppose we have observed S− of size |S−| = s, and for any I ⊆ [n] and any +v ∈ {0, . . . , |I|} we want to check whether the test in (13) rejects all the hypotheses JK +with |K ∩ I| = v at level α (or, equivalently, if Jv(I) : n+(I) = v is rejected at level α): +fv(I) = +max +K:|K∩I|=v f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) ≤ α. +Algorithm 1 computes +fv,u(I) = +max +K:|K∩I|=v, +|K∩Ic|=u +f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) +so that fv(I) = max{fv,u(I), u = 0, . . . , |Ic|}. +The function f(·) in (3) combines 2pi +with i ∈ S− ∩ Kc and 2qi with i ∈ S+ ∩ K. +Writing K = (K ∩ I) ∪ (K ∩ Ic) and +Kc = (Kc ∩ I) ∪ (Kc ∩ Ic) gives +fK(I) += +f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) += +f({2pi, i ∈ S− ∩ Kc ∩ I} ∪ {2pi, i ∈ S− ∩ Kc ∩ Ic} +∪{2qi, i ∈ S+ ∩ K ∩ I} ∪ {2qi, i ∈ S+ ∩ K ∩ Ic}). +Consider Vandermonde’s convolutions: +�|I| +v +� += +v +� +k=0 +�|S− ∩ I| +v − k +��|S+ ∩ I| +k +� +, +�|Ic| +u +� += +u +� +j=0 +�|S− ∩ Ic| +u − j +��|S+ ∩ Ic| +j +� +. +The first convolution states that for each k ∈ {0, . . . , v}, there are +�|S−∩I| +v−k +��|S+∩I| +k +� +sets K +such that |K ∩I| = v with |S+ ∩K ∩I| = k and |S− ∩K ∩I| = v −k. Likewise, the second +convolution states that for each j ∈ {0, . . . , u}, there are +�|S−∩I⌋| +u−j +��|S+∩Ic| +j +� +sets K such that +|K ∩ Ic| = u with |S+ ∩ K ∩ Ic| = j and |S− ∩ K ∩ Ic| = u − j. Then, the maximization +problem becomes +fv,u(I) = +max +k∈{k1,...,k2}, +j∈{j1,...,j2} +max +K:|S+∩K∩I|=k,|S−∩K∩I|=v−k, +|S+∩K∩Ic|=j,|S−∩K∩Ic|=u−j +fK(I) +where k1 = max(0, v − |S− ∩ I|), k2 = min(v, |S+ ∩ I|), j1 = max(0, u − |S− ∩ Ic|) and +j2 = min(u, |S+ ∩ Ic|). +47 + +For any increasing function f(·), the maximum has solution with largest k p-values qi +with i ∈ S+ ∩ I, the largest k − v + |S− ∩ I| p-values pi with i ∈ S− ∩ Ic, the largest j +p-values qi with i ∈ S+ ∩ I and the largest j − u + |S− ∩ Ic| p-values pi with i ∈ S− ∩ I: +fv,u(I) = +max +k∈{k1,...,k2} +j∈{j1,...,j2} +f({a1, . . . , ak−v+|S∩I|}∪{b1, . . . , bj−u+|S∩Ic|}∪{c1, . . . , ck}∪{d1, . . . , dj}) +where a1 ≥ . . . ≥ a|S−∩I|, b1 ≥ . . . ≥ b|S−∩Ic|, c1 ≥ . . . ≥ c|S+∩I| and d1 ≥ . . . ≥ d|S+∩Ic| +denote the sorted values of {2pi : i ∈ S− ∩ I}, {2pi : i ∈ S− ∩ Ic}, {2qi : i ∈ S+ ∩ I} and +{2qi : i ∈ S+ ∩ Ic}, respectively. +Algorithm 1 evaluates fv,u(I) with a nested loop. The outer loop executes min(v, |S+ ∩ +I|)−max(0, v−|S−∩I|) times. Every time the outer loop executes, the inner loop executes +min(v, |S+ ∩ I|) − max(0, u − |S ∩ Ic|) times. As a result, the complexity for evaluating +fv,u(I) is O(|I||Ic|). +The complexity for computing fv(I) for v = 0, . . . , |I| is O(|I|2|Ic|2) because it requires +to compute fv,u(I) for u = 0, . . . , |Ic| and v = 0, . . . , |I|. If I = {i}, it takes O(n2) to +compute the adjusted p-values ¯pi = f0({i}) and ¯qi = f1({i}) for H− +i and Ki, respectively. +48 + diff --git a/ttAzT4oBgHgl3EQfr_2o/content/tmp_files/load_file.txt b/ttAzT4oBgHgl3EQfr_2o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7d64f87d57a9bed77e08bc62d1154a8905503bc --- /dev/null +++ b/ttAzT4oBgHgl3EQfr_2o/content/tmp_files/load_file.txt @@ -0,0 +1,1583 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf,len=1582 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='01653v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='ME] 4 Jan 2023 Simultaneous directional inference Ruth Heller∗ Department of Statistics and Operations Research, Tel-Aviv University ruheller@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='com Aldo Solari Department of Economics, Management and Statistics, University of Milano-Bicocca solari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='aldo@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='com November 2022 Abstract We consider the problem of inference on the signs of n > 1 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Within a simultaneous inference framework, we aim to: identify as many of the signs of the individual parameters as possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' provide confidence bounds on the number of positive (or negative) parameters on subsets of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Our suggestion is as follows: start by using the data to select the direction of the hypothesis test for each parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' then, adjust the one-sided p-values for the selection, and use them for simultaneous inference on the selected n one-sided hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The adjustment is straightforward assuming that the one-sided p-values are conditionally valid and mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Such assumptions are commonly satisfied in a meta-analysis, and we can apply our approach following a test of the global null hypothesis that all parameters are zero, or of the hypothesis of no qualitative interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We consider the use of two multiple testing principles: closed testing and partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The novel procedure based on partitioning is more powerful, but slightly less informative: it only infers on positive and non-positive signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The procedure takes at most a polynomial time, and we show its usefulness on a subgroup analysis of a medical intervention, and on a meta-analysis of an educational intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ∗The authors gratefully acknowledge the Rita Levi-Montalcini prize for Scientific Cooperation between Italy and Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 1 Keywords: conditional inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' directional inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' meta-analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' multiple testing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' partitioning principle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' simultaneous confidence bounds 1 Introduction Let θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , θn) be a vector of n unknown real-valued parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' A conventional analysis is two-sided multiple testing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', testing the family of point null hypotheses {θ1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , θn = 0} with a procedure that guarantees familywise error rate (FWER) control at a pre-specified level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' If the ith null hypothesis is rejected, the conclusion is that θi ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Tukey (1991) argued that such a conclusion is unsatisfactory, and the analysis should aim instead at concluding on the sign of the parameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', that θi > 0 or θi < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Following rejection, it is tempting to conclude for the ith parameter its direction based on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' However, a directional error, referred to often as a type III error (see Finner 1999 and references within) may occur if we decide after rejection of the point null hypothesis θi = 0 that θi > 0 when in fact θi < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore, the probability of making at least one type I or type III error, henceforth referred to as the directional FWER (dFWER), may be larger than α even though FWER ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Existing multiple testing procedures on the family of point null hypotheses have been shown to control the dFWER for independent two-sided p-values satisfying additional dis- tributional assumptions: Holm’s procedure (Shaffer, 1980), Hochberg’s procedure (Finner, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Liu, 1997) and, in general, the closed testing procedure (Finner, 1999), which includes the previous two as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We are interested in simultaneous inference on the signs of the coordinates of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' As in previous works, we would like to control both type I and type III errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' However, we aim not only to identify positive and negative parameters, but also to provide simultaneous confidence bounds on the number of positive and on the number of negative parameters for any subsets of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For this purpose, rather than adopting existing procedures for two-sided tests for the purpose of directional identification, we start by considering the problem of testing n pairs of one-sided hypotheses given by H− i : θi ≤ 0 H+ i : θi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (1) Lehmann (1950, 1957);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Duncan (1955);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Kaiser (1960);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Shaffer (1974) considered (1) refer- ring to it as a directional hypothesis-pair or a three-decision problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Multiple directional hypothesis-pairs were considered in Holm (1979);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Shaffer (1980);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (1986);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Finner 2 (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Shaffer (2002, 2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Guo and Romano (2015), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We suggest the fol- lowing two step approach, referred to henceforth as directional closed testing: first, select from each pair the hypothesis to test (based on the data);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' second, on the selected n one- sided hypotheses, apply an α level closed testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Of course, the second step has to take care to adjust for the first step of selection from the same data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The adjustment is straightforward for independent p-values that are conditionally valid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', for which we can compute valid p-values conditional on the first selection step, see Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Ellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' With this approach, we can provide simultaneous confidence bounds on the number of positive and on the number of negative parameters for any subsets of interest, building upon the work of Goeman and Solari (2011) that showed how to obtain simultaneous bounds for a closed testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This approach can be viewed as a direct generalization of dFWER controlling proce- dures suggested previously, since it provides bounds of interest in addition to identification of positive and negative parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For some local tests used in the α level closed testing procedure in the second step, the bounds can be much tighter than the number of param- eters identified with positive or negative signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In some applications, tight bounds may be more important than identification, so the local test should be selected with care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For example, when evaluating a treatment effect on multiple subgroups (or cohorts), so θi is the average treatment effect for subgroup i, a positive lower bound on both the number of subgroups for which θi > 0, and on the number of subgroups for which θi < 0, conveys the heterogeneity of the treatment effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This can be of major clinical importance, since it provides the researcher with the knowledge that at least for some subgroups the treatment is harmful rather than effective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', that there is a qualitative interaction, Gail and Simon 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' With our approach, we can provide tight lower bounds on the number of positive and on the number of negative parameters, as well as identify positive and negative parameters, with the guarantee that the probability of any wrong inference is at most α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For some applications, it is enough to infer on positive and non-positive findings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', we modify H+ i in (1) by removing the equality sign, and thus exactly one of the hypothesis pair is true: H− i : θi ≤ 0 Ki : θi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2) In such a case, more powerful procedures than typically used with two-sided testing can be devised (Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Guo and Romano, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For example, the Bonferroni proce- dure compares each one sided p-value to α/n instead of to the two-sided threshold α/(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 3 This can be understood from the work of Shaffer (1986), who showed that when performing a Bonferroni procedure, the proper correction factor for FWER control is not the number of hypotheses tested, but the maximum number of hypotheses that can simultaneously be true (Goeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We suggest the following two step approach, referred to henceforth as adaptive partitioning: first, select (based on the data) from each pair the one-sided hypothesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' second, using the selected n one-sided hypotheses, test each parti- tioning hypothesis at level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The first step is identical to the first step in the directional closed testing procedure, and the requirement that the p-values be independent and condi- tionally valid is the same as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The second step applies an α level partitioning procedure (Stefansson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Finner and Strassburger, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Finner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2021) based on the selected p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' With this novel approach, we can provide tight lower and upper bounds on the number of positive parameters (where the upper bound is n minus the lower bound on the number of nonpositive parameters), as well as identify positive and nonpositive pa- rameters, with the guarantee that the probability of any wrong inference is at most α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In systematic reviews in clinical trials, for example, it is important to evaluate the bounds on the number of studies with an effective intervention effect, and a non-trivial upper bound is of concern since it suggests that the intervention effect is null or harmful (IntHout et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We show that the bounds are uniformly tighter than with directional closed testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' So unless it is important to infer on the positive as well as the negative parameters, this approach should always be preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Importantly, it should always be preferred in settings where the point null hypotheses are never true (which is always the case according to Tukey 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Jones and Tukey 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In the next subsection, we illustrate our suggested approach to a small problem of a subgroup analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The results highlight the difference between directional closed testing and adaptive partitioning, and the potential usefulness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Then, we explain in detail directional closed testing in § 2, and adaptive partitioning in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We provide the conditional and unconditional inferential guarantees for each procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In § 4 we compare and contrast the directional closed testing and adaptive partitioning procedures in simulations, using various local tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In § 5 we provide two usages of our suggested methods: for enhancing meta-analysis, and for providing inference following the test of qualitative interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Finally, in § 6 we discuss extensions and some final remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' All proofs are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 4 i 1 2 3 4 di 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='151 sei 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0788 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0547 pi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='9510 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='7780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='9971 Table 1: Analysis of the proportions free of breast cancer at 3 years in 1260 patients divided in four subgroups defined by age and progesterone receptor levels (i = 1: Age < 50, PR < 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' i = 2: Age ≥ 50, PR < 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' i = 3: Age < 50, PR ≥ 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' i = 4: Age ≥ 50, PR ≥ 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Reproduced from Table 2 in Gail and Simon (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The differences, di, in proportions disease-free at 3 years and their relative standard errors, sei, are used to compute the right tailed p-values pi by using the large-sample normal approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 An example: PFT therapy for breast cancer The data of Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (1983), re-analysed in Gail and Simon (1985), consists of the difference in the proportion of patients that are disease free with the PFT treatment versus with the FT treatment, in n = 4 subgroups defined by age and progesterone receptor levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Table 1 provides for each subgroup the difference, standard error, and right-tailed p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Applying the first step of hypothesis selection based on the direction favored by the data, the directional closed testing procedure selects H− 1 , H+ 2 , H+ 3 , H+ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Similarly, the adaptive partitioning procedure selects H− 1 , K2, K3, K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For both procedures, the p-values used for testing in the second step, are therefore: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0193, 1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='9510, 1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='7780, and 1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='99714, for groups 1, 2, 3, and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The procedures are illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Nodes in the graph represent closed testing intersection hypotheses, labelled by their corresponding index set and selected direction, as well as the partition of the parameter space, for which the same local test is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The local test p-value, obtained by the adaptive Simes combination test (described in § 4), is shown for each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Applying directional closed testing at level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='05, we have 0 ≤ n+ and 1 ≤ n− with 95% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Furthermore, we conclude that θ4 < 0, since H+ 4 is rejected by closed testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We obtain a trivial lower bound for n+ because H− 1 is not rejected by closed testing, and a non-trivial lower bound for n− because H+ 2 ∩ H+ 3 ∩ H+ 4 is rejected by closed testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The non-rejected partitioning hypotheses at the 5% level are denoted by − + +−, − − +−, + + +−, + + −−, + − +−, + − −−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For example, − + +− corresponds to testing the intersection hypothesis H− 1 ∩ K2 ∩ K3 ∩ H− 4 , and this is carried out in the adaptive partitioning procedure by a local test of H− 1 ∩ K2 ∩ K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This is valid, since the actual 5 test carried out is for a larger null (H− 1 ∩ K2 ∩ K3 ⊇ H− 1 ∩ K2 ∩ K3 ∩ H− 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The reason for testing H− 1 ∩ K2 ∩ K3 is that the fourth p-value is greater than half (so K4 is selected for testing, not H− 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The local test p-value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0772 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We can extract the confidence set for n+ from these non-rejected partitioning null hypotheses as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Since the number of positive parameters in each non-rejected partition null hypothesis is at least one and at most three, the adaptive partitioning procedure provides bounds 1 ≤ n+ ≤ 3 with 95% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This is an improvement over the directional closed testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Alternatively, we can see that the lower bound is one because H− 1 ∩H− 2 ∩H− 3 ∩H− 4 (denoted by −−−− in Figure 1) is rejected by the partitioning procedure (by a local test of H− 1 , since for the remaining coordinates K2, K3, K4 were selected for testing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Moreover, since the fourth individual hypothesis in each non-rejected partition null hypothesis is non-positive, we further conclude that θ4 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This is a weaker conclusion compared to θ4 < 0 obtained by closed testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We revisit this example in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 6 − + ++ 1−2+3+4+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0115 − + +− 1−2+3+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0772 − + −+ 1−2+4+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0115 − − ++ 1−3+4+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0115 + + ++ 2+3+4+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0115 − + −− 1−2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0115 − − +− 1−3+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0772 − − −+ 1−4+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0115 + + +− 2+3+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1960 + + −+ 2+4+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0115 + − ++ 3+4+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0115 − − −− 1− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0386 + + −− 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0980 + − +− 3+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='4440 + − −+ 4+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0058 + − −− ∅ 1 Figure 1: The example from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Nodes in the graph represent closed testing intersec- tion hypothesis, labeled by their corresponding index set and selected direction (superscript + or −);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 1−2+3+ corresponds to H− 1 ∩ H+ 2 ∩ H+ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Below each intersection hypothesis, the corresponding p-value obtained by the adaptive Simes combination test (described in § 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Above each intersection hypothesis, the corresponding partitioning hypothesis (rep- resented by a sequence of signs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' − + +− corresponds