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+arXiv:2301.04412v1  [stat.ME]  11 Jan 2023
+Observational Studies (2023)
+Submitted ; Published
+RobustIV and controlfunctionIV: Causal Inference for Linear
+and Nonlinear Models with Invalid Instrumental Variables
+Taehyeon Koo
+tk587@stat.rutgers.edu
+Department of Statistics
+Rutgers University
+Piscataway, NJ 08854
+Youjin Lee
+youjin lee@brown.edu
+Department of Biostatistics
+Brown University
+Providence, RI 02912
+Dylan S. Small
+dsmall@wharton.upenn.edu
+Department of Statistics
+The Wharton School, University of Pennsylvania
+Philadelphia, PA 19104
+Zijian Guo
+zijguo@stat.rutgers.edu
+Department of Statistics
+Rutgers University
+Piscataway, NJ 08854
+Abstract
+We present R software packages RobustIV and controlfunctionIV for causal inference
+with possibly invalid instrumental variables. RobustIV focuses on the linear outcome model.
+It implements the two-stage hard thresholding method to select valid instrumental variables
+from a set of candidate instrumental variables and make inferences for the causal effect in
+both low- and high-dimensional settings. Furthermore, RobustIV implements the high-
+dimensional endogeneity test and the searching and sampling method, a uniformly valid
+inference method robust to errors in instrumental variable selection. controlfunctionIV
+considers the nonlinear outcome model and makes inferences about the causal effect based
+on the control function method. Our packages are demonstrated using two publicly avail-
+able economic data sets together with applications to the Framingham Heart Study.
+Keywords: Instrumental variable selection, confidence interval, nonlinear outcome model,
+control function, maximum clique
+©2023 Taehyeon Koo, Youjin Lee, Dylan S. Small, and Zijian Guo.
+
+Koo, Lee, Small, and Guo
+1. Introduction
+A common problem in making causal inferences from observational studies is that there may
+be unmeasured confounders. The instrumental variable (IV) method is one of the most use-
+ful methods to estimate the causal effect when there might exist unmeasured confounding.
+The validity of IV methods relies on that the constructed IVs satisfy the following three
+assumptions simultaneously (Wooldridge, 2010, e.g.): conditioning the measured covariates,
+(A1) the IVs are associated with the treatment;
+(A2) the IVs are independent with the unmeasured confounders;
+(A3) the IVs have no direct effect on the outcome.
+The main challenge of applying IV-based methods in practice is that the proposed IVs
+might not satisfy the above assumptions (A1)-(A3). For example, in studying the causal
+effect of education on earning, the proximity of school (Angrist and Krueger, 1991; Card,
+1999) has been used as an instrumental variable. However, this instrument might be re-
+lated to other factors, such as socioeconomic status, which could affect one’s earnings.
+Also, there might be other advantages due to the proximity; for instance, people living
+close to college could be more likely to be exposed to vocational programs linked to col-
+leges. So, the instrument could have a direct effect on earnings. In addition, the problem
+of IVs not satisfying assumptions (A1) to (A3) is a fundamental problem in Mendelian
+Randomization (MR), whose goal is to estimate the causal effect of exposure on the disease
+by using genetic variants as instruments.
+These genetic variants might violate assump-
+tions (A2) and (A3) due to pleiotropic effects (Bowden, Davey Smith, and Burgess, 2015;
+Bowden, Davey Smith, Haycock, and Burgess, 2016; Kang, Zhang, Cai, and Small, 2016).
+This paper presents the R packages RobustIV and controlfunctionIV, implementing
+robust causal inference approaches proposed in Guo, Kang, Cai, and Small (2018a,b); Guo
+(2021); Guo and Small (2016); Li and Guo (2020).
+The implemented inference methods
+choose the valid IVs among a set of candidate IVs that may violate the assumptions (A2)
+and (A3). The proposed methods target both linear and nonlinear causal effects. We also
+include the algorithm implementation for settings with high-dimensional covariates and IVs.
+In the package RobustIV, we implement robust and high-dimensional IV algorithms for
+models assuming a constant and linear treatment effect.
+We implement the Two Stage
+Hard Thresholding (TSHT) proposed in Guo et al. (2018b), which selects valid IVs based on
+a voting method. The selected IVs are then used to infer the linear treatment effect. Addi-
+tionally, RobustIV implements uniformly valid confidence intervals proposed in Guo (2021),
+which guarantees valid coverage even if there are errors in selecting valid IVs. RobustIV also
+contains the high-dimensional endogeneity test proposed in Guo et al. (2018a), generalizing
+the Durbin-Wu-Hausman test (Durbin, 1954; Wu, 1973; Hausman, 1978).
+2
+
+RobustIV and controlfunctionIV
+We implement several control function methods in the package controlfunctionIV
+to infer causal effects under nonlinear outcome models. We implement the control func-
+tion for the continuous outcome variable by showing it as the two-stage least squares
+(TSLS) estimator with an augmented set of IVs (Guo and Small, 2016). We further follow
+Guo and Small (2016) to test the validity of the augmented set of IVs and construct the
+pretest estimator by comparing the control function estimator and the TSLS estimator. We
+implement the probit control function method for the binary outcome and make inferences
+for the conditional average treatment effect (CATE) with possibly invalid IVs. Moreover,
+the controlfunctionIV package implements the SpotIV method proposed in Li and Guo
+(2020) for the semi-parametric outcome model with possibly invalid IVs.
+In R, there are well-developed IV methods when all IVs are assumed to satisfy the
+assumptions (A1) to (A3), such as AER by Kleiber and Zeileis (2008) and ivmodel by
+Kang, Jiang, Zhao, and Small (2020). The main difference in our packages RobustIV and
+controlfunctionIV is that we allow for invalid IVs and leverage the multiple IVs to
+learn the validity of the candidate IVs. We shall mention other R packages implementing
+causal inference approaches with possibly invalid IVs: sisvive implemented the method
+proposed in Kang et al. (2016) to estimate the treatment effect under the majority rule;
+CIIV by Windmeijer, Liang, Hartwig, and Bowden (2021) considered the causal inference
+approaches with possibly invalid IVs for the low-dimensional linear outcome model. In con-
+trast, our packages RobustIV and controlfunctionIV are designed under a broader frame-
+work by allowing for linear and nonlinear outcome models with low- and high-dimensional
+IVs and covariates. Moreover, our package RobustIV provides a uniformly valid confidence
+interval robust to the errors in separating valid and invalid IVs.
+The GitHub repository at https://github.com/bluosun/MR-GENIUS implemented the
+MR Genius method (Tchetgen, Sun, and Walter, 2021), generalizing the method in Lewbel
+(2012) and leveraging the heteroscedastic regression errors in the treatment model to identify
+the causal parameter. The R package TSCI implemented the two-stage curvature identifica-
+tion method proposed in Guo and B¨uhlmann (2022), which leveraged the machine learning
+methods to capture the nonlinearity in the treatment and identify the treatment effect
+with possibly invalid instruments. In contrast, our packages use the different identification
+conditions from Tchetgen et al. (2021); Lewbel (2012); Guo and B¨uhlmann (2022).
+The paper is organized as follows. In Section 2, we review the methods implemented in
+RobustIV under the linear outcome model; in Section 3, we discuss the inference approaches
+for the nonlinear outcome models implemented in controlfunctionIV. In Section 4, we
+demonstrate the usage of RobustIV and controlfunctionIV by analyzing economics data
+sets from Angrist and Krueger (1991) and Mroz data. In Section 5, we demonstrate our
+packages with an MR application to analyze the data from Framingham Heart Study (FHS).
+3
+
+Koo, Lee, Small, and Guo
+Notation.
+Let Rp be the set of real numbers with dimension p. For any vector v ∈ Rp,
+vj denotes its jth element, v−j denotes whole v except for j-th index, and ∥v∥0 denotes
+the number of non-zero elements in v. For any n × p matrix M, denote the (i, j) entry by
+Mij, the ith row by Mi· , the jth column by M·j, and the transpose of M by MT; also,
+MIJ denotes the submatrix of M consisting of rows specified by the set I ⊂ {1, ..., n} and
+columns specified by the set J ⊂ {1, ..., p}, MI· denotes the submatrix of M consisting of
+rows indexed by the set I and all columns, and M·J denotes the submatrix of M consisting of
+columns specified by the set J and all rows. Ip denotes p × p identity matrix.
+1 denotes the
+indicator function. Φ denotes the CDF of the standard normal distribution. For a sequence
+of random variable Xn, we use Xn
+d→ X to denote that Xn converges to X in distribution.
+2. Linear outcome models
+Throughout the paper, we consider n i.i.d.
+observations.
+For 1 ≤ i ≤ n, let Yi ∈ R,
+Di ∈ R, Zi· ∈ Rpz, and Xi· ∈ Rpx denote the outcome, the treatment, the instruments, and
+the baseline covariates, respectively. This section reviews the robust instrumental variable
+approaches in Guo et al. (2018a,b); Guo (2021), which are implemented in the RobustIV
+package. We demonstrate the usage of RobustIV in Sections 4.1 and 4.2.