to H− 1 ∩ K2 ∩ K3 ∩ H− 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For the closed testing hypotheses only, arrows represent subset relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Hypotheses rejected by the adaptive Simes local test at the 5% level are marked in bold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' hypotheses rejected by the closed testing procedure at 5% level are filled in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 7 2 Directional closed testing Suppose that we have right-tailed p-values p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , pn for the hypotheses H− 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , H− n , and left-tailed p-values q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , qn for the hypotheses H+ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , H+ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Let p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , pn) denote the p-value vector, and [n] the index set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Our assumptions regarding the p-values are as follows: (A0) Each p-value is valid, in the sense ∀x ∈ [0, 1], sup θ∈H− i Pθ(pi ≤ x) ≤ x, sup θ∈H+ i Pθ(qi ≤ x) ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' where the inequality is an equality for θi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (A1) Each p-value is conditionally valid: sup θ∈H− i Pθ(2pi ≤ x | pi ≤ 1/2) ≤ x ∀x ∈ [0, 1], sup θ∈H+ i Pθ(2qi ≤ x | pi > 1/2) ≤ x ∀x ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (A2) p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , pn are mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For continuous exponential families (where the p-value is a monotone trans- formation of the sufficient statistic), the p-values satisfy (A0) as well as uniform validity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', ∀θi ∈ H− i , Pθi(pi/τ ≤ x | pi ≤ τ) ≤ x ∀ 0 ≤ x, τ ≤ 1, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Karlin and Rubin (1956);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We only concentrate on the most natural selection value τ = 1/2, hence assumption (A1) which is slightly more general than uniform validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We address other values of τ in the § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Moreover, for simplicity, we further assume in (A0) that the p-value is uniform when θi = 0, but this assumption can be relaxed for discrete distributions, where the adjustment for selection may be 1/P0(pi ≤ 1/2) instead of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We are interested in simultaneous inference on the selected family of n directional hypotheses {H+ i : pi ≤ 1/2} ∪ {H− i : pi > 1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' A general way to derive procedures with simultaneous error control is provided by closed testing (Goeman and Solari, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Goeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2021), introduced by Marcus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (1976) for FWER control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We propose the directional closed testing procedure, which is the closed testing procedure on the selected one-sided hypotheses: Procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 (Directional closed testing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 8 Step 1 Select the n one-sided hypotheses for testing: � H = {H− i : pi ≤ 1/2}∪{H+ i : pi > 1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Let S− = {i : pi ≤ 1/2} and S+ = {i : pi > 1/2} be, respectively, the indices for which we test H− i and H+ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Step 2 Apply a level α closed testing procedure on the family of hypotheses � H, using the conditional p-values {2pi, i ∈ S−} ∪ {2qj, j ∈ S+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' So, the procedure tests all intersection hypotheses: HI : � � i∈I∩S− H− i � ∩ � � i∈I∩S+ H+ i � , I ⊆ [n], where HI is true if and only if all selected hypotheses with i ∈ I are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Testing HI at level α is done by the local test φI = 1{f({2pi, i ∈ I ∩ S−} ∪ {2qj, j ∈ I ∩ S+}) ≤ α} (3) using a combining function f(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We denote by S the vector of signs that we condition on: S = (sign(p1 − 1/2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , sign(pn − 1/2)), sign(pi − 1/2) = � −1 if pi ≤ 1/2, 1 if pi > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Conditional on the vector of signs S ∈ {−1, 1}n, the probability of rejecting the intersection hypotheses of the true nulls among the selected is at most α, if we use the local test in (3), since we combine (conditional) p-values that have each a distribution that is at least as large as uniform given S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We formalize this in the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Let p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , pn fulfil conditions (A1)−(A2), and let θ be the true unknown vector of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' So among the n selected for testing in Step 1 of Procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1, T − = {i : θi ∈ H− i } ∩ S− and T + = {i : θi ∈ H+ i } ∩ S+ are the true left sided and right sided null hypotheses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' T = T −∪T + is the (unknown) index set of the true null hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Then the test of the intersection of the hypotheses in T satisfies conditional type I error control: sup θ′∈HT Pθ′ � φT = 1 | S � ≤ α, (4) 9 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 implies that all the probability statements that hold for the closed test- ing procedure, will hold conditional on S for Procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' So the conditional guarantees imply the unconditional guarantees, but are stronger since they are conditional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' How does directional closed testing compare with the direct approach of using two-sided p-values in a closed testing procedure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The direct approach with two-sided p-values infers on intersection hypothesis of the form ∩i∈I{θi = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The indices of rejected individual hypotheses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', point null hypotheses θi = 0) is identical to the set of indices identified using the directional closed testing procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' However, using procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 it is clear that we can also infer about the signs (as long as each intersection hypothesis is tested with a valid local test), and it is clear how to provide simultaneous confidence bounds on the number of positive and negative parameters of any subset of hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' To the best of our knowledge, these bounds have not been available up to now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' More generally, all the guarantees provided by closed testing procedures follow immediately using directional closed testing, while many of these guarantees were unclear with the direct approach using two-sided p-values, due to the worry about type III errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Moreover, to the best of our knowledge, the specific assumption (A1) − (A2) are more general than the assumptions considered in Shaffer (1980);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (1986);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Finner (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Guo and Romano (2015) for directional FWER control on individual discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Specifically, conditional validity in (A1) is a weaker condition than assuming the monotone likelihood ratio order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We conclude this section by showing explicitly how to obtain individual discoveries and confidence bounds on subsets of interest with Procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The closed testing procedure corrects the local tests for multiple testing by ¯φI = min{φJ : J ⊇ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Marcus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (1976) showed that the adjusted tests ¯φI have FWER control, so with Pro- cedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 we have: Pθ � ¯φI = 0 for all I ⊆ T | S � ≥ 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Let Dα = D+ α ∪ D− α = {i ∈ S− : ¯φi = 1} ∪ {i ∈ S+ : ¯φi = 1} be the index set of the discoveries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the elementary hypotheses rejected by the closed testing procedure at level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The procedure concludes θi > 0 for all i ∈ D+ α and θi < 0 for all i ∈ D− α while controlling the conditional FWER at level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 10 Let n+(I) and n−(I) be the number of parameters θi, i ∈ I with positive value and negative values, respectively: n+(I) = |{i ∈ I : θi > 0}|, n−(I) = |{i ∈ I : θi < 0}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For simplicity of notation, we use n+ and n− instead of n+([n]) and n−([n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Goeman and Solari (2011) showed that eα(I) = max(|J | : J ⊆ I, ¯φJ = 0) provide false discovery control over all I, so with Procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 we have: Pθ � |I ∩ T | ≤ eα(I) for all I | S � ≥ 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (5) eα(I) is an upper bound for the false discoveries in I, for all I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Let dα(I) = |I| − eα(I) be the lower bound on true discoveries in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' From (5) it follows that � l+ α (I) = dα(I ∩ S−) u+ α(I) = |I| − dα(I ∩ S+) , � l− α (I) = dα(I ∩ S+) u− α(I) = |I| − dα(I ∩ S−) (6) are such that Pθ � l+ α (I) ≤ n+(I) ≤ u+ α(I), l− α (I) ≤ n−(I) ≤ u− α(I), for all I | S � ≥ 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (7) In particular: � l+ α = dα(S−) u+ α = n − dα(S+) , � l− α = dα(S+) u− α = n − dα(S−) , are the lower and upper bound on n+ and n−, the number of parameters θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , θn with positive value and negative values, and Dα = D+ α ∪ D− α = {i : l+ α (i) = u+ α(i) = 1} ∪ {i : l− α (i) = u− α(i) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' A computational problem remains: computing (6) involves the evaluation of exponen- tially many tests, which hinders its practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For specific combining functions, however, there exist shortcuts to derive the closed testing results in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Our assumption regarding the combining function f(·) is as follows: 11 (A3) The combining function f(|I|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (x)i∈I) depends on the size of I and the vector of p-values (x)i∈I, and it satisfies monotonicity: f(|I|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , x|I|)) ≤ f(|I|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (x′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , x′ |I|)) (8) for xi ≤ x′ i for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |I|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' and symmetry: f(|I|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , x|I|)) = f(|I|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (xj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , xj|I|)) (9) for any permutation (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , j|I|) of (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |I|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Quadratic time shortcuts for combining functions satisfying (A3) have been developed for FWER control (Dobriban, 2020) and simultaneous FDP control (Goeman and Solari, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Goeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2019, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Another condition that could reduce computation time is separability (Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 3 Adaptive partitioning If one is interested only in inference on the number of positive parameters, we can provide bounds that are uniformly better than the bounds in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The individual hypotheses we consider for providing the (lower and upper) bounds on the number of positive parameters are the n pairs in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We derive (1 − α) confidence bounds l+ α (I) and u+ α(I) for n+(I) that are simultaneous for all I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', Pθ � l+ α (I) ≤ n+(I) ≤ u+ α(I) for all I | S � ≥ 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (10) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The conditional guarantee in (10) imply of course the unconditional guar- antee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We shall further show that the unconditional coverage is slightly larger than (1−α), due to the fact that there is positive probability that the realized S cannot produce an infer- ential error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This is in contrast with the conditional guarantee in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1, which can reach α when θi = 0 for at least one i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This is due to the fact that zero is included in both H+ i and H− i , so for any realized S there will be a danger of falsely rejecting the intersection null among the selected, HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We start by considering the case of n = 2 parameters in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The inference using adaptive partitioning is easy to explain in this case, since all partitions and all the possibil- ities for the vector of signs S are easily enumerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Moreover, we can visualize the power advantage of adaptive partitioning over directional closed testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Then we proceed to the general case of n > 2 parameters in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 Inference for n = 2 Figure 2 shows directional closed testing and adaptive partitioning procedures as a function of the selection event (Step 1 of the procedures), for the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Possible selections of one-sided hypotheses are {H− 1 , H− 2 }, {H− 1 , H+ 2 }, {H+ 1 , H− 2 } and {H+ 1 , H+ 2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' p1 ≤ 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' p2 ≤ 1 2 ˆH = {H− 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' H− 2 } −− 1−2− f(2p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 2p2) −+ 1+ 2p1 +− 2+ 2p2 ++ ∅ 1 p1 ≤ 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' p2 > 1 2 ˆH = {H− 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' H+ 2 } −+ 1−2+ f(2p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 2q2) −− 1− 2p1 ++ 2+ 2q2 +− ∅ 1 p1 > 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' p2 ≤ 1 2 ˆH = {H+ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' H− 2 } +− 1+2− f(2q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 2p2) ++ 1+ 2q1 −− 2− 2p2 −+ ∅ 1 p1 > 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' p2 > 1 2 ˆH = {H+ 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' H+ 2 } ++ 1+2+ f(2q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 2q2) +− 1+ 2q1 −+ 2+ 2q2 −− ∅ 1 Figure 2: Directional closed testing and adaptive partitioning procedures as a function of the selection event (displayed on top),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' for the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Nodes in the graph show the par- titioning hypothesis (first row, represented by a sequence of signs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' −+ corresponds to H− 1 ∩K2), the closed testing hypothesis (second row, represented by an index set along with the selected direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 1−2+ corresponds to H− 1 ∩ H+ 2 ) and the corresponding p-value (third row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For the closed testing hypotheses only, arrows represent subset relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For example, when ˆH{H− 1 , H+ 2 } is selected, the closed testing intersection hypothe- sis H− 1 ∩ H+ 2 (1−2+) and the partitioning hypothesis H− 1 ∩ K2 (−+) are both tested by the p-value f(2p1, 2p2) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', with Fisher’s combining function, f(2p1, 2p2) = P(χ2 4 ≥ 13 −2(log(2p1) + log(2p2)), H− 1 (1−) and H− 1 ∩ H− 2 (−−) by the p-value 2p1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' and H+ 2 (2+) and K1 ∩ K2 (++) by the p-value 2q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The partitioning hypothesis K1 ∩ H− 2 (+−) is not tested, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the p-value is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Because H− 1 ∩ H− 2 implies both H− 1 and H+ 2 , directional closed testing adjusts the p-values for H− 1 and H+ 2 by max(2p1, f(2p1, 2p2)) and max(2q2, f(2p1, 2p2)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' If e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' f(2p1, 2p2) > α and p1 < α/2, then di- rectional closed testing does not reject any hypothesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' conversely, adaptive partitioning rejects H− 1 ∩H− 2 (−−), which implies n+ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This situation is illustrated in Figure 3 with Fisher’s combining function, where the top-left and top-right plots show the areas leading to inference about n+ with (1−α) confidence for the directional closed testing and adaptive partitioning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 14 Directional Closed Testing α/2 p1 p2 Adaptive Partitioning α/2 p1 p2 Rejection of individual hypotheses 1−, 2− 1−, 2+ 1+, 2+ 1+, 2− 1− 1+ 2− 2+ p1 p2 α/2 Unconditional Adaptive Partitioning 2α/3 p1 p2 n+ = 2 n+ ≥ 1 n+ = 1 n+ ≤ 1 n+ = 0 Figure 3: Inference for n = 2 using Fisher’s combining function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Areas leading to inference about n+ with (1 − α) confidence for the directional closed testing (top-left plot) and the adaptive partitioning (top-right plot with conditional guarantee and bottom-right plot with unconditional guarantee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The gray scale indicates the type of inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Bottom-left plot: areas leading to the rejection of individual hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The dot pattern indicates areas in which one hypothesis is rejected at level α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the north east lines pattern indicates areas in which two hypotheses are rejected at level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Hypotheses are represented by their indices and directions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 1−, 2+ corresponds to the hypotheses H− 1 and H+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Each plot is based on α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 15 We see that the area is larger for the adaptive partitioning procedure when {p1 ≤ 1/2, p2 > 1/2} or {p1 > 1/2, p2 ≤ 1/2}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' adaptive partitioning is uniformly more powerful than directional closed testing for inference on n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' If we are only interested in providing level α unconditional error control, the power gain of adaptive partitioning is even larger, as illustrated in the bottom-right plot of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Indeed, we will see in Section 4 that the adaptive partitioning procedure can be carried out at the larger level α∗ = α/(1 − 2−n) = 4α/3 while providing level α unconditional error control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Figure 4 shows the power gain of using adaptive partitioning over directional closed testing, in discovering that n+ ≥ 1, for a range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The power gain is substantial, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', when θ1 > 0 and θ2 < 0 there can be a power gain of 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In addition, we see the additional gain we get from using α∗ instead of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 16 −3 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5 −2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='8 1 θ2 Power DCT (θ1 = 0) AP (θ1 = 0) UAP (θ1 = 0) DCT (θ1 = 2) AP (θ1 = 2) UAP (θ1 = 2) Figure 4: Power to discover that the lower bound for n+ is at least one with 95% confidence versus θ2 with: directional closed testing (DCT, solid line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' adaptive partitioning (AP, dashed line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the unconditional adaptive partitioning (UAP, dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The test statistics are generated independently from a Gaussian distribution with standard deviation 1 and mean centered at θ1 = 0 (black curves), θ1 = 2 (blue curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Finally, the bottom-left plot of Figure 3 shows the area leading to the rejection of individual hypotheses for the directional closed testing with Fisher’s combining function, which is contained in the area leading to inference about n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Adaptive partitioning has the same rejection region but with H+ i replaced by Ki, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Appendix B discusses the two procedures with Simes’ combining function for n = 2, comparing them to the dFWER controlling procedures of Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (1986) and Guo and Romano (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2 Simultaneous confidence bounds for n+(I) We derive simultaneous (1 − α)-confidence bounds l+ α (I) and u+ α(I) for n+(I) by using the partitioning principle (Stefansson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Finner and Strassburger, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Finner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The main idea is to partition the parameter space Θ into disjoint subspaces: consider the 2n orthants ΘK = {θ ∈ Θ : θi > 0 for all i ∈ K, θj ≤ 0 for all j /∈ K} for all K, so that exactly one ΘK contains the true parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Each orthant ΘK has a corresponding null hypothesis JK : θ ∈ ΘK that the orthant includes the true parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This hypothesis is true if and only if all Ki, i ∈ K are true and all H− j , j /∈ K are true: JK : { � i∈K Ki} ∩ { � i/∈K H− i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Suppose that for every K there is a partitioning local test ψK taking values in {0, 1}, with 1 indicating the rejection of JK at level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The hypotheses JK are all disjoint, and only one of them is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Assume that ψK is a valid statistical test for the true JK, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' sup θ∈ΘK Pθ(ψK = 1) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Consider the hypothesis Jv(I) : n+(I) = v, which can be equivalently expressed as Jv(I) : � K:|K∩I|=v JK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Then Jv(I) is rejected if and only if all JK with |K ∩ I| = v are rejected, and denote by ψv(I) = min{ψK : K ⊆ [n] : |K ∩ I| = v} (11) the statistical test for Jv(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The collection of values v for which we failed to reject Jv(I) at level α: N +(I)α = {v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |I|} : ψv(I) = 0}, constitutes a (1 − α) confidence set for n+(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Pθ(n+(I) ∈ N +(I)α for all I) ≥ 1−α for any θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Furthermore, the lower and upper bounds given by l+ α (I) = min(N +(I)α), u+ α(I) = max(N +(I)α) (12) satisfy Pθ � l+ α (I) ≤ n+(I) ≤ u+ α(I) for all I � ≥ 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 18 However, the computational problem remains: direct application of (11) takes expo- nential time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We propose a procedure that apply to a combining function f(·) satisfying condition (A3) and that can be computed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 (Adaptive partitioning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Step 1 Apply Step 1 of procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 Step 2 Apply Algorithm 1 to obtain the bounds l+ α (I) and u+ α(I) for n+(I) for one or several I ⊆ [n] of your choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Algorithm 1 performs a level α partitioning procedure testing JK by the test for the larger intersection hypothesis with indices in {Kc ∩ S−} ∪ {K ∩ S+}: ψK = φ{Kc∩S−}∪{K∩S+} (13) = 1{f({2pi, i ∈ Kc ∩ S−} ∪ {2qj, j ∈ K ∩ S+}) ≤ α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 19 Algorithm 1: Shortcut for computing the confidence bounds l+ α (I) and u+ α(I) Input : right-tailed p-values p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' combining function f(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' subset I ⊂ [n];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' level α Output : Confidence bounds l+ α (I) and u+ α(I), p-values fv(I) for Jv(I) : n+(I) = v, for v = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |I| Initialize: S− = {i ∈ [n] : pi ≤ 1/2}, |S−| = s, fs(I) = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , a|S−∩I|) with a1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≥ a|S∩I| sorted values of {2pi : i ∈ S− ∩ I} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , b|S−∩Ic|) with b1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≥ b|S−∩Ic| sorted values of {2pi : i ∈ S− ∩ Ic} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' c = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , c|S+∩I|) with c1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≥ c|S+∩I| sorted values of {2qi : i ∈ S+ ∩ I} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' d = (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , d|S+∩Ic|) with d1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≥ d|S+∩Ic| sorted values of {2qi : i ∈ S+ ∩ Ic} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 1 for v ← 0 to |I| do 2 for u ← 0 to |Ic| do 3 for k ← max(0, v − |S− ∩ I|) to min(v, |S+ ∩ I|) do 4 for j ← max(0, u − |S− ∩ Ic|) to min(u, |S+ ∩ Ic|) do 5 fk,j = f({a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , ak−v+|S−∩I|} ∪ {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , bj−u+|S−∩Ic|} ∪ {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , ck} ∪ {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , dj}) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 6 end 7 end 8 fv,u(I) ← max{fk,j} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 9 end 10 fv(I) = max{fv,u(I), u = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |Ic|} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 11 end 12 l+ α (I) ← min(v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , s} : fv(I) > α) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 13 u+ α(I) ← max(v ∈ {s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |I|} : fv(I) > α) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 14 return l+ α (I), u+ α(I), f0(I), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , f|I|(I) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The following proposition asserts that Procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 guarantees conditional coverage and it also improves upon the directional closed testing Procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Let p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , pn fulfil conditions (A1) − (A2), and the combining function f(·) fulfils condition (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Then Procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 returns the bounds l+ α (I) and u+ α(I) for n+(I) with at most O(|I|2 · max(1, |Ic|2)) computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The bounds satisfy the conditional coverage guarantee in (12) and are at least as good as those obtained by the directional closed testing Procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Furthermore, the unconditional coverage guarantee is Pθ � l+ α (I) ≤ n+(I) ≤ u+ α(I) for all I � ≥ 1 − (1 − 2−n)α > 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (14) 20 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For inference on n+ only, in Appendix C we derive (1−α) confidence bounds for n+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', a lower bound l+ α and an upper bound u+ α such that Pθ � l+ α ≤ n+ ≤ u+ α | S � ≥ 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (15) These bounds may be tighter for n+, than taking I = [n] in procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1, but there is no follow-up identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Interestingly, the gap between these bounds and the bounds for I = [n] in the adaptive partitioning procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This finding is based on a wide range of data generations examined (omitted for brevity), including the ones detailed in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Since we are also concerned with follow-up identification and the cost of further inferences is minimal, we suggest using Procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 in order to find the bounds for n+ and identify positive and non-positive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Nevertheless, procedure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 should be used if interest is only in bounds for n+, since it is computationally much simpler and the bounds are at least as tight as with I = [n] in procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' See Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Theorem 2 in Goeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2021) shows that closed testing and partition- ing are equivalent principles, which are also equivalent to sequential rejection (Goeman and Solari, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We employed the closed testing principle in procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 and the partitioning prin- ciple in procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 simply because they seemed the more natural formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 4 Simulations We carry out simulation studies in order to evaluate the performance of our suggested meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Our main focus is on the evaluation of two procedures: the directional closed testing (DCT) procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 and the adaptive partitioning (AP) procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2 states that AP is uniformly better than DCT, but the scientific discovery is slightly weaker: inference is only on the positive and nonpositive parameters, rather than on the positive and negative parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore, our primary aim is to quantify the potential gain in terms of tighter bounds and better identification using AP versus DCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We shall use two representative combining methods for this purpose: Fisher, which is a sum-based method, and Simes, which is quantile based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Sum based tests, such as Fisher, are better than Simes for the bounds on n+ when more than a single parameter is positive or negative, but are worse for identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This finding is demonstrated in our simulations but is also expected based on previous work (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', Benjamini and Heller 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Heard and Rubin-Delanchy 2018 and references within).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 21 For adaptive partitioning, we shall further consider the adaptive likelihood ratio test (ALRT, Al Mohamad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 2020), which is a sum based test for normal test statistics, and the adaptive Simes (Bogomolov, 2021) combination test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We pause to motivate and introduce adaptive Simes, which is superior to Simes for the bounds as well as for identification in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The Simes combining method is essentially based on a single quantile, and thus does not take full advantage of the pooled evidence when the number of positive (or negative) parameters is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Bogomolov (2021) suggested the adaptive Simes combination test, which uses the following combination function: fadaptive Simes(x(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , x(d)) = min 1≤i≤d � 2(�d i=1 1(xi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5) + 1) i x(i) � , where in our setting x(1) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≤ x(d) are the ordered conditional p-values within the selected set for the conditional approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The adaptive Simes p-value may be much smaller than Simes, except in the case where the intersection is only of two hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore, it makes sense to switch to the Simes combining method in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' So we shall denote by adaptive Simes the following combining function to be used in Procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1: f(x(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , x(d)) = � fadaptive Simes(x(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , x(d)) if d > 2, min1≤i≤d �d i x(i) � otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We compare our procedures to the FWER and FDR controlling procedures in Guo and Romano (Guo and Romano, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' These procedures aim to identify the positive and nonpositive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The FWER controlling procedure is a good competitor for identification, but we expect it to do poorly for the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The FDR controlling procedure is less relevant for identification, since it targets a more lenient error guarantee (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', the FDR), whereas all other methods control the FWER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' However, we included it since it is interesting that for the bounds it is not necessarily better, even though there is no confidence guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Our data generation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For a total of n = 50 parameters, there are n+ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n/2} positive parameters and n+ negative parameters (we exclude simulations with a different ratio of the number of positive and negative parameters, since the qualitative conclusions remain unchanged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The remaining parameters are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The test statistics are generated independently from a Gaussian distribution with standard deviation one and mean centered at the parameter value, ˆθi ∼ N(θi, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The right sided p-values are pi = 1 − Φ(ˆθi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Our analysis is based on B = 2000 data generations, and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 22 Figure 5 provides the average length of the bounds and the number of parameters discovered for the settings considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Our key findings are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' First, adaptive partitioning provides much tighter bounds than directional closest testing, with barely any difference in the number of discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Second, the sum-based tests are better for the bounds than the quantile based tests and than the FWER controlling procedure in Guo and Romano (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' They are also much better than the FDR controlling procedure in Guo and Romano (2015) in the data generation with fairly weak signal (in row 1 of Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Third, the adaptive Simes method dominates Simes in the adaptive partitioning proce- dure, and is competitive with the FWER controlling procedure in Guo and Romano (2015) for identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' It (as well as Simes) is also much better than the sum-based tests for identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' From the simulations, we can support the following general guidance: adap- tive Simes is a good choice if identification is important in addition to the bounds, or if it is expected that the fraction of parameters far enough from zero is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Otherwise it is best to use a sum based combining method, such as Fisher or the ALRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 23 0 10 20 25 30 35 40 45 50 n+ Average length u+ − l+ 0 10 20 25 0 5 10 15 20 n+ Average number of hypotheses rejected 0 10 20 25 30 35 40 45 50 n+ Average length u+ − l+ 0 10 20 25 0 5 10 15 20 n+ Average number of hypotheses rejected Figure 5: The average length (left column) and number of individual hypotheses rejected (right column),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' versus n+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' for the following methods: the partitioning algorithm with combining functions Fisher (black circles),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ALRT (black square),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Simes (black triangle),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' adaptive Simes (black diamond);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the CT procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 with combining functions Fisher (blue circles),and Simes (blue triangle);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the Guo-Romano procedure for FWER control (red x’s) and for FDR control (red pluses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The rows differ by the θ configuration: θi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n+, θi = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5, i = n+ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , 2n+, θi = 0, i = 2n+ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n in row 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the nonzero θi’s are generated independently from the absolute value of ∼ N(0, 2) in row 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In the right column, the absence of black circles and triangles is due to the fact that they coincide with their blue counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 24 5 Applications 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 Enhancing meta-analysis The first step in a meta-analysis is often the computation of a global null p-value, which communicates the strength of the evidence against the null hypothesis of no association in any of the studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Rejecting the global null does not rule out the possibility that this is due to a non-zero association only in a single study (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', due to the particulars of the cohort in that study, or due to bias).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore, it is of interest provide tight lower and upper bounds on the number of studies with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', a positive effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' A lower bound greater than one is related to the replicability goal of establishing that the effect is present in at least r (r ≥ 2) of n studies (see Bogomolov and Heller 2022 and references within).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' An upper bound smaller than n conveys the limit on replicability or the specificity of the association to a subset of studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Moreover, for non-trivial bounds it is natural also to try to identify the studies where there is an association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Directional closed testing should be used for this purpose if inference on positive and negative associations are desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Adaptive partitioning should be used if it is enough to infer on n+, and to identify positive and non-positive associations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', when studying an intervention effect, identify studies where the intervention is beneficial and studies where the intervention has no effect or may be harmful).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Since in a typical meta-analysis the studies are independent and the test statistic in each study is approximately normal, the required conditions (A0) − (A2) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Fore a concrete example, we consider the data of Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2003), made public by Konstantopoulos (2011), where the effect of a modified school calendar (with more frequent but shorter breaks) on student achievement is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The meta-analysis uses studies of n = 56 schools in 11 districts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The experimental design is a two-group comparison (modified calendar vs traditional calendar) which involves computing the standardized mean difference Xi ∼ N(θi, 1), where θi > 0 indicates a positive effects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' a higher level of achievement in the group following the modified school calendar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2019) showed that there is evidence of a qualitative interaction at the 5% level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', that the effect of the modified school calendar on student achievement is positive in at least one study, and negative in at least one study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We re-analyze the data by providing the lower and upper bounds on the number of studies with positive/negative effect in all schools (Table 2), and in the (data-driven) top k schools (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Directional closed testing makes the same number of individual discoveries as adaptive partitioning, but the upper bound on n+ is the trivial bound of 25 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' With adaptive partitioning the bounds are much more informative: using Fisher’s combining method, the 95% confidence bound for n+ is 10 to 53, and we have four individual discoveries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' using Adaptive Simes, the 95% confidence bound for n+ is 8 to 54, and we have eight individual discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Using the procedure of Guo and Romano (2015) which is targeted for dFWER control, the 95% confidence bound for n+ is 8 to 55, with nine individual discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Figure 6 shows simultaneous 95% confidence intervals for the top k schools: for example, within the top 38 schools, we have at least 9 positive effects and at least 1 non-positive effect, or within the top 51 schools, we have at least 10 positive effects and at least 2 non-positive effect, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' n+ n− Local test Procedure l+ u+ |D+| l− u− |D−| Fisher DCT 9 56 4 0 47 0 AP 10 53 4 3 46 0 Simes DCT 8 56 8 0 48 0 AP 8 55 8 1 48 0 Adaptive Simes AP 8 54 8 2 48 0 ALRT AP 11 53 5 3 45 0 Guo and Romano (2015) 8 55 8 1 48 1 Table 2: The effect of modified school calendars on student achievement: intervals for n+ and n− and corresponding number of discoveries |D+| and |D−|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' for different combining methods and procedures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' with 95% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' DCT and AP are directional closed testing and adaptive partitioning procedures 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' n− is the number of negative parameters for DCT and non-positive parameters for AP and Guo and Romano (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' |D−| is the number of discoveries of negative parameters for DCT and non-positive parameters for AP and Guo and Romano (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 26 1 56 0 56 k 95% confidence interval for n+(Ik) Figure 6: Simultaneous 95% confidence intervals for n+(Ik) with Fisher’s combining func- tion, where Ik are the indexes of the top k = |Ik| schools with largest (in absolute value) effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For example, within the top 38 schools, we have at least 9 positive effects and at least 1 non-positive effect (red interval);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' or within the top 51 schools, we have at least 10 positive effects and at least 2 non-positive effect (blue interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2 Testing for qualitative interactions with follow-up inference If testing for mixed signs of the parameter vector θ is of primary concern, the approach described in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 can be adjusted so that directional closed is applied only if the hypothesis of no qualitative interaction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', no positive and negative parameters in θ) is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Sev- eral tests have been developed in the literature (Gail and Simon, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Zelterman, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Piantadosi and Gail, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Ciminera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Pan and Wolfe, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Silvapulle, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Li and Chan, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Hudson and Shojaie, 2020) for testing the null hy- 27 pothesis of no qualitative interaction: H0 : � n� i=1 H− i � ∪ � n� i=1 H+ i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Gail and Simon (1985) analyzed the data discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 to see whether they could demonstrate a qualitative interaction at the α = 5% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' However, the likelihood ratio test proposed by Gail and Simon (1985) results in a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2019), when their selection threshold τ = 1/2, uses the p-value of HS− to test �n i=1 H− i and the p-value of HS+ to test �n i=1 H+ i , which are valid tests since HS− ⊇ �n i=1 H− i and HS+ ⊇ �n i=1 H+ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' So the p-value for H0 is the larger of p1/n = f({2pi, i ∈ S−}) (the p-value for testing �n i=1 H− i ) and q1/n = f({2qi, i ∈ S+}) (the p-value