+2.1 Model assumption
+We assume the following outcome model with possibly invalid IVs (Small, 2007; Kang et al.,
+2016; Guo et al., 2018a; Windmeijer et al., 2021)
+Yi = Diβ + ZT
+i·π + XT
+i·φ + ǫi,
+E[ǫiZi·] = 0, E[ǫiXi·] = 0.
+(1)
+This is the linear structural model in econometrics (Wooldridge, 2010). Here, we aim to
+estimate the constant causal effect β ∈ R. If Di is correlated with ǫi in the model (1), we
+say it is an endogenous variable, and we cannot use popular estimators such as the OLS
+estimator. We also assume the linear association model for the treatment
+Di = ZT
+i·γ + XT
+i·ψ + δi,
+E[δiZi·] = 0, E[δiXi·] = 0.
+(2)
+As a remark, the errors in (1) and (2) are allowed to be heteroscedastic. In (1) and (2),
+πj = 0 if j-th IV satisfies the exclusion restriction conditions (A2) and (A3), and γj ̸= 0 if
+it satisfies the strong IV assumption (A1).
+We discuss the causal interpretation of the above model (1) using the potential outcome
+framework (Small, 2007; Kang et al., 2016). Let Y (d,z)
+i
+be the potential outcome if individual
+i were to receive the treatment d and the instruments z. For two possible values of the
+treatment d′, d and instruments z′, z, if we assume the following potential outcomes model
+Y (d′,z′)
+i
+− Y (d,z)
+i
+= (d′ − d)β + (z′ − z)Tκ,
+E[Y (0,0)
+i
+|Zi·, Xi·] = XT
+i·φ + ZT
+i·η,
+(3)
+4
+
+RobustIV and controlfunctionIV
+and define π = κ + η, and ǫi = Y (0,0)
+i
+− E[Y (0,0)
+i
+|Zi·, Xi·], we obtain the model (1).
+By combining (1) and (2), we obtain the reduced form models of Y and D as
+Yi = ZT
+i·Γ + XT
+i·Ψ + ξi,
+E[ξiZi·] = 0, E[ξiXi·] = 0,
+(4)
+Di = ZT
+i·γ + XT
+i·ψ + δi,
+E[δiZi·] = 0, E[δiXi·] = 0.
+(5)
+Here, Γ = βγ +π, Ψ = βψ +φ are reduced form parameters and ξi = βδi +ǫi is the reduced
+form error term.
+We introduce identifiability conditions for models (4) and (5).
+Let S be the set of
+relevant IVs, i.e., S = {1 ≤ j ≤ pz : γj ̸= 0} and V be the set of relevant and valid IVs, i.e.,
+V = {j ∈ S : πj = 0}. The set S contains all candidate IVs that are strongly associated
+with the treatment. The set V is a subset of S, which contains all candidate IVs satisfying
+all classical IV assumptions. The main challenge is that the set V is not known a priori in
+the data analysis. Additional identifiability conditions are needed for identifying the causal
+effect without any prior knowledge of V. The majority rule is introduced to identify causal
+effects with invalid IVs (Bowden et al., 2016; Kang et al., 2016).
+Condition 1 (Majority Rule). More than half of the relevant IVs are valid: |V| > |S|/2.
+The following plurality rule is a weaker identification condition than the majority rule
+(Hartwig, Davey Smith, and Bowden, 2017; Guo, Kang, Cai, and Small, 2018b).
+Condition 2 (Plurality Rule). The valid instruments form a plurality compared to the
+invalid instruments: |V| > maxc̸=0 |{j ∈ S : πj/γj = c}|.
+We present two inference methods for β utilizing the majority and plurality: two stage
+hard thresholding in Section 2.2 and searching and sampling in Section 2.3. To present the
+methods, we consider the reduced form estimators (�Γ⊺, �γ⊺)⊺ satisfying
+√n
+���Γ
+�γ
+�
+−
+�
+Γ
+γ
+��
+d→ N2pz
+�
+02pz,
+�
+VΓ
+C
+CT
+Vγ
+��
+.
+(6)
+We use �VΓ, �C, and �Vγ to denote consistent estimators of asymptotic covariance matrix
+terms. In low dimensions, we estimate the reduced form (Γ⊺, γ⊺)⊺ by the OLS estimator
+(�Γ⊺, �γ⊺)⊺ and estimate the variance covariance matrices by sandwich estimators; see the
+detailed construction in Section 2 of Guo (2021). In high-dimensional settings, we can con-
+struct (�Γ⊺, �γ⊺)⊺ as the debiased Lasso estimator (Belloni, Chernozhukov, and Wang, 2011;
+Javanmard and Montanari, 2014; Guo, Kang, Cai, and Small, 2018b); see more details in
+Section 4.1 of Guo et al. (2018b).
+5
+
+Koo, Lee, Small, and Guo
+2.2 Two stage hard thresholding (TSHT)
+The TSHT consists of two steps: the first step is to screen out the weak IVs, and the
+second step is to screen out invalid IVs. Specifically, the first step of TSHT is to estimate
+the set S of relevant IVs by �S =
+�
+1 ≤ j ≤ pz : |�γj| ≥ λ1
+�
+�Vγ
+jj/n
+�
+, where λ1 > 0 is a tuning
+parameter adjusting the testing multiplicity.
+The second thresholding step estimates the set V of valid instruments. Our main strategy
+is to assume that one IV is valid and evaluate whether the other IVs are valid from the
+point of view of that IV. Particularly, for j ∈ �S, we assume the j-th IV to be valid (i.e.,
+πj = 0) and construct an estimator �π−j of π−j using the equation π−j = Γ−j − β[j]γ−j
+with β[j] = Γj/γj, and get the standard error of the estimator. We test whether π−j = 0
+by comparing �π−j to a threshold, calculated as multiplying the standard error of �π−j by
+a tuning parameter λ2, which is a Bonferroni correction adjusting for testing multiplicity.
+Using the above test procedures, we construct a voting matrix ˜Π ∈ R| �S|×| �
+S| where ˜Πj,k = 1
+indicates that the k-th and j-th IVs agree with each other to be valid. Finally, we get a
+symmetric voting matrix �Π by setting �Πj,k = min{˜Πj,k, ˜Πk,j}.
+Once we get �Π, we estimate V by two options. Let VMk denote the number of votes
+that the kth IV, with k ∈ �S, received from other candidates of IVs. First, we define �V by
+the set of IVs that receive a majority and a plurality of votes (Guo et al., 2018b)
+�VMP := {k ∈ �S : VMk > | �S|/2} ∪ {k ∈ �S : VMk = max
+l∈ �
+S
+VMl}.
+(7)
+The next method is to estimate V by the maximum clique method. We can generate a graph
+G with indexes belonging to �S and the adjacency matrix as �Π. That is, the indexes j, k ∈ �S
+are connected if and only if �Πj,k = 1. Then as suggested in Windmeijer et al. (2021), we can
+estimate �VMC as the maximum clique of the graph G, which is the largest fully connected sub-
+graph of G (Csardi and Nepusz, 2006). Note that there might be several maximum cliques.
+In this case, each maximum clique forms an estimator of V and our proposal reports several
+causal effect estimators based on each maximum clique.
+We further illustrate the definitions of �VMP and �VMC using the following example. Consider
+pz = 8 with {z1, z2, z3, z4} being valid and {z5, z6, z7} being invalid with the same invalidity
+level, and z8 being invalid IV with a different invalidity level.
+The left side of Table 1
+corresponds to an ideal setting where the valid and invalid IVs are well separated and the
+valid IVs {z1, z2, z3, z4} only vote for each other. In this case, �VMC = �VMP = {z1, z2, z3, z4}.
+On the right side of Table 1, we consider the setting that the invalidity level of z5 might be
+mild and the IV z5 receives the votes from three valid IVs {z2, z3, z4}. In this case, �VMP =
+{z2, z3, z4, z5}. In contrast, there are two maximum cliques {z1, z2, z3, z4} and {z2, z3, z4, z5}
+and �VMC can be either of these two.
+6
+
+RobustIV and controlfunctionIV
+z1
+z2
+z3
+z4
+z5
+z6
+z7
+z8
+z1
+✓
+✓
+✓
+✓
+X
+X
+X
+X
+z2
+✓
+✓
+✓
+✓
+X
+X
+X
+X
+z3
+✓
+✓
+✓
+✓
+X
+X
+X
+X
+z4
+✓
+✓
+✓
+✓
+X
+X
+X
+X
+z5
+X
+X
+X
+X
+✓
+✓
+✓
+X
+z6
+X
+X
+X
+X
+✓
+✓
+✓
+X
+z7
+X
+X
+X
+X
+✓
+✓
+✓
+X
+z8
+X
+X
+X
+X
+X
+X
+X
+✓
+Votes
+4
+4
+4
+4
+3
+3
+3
+1
+z1
+z2
+z3
+z4
+z5
+z6
+z7
+z8
+z1
+✓
+✓
+✓
+✓
+X
+X
+X
+X
+z2
+✓
+✓
+✓
+✓
+✓
+X
+X
+X
+z3
+✓
+✓
+✓
+✓
+✓
+X
+X
+X
+z4
+✓
+✓
+✓
+✓
+✓
+X
+X
+X
+z5
+X
+✓
+✓
+✓
+✓
+✓
+✓
+X
+z6
+X
+X
+X
+X
+✓
+✓
+✓
+X
+z7
+X
+X
+X
+X
+✓
+✓
+✓
+X
+z8
+X
+X
+X
+X
+X
+X
+X
+✓
+Votes
+4
+5
+5
+5
+6
+3
+3
+1
+Table 1: The left voting matrix �Π denotes that all valid IVs {z1, z2, z3, z4} vote each other
+but not any other invalid IV. The right voting matrix �Π denotes that the locally
+invalid IV z5 receives votes from valid IVs {z2, z3, z4} and invalid IVs {z6, z7}.
+Once we have �V, we can construct an efficient point estimator �β for β in a low-
+dimensional setting via one-step iteration as follows. First, we construct an initial estimator
+˜β =
+�γT
+�V
+˜
+A�Γ�V
+�γT
+�V
+˜
+A�γ�V , where ˜A = �Σ�V,�V −�Σ�V,�Vc �Σ−1
+�Vc,�Vc �Σ�Vc,�V, �Σ = 1
+n
+�n
+i=1 Wi·WT
+i· , and Wi· = (ZT
+i·, XT
+i·)T.
+Next, we get a point estimator �β by one-step iteration (Holland and Welsch, 1977)
+�β =
+�γT
+�V �A�Γ�V
+�γT
+�V �A�γ�V
+,
+where
+�A = [( �VΓ − 2˜β �C + ˜β2 �Vγ)�V,�V]−1.
+(8)
+Finally, the 1 − α confidence interval for β is
+(�β − z1−α/2 �
+SE, �β + z1−α/2 �
+SE)
+where
+�
+SE =
+�
+�
+�
+��γT
+�V �A( �VΓ − 2�β �C + �β2 �Vγ)�V,�V �A�γ�V
+n(�γT
+�V �A�γ�V)2
+.
+(9)
+As a remark, �VΓ, �Vγ, and �C are heteroscedasticity-robust covariance estimators and hence
+(9) is also robust to heteroscedastic errors in a low-dimensional setting. In a high-dimensional
+setting, we set �A = I in (8), and �VΓ, �Vγ, and �C are constructed under the homoscedastic
+error assumptions; see more details in Guo et al. (2018b).
+2.3 Searching and Sampling
+We now review the searching and sampling method proposed in Guo (2021), which provides
+uniformly valid conference intervals even if there are errors in separating valid and invalid
+IVs. The right-hand side of Table 1 illustrates an example of the invalid IVs not being
+separated from valid IVs in finite samples. In the following, we review the idea of searching
+7
+
+Koo, Lee, Small, and Guo
+and sampling under the majority rule and the more general method with the plurality rule
+can be found in Guo (2021).
+Let α ∈ (0, 1) denote the pre-specified significance level. Given β ∈ R and the reduced
+form estimator �Γ and �γ, we estimate πj with j ∈ �S by
+�πj(β) = (�Γj − β�γj)1(|�Γj − β�γj| ≥ �ρj(β, α)),
+(10)
+where �ρj(β, α) = Φ−1 �
+1 −
+α
+2| �
+S|
+�
+�
+SE(�Γj − β�γj) with �
+SE(�Γj − β�γj) denoting a consistent
+estimator of the standard error of �Γj − β�γj. We search for the value of β leading to enough
+valid IVs and construct the searching confidence interval as
+CIsearch =
+�
+β ∈ R :
+���π �
+S(β)
+��
+0 < | �S|/2
+�
+,
+(11)
+which collects all β values such that more than half of IVs in �S are selected as valid.
+Based on the searching method, Guo (2021) proposed a sampling confidence interval,
+which retains the uniform coverage property and improves the precision of the con��dence
+interval. In particular, we sample
+��Γ[m]
+�γ[m]
+�
+iid
+∼ N
+���Γ
+�γ
+�
+, 1
+n
+� �VΓ
+�C
+�CT
+�Vγ
+��
+,
+for
+1 ≤ m ≤ M.
+For 1 ≤ m ≤ M and j ∈ �S, we modify (10) and define
+�π[m]
+j
+(β, λ) = (�Γ[m]
+j
+− β�γ[m]
+j
+)1(|�Γ[m]
+j
+− β�γ[m]
+j
+| ≥ λ · �ρj(β, α))
+with the shrinkage parameter λ ≍ (log n/M)
+1
+2| �
+S| . A data-dependent way of choosing λ can
+be found in Remark 3 of Guo (2021). For each 1 ≤ m ≤ M, we construct a searching
+interval (β[m]
+min, β[m]
+max) where
+β[m]
+min = min
+β∈B[m]
+λ
+β
+and
+β[m]
+max = max
+β∈B[m]
+λ
+β
+with B[m]
+λ
+=
+�
+β ∈ R :
+����π[m]
+�
+S (β, λ)
+���
+0 < | �S|/2
+�
+. Then the sampling CI is defined as
+CIsample =
+�
+min
+m∈M β[m]
+min, max
+m∈M β[m]
+max
+�
+.
+(12)
+with M = {1 ≤ m ≤ M : (β[m]
+min, β[m]
+max) ̸= ∅}. The sampling confidence intervals in general
+improve the precision of the searching confidence intervals. But both intervals can provide
+uniformly valid coverage robust to the errors in separating valid and invalid IVs.
+8
+
+RobustIV and controlfunctionIV
+2.4 Endogeneity test in high dimensions
+We review the high-dimensional endogeneity test proposed in Guo et al. (2018a). We focus
+on the homoscedastic error setting by writing Θ11 = Var[ξi|Zi·, Xi·], Θ22 = Var[δi|Zi·, Xi·],
+and Θ12 = Cov[ξi, δi|Zi·, Xi·] for the reduced form models (4) and (5).
+With the same
+estimators from TSHT in the high-dimensional settings in Section 2.2, we can estimate the
+covariance σ12 by �σ12 = �Θ12 − �β �Θ22 where �β =
+�
+j∈ �V �γj�Γj
+�
+j∈ �V �γ2
+j
+and �Θ12 and �Θ22 are consistent
+estimators of the reduced form covariance Θ22 and Θ12. We establish the asymptotic nor-
+mality of �σ12 − σ12 in Guo et al. (2018a) and propose a testing procedure for H0 : σ12 = 0.
+3. Nonlinear outcome models
+This section reviews the control function IV methods (Guo and Small, 2016; Li and Guo,
+2020) implemented in the controlfunctionIV package, whose usage is demonstrated in
+Sections 4.3 and 4.4.
+3.1 Control function and pretest estimators
+We consider the following nonlinear outcome and treatment models:
+Yi = G(Di)Tβ + XT
+i·φ + ui, E[uiZi·] = E[uiXi·] = 0,
+(13)
+Di = H(Zi·)Tγ + XT
+i·ψ + vi, E[viZi·] = E[viXi·] = 0,
+(14)
+where G(Di) = (Di, g2(Di), ..., gk(Di))T, H(Zi·) = (Zi·, h2(Zi·), ..., hk(Zi·))T with {gj(·)}2≤j≤k
+and {hj(·)}2≤j≤k denoting the known nonlinear transformations. Under the models (13)
+and (14), the IVs are assumed to be valid and the causal effect of increasing the value of D
+from d2 to d1 is defined as G(d1)Tβ − G(d2)Tβ.
+The control function (CF) method is a two-stage procedure. In the first stage, regress D
+on H(Z) and X, and obtain the predicted value �D and its associated residual �v = D− �D. In
+the second stage, we use �v as the proxies for the unmeasured confounders and regress Y on
+G(D), X, and �v. We use �βCF to denote the estimated regression coefficient corresponding
+to D. Guo and Small (2016) showed that �βCF is equivalent to the TSLS estimator with
+the augmented set of IVs.
+Even if all IVs satisfy the classical assumptions (A1)-(A3),
+there is no guarantee of the validity of the augmented IVs generated by the CF estimator.
+Guo and Small (2016) applied the Hausman test to assess the validity of the augmented set
+of IVs generated by the CF estimator. The test statistic is defined as
+H(�βCF, �βTSLS) = (�βCF − �βTSLS)T[Cov(�βTSLS) − Cov(�βCF)]−(�βCF − �βTSLS),
+(15)
+where �βTSLS is the two stage least square estimator, Cov(�βTSLS) and Cov(�βCF) are the
+covariance matrices of �βTSLS and �βCF, and A− denote the Moore-Penrose pseudoinverse.