for testing �n i=1 H+ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We see from Figure 1 that this approach gives a p-value of max(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0386, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0115) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0386 with the adaptive Simes combination test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (With Fisher, Simes and ALRT combination tests we obtain the same result: p1/4 = 2p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0386 and q1/4 = f(2q2, 2q3, 2q4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0110, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0173 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='0120 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We propose the following closed testing procedure tailored for qualitative interactions: start with Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2019) global test for H0 and, if rejected, continue with the directional closed testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Procedure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 (Closed testing for qualitative interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Step 1 Apply Step 1 of procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 Step 2 Apply a level α closed testing procedure on the family of hypotheses ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Testing HI at level α is done by the local test φI = � 1{max{f({2pi, i ∈ S−}), f({2qi, i ∈ S+})} ≤ α} if I ⊇ S− or I ⊇ S+ 1{f({2pi, i ∈ I ∩ S−} ∪ {2qj, j ∈ I ∩ S+}) ≤ α} otherwise From Figure 1 we see that at level 5% procedure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 rejects all hypotheses but H+ 2 ∩H+ 3 , H+ 2 and H+ 3 , giving 1 ≤ n+ ≤ 3, 1 ≤ n− ≤ 3 and θ1 > 0, θ4 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This result improves the one obtained by the directional closed testing procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 (1 ≤ n− ≤ 4 and θ4 < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' However, with the closed testing for qualitative interactions procedure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1, rejecting H0 is crucial: if H0 is not rejected, the closed testing procedure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 does not reject any intersection hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For instance, at α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5%, procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 gives the same conclusions as at α = 5%, while procedure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 is uninformative since no rejections are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' On the other hand, if H0 is rejected, then the lower bounds for n+ and n− are both informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 28 6 Concluding remarks In Procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1, the selection Step 1 has a cost of inflating the p-values by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' It is a reasonable cost, since it is no worse than the cost of 2-sided testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' It is possible to consider more generally the selection of {H+ i : pi ≤ τ} ∪ {H− i : pi > τ} for a τ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Such a selection step will inflate the right-sided p-value by 1/τ, and the left-sided p-value by 1/(1 − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' It may be useful when it is more important to detect one direction over another, and the inferential tools can easily be adjusted for independent p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Each parameter may be accompanied by its own predefined selection threshold τi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In particular, if it is a-priori known for a parameter that only the right-sided alternative is of interest, set τi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Similarly, set τi = 0 if it is a-priori known for the parameter that only the left-sided alternative is of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The selection step is thus more general, and the following step of directional closed testing or adaptive procedures is carried as described, using the conditional p-values {pi/τi : pi ≤ τi}∪{qi/(1−τi) : pi > τi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Importantly, the same computational shortcuts are carried out on these conditional p-values for the desired inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' An open problem is how to deal with dependent p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Adjusting for the selection event S by conditioning may incur an inflation factor much greater than 2 (or more gen- erally 1/τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For example, using the polyhedral lemma and the data carving methods of Fithian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Currently, it is unclear to us whether using conditional p-values is useful in settings where the p-values are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The conditional (1 − α) confidence bounds for n+ and n+(I) have an unconditional 1−α(1−2−n) confidence guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore, adaptive partitioning can be carried out at level α∗ = α/(1 − 2−n) if the conditional guarantee is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Using α∗ falls within the framework of the holistic approach of Goeman and Solari (2022), that view selection and conditioning as means for useful inferences with unconditional error guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Inter- estingly, our directional closed testing procedure provides a conditional error control that can be equal to the unconditional control if θi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' It is interesting to consider in our setting what can be gained from having a conditional versus an unconditional guarantee, and what are the implications when the conditional and unconditional guarantee coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' As mentioned in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2, if only n+ is of interest, (1 −α) confidence bounds can be achieved with the simpler and (slightly) more powerful procedure than adaptive partition- ing described in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The procedure is in line with all other procedures suggested in this paper, in that the local tests are conditional on the vector of signs S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In Jaljuli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Solari and Goeman (2021) suggested combining α/2 one-sided bounds, that were computed with more general local tests of intersection hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For most data gen- 29 erations encountered in practice, we argue in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1, that combining α one-sided bounds is enough (this is provably so for the conditional procedure, see Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In our numerical experiments (omitted for brevity), we see that the tightest bounds are provided with our conditional approach if there are several positive and several negative parameters, but with the unconditional approach if all parameters are non-positive or non- negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Deriving a data driven approach that adapts to the choice of conditional versus unconditional bounds for n+ is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This work concentrated on simultaneous inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For large scale testing, however, a popular error for identification is the false discovery rate (FDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Overall bounds can be calculated from the FDR controlling procedure by counting the number of discoveries in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Of course, there is no (1 − α) confidence guarantee on the overall bounds, but for large problems knowing that it is based on an FDR guarantee may be enough for some purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' However, it is interesting to note that the bounds can be smaller than with our proposed approach, which provides the desired coverage guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Specifically, we considered the FDR controlling procedure in Guo and Romano (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' They showed that for independent p-values (if U(0, 1) ≤rh pi for null p-values, where ≤rh is the reverse hazard rate order), when considering the family {H− i , Ki : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n}, it is enough to apply the Benjamini Hochberg procedure (Benjamini and Hochberg, 1995) on the set of smallest one-sided p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Bounds on n+ can be calculated from the FDR controlling procedure of Guo and Romano (2015) by counting the number of positive and nonpositive discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' These bounds tended to be wider than with our approach when the signal is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' References Al Mohamad, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', Van Zwet, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', Cator, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', and Goeman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2020).' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Statistics & probability letters, 10(1):59–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 34 Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', Small, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', and Su, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Multiple testing when many p-values are uniformly conservative, with application to testing qualitative interaction in educational interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Journal of the American Statistical Association, 114(527):1291–1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' A Proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' From step 2 of the procedure, it follows that φT is a function of {2pi, i ∈ S− ∩T −}∪ {2qj, j ∈ S− ∩ T −}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Since the p-values are independent (assumption (A2)), it follows that Pθ(φT = 1 | S) = P{θi:i∈T }(φT = 1 | {sign(pi − 1/2) : i ∈ T }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Since φT = 1{f({2pi, i ∈ T −}∩{2qi, i ∈ T +}) ≤ α} is a valid local test, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', the probability of rejection is at most α if the p-values in the intersection test are all valid, it follows from the independence across p-values (assumption (A2)) that the right hand side is at most α if each conditional p-value is (marginally) valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The validity of each conditional p-values is guaranteed by assumption (A1), thus concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Suppose that the true θ lies in ΘK for some K ⊆ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We have ψ|K∩I|(I) ≤ ψK for every I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore, if ψK = 0, then |K ∩ I| = n+(I) ∈ N +(I)α for all I, thus Pθ(l+ α (I) ≤ n+(I) ≤ u+ α(I) ∀I) ≥ Pθ(n+(I) ∈ N +(I)α ∀I) ≥ Pθ(ψK = 0) ≥ 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='3 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2 shows that Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 returns the bounds l+ α (I) and u+ α(I) defined in (12) with at most O(|I|2 · max(1, |Ic|2)) computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Suppose the true θ ∈ ΘK for some K ⊆ [n], and let ˜T = T −∪ ˜T + = {Kc∩S−}∪{K∩S+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The partitioning Procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 tests JK by using ψK = φ ˜T , and sup θ∈ΘK Pθ(φK = 1 | S) ≤ sup θ∈H ˜ T Pθ(φ ˜T = 1 | S) ≤ α 35 where the first inequality follows from JK ⊆ H ˜T and the second inequality follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Furthermore, Pθ( ˜T = ∅) = Pθ(S− = K) = � i∈K Pθ(pi ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5) � j /∈K Pθ(pj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5) ≥ P0(S− = K) ≥ 2−n, thus Pθ(ψK = 1) = � S Pθ(φ ˜T = 1|S)Pθ(S) ≤ α(1 − Pθ( ˜T = ∅)) ≤ α(1 − 2−n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The previous results together with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 imply the conditional and unconditional coverage (12) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' See also § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2 of the Supplementary Material to Al Mohamad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2020) for the unconditional type I error control with the adaptive likelihood ratio test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Now we want to prove that l+ DCT(I) ≤ l+ AP(I) for every I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We proceed by contradiction: assume that ∃ I such that l+ AP(I) < l+ DCT(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' By Lemma 1 in Goeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2021): l+ DCT(I) = min V⊆[n](|(I ∩ S−) \\ V| : φV = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (16) Because the null hypothesis that n+(I) is equal to l+ AP(I) is not rejected at level α, then ∃ K such that |K ∩ I| = l+ AP(I) for which JK is not rejected at level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Equivalently, ∃ U = (Kc ∩ S−) ∪ (K ∩ S+) such that φU = 0 and |(I ∩ S−) \\ U| = |I ∩ S− ∩ K| ≤ l+ AP(I) < l+ DCT(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' On the other hand, from equation (16) it follows that l+ DCT(I) ≤ |(I∩S−)\\U| contradicting the assumption that l+ AP(I) < l+ DCT(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Finally, u+ AP(I) ≤ u+ DCT(I) for every I can be proved in analogous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' B Inference for n = 2 with Simes combining function The top-left plot of Figure 7 displays inference about n+ with (1−α) confidence for the di- rectional closed testing procedure with Simes combining function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For n = 2, the directional closed testing procedure with Simes’ local test is a consonant procedure (Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2011), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the rejection of the intersection hypothesis implies the rejection of at least one of its component hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We see that each area leading to inference about n+ gives also the rejection of one or two individual hypotheses (hypotheses are represented by indices and directions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 1−, 2+ corresponds to the hypotheses H− 1 and H+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 36 Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (1986) proposed a modified Bonferroni procedure for testing the hypotheses pairs (H− i , Ki), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n, comparing each one sided p-value (pi, qi) to α/n instead of α/2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Bauer et al.’s modified Bonferroni procedure requires condition (A0) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In a similar fashion, if p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , pn fulfil conditions (A0) and (A2), we can consider a modified ˇSid´ak procedure comparing (pi, qi) to 1 − (1 − α)1/n instead of 1 − (1 − α)1/(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The bottom-left plot of Figure 7 shows Bauer et al.’s modified ˇSid´ak procedure for n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We see that for inference about n+, Bauer et al.’s modified ˇSid´ak procedure is uniformly more powerful than directional closed testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' However, directional closed testing provides also inference about n−, whereas Bauer et al.’s procedure does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The top-right plot of Figure 7 shows the adaptive partitioning procedure with Simes test performed at the larger level α∗ = 4α/3 providing level α unconditional error control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Note that the adaptive partitioning procedure performed at level α is not admissible because is dominated by Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' modified ˇSid´ak procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The bottom-right plot of Figure 7 shows Guo and Romano (2015) Holm-type proce- dure (Procedure 3 of their paper), which requires assumptions similar to (A0) − (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Guo and Romano (2015) proposed another procedure (Procedure 2 of their paper) that is uniformly more powerful than Bauer et al.’s modified Bonferroni procedure, altough it does not dominate Bauer et al.’s modified ˇSid´ak procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 37 Directional Closed Testing α/2 α/4 p1 p2 1−, 2− 1−, 2+ 1+, 2+ 1+, 2− 1− 1+ 2− 2+ Unconditional Adaptive Partitioning 2α/3 α/3 p1 p2 Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (1986) 1 − (1 − α)1/2 p1 p2 Guo and Romano (2015) α/(1 + α) α/(2 + α) p1 p2 n+ = 2 n+ ≥ 1 n+ = 1 n+ ≤ 1 n+ = 0 Figure 7: Areas leading to inference about n+ with (1 − α) confidence for the di- rectional closed testing procedure with Simes combining function (top-left plot), the unconditional adaptive partitioning procedure with Simes combining function (top- right plot), Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (1986) procedure with ˇSid´ak correction (bottom-left plot), and Guo and Romano (2015) Holm-type procedure (bottom-right plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Each plot is based on α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 38 C Adaptive confidence bounds for n+ only Let Hr/n : n+ ≤ r − 1, be the partial conjunction (PC) null hypothesis that at most r − 1 hypotheses among H− 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , H− n are false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' and Kr/n : n+ ≥ n−r+1, the PC null hypothesis that at most r − 1 hypotheses among K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , Kn are false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For r = 1, Hr/n is the global null hypothesis that none of the parameters are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For r = 2, rejection of Hr/n leads to establishing minimal replicability in the positive direction (Benjamini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Jaljuli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Since Hr/n is false if and only if every intersection hypothesis of size n − r + 1 is false (Benjamini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', 2009), a valid p-value pr/n for Hr/n is the largest intersection hypothesis p-value, over all intersections of n − r + 1 null hypotheses: pr/n = max {I:I⊆[n],|I|=n−r+1}pI, where pI is the p-value for the intersection hypothesis ∩i∈IH− i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Similarly, a valid p-value qr/n for Kr/n is qr/n = max {I:I⊆[n],|I|=n−r+1}qI, where qI is the p-value for the intersection hypothesis ∩i∈IKi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For example, the PC p-values using Fisher’s combining method (Fisher, 1934) are: pr/n = P � χ2 2(n−r+1) ≥ −2 n � k=r log(p(k)) � , qr/n = P � χ2 2(n−r+1) ≥ −2 n � k=r log(q(k)) � where p(1) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≤ p(n) and q(1) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≤ q(n) denote the sorted values of p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , pn and q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , qn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Note that 1 − p(k) = q(n−k+1) for continuous test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For f that satisfies the monotonicity and symmetry conditions (8) and (9), pr/n = f(p(r), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , p(n)) is a valid p-value, satisfying sup θ∈Hr/n Pθ(pr/n ≤ x) ≤ x ∀x ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The inequality is an equality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', pr/n is uniformly distributed, for the least favorable parameter configuration (LFC) θLF C ∈ Hr/n, for which r − 1 parameters have an infinite value and their corresponding p-values are zero (almost surely), and n − r + 1 parameters are zero and their corresponding p-values are uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' By considering only hypotheses in S, we can avoid including p-values that are stochastically much larger than 39 uniform when their null hypotheses are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore, as in § 2, we shall restrict ourselves to the directions guided by the data, so we shall use for testing Hr/n pr/n = max {I:I⊆[n],|I|=n−r+1}f({2pi : i ∈ I ∩ S−}) = � f � 2p(r), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , 2p(|S−|) � if r ≤ |S−|, 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (17) and for testing Kr/n qr/n = max {I:I⊆[n],|I|=n−r+1}f({2qi : i ∈ I ∩ S+}) = � f � 2q(r), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , 2q(|S+|) � if r ≤ |S+|, 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (18) Procedure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 (Adaptive PC testing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Step 1 Apply Step 1 of procedure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Step 2 Test in order {Hr/n : r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |S−|}, using pr/n in (17), at level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Stop at the first non-rejection, pr/n > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Let l+ α be the number of rejections, with l+ α ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |S−|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' If l+ α = n, return l+ α = u+ α = n, otherwise go to the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Step 3 Test in order {Kr/n : r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n − |S−|}, using qr/n in (18), at level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Stop at the first non-rejection, qr/n > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Let n − u+ α be the number of rejections, with n − u+ α ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n − |S−|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Return l+ α and u+ α (with l+ α ≤ u+ α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Note that the bounds of the procedure will be the same if the testing in order is continued until n in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' This is so because p(|S−|+1)/n > α and q(n−|S−|+1)/n > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The guaranteed coverage is formalized in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Let � pr/n : r ∈ [n] � be valid conditional PC p-values for {Hr/n : r ∈ [n]}, and let � qr/n : r ∈ [n] � be valid conditional PC p-values for {Kr/n : r ∈ [n]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Then l+ α and u+ α satisfy (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Furthermore, the unconditional coverage is Pθ � l+ α ≤ n+ ≤ u+ α � ≥ 1 − (1 − 2−n)α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Suppose that θ is the true parameter value with n+(θ) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We have l+ α ≤ t ≤ u+ α if and only if p(t+1)/n > α and q(n−t+1)/n > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The result follows since conditional on S−, it is only possible to make an error in one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Specifically, if t > |S−|, then 40 p(|S−|+1)/n > α, since it is not possible to reject H(|S−|+1)/n, so it is not possible to err with regard to the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore, if