+9
+
+Koo, Lee, Small, and Guo
+If the p-value P
+�
+χ2
+1 ≥ H(�βCF, �βTSLS)
+�
+is less than α = 0.05, then we define the level α
+pretest estimator �βPretest as �βTSLS; otherwise, �βCF defined above (Guo and Small, 2016).
+3.2 Probit CF and SpotIV
+We now consider the binary outcome model and continuous treatment model,
+E [Yi|Di = d, Wi· = w, ui = u] =
+1(dβ + wTκ + u > 0),
+and
+Di = WT
+i· γ + vi,
+(16)
+where Wi· = (ZT
+i·, XT
+i·)T, the errors (ui, vi)⊺ are bivariate normal random variables with
+zero means and independent of Wi·, κ = (κT
+z , κT
+x)T is the coefficient vector of the IVs and
+measured covariates, and γ = (γT
+z , γT
+x)T is a parameter representing the association between
+Di and Wi·. When κz ̸= 0, the instruments are invalid. Since ui and vi are bivariate normal,
+we write ui = ρvi +ei. The model (16) implies E [Yi|Wi·, vi] = Φ(Diβ∗ +W ⊺
+i·Γ∗ +ρ∗vi) where
+β∗ = β/σe, Γ∗ = κ/σe + β∗ · γ, and ρ∗ = ρ/σe + β∗ with σe denoting the standard error
+of ei = ui − ρvi. That is, the conditional outcome model of Yi given Wi,· and vi is a probit
+regression model.
+Our goal is to estimate the conditional average treatment effect (CATE) from d2 to d1
+CATE(d1, d2|w) := E[Yi|Di = d1, Wi· = w] − E[Yi|Di = d2, Wi· = w].
+(17)
+We first construct the OLS estimator �γ of γ. We compute its residual �v = D − W�γ and
+define �Σ = 1
+n
+�n
+i=1 Wi·WT
+i· . We estimate S = {1 ≤ j ≤ pz : (γz)j ̸= 0} by
+�S =
+�
+1 ≤ j ≤ pz : |�γj| ≥ �σv
+�
+2{�Σ−1}j,j log n/n
+�
+(18)
+with �σ2
+v = �n
+i=1 �v2
+i /n. Next, as CF in Section 3.1, we use �v as the proxy for unmeasured con-
+founders and implement the probit regression Y on W and �v. We use �Γ and �ρ to denote the
+probit regression coefficients of W of �v respectively. We apply the majority rule and compute
+�β as the median of (�Γj/�γj)j∈ �S. We then estimate �κ = �Γ − �γ �β. Finally, we estimate CATE
+defined in (17) by the partial mean method (Newey, 1994; Mammen, Rothe, and Schienle,
+2012),
+1
+n
+n
+�
+i=1
+�
+Φ(d1 �β + wT�κ + �vi�ρ)
+�
+− 1
+n
+n
+�
+i=1
+�
+Φ(d2 �β + wT�κ + �vi�ρ)
+�
+and construct the confidence interval by bootstrap (Li and Guo, 2020).
+Li and Guo (2020) has proposed a more general methodology, named SpotIV, to con-
+duct robust causal inference with possibly invalid IVs. The model considered in Li and Guo
+(2020) includes the probit outcome model in (16) as a special case. In particular, Li and Guo
+(2020) replaced the known probit transformation in (16) with the more general non-parametric
+function, which is possibly unknown. Moreover, Li and Guo (2020) allows some instruments
+to be correlated with the unmeasured confounders ui in the outcome model.
+10
+
+RobustIV and controlfunctionIV
+4. RobustIV and controlfunctonIV Usage
+In this section, we illustrate the basic usage of RobustIV with the data set from Angrist and Krueger
+(1991) and simulated high-dimensional data. Also, we use the Mroz data set from Wooldridge
+(2010) to demonstrate the usage of controlfunctionIV.
+4.1 TSHT and SearchingSampling
+In the following, we introduce usages of the R functions TSHT and SearchingSampling
+with the data used in Angrist and Krueger (1991).
+Angrist and Krueger (1991) studied
+the causal effect of the years of education (EDUC) on the log weekly earnings (LWKLYWGE).
+Following Angrist and Krueger (1991), we take 30 interactions (QTR120-QTR129, QTR220-
+QTR229, QTR320-QTR329) between three quarter-of-birth dummies (QTR1-QTR3) and ten year-
+of-birth dummies (YR20-YR29) as the instruments Z. For example, QTR120 is element-wise
+product of QTR1 and YR20. Here, the quarter-of-birth dummies are the indicators of whether
+the observed person was born in the first, second, and third quarter of the year respectively,
+and the year-of-birth dummies are indicators of which year the subject was born from 1940
+to 1949 respectively. We also include the following baseline covariates X: 9 year-of-birth
+dummies (YR20-YR28), a race dummy (RACE), a marital status dummy (MARRIED), a dummy
+for residence in an SMSA (SMSA), and eight region-of-residence dummies (NEWENG, MIDATL,
+ENOCENT, WNOCENT, SOATL, ESOCENT, WSOCENT, MT). We first apply the function TSHT.
+R> Y <- as.vector(LWKLYWGE); D <- as.vector(EDUC)
+R> Z <- sapply(paste0("QTR", c(seq(120,129), seq(220,229), seq(320,329))),
+function(x){get(x)})
+R> X <- cbind(sapply(paste0("YR",seq(20,28)),function(x){get(x)}),RACE, MARRIED,
+SMSA, NEWENG, MIDATL, ENOCENT, WNOCENT, SOATL, ESOCENT, WSOCENT, MT)
+R> pz <- ncol(Z)
+R> out.TSHT <- TSHT(Y=Y,D=D,Z=Z,X=X,
+tuning.1st = sqrt(2.01*log(pz)), tuning.2nd = sqrt(2.01*log(pz)))
+R> summary(out.TSHT)
+betaHat Std.Error CI(2.5%) CI(97.5%) Valid IVs
+0.0874
+0.019
+0.0502
+0.1247
+QTR120 QTR121 QTR122 QTR220 QTR222 QTR227 QTR322
+_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
+Detected invalid IVs: QTR126 QTR226
+Here, tuning.1st and tuning.2nd are tuning parameters λ1 and λ2 used for the thresholds
+to get �S and �V in Section 2.2 respectively. The default values for these parameters are
+√log n in the low-dimensional setting. However, in theory, any value above √2 log p and
+diverging to infinity would suffice. Since the data has 486926 observations, we choose the
+tuning parameters as √2.01 log pz to avoid too conservative threshold levels due to the
+huge sample. Once TSHT is implemented, we can call summary to see the outputs of TSHT
+11
+
+Koo, Lee, Small, and Guo
+including the point estimator, its standard error, confidence interval, and valid IVs as we
+discussed in Section 2.2.
+The above result shows that TSHT selected QTR120, QTR121, QTR122, QTR220, QTR222,
+QTR227, and QTR322 as valid IVs. Thus, valid IVs are interactions with the first quarter
+of birth and dummies representing births in 1940, 1941, and 1942, interactions with the
+second quarter of birth and dummies representing births in 1940, 1942, and 1947, and
+finally the interaction between the third quarter-of-birth and dummy representing births in
+1942. On the other hand, it is reported that QTR126 and QTR226 are invalid IVs. That is,
+interactions with the first and the second quarter of birth and dummy representing births
+in 1946 are relevant but invalid IVs. The remaining IVs have been screened out of the
+first-stage selection as individually weak IVs.
+The detection of invalid IVs implies that using whole Z as valid IVs can cause the
+estimate to be biased. In Angrist and Krueger (1991), the TSLS estimate by using whole Z
+as valid IVs is 0.0393. In contrast, our procedure is more robust to the existence of possibly
+invalid IVs, giving the causal estimate as 0.0874. Our 95% confidence interval is above zero,
+indicating a positive effect of education on earning.
+In addition to the above output, the class object TSHT has other values that are not
+reported by summary, for example, whether the majority rule is satisfied or not, and the
+voting matrix to construct �V in Section 2.2. These can be checked by directly calling TSHT.
+As discussed in Section 2.2, there are different voting options to get �V, where the default
+option voting = ’MaxClique’ stands for �VMC and voting = ’MP’ stands for �VMP in (7).
+If there are several maximum cliques, summary returns results corresponding to each maxi-
+mum clique. Furthermore, since the default argument for which estimator to use is method =
+’OLS’, one can choose other estimators by method = ’DeLasso’ for the debiased Lasso esti-
+mator with SIHR R package (Rakshit, Cai, and Guo, 2021) and method = ’Fast.DeLasso’
+for the fast computation of the debiased Lasso estimator (Javanmard and Montanari, 2014).
+The above methods are useful in a high-dimensional setting.
+Next, we implement the uniformly valid confidence intervals by calling the function
+SearchingSampling. We start with the searching CI defined in (11) with the argument
+Sampling = FALSE.
+R> out1 = SearchingSampling(Y=Y, D=D, Z=Z, X=X, Sampling=FALSE,
+tuning.1st = sqrt(2.01*log(pz)), tuning.2nd = sqrt(2.01*log(pz)))
+R> summary(out1)
+Confidence Interval for Causal Effect: [-0.0964,0.2274]
+With the default argument Sampling = TRUE, one can use the following code to implement
+the more efficient sampling CI in (12).
+12
+
+RobustIV and controlfunctionIV
+R> set.seed(1)
+R> out.SS = SearchingSampling(Y=Y, D=D, Z=Z, X=X,
+tuning.1st = sqrt(2.01*log(pz)), tuning.2nd = sqrt(2.01*log(pz)))
+R> summary(out.SS)
+Confidence Interval for Causal Effect: [0.0135,0.1775]
+The SearchingSampling confidence intervals are generally wider than that of the TSHT
+since they are robust to the IV selection errors. The function summary displays confidence
+interval for β, which are discussed in Section 2.3. As in TSHT, one can use the argument
+method to employ the high-dimensional debiased estimators instead of OLS.
+4.2 endo.test
+In the following, we show the usage of endo.test, a function for the endogeneity test in high
+dimension with a simulated example. The corresponding model and method are presented
+in Section 2.4. We consider the models (1) and (2) and set pz = 600 with only the first
+10 IVs being relevant. Among these 10 IVs, the first 3 IVs are invalid but the remaining
+IVs are valid. Moreover, we set Corr (ǫi, δi) = 0.8, which indicates a level of endogeneity.
+The function endo.test generates a class object with same arguments in TSHT. The class
+object from endo.test can be used by calling summary function, which enable us to see a
+brief result of ento.test.
+R> set.seed(5)
+R> n = 500; L = 600; s = 3; k = 10; px = 10; epsilonSigma = matrix(c(1,0.8,0.8,1),2,2)
+R> beta = 1; gamma = c(rep(1,k),rep(0,L-k))
+R> phi = (1/px)*seq(1,px)+0.5; psi = (1/px)*seq(1,px)+1
+R> Z = matrix(rnorm(n*L),n,L); X = matrix(rnorm(n*px),n,px);
+R> epsilon = MASS::mvrnorm(n,rep(0,2),epsilonSigma)
+R> D = 0.5 + Z %*% gamma + X %*% psi + epsilon[,1]
+R> Y = -0.5 + Z %*% c(rep(1,s),rep(0,L-s)) + D * beta + X %*% phi + epsilon[,2]
+R> endo.test.model <- endo.test(Y,D,Z,X, invalid = TRUE)
+R> summary(endo.test.model)
+P-value Test
+Valid IVs
+0
+H0 rejected Z4 Z5 Z6 Z7 Z8 Z9 Z10
+_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
+Detected invalid IVs: Z1 Z2 Z3
+When we call summary function, p-value, it reports the test result with significance level
+α (default alpha = 0.05), the valid IVs, and detected invalid IVs. H0 rejected means
+that the treatment is endogenous, otherwise not. Since we set invalid = TRUE, ento.test
+allows some of IVs to be invalid and conducts the endogeneity test with the selected �V
+defined in Section 2.2. With invalid = FALSE, the function assumes that all IVs are valid.
+13
+
+Koo, Lee, Small, and Guo
+As in the above sections, one can use method argument to employ other estimators both in
+the low and high dimensions.
+4.3 cf and pretest
+In this section, we introduce usages of cf and pretest in the package controlfunctionIV.
+The Mroz data was introduced in Mroz (1987) and then used in various works of literature
+including Wooldridge (2010), which has n = 428 individuals after removing the data with
+NA. Following Wooldridge (2010), we estimate the causal effect of education on the log
+earnings of married working women. The data is available in the Wooldridge package.
+Here, the outcome Y is log earnings (lwage), and the exposure D is years of schooling
+(educ). Moreover, there are other variables such as the father’s education (fatheduc), the
+mother’s education (motheduc), the husband’s education (huseduc), actual labor market
+experience (exper), its square (expersq), and the women’s age (age).
+Following Example 5.3 in Wooldridge (2010), we assume motheduc, fatheduc, and
+huseduc to be valid IVs, denoted as Zi = (Zi1, Zi2, Zi3)T; we use and exper, expersq,
+and age as baseline covariates, denoted as Xi = (Xi1, Xi2, Xi3)T. Also assume that the
+outcome and treatment models are (13) and (14) respectively with G(Di) = (Di, D2
+i )T and
+H(Zi·) = (Zi1, Zi2, Zi3, Z2
+i1, Z2
+i2, Z2
+i3)T.
+Then we can implement the cf function by inputting a formula object, which has the
+same form as that of ivreg in AER package. The function summary gives us information on
+coefficients of the control function estimators, including the point estimator, its standard
+error, t value, and p value.
+R> library(wooldridge); library(controlfunctionIV); data(mroz); mroz <- na.exclude(mroz)
+R> Y <- mroz[,"lwage"]; D <- mroz[,"educ"]
+R> Z <- as.matrix(mroz[,c("motheduc","fatheduc","huseduc")])
+R> X <- as.matrix(mroz[,c("exper","expersq","age")])
+R> cf.model <- cf(Y~D+I(D^2)+X|Z+I(Z^2)+X)
+R> summary(cf.model)
+_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
+Coefficients of the control function estimators:
+Estimate
+Std.Error t value Pr(>|t|)
+(Intercept)
+1.2573907
+0.7871438
+1.597 0.055457 .
+D
+-0.1434395
+0.1102058
+1.302 0.096884 .
+I(D^2)
+0.0086426
+0.0041004
+2.108 0.017817 *
+Xexper
+0.0438690
+0.0131574
+3.334 0.000465 ***
+Xexpersq
+-0.0008713
+0.0003984
+2.187 0.014631 *
+Xage
+-0.0011636
+0.0048634
+0.239 0.405511
+---
+Signif. codes:
+0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+14
+
+RobustIV and controlfunctionIV
+The following code infers the causal effect G(d1)Tβ−G(d2)Tβ by changing the treatment
+level from d2 to d1 = d2 + 1. Since the second and third coefficients are related to D, we
+use the second and third index to get the causal effect and its standard error.
+R> d2 = median(D); d1 = median(D)+1;
+R> D.diff <- c(d1,d1^2)-c(d2,d2^2); CE <- (D.diff)%*%cf.model$coefficients[c(2,3)]
+R> CE.sd <-sqrt(D.diff%*%cf.model$vcov[c(2,3),c(2,3)]%*%D.diff)
+R> CE.ci <- c(CE-qnorm(0.975)*CE.sd,CE+qnorm(0.975)*CE.sd)
+R> cmat <- cbind(CE,CE.sd,CE.ci[1],CE.ci[2])
+R> colnames(cmat)<-c("Estimate","Std.Error","CI(2.5%)","CI(97.5%)"); rownames(cmat)<- "CE"
+R>
+print(cmat, digits = 4)
+Estimate Std.Error CI(2.5%) CI(97.5%)
+CE
+0.07263
+0.02171
+0.03007
+0.1152
+The function pretest can be used to choose between the TSLS or the control function
+method. If we run pretest with the same argument above and call summary, it will output
+the following result:
+R> pretest.model <- pretest(Y~D+I(D^2)+X|Z+I(Z^2)+X)
+R> summary(pretest.model)
+Level 0.05 pretest estimator is control function estimator.
+_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
+Coefficients of the pretest estimators:
+Estimate
+Std.Error t value Pr(>|t|)
+(Intercept)
+1.2573907
+0.7871438
+1.597 0.055457 .
+D
+-0.1434395
+0.1102058
+1.302 0.096884 .
+I(D^2)
+0.0086426
+0.0041004
+2.108 0.017817 *
+Xexper
+0.0438690
+0.0131574
+3.334 0.000465 ***
+Xexpersq
+-0.0008713
+0.0003984
+2.187 0.014631 *
+Xage
+-0.0011636
+0.0048634
+0.239 0.405511
+---
+Signif. codes:
+0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+The first section of the output of summary reports which estimator is chosen after the
+pretesting step. The second section lists brief information on coefficients of pretest estima-
+tors including the point estimator, its standard error, t value, and p-value, similar to cf.
+Since the pretest estimator is the control function estimator, the second section of summary
+is the same as that of summary(cf.model).
+4.4 Probit.cf
+Finally, we conclude the usage part by looking at the usage of Probit.cf, which is designed
+for the binary outcome with unmeasured confounders and possibly invalid IVs. For illus-
+15
+
+Koo, Lee, Small, and Guo
+tration, we use the Mroz data and define the binary outcome variable Y0 to take the value
+1 if the continuous outcome Y is greater than the median of Y and 0 otherwise. We use the
+same treatment variable D as in the cf example. Contrary to the cf example, we set the
+candidates of IVs Z as motheduc, fatheduc, huseduc, exper, and expersq, and assume
+that we have covariates X as age.