t > |S−|, then Pθ(t /∈ [l+ α (p), u+ α(p)] | S) = Pθ(t > u+ α(p) | S) ≤ Pθ(q(n−t+1)/n ≤ α | S) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Similarly, if t < |S−|, then q(n−|S−|+1)/n > α, since it is not possible to reject K(n−|S−|+1)/n, so it is not possible to err with regard to the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore, if t < |S−|, then Pθ(t /∈ [l+ α (p), u+ α(p)] | S) = Pθ(t < l+ α (p) | S) ≤ Pθ(p(t+1)/n ≤ α | S) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' If t = |S−|, then since the lower bound is at most |S−| and the upper bound is at least n − |S+|, it is not possible to make an error on either bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore, the unconditional error of non-covering n+ is Pθ(t /∈ [l+ α (p), u+ α(p)]) = E � Pθ(t /∈ [l+ α (p), u+ α(p)] | S) � = Eθ � I(t > |S−|)Pθ(t > u+ α(p) | S−) + I(t < |S−|)Pθ(t < l+ α (p) | S−) � ≤ αEθ � I(t > |S−|) + I(t < |S−|) � = αPθ(t ̸= |S−|) ≤ α(1 − 2−n), where the last inequality follows since Pθ(t = |S−|) is bounded below by � {i:θi>0} Pθ(pi ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5) � {j:θj≤0} Pθ(pj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5) ≥ � {i:θi>0} P0(pi ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5) � {j:θj≤0} P0(pj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='5) = 2−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The number of nonpositive parameters, n − n+, can be decomposed into the number of negative parameters n− and the number of parameters with value zero n0, so n = n+ +n− +n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The 1-α confidence bound on the number of positive parameters, [l+ α , u+ α] has also the following interpretation if n0 = 0: with (1 − α) confidence, the number of positive parameters is at least l+ α and the number of negative parameters is at least n − u+ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' However, if n0 > 0 then the probability that the lower bounds do not cover at least one of n+, n− may exceed α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' If n0 = n and all p values are uniform then P(p1/n ≤ α ∪ q1/n ≤ α) ≈ 2α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' as n0 decreases the probability that the lower bounds do not cover at least one parameter decreases from 2α to α (for n0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For a combination function f(), the test for qualitative interactions in Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2019) is rejected at level α if and only if l+ α ≥ 1 and u+ α ≤ n − 1 in the above procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore, if the assumption n0 = 0 is reasonable, then the above procedure 41 complements nicely a conclusion that there is qualitative interaction, by providing with (1 − α) confidence the (interval) estimate of the parameter tested, n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' More generally, a level α test of a generalized qualitative interaction null hypothesis that n+ < a or n+ > b, for predefined 1 ≤ a < b ≤ n − 1, has the following rejection rule: reject if l+ α ≥ a and u+ α ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' To see that this is an α level test, consider the null value θ such that n+(θ) /∈ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Without loss of generality, suppose n+(θ) > b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Then the probability of falsely rejecting the generalized qualitative interaction true null hypothesis is Pθ(l+ α ≥ a and u+ α ≤ b) ≤ Pθ(u+ α ≤ b) ≤ Pθ(u+ α < n+) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 A note on general confidence bounds for n+ If pr/n is a valid p-value for testing Hr/n for r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n, then lα(p) = max{l ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n} : pr/n ≤ α for r = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , l} (20) satisfies Pθ � lα(p) ≤ n+� ≥ 1−α for all θ ∈ Θ, where p0/n ≡ 0 since H0/n : n+ < 0 is always false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Analogously, if qr/n is a valid p-value for testing Kr/n for r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n, then uα(p) = n − max{u ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n} : qr/n ≤ α for r = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , u} (21) satisfies Pθ � n+ ≤ uα(p) � ≥ 1 − α for all θ ∈ Θ, where q0/n ≡ 0 since K0/n : n+ > n is always false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For notational simplicity, we shall often write lα and uα instead of lα(p) and uα(p), but of course these bounds are functions of the p-value vector p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' A straightforward application of the Bonferroni inequality shows that if level α/2 is used for each bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', lα/2 in (20) and uα/2 in (21), then Pθ � lα/2 ≤ n+ ≤ uα/2 � ≥ 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' These bounds where used in Jaljuli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (2022) in order to complement meta-analyses in systematic reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The correction of using α/2 in each direction (instead of α, as in the adaptive PC testing procedure ) is, however, conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Intuitively, the correction should be less severe since in a given configuration, the probability of erring by exceeding one bound is much larger than the probability of erring by exceeding the other bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' To see this, note that Pθ(n+ /∈ [lα/2, uα/2] = Pθ(n+ < lα/2) + Pθ(n+ > uα/2) has value α/2 for the following least favorable parameter configurations (LFCs) when test- ing PC null hypotheses: the positive parameter value is θi = ∞ and the non-positive parameter value is θi = 0 in θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' or the positive parameter value is θi = 0+ (where 0+ is 42 an arbitrary fixed small positive value, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', the machine precision) and the non-positive parameter value is θi = −∞ in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' To see this, note that for the LFC with θi = ∞ for n+ parameters, p(n++1)/n ∼ U(0, 1) and q(n−n++1)/n = 1 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For the LFC with θi = −∞ for n − n+ parameters, q(n−n++1)/n ∼ U(0, 1) and p(n++1)/n = 1 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Non-coverage can occur if the lower bound is violated, so p(n++1)/n ≤ α/2, or if the upper bound is violated, so q(n−n++1)/n ≤ α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore PLF C(n+ /∈ [lα/2, uα/2]) = P(U ≤ α/2) = α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' We conjecture that the coverage guarantee is typically (1 − α) if the lower and upper confidence bounds are lα and uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In particular, whenever the test statistics are continuous, from one dimensional exponential families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Without loss of generality, suppose the first t coordinates are positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , θt > 0 and θt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , θn ≤ 0, and n+ = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Our conjecture is thus that the solution to the following optimization problem is α for a large class of valid PC p-values: max θ Pθ(p(t+1)/n ≤ α) + Pθ(q(n−t+1)/n ≤ α) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' θi > 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , t, θj ≤ 0, j = t + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' To see that this is not true in general for unconditional combination tests (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', using local tests that do not condition on the vector of signs S), consider the following stylized example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Let n = 2 and n+ = 1 be such that θ1 = 0 and θ2 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Assume that the distribution of a p-value from Hi, xi, has the following distribution: P(xi = α) = α, P(xi = 1) = 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Similarly, the distribution of a p-value from Ki, yi = 1 − xi, has distribution: P(yi = α) = α, P(yi = 1) = 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' These p-values are valid since PHi(pi ≤ a) ≤ a, PKi(qi ≤ a) ≤ a, ∀a ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The unconditional PC p-value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=', it is derived from a local test that does not condition on the vector of signs S) for H2/2 and K2/2 is, respectively, p2/2 = max(p1, p2) which has distribution max(1−x1, x2), and q2/2 = max(q1, q2) which has distribution max(x1, 1−x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Therefore: Pθ(1 /∈ [lα, uα] = Pθ(1 < lα) + Pθ(1 > uα) = Pθ(max(p1/2, p2/2) < α) + Pθ(max(q1/2, q2/2) < α) = P((1 − x1, x2) = (0, α)) + P((x1, 1 − x2) = (α, 0)) = 2 × α × (1 − α) > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 43 D Exact shortcuts for Procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1 We first discuss the computation of the confidence bounds l+ α and u+ α for n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Suppose we have observed S− of size |S−| = s, and we want to check whether the test in (13) rejects all the hypotheses JK with |K| = k at level α: fk = max K:|K|=k f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Vandermonde’s convolution �n k � = k � v=0 � s k − v ��n − s v � states that for each v = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , k there are � s k−v ��n−s v � sets K of size k such that |S+ ∩K| = v and |S− ∩ K| = k − v (or equivalently, |S− ∩ Kc| = s − k + v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Then, the maximization problem becomes fk = max v∈{max(0,k−s),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=',min(k,n−s)} max K:|S+∩K|=v, |S−∩K|=k−v f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) For any increasing function f, the maximum has solution with the v largest p-values qi with i ∈ S+ and the s − k + v largest p-values pi with i ∈ S−, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' fk = max v∈{max(0,k−s),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=',min(k,n−s)} f({2p(k−v+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , 2p(s)} ∪ {2q(n−s−v+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , 2q(n−s)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' (22) In order to compute fk in (22) for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n, we can use a nested loop, where the number of iterations of the inner loop (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the index v in (22) from max(0, k − s) to min(k, n − s)) depends on the value of the outer loop’s index (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' k from 0 to n) and the size s of S−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The total complexity for the two loops is O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The maximum number of iterations happens when s ∈ {n/2 − 1, n/2, n/2 + 1} if n is even, and s ∈ {(n − 1)/2, (n + 1)/2} if n is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the mininum number of iterations happens when s ∈ {0, n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Thus we proved the first part of the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Algorithm 2 returns the (1 − α) confidence bounds l+ α and u+ α based on the test in (13), with at most O(n2) computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The bounds are at most as good as those obtained by the Procedure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 44 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For prove the second part of the proposition, it is sufficient to note that if k < s, then the index v in (22) starts at max(0, k − s) = 0 by computing the PC conditional p-value pk+1/n = f({2p(k+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , 2p(s)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Then pk+1/n > α implies fk = max K:|K|=k f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) ≥ pk+1/n > α for any k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , s − 1}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the lower bound l+ α from Algorithm 2 is smaller than or equal to the lower bound of Procedure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Likewise, if k > s then the index v in (22) starts at max(0, k − s) = k − s by computing the PC conditional p-value qn−k+1/n = f({2q(n−k+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , 2q(n−s)}), thus qn−k+1/n > α implies fk = max K:|K|=k f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) ≥ qn−k+1/n > α for any k ∈ {s + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' the upper bound u+ α from Algorithm 2 is larger than or equal to the upper bound of Procedure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 45 Algorithm 2: Shortcut for computing the confidence bounds l+ α and u+ α Input : right-tailed p-values p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , pn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' combining function f(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' level α Output : Confidence bounds lα and uα, p-values fk for Jk : n+ = k, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n Initialize: S− = {i : pi ≤ 1/2}, |S−| = s, fs = 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , as) with a1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≥ as sorted values of {2pi : i ∈ S−} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , bn−s) with b1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≥ bn−s sorted values of {2qi : i ∈ S+} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 1 if s > 0 then 2 for k ← 0 to s − 1 do 3 A ← {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , s − k}, B ← ∅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 4 for v ← 1 to min(k, n − s) + 1 do 5 fk,v ← f({ai, i ∈ A} ∪ {bi, i ∈ B}) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 6 A ← A ∪ {s − k + v}, B ← B ∪ {v} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 7 end 8 fk = maxv{fk,v} 9 end 10 else if s < n then 11 for k ← n to s + 1 do 12 A ← ∅, B ← {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , k − s} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 13 for v ← 1 to min(k, n − s) − (k − s) + 1 do 14 fk,v ← f({ai, i ∈ A} ∪ {bi, i ∈ B}) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 15 A ← A ∪ {v}, B ← B ∪ {k − s + v} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 16 end 17 fk = maxv{fk,v} 18 end 19 l+ α ← min(k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , s} : fk > α) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 20 u+ α ← max(k ∈ {s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , n} : fk > α) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 21 return l+ α , u+ α, f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , fn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Algorithm 2 is just a special case of the following Algorithm for the derivation of the confidence bounds l+ α (I) and u+ α(I) for a generic subset I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' For any I ⊆ [n], Algorithm 1 returns the simultaneous (1−α) confidence bounds l+ α (I) and u+ α(I) in (12) based on the test in (13), with at most O(|I|2·max(1, |Ic|2)) computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' In particular, calculation of the adjusted p-values for Hi and Ki requires O(n2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 46 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Suppose we have observed S− of size |S−| = s, and for any I ⊆ [n] and any v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |I|} we want to check whether the test in (13) rejects all the hypotheses JK with |K ∩ I| = v at level α (or, equivalently, if Jv(I) : n+(I) = v is rejected at level α): fv(I) = max K:|K∩I|=v f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Algorithm 1 computes fv,u(I) = max K:|K∩I|=v, |K∩Ic|=u f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) so that fv(I) = max{fv,u(I), u = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |Ic|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The function f(·) in (3) combines 2pi with i ∈ S− ∩ Kc and 2qi with i ∈ S+ ∩ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Writing K = (K ∩ I) ∪ (K ∩ Ic) and Kc = (Kc ∩ I) ∪ (Kc ∩ Ic) gives fK(I) = f({2pi, i ∈ S− ∩ Kc} ∪ {2qi, i ∈ S+ ∩ K}) = f({2pi, i ∈ S− ∩ Kc ∩ I} ∪ {2pi, i ∈ S− ∩ Kc ∩ Ic} ∪{2qi, i ∈ S+ ∩ K ∩ I} ∪ {2qi, i ∈ S+ ∩ K ∩ Ic}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Consider Vandermonde’s convolutions: �|I| v � = v � k=0 �|S− ∩ I| v − k ��|S+ ∩ I| k � , �|Ic| u � = u � j=0 �|S− ∩ Ic| u − j ��|S+ ∩ Ic| j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The first convolution states that for each k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , v}, there are �|S−∩I| v−k ��|S+∩I| k � sets K such that |K ∩I| = v with |S+ ∩K ∩I| = k and |S− ∩K ∩I| = v −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Likewise, the second convolution states that for each j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , u}, there are �|S−∩I⌋| u−j ��|S+∩Ic| j � sets K such that |K ∩ Ic| = u with |S+ ∩ K ∩ Ic| = j and |S− ∩ K ∩ Ic| = u − j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Then, the maximization problem becomes fv,u(I) = max k∈{k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=',k2}, j∈{j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=',j2} max K:|S+∩K∩I|=k,|S−∩K∩I|=v−k, |S+∩K∩Ic|=j,|S−∩K∩Ic|=u−j fK(I) where k1 = max(0, v − |S− ∩ I|), k2 = min(v, |S+ ∩ I|), j1 = max(0, u − |S− ∩ Ic|) and j2 = min(u, |S+ ∩ Ic|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 47 For any increasing function f(·), the maximum has solution with largest k p-values qi with i ∈ S+ ∩ I, the largest k − v + |S− ∩ I| p-values pi with i ∈ S− ∩ Ic, the largest j p-values qi with i ∈ S+ ∩ I and the largest j − u + |S− ∩ Ic| p-values pi with i ∈ S− ∩ I: fv,u(I) = max k∈{k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=',k2} j∈{j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=',j2} f({a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , ak−v+|S∩I|}∪{b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , bj−u+|S∩Ic|}∪{c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , ck}∪{d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , dj}) where a1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≥ a|S−∩I|, b1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≥ b|S−∩Ic|, c1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≥ c|S+∩I| and d1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' ≥ d|S+∩Ic| denote the sorted values of {2pi : i ∈ S− ∩ I}, {2pi : i ∈ S− ∩ Ic}, {2qi : i ∈ S+ ∩ I} and {2qi : i ∈ S+ ∩ Ic}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Algorithm 1 evaluates fv,u(I) with a nested loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The outer loop executes min(v, |S+ ∩ I|)−max(0, v−|S−∩I|) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' Every time the outer loop executes, the inner loop executes min(v, |S+ ∩ I|) − max(0, u − |S ∩ Ic|) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' As a result, the complexity for evaluating fv,u(I) is O(|I||Ic|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' The complexity for computing fv(I) for v = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |I| is O(|I|2|Ic|2) because it requires to compute fv,u(I) for u = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |Ic| and v = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' , |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' If I = {i}, it takes O(n2) to compute the adjusted p-values ¯pi = f0({i}) and ¯qi = f1({i}) for H− i and Ki, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} +page_content=' 48' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf'} diff --git a/y9FRT4oBgHgl3EQfjTfQ/content/tmp_files/2301.13590v1.pdf.txt b/y9FRT4oBgHgl3EQfjTfQ/content/tmp_files/2301.13590v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..46c1f5df5ff41b5fced67465eae39e0caf2907b5 --- /dev/null +++ b/y9FRT4oBgHgl3EQfjTfQ/content/tmp_files/2301.13590v1.pdf.txt @@ -0,0 +1,2242 @@ +arXiv:2301.13590v1 [math.DS] 31 Jan 2023 +Universal frequency-preserving KAM persistence via modulus of continuity +Zhicheng Tonga 1 , Yong Lia,b,∗ 2 +aCollege of Mathematics, Jilin University, Changchun 130012, P. R. China. +bSchool of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, +Northeast Normal University, Changchun, 130024, P. R. China. +Abstract +In this paper, we study the persistence and remaining regularity of KAM invariant torus under sufficiently small +perturbations of a Hamiltonian function together with its derivatives, in sense of finite smoothness with modulus +of continuity, as a generalization of classical H¨older continuous circumstances. To achieve this goal, we extend the +Jackson approximation theorem to the case of modulus of continuity, and establish a corresponding regularity theorem +adapting to the new iterative scheme. Via these tools, we establish a KAM theorem with sharp differentiability +hypotheses, which asserts that the persistent torus keeps prescribed universal Diophantine frequency unchanged and +reaches the regularity for persistent KAM torus beyond H¨older’s type. +Keywords: Hamiltonian system, KAM torus, frequency-preserving, modulus of continuity, Jackson approximation +theorem. +2020 MSC: 37J40, 70K60 +1. Introduction +The KAM theory mainly concerns the preservation of invariant tori of a Hamiltonian function H(y) under small +perturbations (i.e., H(y) → H (x, y, ε) of freedom n ∈ N+ with ε > 0 sufficiently small), which has a history of more +than sixty years. See, for instance, Kolmogorov and Arnold [2, 3, 4], Moser [13, 12], P¨oschel [16, 17] and etc. As is +known to all, for frequency ω = Hy (y) of the unperturbed system, we often require it to satisfy the following classical +Diophantine condition (or be of Diophantine class τ) +|⟨˜k, ω⟩| ≥ α∗|˜k| +−τ, ∀0 � ˜k ∈ Zn +(1.1) +with respect to τ ≥ n − 1 and some α∗ > 0, where |˜k| := �n +j=1 |˜k j|. Otherwise, the torus may break no matter how +small the perturbation is. Furthermore, to ensure the KAM persistence one also is interested in the minimal order +of derivatives required for H (x, y, ε). Much effort has been devoted on this problem in terms of H¨older continuity, +including constructing counterexamples and reducing the differentiability hypotheses. For some classic foundational +work, see Moser [14], Jacobowitz [9], Zehnder [22, 23], Mather [11], Herman [7, 8], Salamon [19] and etc. It is +worth mentioning that, very recently P¨oschel [18] obtained a KAM theorem on n-dimensional torus (without action +variables) based on a frequency being of Diophantine class τ = n − 1 in (1.1). Specially, he pointed out that the +derivatives of order n need not be continuous, but rather L2 in a certain strong sense. +Back to our concern on Hamiltonian systems with action-angular variables, it is always conjectured that the min- +imum regularity requirement for the Hamiltonian function H is at least C2n. Along with the idea of Moser, the best +known H¨older case Cℓ with ℓ > 2τ + 2 > 2n has been established by Salamon in [19], where the prescribed frequency +is of Diophantine class τ > n − 1 in (1.1) (with full Lebesgue measure and thus reveals the universality of the KAM +persistence), and the remaining regularity of the KAM torus is also showed to be H¨older’s type. More precisely, the +∗Corresponding author at: School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China +1E-mail address : tongzc20@mails.jlu.edu.cn +2E-mail address : liyong@jlu.edu.cn +Preprint submitted to +February 1, 2023 + +resulting solutions are of class Cm with 0 < m < 2ℓ − 2τ − 2, and the function whose graph is the invariant torus is +of class Cm+τ+1. Besides, the differentiability hypotheses is sharp due to the counterexample work of Herman [7, 8] et +al., which will be explained later in section 3.2.1. In the