+We implement the Probit.cf function to estimate the CATE by increasing the treat-
+ment value from the median of D to the median plus one. We can call summary to see the
+result of Probit.cf. The function summary provides information on the valid IVs �V, the
+point estimator, standard error, and 95% confidence interval for β in (16), and the point
+estimator, the standard error, and 95% confidence interval of CATE.
+R> Z <- as.matrix(mroz[,c("motheduc","fatheduc","huseduc","exper","expersq")])
+R> Y0 <- as.numeric((Y>median(Y)))
+R> d2 = median(D); d1 = d2+1; w0 = apply(cbind(Z,X)[which(D == d2),], 2, mean)
+R> Probit.model <- Probit.cf(Y0,D,Z,X,d1 = d1,d2 = d2,w0 = w0)
+R> summary(Probit.model)
+Estimate Std.Error CI(2.5%) CI(97.5%) Valid IVs
+Beta 0.2119
+0.092
+0.0316
+0.3922
+motheduc fatheduc huseduc
+CATE 0.0844
+0.033
+0.0198
+0.1489
+motheduc fatheduc huseduc
+_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
+No invalid IV is detected
+With the option invalid = TRUE, we allow invalid IVs and choose the valid IVs among all
+provided IVs. If one wants to assume all IVs are valid, one can set invalid = FALSE.
+5. Application to Framingham Heart Study
+We analyze the Framingham Heart Study (FHS) data and illustrate our package using ge-
+netic variants as IVs. The FHS is an ongoing cohort study of participants from the town of
+Framingham, Massachusetts, that has grown over the years to include five cohorts with a
+total sample of over 15,000. The FHS, initiated in 1948, is among the most critical sources
+of data on cardiovascular epidemiology (Sytkowski, Kannel, and D’Agostino, 1990; Kannel,
+2000; Mahmood, Levy, Vasan, and Wang, 2014). Since the late 1980s, researchers across
+human health-related fields have used genetic factors underlying cardiovascular diseases
+and other disorders. Over the last two decades, DNA has been collected from blood sam-
+ples and immortalized cell lines from members of the Original Cohort, the Offspring Cohort,
+and the Third Generation Cohort (Govindaraju et al., 2008). Several large-scale genotyping
+projects and genome-wide linkage analysis have been conducted, and several other recent
+collaborative projects have completed thousands of SNP genotypes for candidate gene re-
+gions in subsets of FHS subjects with available DNA. The FHS has recently been used for
+Mendelian Randomization to determine causal relationships even in the presence of unmea-
+16
+
+RobustIV and controlfunctionIV
+sured confounding thanks to the availability of genotype and phenotype data (Holmes et al.,
+2014; Dalbeth et al., 2015; Hughes et al., 2014). As candidate IVs, we will use genotype
+data from the FHS associated with the phenotype of interest and apply the proposed meth-
+ods described above.
+We apply the RobustIV package to investigate the effect of low-density lipoprotein (LDL-
+C) on globulin levels among individuals in the Framingham Heart Study (FHS) Offspring
+Cohort, as was studied in Kang et al. (2020).
+We use eight SNP genotypes (rs646776,
+rs693, rs2228671, rs2075650, rs4299376, rs3764261, rs12916, rs2000999) that are known to be
+significantly associated with LDL-C measured in mg/dL as candidate IVs (Kathiresan et al.,
+2007; Ma et al., 2010; Smith et al., 2014). See Table 2 for details. The outcome of interest
+Yi is a continuous globulin level (g/L) and the exposure variable Di is the LDL-C level.
+Globulin is known to play a crucial role in liver function, clotting, and the immune system.
+We also use the age and sex of the subjects as covariates Xi·. The study includes n = 1445
+subjects, with an average globulin level of 27.27 (SD: 3.74) and an average LDL-C of 1.55
+(SD: 0.50). An average age is 35.58 (SD: 9.74) and 54.95% are males.
+Zj
+SNP
+Position
+lm(D ∼ Z)
+lm(Y ∼ Z)
+Estimate (Std. Error)
+t-statistic (p-value)
+Estimate (Std. Error)
+t-statistic (p-value)
+Z1
+rs646776
+chr1:109275908
+-5.160 (1.610)
+-3.205 (0.001)
+-0.001 (0.170)
+-0.007 (0.994)
+Z2
+rs693
+chr2:21009323
+-3.600 (1.286)
+-2.799 (0.005)
+0.318 (0.135)
+2.349 (0.019)
+Z3
+rs2228671
+chr19:11100236
+7.138 (2.029)
+3.518 (<0.001)
+0.529 (0.214)
+2.474 (0.014)
+Z4
+rs2075650
+chr19:44892362
+8.451 (2.021)
+4.183 (<0.001)
+0.471 (0.213)
+2.208 (0.027)
+Z5
+rs4299376
+chr2:43845437
+3.847 (1.387)
+2.773 (0.006)
+0.110 (0.146)
+0.752(0.452)
+Z6
+rs3764261
+chr16:56959412
+3.651 (1.429)
+2.555 (0.011)
+0.275 (0.151)
+1.829 (0.067)
+Z7
+rs12916
+chr5:75360714
+3.363 (1.365)
+2.463 (0.014)
+-0.195 (0.144)
+-1.357 (0.175)
+Z8
+rs2000999
+chr16:72074194
+-2.961 (1.629)
+-1.818 (0.069)
+-0.119 (0.172)
+-0.691 (0.489)
+Table 2: Summary of the relationship between the single nucleotide polymorphisms (SNPs)
+and low-density lipoprotein. The point estimator, its standard error, t value, and
+p-value are summary statistics from running a marginal regression model specified
+in the column title. Position refers to the position of the SNP in the chromosome,
+denoted as chr.
+By applying endo.test, we detect one invalid IV and observe the evidence for the
+existence of unmeasured confounders since the null hypothesis H0 : σ12 = 0 is rejected.
+R> pz <- ncol(Z)
+R> globulin.endo2 <- endo.test(Y,D,Z,X, invalid = TRUE,
+tuning.1st = sqrt(2.01*log(pz)), tuning.2nd = sqrt(2.01*log(pz)))
+R> summary(globulin.endo2)
+P-value Test
+Valid IVs
+0.0091
+H0 rejected Z.1 Z.3 Z.4 Z.5 Z.6 Z.8
+17
+
+Koo, Lee, Small, and Guo
+_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
+Detected invalid IVs: Z.2
+Next, we implement TSHT with the default method of "OLS" under the low-dimensional
+setting. Again, the same invalid IV is detected and the confidence interval is above zero,
+indicating a positive effect of LDL on the glucose level.
+R> pz <- ncol(Z)
+R> TSHT2 <- TSHT(Y, D, Z, X,
+tuning.1st = sqrt(2.01*log(pz)), tuning.2nd = sqrt(2.01*log(pz)))
+R> summary(TSHT2)
+betaHat Std.Error CI(2.5%) CI(97.5%) Valid IVs
+0.0529
+0.0146
+0.0243
+0.0814
+Z.1 Z.3 Z.4 Z.5 Z.6 Z.8
+_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
+Detected invalid IVs: Z.2
+We also constructed the confidence interval using the searching method, which provides
+robustness to the IV selection errors.
+R> SS1 <- SearchingSampling(Y, D, Z, X, tuning.1st = sqrt(2.01*log(pz)),
+tuning.2nd = sqrt(2.01*log(pz)), Sampling = FALSE)
+R> summary(SS1)
+Confidence Interval for Causal Effect: [-0.2427,0.1894]
+We further implement the sampling method, which leads to a shorter uniformly valid CI
+than the searching method.
+R> SS2 <- SearchingSampling(Y, D, Z, X, tuning.1st = sqrt(2.01*log(pz)),
+tuning.2nd = sqrt(2.01*log(pz)), Sampling = TRUE)
+R> summary(SS2)
+Confidence Interval for Causal Effect: [-0.0521,0.1259]
+In the following, we study nonlinear causal relationships using the controlfunctionIV
+package. Burgess, Davies, and Thompson (2014) investigated a nonlinear causal relation-
+ship between BMI and diverse cardiovascular risk factors. Here we examine BMI’s possibly
+nonlinear causal effect on the insulin level. Among n = 3733 subjects, we excluded 618
+subjects with missing information on insulin level, and 50 subjects whose insulin level is
+greater than 300pmol/L and whose BMI is greater than 45kg/m2. We use log-transformed
+insulin as the outcome of interest Yi measured at Exam 2. The exposure Di denotes the BMI
+measures at Exam 1. The covariates Xi· that we adjusted for are age and sex. As valid IVs
+Zi·, we propose using four SNP genotypes known to be significantly associated with obesity.
+In our analysis, we include I(D^2) and I(X^2) to account for quadratic effects of BMI, age,
+and sex on the outcome. We also include I(Z^2) to account for possible quadratic effects
+of SNPs on the exposure. The result from the pretest estimator is as follows:
+18
+
+RobustIV and controlfunctionIV
+R> insulin.pretest = pretest( Y ~ D + I(D^2) + X
++ I(X^2) | Z + I(Z^2) + X + I(X^2))
+R> summary(insulin.pretest)
+Level 0.05 pretest estimator is control function estimator.
+_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
+Coefficients of Pretest Estimators:
+Estimate
+Std.Err t value Pr(>|t|)
+(Intercept)
+2.674e+00
+6.006e-01
+4.453 4.39e-06 ***
+D
+8.295e-02
+2.828e-02
+2.933 0.001690 **
+I(D^2)
+-7.784e-04
+2.742e-04
+2.839 0.002276 **
+X1
+-1.780e-02
+7.816e-03
+2.277 0.011427 *
+I(X1^2)
+2.954e-04
+8.852e-05
+3.337 0.000428 ***
+X2
+-1.361e-01
+5.654e-02
+2.406 0.008087 **
+---
+Signif. codes:
+0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+The pretest estimator chooses the control function over the standard TSLS. The results also
+show that BMI has a positive linear effect on the outcome but a negative quadratic effect
+on the outcome.
+Acknowledgement
+The research of T. Koo was supported in part by NIH grants R01GM140463 and R01LM013614.
+The research of D. Small was supported in part by NIH grant 5R01AG065276-02.The re-
+search of Z. Guo was partly supported by the NSF grants DMS 1811857 and 2015373 and
+NIH grants R01GM140463 and R01LM013614. Z. Guo is grateful to Dr. Frank Windmeijer
+for bringing up the maximum clique method.
+The Framingham Heart Study is conducted and supported by the National Heart, Lung,
+and Blood Institute (NHLBI) in collaboration with Boston University (Contract No. N01-
+HC-25195, HHSN268201500001I, and 75N92019D00031). This manuscript was not prepared
+in collaboration with investigators of the Framingham Heart Study and does not necessarily
+reflect the opinions or views of the Framingham Heart Study, Boston University, or NHLBI.
+Funding for SHARe Affymetrix genotyping was provided by NHLBI Contract N02-HL64278.
+SHARe Illumina genotyping was provided under an agreement between Illumina and Boston
+University. Funding for Affymetrix genotyping of the FHS Omni cohorts was provided by
+Intramural NHLBI funds from Andrew D. Johnson and Christopher J. O’Donnell.
+19
+
+Koo, Lee, Small, and Guo
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+23
+

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+Preprint. Under review.
+LEARNING TO UNLEARN: INSTANCE-WISE UNLEARN-
+ING FOR PRE-TRAINED CLASSIFIERS
+Sungmin Cha1,2*, Sungjun Cho1*, Dasol Hwang1*, Honglak Lee1,4, Taesup Moon2, and Moontae Lee1,3
+1LG AI Research
+2Seoul National University
+3University of Illinois Chicago
+4University of Michigan
+sungmin.cha@snu.ac.kr, {sungjun.cho, dasol.hwang, honglak.lee}@lgresearch.ai,
+tsmoon@snu.ac.kr, moontae.lee@lgresearch.ai
+* denotes equal contribution
+ABSTRACT
+Since the recent advent of regulations for data protection (e.g., the General Data
+Protection Regulation), there has been increasing demand in deleting information
+learned from sensitive data in pre-trained models without retraining from scratch.
+The inherent vulnerability of neural networks towards adversarial attacks and un-
+fairness also calls for a robust method to remove or correct information in an
+instance-wise fashion, while retaining the predictive performance across remain-
+ing data. To this end, we define instance-wise unlearning, of which the goal is to
+delete information on a set of instances from a pre-trained model, by either mis-
+classifying each instance away from its original prediction or relabeling the in-
+stance to a different label. We also propose two methods that reduce forgetting on
+the remaining data: 1) utilizing adversarial examples to overcome forgetting at the
+representation-level and 2) leveraging weight importance metrics to pinpoint net-
+work parameters guilty of propagating unwanted information. Both methods only
+require the pre-trained model and data instances to forget, allowing painless ap-
+plication to real-life settings where the entire training set is unavailable. Through
+extensive experimentation on various image classification benchmarks, we show
+that our approach effectively preserves knowledge of remaining data while un-
+learning given instances in both single-task and continual unlearning scenarios.
+1
+INTRODUCTION
+Humans remember and forget: efficiently learning useful knowledge yet regulating privately sensi-
+tive information and protecting from malicious attacks. Recent advances in large-scale pre-training
+enable models to memorize massive information for intelligent operations (Radford et al., 2019),
+but there is a cost. Language models trained on indiscriminately collected data often disclose pri-
+vate information such as occupations, phone numbers, and family background during text genera-
+tion (Heikkil¨a, 2022). Vision models trained on numerous image data sometimes misclassify natu-
+rally adversarial or adversarially attacked examples with high-confidence (Hendrycks et al., 2021).
+A na¨ıve solution is to retrain these models from scratch after refining or reweighting their training
+datasets (Lison et al., 2021; Zemel et al., 2013; Lahoti et al., 2020). However, such post-hoc process-
+ing is impractical due to growing volumes of data and substantial cost of large-scale training: while
+exercising the Right to be Forgotten (Rosen, 2011; Villaronga et al., 2018) may be straightforward
+to humans, it is not so straightforward in the context of machine learning. This has sparked the field
+of machine unlearning, in which the main goal is to efficiently delete information while preserving
+information on the remaining data.
+While many machine unlearning approaches have shown promising results deleting data from tradi-
+tional machine learning algorithms (Mahadevan & Mathioudakis, 2021; Ginart et al., 2019; Brophy
+& Lowd, 2021) as well as DNN-based classifiers (Tarun et al., 2021; Chundawat et al., 2022; Ye
+et al., 2022; Yoon et al., 2022; Golatkar et al., 2020; Kim & Woo, 2022; Mehta et al., 2022), existing
+work are built upon assumptions far too restrictive compared to real-life scenarios. First off, many
+approaches assume a class-wise unlearning setup, where the task is to delete information from all
+data points that belong to a particular class or set of classes. However, data deletion requests are
+1
+arXiv:2301.11578v1  [cs.LG]  27 Jan 2023
+
+Preprint. Under review.
+practically received at a per-instance basis, potentially resulting in a set of data points with a mix-
+ture of class labels (Heikkil¨a, 2022; Mehrabi et al., 2021). Another widely used assumption is that
+at least a subset of the original training data is available at the time of unlearning. In real settings,
+however, loading the original dataset may not be an option due to data expiration policies or lack
+of storage for large amounts of data. Lastly, many approaches consider the main objective as re-
+moving the previous effect of the deleting data during training. While this is indeed the ideal case,
+recent work have shown that fulfilling the objective can still lead to information leakage (Suriyaku-
+mar & Wilson, 2022), and unlearning mechanisms must explicitly enforce misprediction for tighter
+security (Graves et al., 2021).
+In light of aforementioned limitations, we propose a framework for instance-wise unlearning that
+deletes information with access only to the pre-trained model and the data points requested for
+unlearning. Instead of undoing the previous influence of deleting data, we pursue a stronger goal
+where all requested data points are misclassified, preventing collection of information via interpo-
+lation of nearby data points. Inspired by work in the continual learning literature (Ebrahimi et al.,
+2020; Aljundi et al., 2018), we propose two regularization methods that minimize loss in predic-
+tive performance on the remaining data, while completely forgetting information on deleting data.
+Specifically, we 1) generate adversarial examples by attacking each deleting data point with the
+pre-trained model and retrain on these examples to prevent representation-level forgetting and 2)
+use weight importance measures from unlearning instances to focus gradient updates more towards
+parameters responsible for the originally correct classification of such instances. Extensive experi-
+ments on CIFAR-10/-100 (Krizhevsky et al., 2009) and ImageNet-1K (Deng et al., 2009) datasets
+show that our proposed method effectively preserves overall predictive performance, while com-
+pletely misclassifying images chosen for deletion. Our qualitative analyses also reveal interesting
+insights, including lack of any discernible pattern in misclassification that may be exploited by
+adversaries, preservation of the previously learned decision boundary, and forgetting of high-level
+features within deleted images. In summary, our main contributions are as follows:
+• We propose instance-wise unlearning through intended misclassification, under the as-
+sumption that only the pre-trained model and data to forget are available at hand.
+• We present two model-agnostic regularization methods that reduce forgetting on remaining
+data while misclassifying data for deletion.
+• Empirical evaluations on well-known image classification benchmarks show that our pro-
+posed method significantly boosts predictive performance after unlearning.
+2
+RELATED WORK
+Machine unlearning.
+Machine unlearning (Cao & Yang, 2015) is a field that makes a pre-trained
+model forget information learned from a specified subset of data. For this, the existing studies have
+taken an approach that deletes the influence of unwanted data points (denoted as Df) from the model
+while retaining the predictive performance on the rest of the data (denoted as Dr). Mahadevan &
+Mathioudakis (2021); Ginart et al. (2019); Brophy & Lowd (2021) proposed unlearning methods for
+a linear/logistic regression, k-means clustering, and random forests, respectively. These methods are
+specifically designed for simple machine learning models, not for neural networks.