aspect of H¨older’s type, see Bounemoura [5] and Koudjinan +[10] for some new developments. Strictly weaker than H¨older continuity, Albrecht [1] proved a KAM theorem via a +strong Diophantine frequency of class τ = n − 1 in (1.1), which claimed that C2n plus certain modulus of continuity +̟ satisfying the classical Dini condition +� 1 +0 +̟ (x) +x +dx < +∞ +(1.2) +is enough for the KAM persistence. Such strong Diophantine frequencies are continuum many and form a set of zero +Lebesgue measure, see details from [15], therefore the corresponding KAM preservation is usually said to be non- +universal. To the best of our knowledge, there is no other work on KAM via only modulus of continuity except for [1]. +Back to our concern on universal KAM persistence in this paper, the best result so far still requires C2n plus certain +H¨older continuity depending on the Diophantine nonresonance. It is therefore natural that ones should consider the +following questions: +• Can H¨older smoothness in Salamon’s KAM be further weakened into a general form of modulus of continuity? +• If the invariant KAM torus persists, then what kind of smoothness does the torus have (H¨older continuity, or +more general modulus of continuity)? +• Could the prescribed universal Diophantine frequency to be kept unchanged? +• Does there exist a Dini type integrability condition similar to (1.2) that reveals the explicit relation between +nonresonance and regularity? +To answer the above questions, there are at least four difficulties to overcome. Firstly, note that the Jackson +approximation theorem for classical H¨older continuity is no longer valid at present, hence it must be developed to +approximate the perturbed Hamiltonian function H (x, y, ε) in the sense of modulus of continuity, as a crucial step. +Secondly, it is also basic how to establish a corresponding regularity iteration lemma to study the regularity of the +invariant torus and the solution beyond H¨older’s type. Thirdly, we need to set up a new KAM iterative scheme and +prove its uniform convergence via these tools. Fourthly, it is somewhat difficult to extract an equilibrium integrability +condition of nonresonance and regularity from KAM iteration, as well as further touch the remaining regularity. +Indeed, to achieve the main result theorem 2, we apply theorem 1 to construct a series of analytic approximations to +H (x, y, ε) with modulus of continuity, and prove the persistence and regularity of invariant torus via a modified KAM +iteration as well as a generalized Dini type condition. It should be pointed out that our results still admit sharpness +on differentiability C2n due to Herman’s work [7, 8], where he considered the nonexistence of an invariant curve for +an annulus mapping being of H¨older regularity C3−ǫ with any ǫ close to 0+, i.e., C2n = C4 minus arbitrary H¨older +continuity cannot admit KAM persistence when n = 2. +As some new efforts, our theorem 2 applies to a wide range, including non-universal and universal KAM persis- +tence, and reveals the integral relation between regularity and nonresonance. Apart from above, it is well known that +small divisors must lead to the loss of regularity, and our approach gives general estimates of the KAM remaining +regularity without H¨older continuity for the first time. Particularly, as a direct application, our theorem 2 could deal +with the case of general modulus of continuity for H (x, y, ε), such as Logarithmic H¨older continuity case, i.e., for all +0 < |x − ξ| + |y − η| ≤ 1/2, +|∂αH (x, y, ε) − ∂αH (ξ, η, ε)| ≤ +c +(− ln (|x − ξ| + |y − η|))λ +with respect to all α ∈ N2n with |α| = 2n, where n ≥ 2, λ > 1, c, ε > 0 are sufficiently small, (x, y) ∈ Tn × G with +Tn := Rn/Zn, and G ⊂ Rn is a connected closed set with interior points. See section 3 for more details. +This paper is organized as follows. In section 2, we first introduce some notions and properties for modulus +of continuity, and establish a Jackson type approximation theorem based on them (the proof will be postponed to +appendix Appendix B). Then we state our main results in this paper. Namely, considering that the higher-order +2 + +derivatives of Hamiltonian function H with respect to the action-angular variables are only continuous, we present a +KAM theorem (theorem 2) with sharp differentiability hypotheses under certain assumptions, involving a generalized +Dini type integrability condition (H1). The applications of this theorem are given in section 3, including non-universal +(theorem 4) and universal (theorems 5 and 6) KAM persistence. For the former, we reach a conclusion similar to that +in [1]. As to the latter, we provide H¨older and H¨older plus Logarithmic H¨older circumstances, aiming to show the +importance and universality of theorem 2. In particular, an explicit Hamiltonian function H is constructed, which +cannot be studied by KAM theorems for finite smoothness via classical H¨older continuity, but the work generalized +in this paper can be applied. section 4 provides the proof of theorem 2 and is mainly divided into two parts: the first +part deals with the modified KAM steps via only modulus of continuity, while the second part is devoted to giving +an iteration theorem (theorem 7) on regularity, which is used to analyze the remaining smoothness for the persistent +invariant torus. sections 5 to 7 present the proof of theorems 4 to 6 in section 3, respectively. +2. Statement of results +We first give some notions, including the modulus of continuity along with the norm based on it, the semi separa- +bility which will be used in theorem 1, as well as the weak homogeneity which will appear in theorem 2. +Denote by | · | the sup-norm in Rd and the dimension d ∈ N+ may vary throughout this paper. We formulate that in +the limit process, f1(x) = O# (f2(x)) means there are absolute positive constants ℓ1 and ℓ2 such that ℓ1 f2 (x) ≤ f1 (x) ≤ +ℓ2 f2 (x), and f1(x) = O (f2(x)) implies that there exists an absolute positive constant ℓ3 such that |f1(x)| ≤ ℓ3 f2(x), and +finally f1(x) ∼ f2(x) indicates that f1(x) and f2(x) are equivalent. +Definition 2.1. Let ̟(t) > 0 be a nondecreasing continuous function on the interval (0, δ] with respect to some +δ > 0 such that lim +x→0+ ̟ (x) = 0 and lim +x→0+ x/̟ (x) < +∞. Next, we define the following semi norm and norm for a +continuous function f on Rn (f ∈ C0, for short) +�f� +̟ := +sup +x,y∈Rn, 0<|x−y|≤δ +|f (x) − f (y)| +̟ (|x − y|) , |f|C0 := sup +x∈Rn |f (x)| . +We say that f is of Ck,̟ continuous if f has partial derivatives ∂α f for |α| ≤ k ∈ N and satisfies +∥f∥̟ := +� +|α|≤k +�|∂α f|C0 + �∂α f � +̟ +� < +∞. +(2.3) +Denote by Ck,̟ (Rn) the space composed of all functions f satisfying (2.3). +Such a function ̟ is usually referred to as the modulus of continuity of f. It can be seen that the well-known +Lipschitz continuity and H¨older continuity are special cases in the above definition. In particular, for 0 < ℓ � N+, +we denote by f ∈ Cℓ (Rn) the function space in which the higher derivatives in Rn are H¨older continuous, i.e., +the modulus of continuity is of the form ̟{ℓ} +H (x) ∼ xℓ, where {ℓ} ∈ (0, 1) denotes the fractional part of ℓ. As a +generalization of classical H¨older continuity, we define the Logarithmic H¨older continuity with index λ > 0, where +̟λ +LH (x) ∼ 1/(− ln x)λ, and we omit the the range 0 < x ≪ 1 without causing ambiguity. +Remark 2.1. For f : Rn → Ω ⊂ Rd with a modulus of continuity ̟, we modify the above designation to +Ck,̟ (Rn, Ω). +Remark 2.2. It is well known that a mapping defined on a bounded connected closed set in a finite dimensional +space must have a modulus of continuity, see [6]. For example, for a function f(x) defined on [0, 1] ⊂ R1, it automat- +ically admits a modulus of continuity +ωf,δ (x) := +sup +y∈[0,1],0<|x−y|≤δ +|f (x) − f (y)| . +Definition 2.2. Let ̟1 and ̟2 be modulus of continuity on interval (0, δ]. We say that ̟1 is weaker (strictly +weaker) than ̟2 if lim +x→0+ ̟2 (x) /̟1 (x) < +∞ (= 0). +3 + +Remark 2.3. Obviously any modulus of continuity is weaker than Lipschitz’s type, and the Logarithmic H¨older’s +type ̟λ +LH (x) ∼ 1/(− ln x)λ with any λ > 0 is strictly weaker than arbitrary H¨older’s type ̟α +H (x) ∼ xα with any +0 < α < 1. +Definition 2.3 (Semi separability). We say that ̟ in definition 2.1 is semi separable, if for x ≥ 1, there holds +ψ (x) := +sup +0 0, one +verifies that ψ (x) ∼ (ln x)λ = O (x) as x → +∞ in (2.4), and lim +x→0+ ̟λ +LH (x) /̟λ +LH (ax) = 1 < +∞ with all 0 < a < 1 in +(2.5). See more implicit examples in lemmas Appendix A.1 and Appendix A.2, in particular, it is pointed out that a +convex modulus of continuity naturally possesses these two properties. +Next, we give a Jackson type approximation theorem beyond H¨older’s type and some related corollaries based on +definitions 2.1 and 2.3, their proof will be postponed to appendices Appendix B to Appendix D, respectively. +Theorem 1. There is a family of convolution operators +S r f (x) = r−n +� +Rn K +� +r−1 (x − y) +� +f (y) dy, 0 < r ≤ 1, +from C0 (Rn) into the space of entire functions on Cn with the following property. For every k ∈ N, there exists +a constant c (n, k) > 0 such that, for every f ∈ Ck,̟ (Rn) with a semi separable modulus of continuity ̟, every +multi-index α ∈ Nn with |α| ≤ k, and every x ∈ Cn with |Im x| ≤ r, we have +���∂αS r f (x) − P∂α f,k−|α| (Re x; i Im x) +��� ≤ c (n, k) ∥f∥̟rk−|α|̟(r), +(2.6) +where the Taylor polynomial P is defined as follows +P f,k (x; y) := +� +|β|≤k +1 +α!∂β f (x) yα. +Moreover, S r f is real analytic whenever f is real valued. +As a direct consequence of theorem 1, we give the following corollaries 2.1 and 2.2. These results have been +widely used in H¨older’s case, see for instance, [10, 19]. +Corollary 2.1. The approximation function S r f (x) in theorem 1 satisfies +|∂α (S r f (x) − f (x))| ≤ c∗∥f∥̟rk−|α|̟(r) +and +|∂αS r f (x)| ≤ c∗∥f∥̟ +for x ∈ Cn with |Im x| ≤ r, |α| ≤ k, where c∗ = c∗ (n, k) > 0 and c∗ = c∗ (n, k, ̟) > 0 are some universal constants. +Corollary 2.2. If the function f (x) in theorem 1 also satisfies that the period of each variables x1, . . . , xn is 1 and +the integral on Tn is zero, then the approximation function S r f (x) also satisfies these properties. +4 + +We are now in a position to give the frequency-preserving KAM theorem via only modulus of continuity in this +paper. Before this, let’s start with our parameter settings. Let n ≥ 2 (degree of freedom), τ ≥ n − 1 (Diophantine +index), 2τ+2 ≤ k ∈ N+ (differentiable order) and a sufficiently large number M > 0 be given. Consider a Hamiltonian +function H(x, y) : Tn ×G → R with Tn := Rn/Zn, and G ⊂ Rn is a connected closed set with interior points. It follows +from remark 2.2 that H automatically has a modulus of continuity ̟. In view the comments below definition 2.4, +we assume that ̟ admits semi separability (definition 2.3) and weak homogeneity (definition 2.4) without loss of +generality. Besides, we make the following assumptions: +(H1) Integrability condition for modulus of continuity: Assume that H ∈ Ck,̟ (Tn × G) with the above modulus +of continuity ̟. In other words, H at least has derivatives of order k, and the highest derivatives admit the +regularity of ̟. Moreover, ̟ satisfies the Dini type integrability condition +� 1 +0 +̟ (x) +x2τ+3−k dx < +∞. +(2.7) +(H2) Boundedness and nondegeneracy: +∥H∥̟ ≤ M, +������� +�� +Tn Hyy (ξ, 0) dξ +�−1������� ≤ M. +(H3) Diophantine condition: For some α∗ > 0, the frequency ω ∈ Rn satisfies +|⟨˜k, ω⟩| ≥ α∗|˜k| +−τ, ∀0 � ˜k ∈ Zn, |˜k| := +n +� +j=1 +|˜k j|. +(H4) KAM smallness: There holds +� +|α|≤k +�����∂α� +H (x, 0) − +� +Tn H (ξ, 0) dξ +������ ε|α| ++ +� +|α|≤k−1 +����∂α � +Hy (x, 0) − ω +����� ε|α|+τ+1 ≤ Mεk̟ (ε) +(2.8) +for every x ∈ Rn and some constant 0 < ε ≤ ε∗. +(H5) Criticality: For ϕi(x) := xk−(3−i)τ−1̟(x) with i = 1, 2, there exist critical k∗ +i ∈ N+ such that +� 1 +0 +ϕi (x) +xk∗ +i +1 dx < +∞, +� 1 +0 +ϕi (x) +xk∗ +i +2 dx = +∞. +Let us make some comments. +(C1) There seems to be a large number of assumptions above, but they are important conditions abstracted from the +H¨older continuous case, and we have to do so in order to give the KAM theorem in the case of only modulus of +continuity. However, some of such conditions, e.g. (H2)-(H3), are classical, while some are ordinary. +(C2) In view of remark 2.2, H automatically admits a modulus of continuity. The Dini type integrability condition +(2.7) in (H1) is a direct generalization of H¨older’s type, which can be seen in theorem 5. Interestingly, it +becomes the classical Dini condition (1.2) if τ = n − 1 and k = 2τ + 2 = 2n. +(C3) There is a large family of modulus of continuity satisfying the classical Dini condition (1.2), such as the Log- +arithmic H¨older’s type ̟λ +LH (x) ∼ 1/(− ln x)λ with λ > 1, and even more complicated case: the generalized +Logarithmic H¨older’s type +̟̺,λ +GLH (x) ∼ +1 +(ln(1/x))(ln ln(1/x)) · · ·(ln · · · ln +�������� +̺ +(1/x))λ +(2.9) +5 + +with any ̺ ∈ N+ and λ > 1. In particular, ̟λ +LH(x) ∼ ̟1,λ +GLH(x). Note that the above λ > 1 cannot degenerate to +1, otherwise the Dini integral (1.2) diverges. +(C4) According to the properties of Banach algebra, for the H¨older’s type, it is assumed that (H4) only needs the +term of |α| = 0, and does not need higher-order derivatives to satisfy the condition. However, for general +modulus of continuity, it seems not easy to establish the corresponding Banach algebraic properties, we thus +add higher-order derivatives in (H4). Sometimes they can be removed correspondingly. +(C5) The existence of k∗ +i in (H5) is directly guaranteed by (H1), actually this assumption is proposed to investigate +the higher regularity of the persistent KAM torus, that is, the regularity to Ck∗ +i plus certain modulus of continuity. +In general, given an explicit modulus of continuity ̟, such k∗ +i in (H5) are automatically determined by using +asymptotic analysis, see section 3. +Finally, we state the following frequency-preserving KAM theorem under sharp differentiability via only modulus +of continuity: +Theorem 2 (Main Theorem). Assume (H1)-(H4). Then there is a solution +x = u (ξ) , y = v (ξ) +of the following equation with the operator D := +n� +ν=1 +ων ∂ +∂ξν +Du = Hy (u, v) , Dv = −Hx (u, v) , +such that u (ξ) − ξ and v (ξ) are of period 1 in all variables, where u and v are at least C1. +In addition, assume (H5), then there exist ̟i (i = 1, 2) such that u ∈ Ck∗ +1,̟1 (Rn, Rn) and v ◦ u−1 ∈ Ck∗ +2,̟2 (Rn,G). +Particularly, ̟i can be determined as follows +̟i (γ) ∼ γ +� ε +Li(γ) +ϕi (t) +tk∗ +i +2 dt = O# +�� Li(γ) +0 +ϕi (t) +tk∗ +i +1 dt +� +, γ → 0+, +(2.10) +where Li(γ) → 0+ are some functions such that the second relation in (2.10) holds for i = 1, 2. +Remark 2.5. We call such a solution x = u(ξ), y = v(ξ) the KAM one. +Remark 2.6. With the same as in [19], the unperturbed systems under consideration might be non-integrable +(e.g., H = ⟨ω, y⟩ + ⟨A (x) y, y⟩ + · · · ), and the KAM persistence is of frequency-preserving. The main difference from +[19] is that the regularity of the high-order derivatives and the derived smoothness for persistent torus is weakened +to only modulus of continuity from the H¨older’s type. +Remark 2.7. Actually theorem 2 provides a method for determining ̟i with i = 1, 2, see (2.10). For the prescribed +modulus of continuity to Hamiltonian, such as the H¨older and Logarithmic H¨older type, we have to use asymptotic +analysis to derive the concrete continuity of the KAM torus in section 3. +As mentioned forego, the H¨older’s type H ∈ Cℓ(Tn,G) with ℓ > 2τ + 2 (where τ > n − 1 is the Diophantine +exponent) is always regarded as the critical case. Let k = [ℓ]. Then k = 2τ + 2 = 2n (τ = n − 1 at present) +seems to be the critical case in our setting, and our Dini type integrability condition (2.7) becomes the classical Dini +condition (1.2)! But it should be noted that, such Diophantine frequencies with τ = n − 1 can only form a set of +zero Lebesgue measure and are therefore not enough to represent almost all frequencies. In other words, for universal +KAM persistence, we may have to require the generalized Dini condition in (H1), which reveals the deep relationship +between the irrationality for frequency ω, order and continuity of the highest derivatives for the Hamiltonian H. +Obviously, if the highest differentiable order k of H satisfies k ≥ 2τ + 3 or even larger, then (H1) will become trivial +because ̟ does not have a singularity at 0. But our KAM theorem still makes sense, because the regularity of the +persistent torus will also increase. +6 + +3. Applications +In this section, we show certain detailed regularity about KAM torus such as H¨older and Logarithmic H¨older ones +etc. Denote by {a} and [a] the fractional part and the integer part of a ≥ 0, respectively. It should be emphasized +that the Dini type integrability condition (2.7) in (H1) is easy to verify, that is, the KAM persistence is easy to obtain. +However, some techniques of asymptotic analysis are needed to investigate the specific regularity of KAM torus, +which is mainly reflected in the selection of functions Li(γ) (i = 1, 2) in (2.10). In particular, we will explicitly see +the degree of regularity loss caused by small divisors, see for instance, theorems 4 to 6 and the example shown in +section 3.3. +We apply our theorem 2 from two different perspectives. In section 3.1, for the minimum regularity C2n that is +critical under our approach, we investigate KAM preservation in the sense of zero Lebesgue measure (corresponds to +non-universal), i.e., first let k = 2n, then determine Diophantine nonresonance τ = n − 1; while in section 3.2, for the +given Diophantine nonresonance τ > n − 1 of full Lebesgue measure in advance (corresponds to universal), we study +the minimum regularity requirement under our method. In what follows, the modulus of continuity under consider- +ation are always convex near 0+ and therefore automatically admit semi separability as well as weak homogeneity +which we forego. +3.1. Non-universal KAM persistence +Focusing on non-universal KAM persistence for Hamiltonian systems with action-angular variables of freedom +n, Albrecht [1] proved that C2n plus certain modulus of continuity satisfying the classical Dini condition (1.2) for +regularity requirement is enough. The frequencies he used are of Diophantine class τ = n − 1 in (H3), i.e., of zero +Lebesgue measure. However, it is still interesting to study the remaining regularity of the KAM torus, which is still +unknown so far. By applying theorem 2 we directly obtain the following theorem 3 similar to that in [1], therefore +the proof is omitted here. To illustrate our results, we provide an explicit example in theorem 4, and the proof will be +postponed to section 5. +Theorem 3. Let k = 2n and τ = n − 1 be given. Assume that (H1) (H2), (H3) and (H4) hold with a convex +modulus of continuity ̟. That is, the Hamiltonian H only has derivatives of order 2n, the prescribed frequency is of +Diophantine class n − 1, and (H1) turns to the classical Dini condition (1.2). Then the KAM persistence in theorem 2 +could be admited. +Theorem 4. In view of Comment (C3), let the modulus of continuity in theorem 3 be of the generalized Logarithmic +H¨older’s type in (2.9), i.e., +̟̺,λ +GLH (x) ∼ +1 +(ln(1/x))(lnln(1/x)) · · ·(ln · · ·ln +�������� +̺ +(1/x))λ +(3.11) +with ̺ ∈ N+ and λ > 1. Then the remaining regularity in theorem 2 is u ∈ C1,̟1 (Rn, Rn) and v ◦ u−1 ∈ Cn,̟2 (Rn,G), +where +̟1 (x) ∼ ̟2 (x) ∼ +1 +(ln · · · ln +�������� +̺ +(1/x))λ−1 . +(3.12) +Remark 3.1. Particularly (3.11) reduces to the Logarithmic H¨older’s type ̟λ +LH(x) ∼ 1/(− ln x)λ with λ > 1 as +long as ̺ = 1. As can be seen that, the remaining regularity in (3.12) is much weaker than that in (3.11), and it +is indeed very weak if λ > 1 is sufficiently close to 1 (but cannot degenerate to 1, see Comment (C3)), because the +explicit modulus of continuity in (3.12) tends to 0 quite slowly as x → 0+. +3.2. Universal KAM persistence +In this subsection, we always assume that the prescribed Diophantine frequencies ω are of full Lebesgue measure, +that is, τ > n − 1 in (H3). Note that for fixed n, the parameter τ might be very large, and the frequencies being of +Diophantine class τ are at least continuum many. Under such setting, the known minimum regularity requirement +for Hamiltonian H is H¨older’s type Cℓ with ℓ > 2τ + 2, see Salamon [19] and theorem 5 below. Interestingly, if one +considers weaker modulus of continuity, such as C2τ+2 plus Logarithmic H¨older’s type, the above regularity could be +weakened, see our new theorem 6. +7 + +3.2.1. H¨older continuous case +Theorem 5. Let H ∈ Cℓ(Tn,G) with ℓ > 2τ + 2, where ℓ � N+, ℓ − τ � N+ and ℓ − 2τ � N+. That is, H is of Ck,̟ +with k = [ℓ] and ̟(x) ∼ ̟ℓ +H(x) ∼ x{ℓ}. Assume (H2), (H3) and (H4). Then there is a solution x = u (ξ) , y = v (ξ) of +the following equation with the operator D := +n� +ν=1 +ων ∂ +∂ξν +Du = Hy (u, v) , Dv = −Hx (u, v) +such that u (ξ) − ξ and v (ξ) are of period 1 in all variables. In addition, u ∈ Cℓ−2τ−1 (Rn, Rn) and v ◦ u−1 ∈ +Cℓ−τ−1 (Rn,G). +theorem 5 has been completely proved in [19]. Significantly, the differentiability hypotheses under consideration +is sharp, i.e., it is close to the optimal one as in [7, 8], where Herman gave a counterexample about the nonexistence +of an invariant curve for an annulus mapping of C3−ǫ with 0 < ǫ ≪ 1 corresponds to the case n = 2, ℓ = 4 − ε in our +setting, which implies the sharpness of theorem 5. See more from [11, 21]. +3.2.2. H¨older plus Logarithmic H¨older continuous case +To show different modulus of continuity weaker than H¨older’s type, we establish the following theorem 6. One +will see later that theorem 6 employs more complicated asymptotic analysis than theorem 4, and interestingly, the +remaining regularity ̟1 and ̟2 admit different forms. In fact, theorem 6 can completely contain the case of theorem 4, +that is, τ = n − 1, and ̺ = 1 in (3.11). However, in order to distinguish the full Lebesgue measure and zero Lebesgue +measure of Diophantine nonresonance, we show them separately. +Theorem 6. Let τ > n−1 be given and let H ∈ C[2τ+2],̟, where ̟ (x) ∼ x{2τ+2}/(− ln x)λ with λ > 1. Assume (H2), +(H3) and (H4). That is, H is of Ck plus the above ̟ with k = [2τ + 2]. Then there is a solution x = u (ξ) , y = v (ξ) of +the following equation with the operator D := +n� +ν=1 +ων ∂ +∂ξν +Du = Hy (u, v) , Dv = −Hx (u, v) +such that u (ξ) − ξ and v (ξ) are of period 1 in all variables. In addition, letting +̟1 (x) ∼ +1 +(− ln x)λ−1 ∼ ̟λ−1 +LH (x) , +and +̟2 (x) ∼ + +1 +(− ln x)λ−1 ∼ ̟λ−1 +LH (x), +n − 1 < τ ∈ N+, +x{τ} +(− ln x)λ ∼ x{τ}̟λ +LH (x), +n − 1 < τ � N+, +one has that u ∈ C1,̟1 (Rn, Rn) and v ◦ u−1 ∈ C[τ+1],̟2 (Rn,G). +Remark 3.2. Similar to theorem 4, one can also consider the generalized Logarithmic H¨older’s type (3.11) instead +of the Logarithmic H¨older one. Only the latter is presented here for simplicity. +3.3. An explicit example of Logarithmic H¨older’s type +To illustrate the wider applicability of our theorems, we shall present an explicit example strictly beyond H¨older’s +type. Note that the H¨older plus Logarithmic H¨older regularity for H in theorem 6 becomes simpler Logarithmic +H¨older’s type for 2n < 2τ + 2 ∈ N+ (because {2τ + 2} = 0), we therefore consider the following setting. +Recall theorem 6. Let n = 2, τ = 2, k = 6 = [2τ+2], α∗ > 0, λ > 1 and M > 0 be given. Assume that (x, y) ∈ T2×G +with G := {y ∈ R2 : |y| ≤ 1}, and the frequency ω = (ω1, ω2)T ∈ R2 satisfies +|⟨˜k, ω⟩| ≥ α∗|˜k| +−2, ∀0 � ˜k ∈ Z2, |˜k| := |k1| + |k2|, +8 + +i.e., with full Lebesgue measure. Now we shall construct a function for finite smooth perturbation, whose regularity +is C6 plus Logarithmic H¨older’s type ̟λ +LH(r) ∼ 1/(− lnr)λ with index λ > 1. Namely, define +P(r) := + +� r +0 +· · · +� s2 +0 +1 +(1 − ln |s1|)λ ds1 · · ·ds6, +0 < |r| ≤ 1, +0, +r = 0. +Obviously P(r) ∈ C6,̟λ +LH([−1, 1]). Let us consider the perturbed Hamiltonian function below with some constant +0 < ε < ε∗ sufficiently small (ε∗ depends on the constants given above): +H(x, y, ε) = ω1y1 + ω2y2 + 1 +M (y2 +1 + y2 +2) + ε (sin(2πx1) + sin(2πx2) + P(y1) + P (y2)) . +(3.13) +At this point, we have +������� +�� +T2 Hyy (ξ, 0) dξ +�−1������� = +������� +�� +T2 +� 2M−1 +0 +0 +2M−1 +� +dξ +�−1������� += +������ +� 2−1M +0 +0 +2−1M +������� ≤ M < +∞. +In addition, one can verify that H ∈ C6,̟λ +LH(T2 × G) with ̟λ +LH(r) ∼ 1/(− ln r)λ. +However, for ˜α = (0, 0, 6, 0)T with |˜α| = 6 = k, we have +����∂˜αH +� +(0, 0)T, (y1, 0)T, ε +� +− ∂˜αH +� +(0, 0)T, (0, 0)T, ε +����� = +ε +(1 − ln |y1|)λ ≥ εcλ,ℓ|y1|ℓ +for any 0 < ℓ ≤ 1, where cλ,ℓ > 0 is a constant that only depends on λ and ℓ. This implies that H � C6,̟ℓ +H(T2 × G) with +̟ℓ +H(r) ∼ rℓ, i.e., H � C6+ℓ(T2 × G) with any 0 < ℓ ≤ 1, because ̟λ +LH is strictly weaker than ̟ℓ +H, see also remark 2.3. +In other words, the highest derivatives (of order k = 6) of H in (3.13) can be rigorously proved to be Logarithmic +H¨older continuous with index λ > 1, but not any H¨older’s type. Therefore, the finite smooth KAM theorems via clas- +sical H¨older continuity cannot be applied. But, all the assumptions of theorem 6 can be verified to be satisfied, then +the invariant torus persists, and the frequency ω = (ω1, ω2)T for the unperturbed system can remain unchanged. More- +over, the remaining regularity for mappings u and v◦u−1 in theorem 6 could also be determined as u ∈ C1,̟λ−1 +LH (Rn, Rn) +and v ◦ u−1 ∈ C3,̟λ−1 +LH (Rn,G), where ̟λ−1 +LH (r) ∼ 1/(− ln r)λ−1. More precisely, u is at least C1, while v ◦ u−1 is least +C3, and the higher regularity for them is still not any H¨older’s type, but Logarithmic H¨older one with index λ − 1, i.e., +lower than the original index λ > 1, this is because the small divisors causes the loss of regularity. +4. Proof of theorem 2 +Now let us prove theorem 2 by separating two subsections, namely frequency-preserving KAM persistence (sec- +tion 4.1) and further regularity (section 4.2) for KAM torus. For the former, the overall process is similar to that in +[19], but the key points to weaken the H¨older regularity to only modulus of continuity are using theorem 1 and proving +the uniform convergence of the transformation mapping, that is, the convergence of the upper bound series (see (4.24) +and (4.26)). As we will see later, the Dini type integrability condition (2.7) in (H1) guarantees this. As to the latter, +we have to establish a more general regularity iterative theorem (theorem 7) which is not trivial since the resulting +regularity might be somewhat complicated due to asymptotic analysis. +4.1. Frequency-preserving KAM persistence +The proof of the frequency-preserving KAM persistence is organized as follows. Firstly, we construct a series of +analytic approximation functions Hν of H by using theorem 1 and considering (H1) and (H2). Secondly, we shall +construct a sequence of frequency-preserving analytic and symplectic transformations ψν by induction. According +to (H2), (H3) and (H4), the first step of induction is established by applying theorem 8 in appendix Appendix F (or +9 + +Theorem 1 in [19]). Then, combining with weak homogeneity and certain specific estimates we complete the proof +of induction and obtain the uniform convergence of the composite transformations. Finally, in the light of (H5), the +regularity of the KAM torus is guaranteed by theorem 7. +Step1: In view of theorem 1 (we have assumed that the modulus of continuity ̟ admits semi separability and thus +theorem 1 could be applied here), one could approximate H(x, y) by a sequence of real analytic functions Hν(x, y) for +ν ≥ 0 in the strips +|Im x| ≤ rν, |Im y| ≤ rν, rν := 2−νε +around |Re x| ∈ Tn, |Re y| ≤ ρ, such that +�������� +Hν (z) − +� +|α|≤k +∂αH (Re z) (i Im z)α +α! +�������� +≤ c1∥H∥̟rk +ν̟ (rν) , +�������� +Hν +y (z) − +� +|α|≤k−1 +∂αHy (Re z) (i Im z)α +α! +�������� +≤ c1∥H∥̟rk−1 +ν +̟ (rν) , +�������� +Hν +yy (z) − +� +|α|≤k−2 +∂αHyy (Re z) (i Im z)α +α! +�������� +≤ c1∥H∥̟rk−2 +ν +̟ (rν) +(4.14) +for |Im x| ≤ rν, |Im y| ≤ rν, and c1 = c(n, k) is the constant provided in (2.6). +Fix θ = 1/ +√ +2. In what follows, we will construct a sequence of real analytic symplectic transformations z = +(x, y), ζ = (ξ, η), z = φν (ζ) of the form +x = uν (ξ) , y = vν (ξ) + (uν +ξ)T(ξ)−1η +(4.15) +by induction, such that uν (ξ) − ξ and vν (ξ) are of period 1 in all variables, and φν maps the strip |Im ξ| , |η| ≤ θrν+1 into +|Im x| , |y| ≤ rν, |Re y| ≤ ρ, and the transformed Hamiltonian function Kν := Hν ◦ φν satisfies +Kν +ξ (ξ, 0) = 0, Kν +η (ξ, 0) = ω, +(4.16) +i.e., with prescribed frequency-preserving. Namely by verifying certain conditions we obtain z = ψν(ζ) of the form +(4.15) from theorem 8 by induction, mapping |Im ξ| , |η| ≤ rν+1 into |Im x| , |y| ≤ θrν, and ψν (ξ, 0) − (ξ, 0) is of period +1, and (4.16) holds. Here we denote φν := φν−1 ◦ ψν with φ−1 := id (where id denotes the 2n-dimensional identity +mapping and therefore φ0 = ψ0). Further more, theorem 8 will lead to +|ψν (ζ) − ζ| ≤ c (1 − θ) rk−2τ−1 +ν +̟ (rν) , +(4.17) +���ψν +ζ (ζ) − I +��� ≤ crk−2τ−2 +ν +̟ (rν) , +(4.18) +���Kν +ηη (ζ) − Qν (ζ) +��� ≤ crk−2τ−2 +ν +̟ (rν) /2M, +(4.19) +���Uν +x (x) +��� ≤ crk−τ−1 +ν +̟ (rν) , +(4.20) +on |Im ξ| , |η| , |Im x| ≤ rν+1, where S ν (x, η) = Uν (x) + ⟨Vν (x) , η⟩ is the generating function for ψν, and Qν := Kν−1 +ηη , +and I denotes the 2n × 2n-dimensional identity mapping, and +Q0 (z) := +� +|α|≤k−2 +∂αHyy (Re z) (i Im x)α +α! +. +(4.21) +Step2: Here we show that ψ0 = φ0 exists, and it admits the properties mentioned in Step 1. Denote +h(x) := H (x, 0) − +� +Tn H (ξ, 0) dξ, x ∈ Rn. +Then by the first term in (2.8), we have +� +|α|≤k +|∂αh| ε|α| < Mεk̟ (ε) . +(4.22) +10 + +Note that +H0 (x, 0) − +� +Tn H0 (ξ, 0) dξ = H0 (x, 0) − +� +|α|≤k +∂α +xH (Re x, 0) (i Im x)α +α! ++ +� +Tn +� +H (ξ, 0) − H0 (ξ, 0) +� +dξ ++ +� +|α|≤k +∂αh (Re x) (i Im x)α +α! +. +Hence, for |Im x| ≤ θr0 = θε, by using theorem 1, corollary 2.1 and (4.22) we arrive at +�����H0 (x, 0) − +� +Tn H0 (ξ, 0) dξ +����� ≤ 2c1∥H∥̟εk̟ (ε) + Mεk̟ (ε) +≤ cεk̟ (ε) ≤ cεk−2τ−2̟ (ε) · (θε)2τ+2. +Now consider the vector valued function f (x) := Hy (x, 0) − ω for x ∈ Rn. In view of the second term in (2.8), we +have +� +|α|≤k−1 +|∂α f | ε|α| ≤ Mεk−τ−1̟ (ε) . +(4.23) +Note that +H0 +y (x, 0) − ω = H0 +y (x, 0) − +� +|α|≤k−1 +∂α +xHy (Re x, 0) (i Im x)α +α! ++ +� +|α|≤k−1 +∂α f (Re x) (i Im x)α +α! +. +Therefore, for |Im x| ≤ θε, by using (4.14) and (4.23) we obtain that +���H0 +y (x, 0) − ω +��� ≤ c1∥H∥̟εk−1̟ (ε) + Mεk−τ−1̟ (ε) +≤ cεk−τ−1̟ (ε) ≤ cεk−2τ−2̟ (ε) · (θε)τ+1. +Recall (4.21). Then it follows from (4.14) that +���H0 +yy (z) − Q0 (z) +��� ≤ c1∥H∥̟εk−2̟ (ε) ≤ +c +4M εk−2̟ (ε) +≤ +c +4M εk−2τ−2̟ (ε) , |Im x| , |y| ≤ θε, +and +���Q0 (z) +��� ≤ +� +|α|≤k−2 +∥H∥̟ +ε|α| +α! ≤ ∥H∥̟ +� +α∈N2n +ε|α| +α! = ∥H∥̟e2nε ≤ 2M, |Im z| ≤ ε. +Now, by taking r∗ = θε, δ∗ = εk−2τ−2̟ (ε) and using theorem 8 there exists a real analytic symplectic transforma- +tion z = φ0 (ζ) of the form (4.15) (with ν = 0) mapping the strip |Im ξ| , |η| ≤ r1 = r0/2 into |Im x| , |y| ≤ θr0 = r0/ +√ +2, +such that u0 (ξ) − ξ and v0 (ξ) are of period 1 in all variables and the Hamiltonian function K0 := H0 ◦ φ0 satisfies +(4.16) (with ν = 0). Moreover, (4.17)-(4.19) (with ν = 0) hold. +Also assume that +���Kν−1 +ηη (ζ) +��� ≤ Mν−1, +������� +�� +Tn Kν−1 +ηη (ξ, 0) dξ +�−1������� ≤ Mν−1, Mν ≤ M +for |Im x| , |y| ≤ rν. Finally, define +˜H (x, y) := Hν ◦ φν−1 (x, y) +11 + +with respect to |Im x| , |y| ≤ rν. One can verify that ˜H is well defined. +Next we assume that the transformation z = φν−1 (ζ) of the form (4.15) has been constructed, mapping |Im ξ| , |η| ≤ +θrν into |Im x| , |Im y| ≤ rν−1, |Re y| ≤ ρ, and uν−1 (ξ) − ξ, vν−1 (ξ) are of period 1 in all variables, and Kν−1 +ξ +(ξ, 0) = +0, Kν−1 +η +(ξ, 0) = ω. In addition, we also assume that (4.17)-(4.20) hold for 0, . . ., ν − 1. In the next Step 3, we will +verify that the above still hold for ν, which establishes a complete induction. +Step3: We will prove the existence of transformation φν in each step according to the specific estimates below and +theorem 8. +Let |Im x| ≤ θrν. Then φν−1(x, 0) lies in the region where the estimates in (4.14) hold for both Hν and Hν−1. Note +that x �→ Hν−1(φν−1(x, 0)) is constant by (4.16). Then by (4.14), we arrive at the following for |Im x| ≤ θrν +����� ˜H (x, 0) − +� +Tn +˜H (ξ, 0) dξ +����� ≤ 2 +sup +|Im ξ|≤θrν +����Hν � +φν−1 (ξ, 0) +� +− Hν−1 � +φν−1 (ξ, 0) +����� +≤ 2c1∥H∥̟rk +ν̟ (rν) + 2c1∥H∥̟rk +ν−1̟ (rν−1) +≤ crk−2τ−2 +ν +̟ (rν) · r2τ+2 +ν +, +where the weak homogeneity of ̟ with respect to a = 1/2 (see definition 2.4) has been used in the last inequality, +because ̟(rν−1) = ̟(2rν) ≤ c̟(rν) (thus c is independent of ν). For convenience we may therefore not mention it in +the following. +Taking η = 0 in (4.18) we have +���uν−1 +ξ +(ξ) − I +��� ≤ +ν−1 +� +µ=0 +����uµ +ξ (ξ) − uµ−1 +ξ +(ξ) +���� ≤ c +ν−1 +� +µ=0 +rk−2τ−2 +µ +̟ +� +rµ +� +≤ c +∞ +� +µ=0 +� ε +2µ +�k−2τ−2 +̟ +� ε +2µ +� +≤ c +∞ +� +µ=0 +� ε +2µ−1 − ε +2µ +� � ε +2µ +�k−2τ−3 +̟ +� ε +2µ +� +≤ c +∞ +� +µ=0 +� ε/2µ−1 +ε/2µ +̟ (x) +x2τ+3−k dx ≤ c +� 2ε +0 +̟ (x) +x2τ+3−k dx ≤ 1 − θ +(4.24) +for |Im ξ| ≤ θrν, and the Dini type condition (2.7) in (H1) together with Cauchy Theorem are used since ε > 0 is +sufficiently small. Then it leads to +���uν−1 +ξ +(ξ)−1��� ≤ θ−1, |Im ξ| ≤ θrν. +(4.25) +Finally, by (4.25) and (4.14) we obtain that +��� ˜Hy (x, 0) − ω +��� = +����uν−1 +ξ +(x)−1 � +Hν +y +� +φν−1 (x, 0) +� +− Hν−1 +y +� +φν−1 (x, 0) +������ +≤ θ−1 ����Hν +y +� +φν−1 (x, 0) +� +− Hν−1 +y +� +φν−1 (x, 0) +����� +≤ θ−1 � +c1∥H∥̟rk−1 +ν +̟ (rν) + c1∥H∥̟rk−1 +ν−1̟ (rν−1) +� +≤ crk−1 +ν +̟ (rν) +≤ crk−τ−2 +ν +̟ (rν) · rτ+1 +ν +, +and +��� ˜Hyy (z) − Qν (z) +��� = +����uν−1 +ξ +(x)−1 � +Hν +yy +� +φν−1 (z) +� +− Hν−1 +yy +� +φν−1 (z) +�� � +uν−1 +ξ +(x)−1�T���� +≤ θ−2 ����Hν +yy +� +φν−1 (z) +� +− Hν−1 +yy +� +φν−1 (z) +����� +≤ θ−2 � +c1∥H∥̟rk−2 +ν +̟ (rν) + c1∥H∥̟rk−2 +ν−1̟ (rν−1) +� +≤ crk−2τ−2 +ν +̟ (rν) /2M +12 + +for |Im x| , |y| ≤ θrν. Then denote r∗ := rν and δ∗ := crk−2τ−2 +ν +̟ (rν) in theorem 8, we obtain the analytic symplectic +preserving transformation φν of each step, mapping the strip |Im ξ| ≤ θrν, |η| ≤ θrν into |Im x| ≤ rν, |y| ≤ rν, such that +uν (ξ) − ξ and vν (ξ) are of period 1 in all variables, and the transformed Hamiltonian function Kν = Hν ◦ φν satisfies +Kν +ξ (ξ, 0) = 0, Kν +η (ξ, 0) = ω. +Moreover, (4.17)-(4.20) are valid for |Im ξ| , |η| , |Im x| ≤ θrν. +Step4: By (4.18) for 0, . . ., ν − 1 and the arguments in (4.24), there holds +���φν−1 +ζ +(ζ) +��� ≤ 1 + +ν−1 +� +µ=0 +����φµ +ζ (ζ) − φµ−1 +ζ +(ζ) +���� ≤ 1 + +ν−1 +� +µ=0 +�����φµ +ζ (ζ) − I +���� + +����φµ−1 +ζ +(ζ) − I +���� +� +≤ 1 + c +∞ +� +µ=0 +� ε +2µ +�k−2τ−2 +̟ +� ε +2µ +� +≤ 1 + c +� 2ε +0 +̟ (x) +x2τ+3−k dx ≤ 2 +(4.26) +for |Im ξ| , |η| ≤ θrν as long as ε > 0 is sufficiently small, which leads to +���φν (ζ) − φν−1 (ζ) +��� = +���φν−1 (ψν (ζ)) − φν−1 (ζ) +��� +≤ 2 |ψν (ζ) − ζ| ≤ c (1 − θ) rk−2τ−1 +ν +̟ (rν) +for |Im ξ| , |η| ≤ rν+1. Then by Cauchy’s estimate, we obtain that +���φν +ζ (ζ) − φν−1 +ζ +(ζ) +��� ≤ crk−2τ−2 +ν +̟ (rν) , |Im ξ| , |η| ≤ rν+1. +It can be proved in the same way that |φν +ζ (ζ)| ≤ 2 for |Im ξ| , |η| ≤ θrν+1, which implies +|Im z| ≤ 2 |Im ζ| ≤ 2 +� +|Im ξ|2 + |Im η|2 ≤ 2 +� +θ2r2 +ν+1 + θ2r2 +ν+1 = 2rν+1 = rν. +Besides, we have |Re y| ≤ ρ. +Note that +vν ◦ (uν)−1 (x) − vν−1 ◦ +� +uν−1�−1 (x) = +� +uν−1 +ξ +(ξ)−1�TUν +x (ξ) , x := uν−1 (ξ) . +Recall (4.24), by employing the contraction mapping principle we have |Im ξ| ≤ rν+1 if |Im x| ≤ θrν+1 with respect to +x defined above. Then from (4.20) and (4.25) one can verify that +���� +�uν−1 +ξ +(ξ)−1�TUν +x (ξ) +���� ≤ crk−τ−1 +ν +̟ (rν) . +(4.27) +Step5: Finally, we are in a position to prove the convergence of uν and vν, and the regularity of their limit functions. +Note (4.27). Then we have the following analytic iterative scheme +���uν (ξ) − uν−1 (ξ) +��� ≤ crk−2τ−1 +ν +̟ (rν) , |Im ξ| ≤ rν+1, +(4.28) +and +����vν ◦ (uν)−1 (x) − vν−1 ◦ +� +uν−1�−1 (x) +���� ≤ crk−τ−1 +ν +̟ (rν) , |Im x| ≤ θrν+1. +(4.29) +And especially, (4.28) and (4.29) hold when ν = 0 since u0−1 = id and v0−1 = 0. It is obvious to see that the uniform +limits u and v ◦ u−1 of uν and vν ◦ (uν)−1 are at least C1 (in fact, this is implied by the higher regularity studied later +in section 4.2). In addition, the persistent invariant torus possesses the same frequency ω as the unperturbed torus by +(4.16). +13 + +4.2. Iteration theorem on regularity without H¨older’s type +To obtain accurate regularity for u and v ◦ u−1 from the analytic iterative scheme (4.28) and (4.29), we shall +along with the idea of Moser and Salamon to establish an abstract iterative theorem, which provides the modulus of +continuity of the integral form. +Theorem 7. Let n ∈ N+, ε > 0 and {rν}ν∈N = {ε2−ν}n∈N be given, and denote by f : Rn → R the limit of a sequence +of real analytic functions fν (x) in the strips |Im x| ≤ rν such that +f0 = 0, |fν (x) − fν−1 (x)| ≤ ϕ (rν) , ν ≥ 1, +(4.30) +where ϕ is a nondecreasing continuous function satisfying ϕ (0) = 0. Assume that there is a critical k∗ ∈ N such that +� 1 +0 +ϕ (x) +xk∗+1 dx < +∞, +� 1 +0 +ϕ (x) +xk∗+2 dx = +∞. +(4.31) +Then there exists a modulus of continuity ̟∗ such that f ∈ Ck∗,̟∗ (Rn). In other words, the regularity of f is at least +of Ck∗ plus ̟∗. In particular, ̟∗ could be determined as +̟∗ (γ) ∼ γ +� ε +L(γ) +ϕ (t) +tk∗+2 dt = O# +�� L(γ) +0 +ϕ (t) +tk∗+1 dt +� +, γ → 0+, +(4.32) +where L(γ) → 0+ is some function such that the second relation in (4.32) holds. +Proof. Define gν(x) := fν (x) − fν−1 (x) for ν ∈ N+. Determine an integer function �N(γ) : [0, 1] → N+ (note that �N(γ) +can be extended to R+ due to the arguments below, we thus assume that it is a continuous function). Then for the +given critical k∗ ∈ N and x, y ∈ Rn, we obtain the following for all multi-indices α = (α1, . . . , αn) ∈ Nn with |α| = k∗: +�N(|x−y|)−1 +� +ν=1 +|∂αgν (x) − ∂αgν (y)| ≤ |x − y| +�N(|x−y|)−1 +� +ν=1 +|∂αgνx|C0(Rn) ≤ |x − y| +�N(|x−y|)−1 +� +ν=1 +1 +rk∗+1 +ν +ϕ (rν) += 2 |x − y| +�N(|x−y|)−1 +� +ν=1 +� ε +2ν − +ε +2ν+1 +� �2ν +ε +�k∗+2 +ϕ +� ε +2ν +� +≤ c |x − y| +� ε +ε2−�N(|x−y|) +ϕ (t) +tk∗+2 dt, +(4.33) +where Cauchy’s estimate and (4.30) are used in the second inequality, and arguments similar to (4.24) are employed +in (4.33), c > 0 is a universal constant. Besides, we similarly get +∞ +� +ν=�N(|x−y|) +|∂αgν (x) − ∂αgν (y)| ≤ +∞ +� +ν=�N(|x−y|) +2|∂αgν|C0(Rn) ≤ 2 +∞ +� +ν=�N(|x−y|) +1 +rk∗ +ν +ϕ (rν) += 2 +∞ +� +ν=�N(|x−y|) +� ε +2ν − +ε +2ν+1 +� �2ν +ε +�k∗+1 +ϕ +� ε +2ν +� +≤ c +� ε2−�N(|x−y|) +0 +ϕ (t) +tk∗+1 dt. +(4.34) +Now choose �N(γ) → +∞ as γ → 0+ such that +γ +� ε +L(γ) +ϕ (t) +tk∗+2 dt = O# +�� L(γ) +0 +ϕ (t) +tk∗+1 dt +� +:= ̟∗ (γ) , γ → 0+, +(4.35) +where ε2− ˜N(γ)−1 := L (γ) → 0+. This is achievable due to assumption (4.31), Cauchy Theorem and The Intermediate +Value Theorem. Note that the choice of L(γ) (i.e., �N) and ̟∗ is not unique (may up to a constant), and ̟∗ could be +14 + +continuously extended to some given interval (e.g., [0, 1]), but this does not affect the qualitative result. Combining +(4.33), (4.34) and (4.35) we finally arrive at f ∈ Ck∗,̟∗ (Rn) because +|∂α f (x) − ∂α f (y)| ≤ +�N(|x−y|) +� +ν=1 ++ +∞ +� +ν=�N(|x−y|)+1 +|∂αgν (x) − ∂αgν (y)| +≤ c +|x − y| +� ε +ε2−�N(|x−y|)−1 +ϕ (t) +tk∗+2 dt + +� ε2−�N(|x−y|)−1 +0 +ϕ (t) +tk∗+1 dt + +≤ c̟∗ (|x − y|) . +theorem 7 can be extended to the case f : Rn → Rm with n, m ∈ N+ since the analysis is completely the same, +and the strip |Im x| ≤ rν can also be replaced by |Im x| ≤ rν+1 (or ≤ θrν+1). theorem 7 can also be used to estimate the +regularity of solutions of finite smooth homological equations, thus KAM uniqueness theorems in some cases might +be derived, see Section 4 in [19] for instance. However, in order to avoid too much content in this paper, it is omitted +here. +Recall (4.28) and (4.29). Then one can apply theorem 7 on {uν − id}ν (because theorem 7 requires that the initial +value vanishes) and {vν ◦ (uν)−1}ν to directly analyze the regularity of the KAM torus according to (H5), i.e., there +exist ̟i (i = 1, 2) such that u ∈ Ck∗ +1,̟1 (Rn, Rn) and v ◦ u−1 ∈ Ck∗ +2,̟2 (Rn,G). This completes the proof of theorem 2. +5. Proof of theorem 4 +Only need to determine k∗ +i in (H5) and choose functions Li(γ) → 0+ (as γ → 0+) to obtain the modulus of +continuity ̟i in (2.10) for i = 1, 2. Obviously k∗ +1 = 1 and k∗ +2 = n because +� 1 +0 +̟̺,λ +GLH (x) +x +dx < +∞, +� 1 +0 +̟̺,λ +GLH (x) +x2 +dx = +∞. +In view of ϕi(x) in (H5), then by applying lemma Appendix E.1 we get +γ +� ε +Li(γ) +ϕi (t) +tk∗ +i +2 dt = O# +� +γ +� ε +Li(γ) +1 +t2(ln(1/t)) · · ·(ln · · · ln +�������� +̺ +(1/t))λ dt +� += O# +� +γ +� 1/Li(γ) +1/ε +1 +(ln z) · · · (ln · · · ln +�������� +̺ +z)λ dz +� += O# +� +γ +Li (γ) (ln(1/Li (γ)) · · ·(ln · · · ln +�������� +̺ +(1/Li (γ)))λ +� +, +(5.36) +and by direct calculation one arrives at +� Li(γ) +0 +ϕi (t) +tk∗ +i +1 dt = O# +�� ε +Li(γ) +1 +t(ln(1/t)) · · ·(ln · · · ln +�������� +̺ +(1/t))λ dt +� += O# +� +1 +(ln · · · ln +�������� +̺ +(1/Li (γ)))λ−1 +� +. +(5.37) +15 + +Finally, choosing +Li (γ) ∼ +γ +(ln(1/γ)) · · ·(ln · · · ln +�������� +̺ +(1/γ)) → 0+, γ → 0+ +(5.38) +will lead to the second relation in (2.10) for i = 1, 2, and substituting Li(γ) into (5.36) or (5.37) yields that +̟1 (γ) ∼ ̟2 (γ) ∼ +1 +(ln · · · ln +�������� +̺ +(1/γ))λ−1 . +(5.39) +in theorem 2, see (2.10). This proves theorem 4. +6. Proof of theorem 5 +Note that ℓ � N+ implies {ℓ} ∈ (0, 1). Then k = [ℓ] and ̟(x) ∼ ̟ℓ +H(x) ∼ x{ℓ}, i.e., modulus of continuity of +H¨older’s type. Consequently, (H1) can be directly verified because of ℓ > 2τ + 2: +� 1 +0 +̟ (x) +x2τ+3−k dx = +� 1 +0 +x{ℓ} +x2τ+3−[ℓ] dx = +� 1 +0 +1 +x1−(ℓ−2τ−2) dx < +∞. +Here and below, let i be 1 or 2 for simplicity. Recall that ϕi (x) = xk−(3−i)τ−1̟ (x) = x[ℓ]−(3−i)τ−1 · x{ℓ} = xℓ−(3−i)τ−1, and +let +� 1 +0 +ϕi (x) +xk∗ +i +1 dx = +� 1 +0 +1 +xk∗ +i −(ℓ−(3−i)τ−2) dx < +∞, +(6.40) +� 1 +0 +ϕi (x) +xk∗ +i +2 dx = +� 1 +0 +1 +xk∗ +i −(ℓ−(3−i)τ−2)+1 dx = +∞. +(6.41) +Then the critical k∗ +i in (H5) could be uniquely chosen as k∗ +i := [ℓ − (3 − i) τ − 1] ∈ N+ since ℓ − (3 − i) τ − 1 � N+. +Further, letting Li (γ) = γ → 0+ yields that +� Li(γ) +0 +ϕi (t) +tk∗ +i +1 dt = O# +�� γ +0 +1 +t1−{ℓ−(3−i)τ−2} dt +� += O# � +γ{ℓ−(3−i)τ−2}� +and +γ +� ε +Li(γ) +ϕi (t) +tk∗ +i +2 dt = O# +� +γ +� ε +γ +1 +t2−{ℓ−(3−i)τ−2} dt +� += O# � +γ{ℓ−(3−i)τ−2}� +. +This leads to H¨older’s type +̟i (γ) ∼ (Li (γ)){ℓ−(3−i)τ−2} ∼ γ{ℓ−(3−i)τ−2} ∼ ̟{ℓ−(3−i)τ−2} +H +(γ) +due to (2.10) in theorem 2. +By observing k∗ +i + {ℓ − (3 − i) τ − 2} = ℓ − (3 − i) τ − 1 we finally arrive at u ∈ +Cℓ−2τ−1 (Rn, Rn) and v ◦ u−1 ∈ Cℓ−τ−1 (Rn,G). This proves theorem 5. +7. Proof of theorem 6 +Firstly, note that k = [2τ + 2] and ̟ (x) ∼ x{2τ+2}/(− ln x)λ with λ > 1, then (H1) holds since +� 1 +0 +̟ (x) +x2τ+3−k dx = O# +�� 1/2 +0 +x{2τ+2} +x2τ+3−[2τ+2](− ln x)λ dx +� += O# +�� 1/2 +0 +1 +x(− ln x)λ dx +� +< +∞. +16 + +Secondly, in view of ϕi(x) in (H5), we have +� 1 +0 +ϕi (x) +xk∗ +i +1 dx = O# +�� 1/2 +0 +1 +xk∗ +i −(i−1)τ(− ln x)λ dx +� +i = 1, 2. +This leads to critical k∗ +1 = 1 and k∗ +2 = [τ + 1] in (H5). Here one uses the following fact: for given λ > 1, +� 1/2 +0 +1 +xι(− ln x)λ dx < +∞, +� 1/2 +0 +1 +xι+1(− ln x)λ dx = +∞ +if and only if ι ∈ (0, 1]. +Next, we investigate the KAM remaining regularity through certain complicated asymptotic analysis. One notices +that the analysis of ̟1 with all τ > n − 1 and ̟2 with n − 1 < τ � N+ is the same as ̺ = 1 in theorem 4, i.e., L1(γ) +and L2(γ) could be chosen as γ/(− ln γ) → 0+, see (5.38) with ̺ = 1. Therefore, in view of (5.39), we arrive at +̟1 (γ) ∼ +1 +(− ln γ)λ−1 ∼ ̟λ−1 +LH (γ) , τ > n − 1, +and +̟2 (γ) ∼ +1 +(− ln γ)λ−1 ∼ ̟λ−1 +LH (γ) , n − 1 < τ � N+. +However, the asymptotic analysis for ̟2 becomes different when n − 1 < τ ∈ N+. Note that {τ} ∈ (0, 1) and +[τ + 1] − τ = [τ] + 1 − τ = 1 − {τ} at present. Hence, by applying (E.1) in lemma Appendix E.2 we get +� L2(γ) +0 +ϕ2 (t) +tk∗ +2+1 dt = O# +�� L2(γ) +0 +1 +t[τ+1]−τ(− ln t)λ dt +� += O# +�� L2(γ) +0 +1 +t1−{τ}(− ln t)λ dt +� += O# +�� +∞ +1/L2(γ) +1 +z1+{τ}(ln z)λ dz +� += O# +� +(L2 (γ)){τ} +(ln (1/L2 (γ)))λ +� +, +(7.42) +and similarly according to (E.2) in lemma Appendix E.2 we have +γ +� ε +L2(γ) +ϕ2 (t) +tk∗ +2+2 dt = O# +� +γ +� 1/L2(γ) +1/ε +1 +z{τ}(ln z)λ dz +� += O# +� γ(L2 (γ)){τ}−1 +(ln (1/L2 (γ)))λ +� +. +(7.43) +Now let us choose L2(γ) ∼ γ → 0+, i.e., different from that when n − 1 < τ ∈ N+, one verifies that the second relation +in (2.10) holds for i = 2, and substituting L2(γ) into (7.42) or (7.43) yields that +̟2 (γ) ∼ +γ{τ} +(− ln γ)λ ∼ γ{τ}̟λ +LH (γ) , n − 1 < τ � N+ +due to (2.10) in theorem 2. This proves theorem 6. +Appendix A. Semi separability and weak homogeneity for modulus of continuity +Lemma Appendix A.1. Let a modulus continuity ̟ be given. If ̟ is piecewise continuously differentiable and +̟′ ≥ 0 is nonincreasing, then ̟ admits semi separability in definition 2.3. As a consequence, if ̟ is convex near 0+, +then it is semi separable. +Proof. Assume that ̟ is continuously differentiable without loss of generality. Then we obtain semi separability due +to +sup +0 0 sufficiently small, one verifies that +̟ (x) = x · ̟ (x) − ̟ (0+) +x − 0 +≤ x · ̟ (ax) − ̟ (0+) +ax − 0 += a−1̟ (ax) , +for 0 < a < 1, which leads to weak homogeneity +lim +x→0+ +̟ (x) +̟ (ax) ≤ a−1 < +∞. +Appendix B. Proof of theorem 1 +Proof. For the completeness of the analysis we give a very detailed proof. An outline of the strategy for the proof is +provided: we firstly construct an approximation integral operator by the Fourier transform of a compactly supported +function, and then present certain properties of the operator (note that these preparations are classical); finally, we +estimate the approximation error in the sense of modulus of continuity. +Let +K (x) = +1 +(2π)n +� +Rn +�K (ξ) ei⟨x,ξ⟩dξ, x ∈ Cn +be an entire function whose Fourier transform +�K (ξ) = +� +Rn K (x) e−i⟨x,ξ⟩dx, ξ ∈ Rn +is a smooth function with compact support, contained in the ball |ξ| ≤ 1, that satisfies �K (ξ) = �K (−ξ) and +∂α�K (0) = + +1, α = 0, +0, α � 0. +(B.1) +Next, we assert that +����∂βF ��K (ξ)� (z) +���� ≤ +c (β, p) +(1 + |Re z|)p e|Im z|, max {1, |β|} ≤ p ∈ R. +(B.2) +Note that we assume �K ∈ C∞ +0 (Rn) and supp �K ⊆ B (0, 1), thus +����(1 + |z|)k∂βF +��K (ξ) +� +(z) +���� ≤ +� +|γ|≤k +����zγ∂βF ��K (ξ)� (z) +���� = +� +|γ|≤k +����∂β+γF ��K (ξ)� (z) +����, +(B.3) +where F represents the Fourier transform. Since ∂β+γF ��K (ξ)� (z) ∈ C∞ +0 (B (0, 1)) does not change the condition, we +only need to prove that +����F ��K (ξ)� (z) +���� ≤ cke|Im z|. +Obviously +����F ��K (ξ)� (z) +���� ≤ +1 +(2π)n +� +Rn +����K (ξ) +���e−⟨Im z,ξ⟩dξ ≤ +c +(2π)n +� +B(0,1) +e|⟨Im z,ξ⟩|dξ ≤ ce|Im z|, +where c > 0 is independent of n. Then assertion (B.2) is proved by recalling (B.3). +The inequality in (B.2) is usually called the Paley-Wiener Theorem, see also Chapter III in [20]. As we will see +later, it plays an important role in the subsequent verification of definitions, integration by parts and the translational +feasibility according to Cauchy’s integral formula. +18 + +Next we assert that K : Cn → R is a real analytic function with the following property +� +Rn (u + iv)α∂βK (u + iv) du = + +(−1)|α|α!, +α = β, +0, +α � β, +(B.4) +for u, v ∈ Rn and multi-indices α, β ∈ Nn. In order to prove assertion (B.4), we first consider proving the following for +x ∈ Rn: +� +Rn xα∂βK (x) dx = + +(−1)|α|α!, +α = β, +0, +α � β. +(B.5) +Case1: If α = β, then +� +Rn xα∂βK (x) dx = +� +Rn +� n +� +j=1 +x +αj +j +�� n +� +j=1 +∂ +αj +xj +� +K (x) dx += (−1)α1α1! +� +Rn−1 +� n +� +j=2 +x +αj +j +�� n +� +j=2 +∂ +αj +xj +� +K (x2, · · · , xn) dx2 · · ·dxn += · · · = (−1)α1+···+αnα1! · · ·αn! +� +R +K (xn) dxn += (−1)|α|α!�K (0) = (−1)|α|α!. +Case2: There exists some α j ≤ β j − 1, let j = 1 without loss of generality. Then +� +Rn xα∂βK (x) dx = +� +Rn +� n +� +j=1 +x +αj +j +�� n +� +j=1 +∂ +β j +xj +� +K (x) dx += (−1)β1−α1 +� +Rn +� n +� +j=2 +x +αj +j +�� n +� +j=2 +∂ +β j +xj +� +∂β1−α1 +x1 +K (x) dx = 0. +Case3: Now we have α1 ≥ β1, and some α j ≥ β j + 1 (otherwise α = β). Let j = 1 without loss of generality. At this +time we first prove a conclusion according to (B.1). Since +∂α�K (0) = (−i)|α| +� +Rn xαK (x) dx = 0, α � 0, +then it follows that +� +Rn xαK (x) dx = 0, α � 0. +Hence, we arrive at +� +Rn xα∂βK (x) dx = (−1) +n� +j=1(β j−αj) � +Rn +� +xα1−β1 +1 +� � n +� +j=2 +x +αj−β j +j +� +K (x) dx = 0. +This proves (B.5). Next, we will consider a complex translation of (B.5) and prove that +� +Rn (u + iv)α∂βK (u + iv) du = + +(−1)|α|α!, +α = β, +0, +α � β. +Actually one only needs to pay attention to (B.2), and the proof is completed according to Cauchy’s integral formula. +Finally, we only prove that +S rp = p, p : Rn → R. +(B.6) +19 + +In fact, only real polynomials need to be considered +p = xα = +n +� +j=1 +x +αj +j . +It can be obtained by straight calculation +S rp = r−n +� +Rn K +� x − y +r +� +n +� +j=1 +yαj +j dy = +� +Rn K (z) +n +� +j=1 +� +rzj + x j +�αjdz += +� n +� +j=1 +xαj +j +� � +Rn K (z) dz + +� +γ +ϕγ (r, x) +� +Rn zγK (z) dz = +n +� +j=1 +xαj +j = p, +where ϕγ (r, x) are coefficients independent of z. +As to the complex case one only needs to perform complex translation to obtain +p (u; iv) = S rp (u + iv) = +� +Rn K +� +ir−1v − η +� +p (u; rη) dη. +The above preparations are classic, see also [19]. Next we begin to prove the Jackson type approximation theorem +via only modulus of continuity. We will make use of (B.6) in case of the Taylor polynomial +pk (x; y) := P f,k (x; y) = +� +|α|≤k +1 +α!∂α f (x) yα +of f with k ∈ N. Note that +|f (x + y) − pk (x; y)| = +������ +� 1 +0 +k(1 − t)k−1 � +|α|=k +1 +α! (∂α f (x + ty) − ∂α f (x)) yαdt +������ +for every x, y ∈ Rn. +Define the following domains to partition Rn: +Ω1 := +� +η ∈ Rn : |η| < δr−1� +, Ω2 := +� +η ∈ Rn : |η| ≥ δr−1� +. +We have to use different estimates in the above two domains, which are abstracted as follows. If 0 < |y| < δ, we obtain +that +|f (x + y) − pk (x; y)| ≤ +� 1 +0 +k(1 − t)k−1 � +|α|=k +1 +α! · �∂α f � +̟̟ (t |y|) · |yα| dt +≤ c (n, k) ∥ f∥̟ +� 1 +0 +̟ (t |y|) dt · |yα| +≤ c (n, k) ∥ f∥̟̟ (|y|) |yα| +≤ c (n, k) ∥ f∥̟̟ (|y|) |y|k. +(B.7) +If |y| ≥ δ, one easily arrives at +|f (x + y) − pk (x; y)| ≤ +� 1 +0 +k(1 − t)k−1 � +|α|=k +1 +α! · 2|∂α f |C0 · |yα| dt +≤ c (n, k) ∥f∥̟ |yα| +≤ c (n, k) ∥f∥̟|y|k. +(B.8) +20 + +The H¨older inequality has been used in (B.7) and (B.8) with k ≥ 1, αi ≥ 1, µi = k/αi ≥ 1 without loss of generality: +|yα| = +n +� +i=1 +|yi|αi ≤ +n +� +i=1 +1 +µi +|yi|αiµi ≤ +n +� +i=1 +|yi|k ≤ +n +� +i=1 +|y|k = n|y|k. +Now let x = u + iv with u, v ∈ Rn and |v| ≤ r. Fix p = n + k + 2, and let c = c (n, k) > 0 be a universal constant, +then it follows that +|S r f (u + iv) − pk (u; iv)| ≤ +� +Rn K +� +ir−1v − η +� +|f (u + rη) − pk (u; rη)| dη +≤ c +� +Rn +er−1v +(1 + |η|)p |f (u + rη) − pk (u; rη)| dη +≤ c +� +Rn +1 +(1 + |η|)p |f (u + rη) − pk (u; rη)| dη += c +� +Ω1 ++ +� +Ω2 +1 +(1 + |η|)p |f (u + rη) − pk (u; rη)| dη +:= c (I1 + I2) . +As it can be seen later, I1 is the main part while I2 is the remainder. +Recall remark 2.4 and (2.4). Hence the following holds due to (B.7) +I1 = +� +Ω1 +1 +(1 + |η|)p |f (u + rη) − pk (u; rη)| dη +≤ +� +|η|<δr−1 +1 +(1 + |η|)p · c∥f∥̟̟ (|rη|) |rη|kdη +≤ +� +|η|<δr−1 +1 +(1 + |η|)p · c∥f∥̟̟ (r) ψ(|η|)|rη|kdη +≤ c∥ f∥̟rk̟(r) +� δr−1 +0 +wk+n +(1 + w)p dw +≤ c∥ f∥̟rk̟(r) +� +∞ +0 +wk+n +(1 + w)p dw +≤ c∥ f∥̟rk̟(r). +(B.9) +In view of (B.8), we have +I2 = +� +Ω2 +1 +(1 + |η|)p |f (u + rη) − pk (u; rη)| dη +≤ +� +|η|≥δr−1 +1 +(1 + |η|)p · c∥f∥̟|rη|kdη +≤ c∥ f∥̟rk +� +∞ +δr−1 +wk+n−1 +(1 + w)p dw +≤ c∥ f∥̟rk +� +∞ +δr−1 +1 +wp−k−n+1 dw +≤ c∥ f∥̟rk+2. +(B.10) +By (B.9) and (B.10), we finally arrive at +|S r f (u + iv) − pk (u; iv)| ≤ c∥f∥̟rk̟(r) +due to lim +r→0+ r/̟ (r) < +∞ in definition 2.1. This proves theorem 1 for |α| = 0. As to |α| � 0, the result follows from +the fact that S r commutes with ∂α. We therefore finish the proof of theorem 1. +21 + +Appendix C. Proof of corollary 2.1 +Proof. Only the analysis of case |α| = 0 is given. In view of theorem 1 and (B.7), we obtain that +|S r f (x) − f (x)| ≤ +���S r f (x) − P f,k (Re x; i Im x) +��� + +���P f,k (Re x; i Im x) − f (x) +��� +≤ c∗∥f∥̟rk̟(r), +(C.1) +where the constant c∗ > 0 depends on n and k. Further, by (C.1) we have +|S r f (x)| ≤ |S r f (x) − f (x)| + |f (x)| +≤ c∗∥ f∥̟rk̟(r) + ∥f∥̟ ≤ c∗∥f∥̟, +provided a constant c∗ > 0 depending on n, k and ̟. This completes the proof. +Appendix D. Proof of corollary 2.2 +Proof. It is easy to verify that +S r f (x + 1) = 1 +rn +� +Rn K +� x − (y − 1) +r +� +f (y) dy = 1 +rn +� +Rn K +� x − u +r +� +f (u + 1) du += 1 +rn +� +Rn K +� x − u +r +� +f (u) du = S r f (x) . +According to Fubini’s theorem, we obtain +� +Tn S r f (x) dx = 1 +rn +� +Rn +� +Tn K +� x − y +r +� +f (y) dy += 1 +rn +� +Rn K +�m +r +� �� +Tn f (x + m) dx +� +dm = 0. +This completes the proof. +Appendix E. Asymptotic analysis in estimates +Here we provide some useful asymptotic results, all of which can be proved by L’Hopital’s rule or by integration +by parts, thus the proof is omitted here. +Lemma Appendix E.1. Let ̺ ∈ N+, λ > 1 and some M > 0 sufficiently large be fixed. Then for X → +∞, there +holds +� X +M +1 +(ln z) · · · (ln · · ·ln +�������� +̺ +z)λ dz = O# +� +X +(ln X) · · ·(ln · · · ln +�������� +̺ +X)λ +� +. +Lemma Appendix E.2. Let 0 < σ < 1, λ > 1 and some M > 0 sufficiently large be fixed. Then for X → +∞, we +have +� X +M +1 +zσ(ln z)λ dz = O# +� X1−σ +(ln X)λ +� +, +(E.1) +and +� +∞ +X +1 +z1+σ(ln z)λ dz = O# +� +1 +Xσ(ln X)λ +� +. +(E.2) +22 + +Appendix F. KAM theorem for quantitative estimates +Here we give a KAM theorem for quantitative estimates, which is used in theorem 2 in this paper. See Theorem 1 +in Salamon’s paper [19] for case τ > n − 1; as to τ = n − 1, the proof is relatively trivial (in fact, just slightly modify +Lemma 2 in [19]). +Theorem 8. Let n ≥ 2, τ ≥ n − 1, 0 < θ < 1, and M ≥ 1 be given. Then there are positive constants δ∗ and c such +that cδ∗ ≤ 1/2 and the following holds for every 0 < r∗ ≤ 1 and every ω ∈ Rn that satisfies (1.1). +Suppose that H(x, y) is a real analytic Hamiltonian function defined in the strip |Im x| ≤ r∗, |y| ≤ r∗, which is of +period 1 in the variables x1, . . ., xn and satisfies +�����H (x, 0) − +� +Tn H (ξ, 0) dξ +����� ≤ δ∗r∗2τ+2, +���Hy (x, 0) − ω +��� ≤ δ∗r∗τ+1, +���Hyy (x, y) − Q (x, y) +��� ≤ cδ∗ +2M , +for |Im x| ≤ r∗, |y| ≤ r∗, where 0 < δ∗ ≤ δ∗, and Q (x, y) ∈ Cn×n is a symmetric (not necessarily analytic) matrix valued +function in the strip |Im x| ≤ r, |y| ≤ r and satisfies in this domain +|Q (z)| ≤ M, +������� +�� +Tn Q (x, 0) dx +�−1������� ≤ M. +Then there exists a real analytic symplectic transformation z = φ (ζ) of the form +z = (x, y) , ζ = (ξ, η) , x = u (ξ) , y = v (ξ) + uT +ξ (ξ)−1η +mapping the strip |Im ξ| ≤ θr∗, |η| ≤ θr∗ into |Im x| ≤ r∗, |y| ≤ r∗, such that u (ξ) − ξ and v (ξ) are of period 1 in all +variables and the Hamiltonian function K := H ◦ φ satisfies +Kξ (ξ, 0) = 0, Kη (ξ, 0) = ω. +Moreover, φ and K satisfy the estimates +|φ (ζ) − ζ| ≤ cδ∗ (1 − θ) r∗, +���φζ (ζ) − I +��� ≤ cδ∗, +���Kηη (ζ) − Q (ζ) +��� ≤ cδ∗ +M , +���v ◦ u−1 (x) +��� ≤ cδ∗r∗τ+1, +for |Im ξ| ≤ θr∗, |η| ≤ θr∗, and |Im x| ≤ θr∗. +Acknowledgments +This work was supported in part by National Basic Research Program of China (Grant No. 2013CB834100), +National Natural Science Foundation of China (Grant No. 12071175, Grant No. 11171132, Grant No. 11571065), +Project of Science and Technology Developmentof Jilin Province (Grant No. 2017C028-1,Grant No. 20190201302JC), +and Natural Science Foundation of Jilin Province (Grant No. 20200201253JC). +References +[1] J. 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P¨oschel, ¨Uber invariante Tori in differenzierbaren Hamiltonschen Systemen, Beitr¨age zur Differentialgeometrie [Contributions to Differ- +ential Geometry], 3 (1980), p. 103. +[17] J. +P¨oschel, +Integrability +of +Hamiltonian +systems +on +Cantor +sets, +Comm. +Pure +Appl. +Math., +35 +(1982), +pp. +653–696, +https://doi.org/10.1002/cpa.3160350504. +[18] J. P¨oschel, KAM below Cn, https://arxiv.org/abs/2104.01866. +[19] D. A. Salamon, The Kolmogorov-Arnold-Moser theorem, Math. Phys. Electron. J., 10 (2004), pp. Paper 3, 37. +[20] E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32. Princeton University +Press, Princeton, N.J., (1971), pp. x+297. +[21] F. Takens, A C1 counterexample to Moser’s twist theorem, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math., 33 (1971), pp. 378–386. +[22] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math., 28 (1975), +pp. 91–140, https://doi.org/10.1002/cpa.3160280104. +[23] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. II, Comm. Pure Appl. Math., 29 (1976), +pp. 49–111, https://doi.org/10.1002/cpa.3160290104. +24 + diff --git a/y9FRT4oBgHgl3EQfjTfQ/content/tmp_files/load_file.txt b/y9FRT4oBgHgl3EQfjTfQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..edb3772119d5d3170f479ede38968244ae327300 --- /dev/null +++ b/y9FRT4oBgHgl3EQfjTfQ/content/tmp_files/load_file.txt @@ -0,0 +1,956 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf,len=955 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='13590v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='DS] 31 Jan 2023 Universal frequency-preserving KAM persistence via modulus of continuity Zhicheng Tonga 1 , Yong Lia,b,∗ 2 aCollege of Mathematics, Jilin University, Changchun 130012, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' bSchool of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Abstract In this paper, we study the persistence and remaining regularity of KAM invariant torus under sufficiently small perturbations of a Hamiltonian function together with its derivatives, in sense of finite smoothness with modulus of continuity, as a generalization of classical H¨older continuous circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' To achieve this goal, we extend the Jackson approximation theorem to the case of modulus of continuity, and establish a corresponding regularity theorem adapting to the new iterative scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Via these tools, we establish a KAM theorem with sharp differentiability hypotheses, which asserts that the persistent torus keeps prescribed universal Diophantine frequency unchanged and reaches the regularity for persistent KAM torus beyond H¨older’s type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Keywords: Hamiltonian system, KAM torus, frequency-preserving, modulus of continuity, Jackson approximation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' 2020 MSC: 37J40, 70K60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Introduction The KAM theory mainly concerns the preservation of invariant tori of a Hamiltonian function H(y) under small perturbations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=', H(y) → H (x, y, ε) of freedom n ∈ N+ with ε > 0 sufficiently small), which has a history of more than sixty years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' See, for instance, Kolmogorov and Arnold [2, 3, 4], Moser [13, 12], P¨oschel [16, 17] and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' As is known to all, for frequency ω = Hy (y) of the unperturbed system, we often require it to satisfy the following classical Diophantine condition (or be of Diophantine class τ) |⟨˜k, ω⟩| ≥ α∗|˜k| −τ, ∀0 � ˜k ∈ Zn (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='1) with respect to τ ≥ n − 1 and some α∗ > 0, where |˜k| := �n j=1 |˜k j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Otherwise, the torus may break no matter how small the perturbation is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Furthermore, to ensure the KAM persistence one also is interested in the minimal order of derivatives required for H (x, y, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Much effort has been devoted on this problem in terms of H¨older continuity, including constructing counterexamples and reducing the differentiability hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' For some classic foundational work, see Moser [14], Jacobowitz [9], Zehnder [22, 23], Mather [11], Herman [7, 8], Salamon [19] and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' It is worth mentioning that, very recently P¨oschel [18] obtained a KAM theorem on n-dimensional torus (without action variables) based on a frequency being of Diophantine class τ = n − 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Specially, he pointed out that the derivatives of order n need not be continuous, but rather L2 in a certain strong sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Back to our concern on Hamiltonian systems with action-angular variables, it is always conjectured that the min- imum regularity requirement for the Hamiltonian function H is at least C2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Along with the idea of Moser, the best known H¨older case Cℓ with ℓ > 2τ + 2 > 2n has been established by Salamon in [19], where the prescribed frequency is of Diophantine class τ > n − 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='1) (with full Lebesgue measure and thus reveals the universality of the KAM persistence), and the remaining regularity of the KAM torus is also showed to be H¨older’s type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' More precisely, the ∗Corresponding author at: School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China 1E-mail address : tongzc20@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='jlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='cn 2E-mail address : liyong@jlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='cn Preprint submitted to February 1, 2023 resulting solutions are of class Cm with 0 < m < 2ℓ − 2τ − 2, and the function whose graph is the invariant torus is of class Cm+τ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Besides, the differentiability hypotheses is sharp due to the counterexample work of Herman [7, 8] et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=', which will be explained later in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' In the aspect of H¨older’s type, see Bounemoura [5] and Koudjinan [10] for some new developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Strictly weaker than H¨older continuity, Albrecht [1] proved a KAM theorem via a strong Diophantine frequency of class τ = n − 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='1), which claimed that C2n plus certain modulus of continuity ̟ satisfying the classical Dini condition � 1 0 ̟ (x) x dx < +∞ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='2) is enough for the KAM persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Such strong Diophantine frequencies are continuum many and form a set of zero Lebesgue measure, see details from [15], therefore the corresponding KAM preservation is usually said to be non- universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' To the best of our knowledge, there is no other work on KAM via only modulus of continuity except for [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Back to our concern on universal KAM persistence in this paper, the best result so far still requires C2n plus certain H¨older continuity depending on the Diophantine nonresonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' It is therefore natural that ones should consider the following questions: Can H¨older smoothness in Salamon’s KAM be further weakened into a general form of modulus of continuity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' If the invariant KAM torus persists, then what kind of smoothness does the torus have (H¨older continuity, or more general modulus of continuity)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Could the prescribed universal Diophantine frequency to be kept unchanged?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Does there exist a Dini type integrability condition similar to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='2) that reveals the explicit relation between nonresonance and regularity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' To answer the above questions, there are at least four difficulties to overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Firstly, note that the Jackson approximation theorem for classical H¨older continuity is no longer valid at present, hence it must be developed to approximate the perturbed Hamiltonian function H (x, y, ε) in the sense of modulus of continuity, as a crucial step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Secondly, it is also basic how to establish a corresponding regularity iteration lemma to study the regularity of the invariant torus and the solution beyond H¨older’s type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Thirdly, we need to set up a new KAM iterative scheme and prove its uniform convergence via these tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Fourthly, it is somewhat difficult to extract an equilibrium integrability condition of nonresonance and regularity from KAM iteration, as well as further touch the remaining regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Indeed, to achieve the main result theorem 2, we apply theorem 1 to construct a series of analytic approximations to H (x, y, ε) with modulus of continuity, and prove the persistence and regularity of invariant torus via a modified KAM iteration as well as a generalized Dini type condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' It should be pointed out that our results still admit sharpness on differentiability C2n due to Herman’s work [7, 8], where he considered the nonexistence of an invariant curve for an annulus mapping being of H¨older regularity C3−ǫ with any ǫ close to 0+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=', C2n = C4 minus arbitrary H¨older continuity cannot admit KAM persistence when n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' As some new efforts, our theorem 2 applies to a wide range, including non-universal and universal KAM persis- tence, and reveals the integral relation between regularity and nonresonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Apart from above, it is well known that small divisors must lead to the loss of regularity, and our approach gives general estimates of the KAM remaining regularity without H¨older continuity for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Particularly, as a direct application, our theorem 2 could deal with the case of general modulus of continuity for H (x, y, ε), such as Logarithmic H¨older continuity case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=', for all 0 < |x − ξ| + |y − η| ≤ 1/2, |∂αH (x, y, ε) − ∂αH (ξ, η, ε)| ≤ c (− ln (|x − ξ| + |y − η|))λ with respect to all α ∈ N2n with |α| = 2n, where n ≥ 2, λ > 1, c, ε > 0 are sufficiently small, (x, y) ∈ Tn × G with Tn := Rn/Zn, and G ⊂ Rn is a connected closed set with interior points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' See section 3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' In section 2, we first introduce some notions and properties for modulus of continuity, and establish a Jackson type approximation theorem based on them (the proof will be postponed to appendix Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Then we state our main results in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Namely, considering that the higher-order 2 derivatives of Hamiltonian function H with respect to the action-angular variables are only continuous, we present a KAM theorem (theorem 2) with sharp differentiability hypotheses under certain assumptions, involving a generalized Dini type integrability condition (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' The applications of this theorem are given in section 3, including non-universal (theorem 4) and universal (theorems 5 and 6) KAM persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' For the former, we reach a conclusion similar to that in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' As to the latter, we provide H¨older and H¨older plus Logarithmic H¨older circumstances, aiming to show the importance and universality of theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' In particular, an explicit Hamiltonian function H is constructed, which cannot be studied by KAM theorems for finite smoothness via classical H¨older continuity, but the work generalized in this paper can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' section 4 provides the proof of theorem 2 and is mainly divided into two parts: the first part deals with the modified KAM steps via only modulus of continuity, while the second part is devoted to giving an iteration theorem (theorem 7) on regularity, which is used to analyze the remaining smoothness for the persistent invariant torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' sections 5 to 7 present the proof of theorems 4 to 6 in section 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Statement of results We first give some notions, including the modulus of continuity along with the norm based on it, the semi separa- bility which will be used in theorem 1, as well as the weak homogeneity which will appear in theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Denote by | · | the sup-norm in Rd and the dimension d ∈ N+ may vary throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' We formulate that in the limit process, f1(x) = O# (f2(x)) means there are absolute positive constants ℓ1 and ℓ2 such that ℓ1 f2 (x) ≤ f1 (x) ≤ ℓ2 f2 (x), and f1(x) = O (f2(x)) implies that there exists an absolute positive constant ℓ3 such that |f1(x)| ≤ ℓ3 f2(x), and finally f1(x) ∼ f2(x) indicates that f1(x) and f2(x) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Let ̟(t) > 0 be a nondecreasing continuous function on the interval (0, δ] with respect to some δ > 0 such that lim x→0+ ̟ (x) = 0 and lim x→0+ x/̟ (x) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Next, we define the following semi norm and norm for a continuous function f on Rn (f ∈ C0, for short) �f� ̟ := sup x,y∈Rn, 0<|x−y|≤δ |f (x) − f (y)| ̟ (|x − y|) , |f|C0 := sup x∈Rn |f (x)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' We say that f is of Ck,̟ continuous if f has partial derivatives ∂α f for |α| ≤ k ∈ N and satisfies ∥f∥̟ := � |α|≤k �|∂α f|C0 + �∂α f � ̟ � < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='3) Denote by Ck,̟ (Rn) the space composed of all functions f satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Such a function ̟ is usually referred to as the modulus of continuity of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' It can be seen that the well-known Lipschitz continuity and H¨older continuity are special cases in the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' In particular, for 0 < ℓ � N+, we denote by f ∈ Cℓ (Rn) the function space in which the higher derivatives in Rn are H¨older continuous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=', the modulus of continuity is of the form ̟{ℓ} H (x) ∼ xℓ, where {ℓ} ∈ (0, 1) denotes the fractional part of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' As a generalization of classical H¨older continuity, we define the Logarithmic H¨older continuity with index λ > 0, where ̟λ LH (x) ∼ 1/(− ln x)λ, and we omit the the range 0 < x ≪ 1 without causing ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' For f : Rn → Ω ⊂ Rd with a modulus of continuity ̟, we modify the above designation to Ck,̟ (Rn, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' It is well known that a mapping defined on a bounded connected closed set in a finite dimensional space must have a modulus of continuity, see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' For example, for a function f(x) defined on [0, 1] ⊂ R1, it automat- ically admits a modulus of continuity ωf,δ (x) := sup y∈[0,1],0<|x−y|≤δ |f (x) − f (y)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Let ̟1 and ̟2 be modulus of continuity on interval (0, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' We say that ̟1 is weaker (strictly weaker) than ̟2 if lim x→0+ ̟2 (x) /̟1 (x) < +∞ (= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' 3 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Obviously any modulus of continuity is weaker than Lipschitz’s type, and the Logarithmic H¨older’s type ̟λ LH (x) ∼ 1/(− ln x)λ with any λ > 0 is strictly weaker than arbitrary H¨older’s type ̟α H (x) ∼ xα with any 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='3 (Semi separability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content=' We say that ̟ in definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FRT4oBgHgl3EQfjTfQ/content/2301.13590v1.pdf'} +page_content='1 is semi separable, if for x ≥ 1, there holds ψ (x) := sup 0