+Table 1: Comparison between existing unlearning methods.
+Methods
+Unit
+Goal
+Dr
+Df
+Tarun et al. (2021)
+class
+undo
+
+
+Chundawat et al. (2022)
+class
+undo
+
+
+Ye et al. (2022)
+class
+undo
+
+
+Yoon et al. (2022)
+class
+undo
+
+
+Golatkar et al. (2020)
+instance
+undo
+
+
+Kim & Woo (2022)
+instance
+undo
+
+
+Mehta et al. (2022)
+instance
+undo
+
+
+Our methods
+instance
+misclassify
+
+
+Recently, the machine unlearning for
+neural networks have been studied in
+different settings, shown in Table 1.
+These methods can be categorized
+into two approaches: class-wise and
+instance-wise unlearning. The class-
+wise unlearning is to forget a cer-
+tain class (e.g., 9-th class of CIFAR-
+10) while retaining the performance
+on the remaining class (Tarun et al.,
+2021; Chundawat et al., 2022; Ye
+et al., 2022; Yoon et al., 2022). On
+the other hand, the instance-wise un-
+learning is designed to delete instance-wise information (e.g., several images of CIFAR-10) from
+2
+
+Preprint. Under review.
+the pre-trained model (Golatkar et al., 2020; Kim & Woo, 2022; Mehta et al., 2022). In other words,
+only instances that are requested to be forgotten should be deleted and the others from the same class
+should be remembered.
+The goal of the existing methods is to make the already trained model identical to the model trained
+on the dataset with unwanted instances removed (denote as undo). Unfortunately, even if the model
+is trained on the removed dataset, the interpolation capabilities of the neural networks may correctly
+predict even that we want to erase. This does not lead to complete unlearning in practical applica-
+tions. Therefore, we define the goal of unlearning as to make the already trained model completely
+misclassifies the set of instances that should be forgotten (denote as misclassify).
+Also, the existing methods have different access level to the unlearning data Df and the rest Dr. The
+existing solutions for instance-wise unlearning require access to the entire dataset (i.e., Dr ∪ Df).
+These methods which rely on the availability of the entire data are very far from real-world scenarios.
+On the other hand, our proposed methods only need to the unlearning dataset (i.e., Df).
+Adversarial examples.
+Since the vulnerability of neural networks has been revealed (Szegedy
+et al., 2013), various methods have been proposed to generate adversarial examples that can de-
+ceive neural networks (Goodfellow et al., 2014; Kurakin et al., 2016; Madry et al., 2017; Carlini &
+Wagner, 2017). In the case of white-box attack, an adversarial example can be generated by adding
+a hardly visible perturbation on a given image based on the gradient information from the model,
+making the model classify the image to a wrong class. The injected noise of the example is hard
+to distinguish visually but it causes a serious misclassification of the model. Recently, (Ilyas et al.,
+2019) experimentally demonstrates that those noise is not meaningless but it rather contains (attack)
+target label’s features for the model.
+Weight importance.
+Weight importance is a measure of how important each weight is when the
+model predicts an output for a given input data, and it has been used for different purposes, such
+as weight pruning (Molchanov et al., 2019; Liu et al., 2017; Wen et al., 2016; Alvarez & Salz-
+mann, 2016; Li et al., 2016) and regularization-based continual learning (Kirkpatrick et al., 2017;
+Aljundi et al., 2018; Chaudhry et al., 2018; Jung et al., 2020; Aljundi et al., 2019). Among them,
+regularization-based continual learning has actively proposed various methods for measuring the
+weight importance. For overcoming catastrophic forgetting of previous tasks, the weight importance
+is utilized as the strength of the L2 regularization between a current model’s weight and the model’s
+weight trained up to the previous task. Most methods estimate the weight-level importance based on
+a gradient of a given input data (Kirkpatrick et al., 2017; Aljundi et al., 2018).
+3
+METHOD
+3.1
+PRELIMINARIES AND NOTATIONS
+Dataset and pre-trained model.
+Let Dtrain be the entire training dataset used to pre-train a clas-
+sification model gθ : X → Y. We denote Df ⊂ Dtrain as the unlearning dataset that we want to
+intentionally forget from the pre-trained model and Dr as the remaining dataset on which we wish
+to maintain predictive accuracy (Dr := Dtrain \ Df). We denote a pair of an input image x ∈ X
+and its ground-truth label y ∈ Y from Dtrain as (x, y) ∼ Dtrain, similarly (xf, yf) ∼ Df and
+(xr, yr) ∼ Dr. Also, Dtest denotes the test dataset used for evaluation. Note that our approaches
+assumes access to only the pre-trained model gθ and the unlearning dataset Df during unlearning.
+Adversarial examples.
+The goal of an adversarial attack on an input (x, y) is to generate an
+adversarial example x′ that is similar to x, but leads to misclassification (gθ(x′) ̸= y) when fed
+to the pre-trained model gθ. In the case of targeted adversarial attack, it makes the model predict a
+specific class different from the true class (gθ(x′) = ¯y). The typical optimization form of generating
+adversarial examples in targeted attack is denoted as
+x′ =
+arg min
+z:∥z−x∥p≤ϵ
+LCE(gθ(z), ¯y; θ)
+(1)
+where LCE stands for the cross-entropy loss. The ∥z − x∥p ≤ ϵ condition requires that the Lp-
+norm is less than a perturbation budget ϵ. The optimization above is intractable in general, and
+3
+
+Preprint. Under review.
+Figure 1: Illustrations of our approaches that reduce forgetting on the remaining data. (Top) Aug-
+menting adversarial examples from unlearning data provides support for preserving the overall de-
+cision boundary. (Bottom) Weight importance measures allow us to pinpoint weights we should
+change to induce misclassification while maintaining other weights to mitigate forgetting.
+thus several papers have proposed approximations that can generate adversarial examples without
+directly solving it (Goodfellow et al., 2014; Kurakin et al., 2016; Carlini & Wagner, 2017; Madry
+et al., 2017). In this paper, we make use of L2-PGD targeted attacks Madry et al. (2017) for all
+experiments.
+Measuring weight importance with MAS.
+To measure weight importance Ω, we consider
+MAS (Aljundi et al., 2018), an algorithm that estimates weight importance by finding parameters
+that brings a significant change in the output when perturbed slightly. It estimates the weight impor-
+tance via a sum of gradients on the L2 norm of the outputs:
+Ωi = 1
+N
+N
+�
+n=1
+����
+∂∥gθ(x(n); θ)∥2
+2
+∂θi
+����
+(2)
+where i stands for the index of network parameter weights and x(n) denotes n-th input image from
+a total of N numbers of images. Note that each Ωi can be interpreted as a measure of influence or
+importance of θi in producing the output of given N input images.
+3.2
+INSTANCE-WISE UNLEARNING FOR PRE-TRAINED CLASSIFIERS
+Definition of instance-wise unlearning.
+Let ˆgθ denote the model after unlearning. We consider
+two types of goals for instance-wise unlearning: (i) misclassifying all data points in Df, (i.e.,
+ˆgθ(xf) ̸= yf). (ii) relabeling (or correcting) the predictions of Df (i.e., ˆgθ(xf) = y∗
+f) where
+y∗
+f ̸= yf is chosen individually for each input xf. Let LUL denote a loss function used for unlearn-
+ing on a classification model. The above two goals can be realized with the following loss functions:
+LMS
+UL(Df; θ) = −LCE(gθ(xf), yf; θ)
+(3)
+LCor
+UL (Df; θ) = LCE(gθ(xf), y∗
+f; θ)
+(4)
+When unlearning solely based on the two loss functions above, the model is likely to suffer from
+significant forgetting on Dr. Therefore, a crucial objective shared across both unlearning goals is to
+overcome forgetting of previously learning knowledge, and maintain as much classification accuracy
+as possible on Dr.
+When both Df and Dr are available, we can easily obtain an oracle model that satisfies the objec-
+tive by re-training the model with the following loss function: Loracle(Df, Dr; θ) = LUL(Df; θ) +
+LCE(Dr; θ). However in real-settings, access to Dr may not be an option due to high cost in data
+4
+
+Unlearning dataset D
+Adversarial examples D
+★:(★,★)
+Remaining dataset DrPreprint. Under review.
+Algorithm 1 Generate adversarial examples
+Input: Forgetting data Df, Model gθ
+Output: Adversarial examples ¯Dr
+1: ¯Dr ← ∅
+2: for i in range Nf do
+3:
+(x(i), y(i)) ∼ Df
+4:
+Randomly sample ¯y(i) ̸= y(i)
+5:
+for j in range Nadv do
+6:
+x′(j)
+f
+← L2-PGD(x(i), ¯y(i)) (Eq. 1)
+7:
+¯Dr ← ¯Dr ∪ {(x′(j)
+f
+, ¯y(j))}
+8:
+end for
+9: end for
+10: return ¯Dr
+Algorithm 2 Measure weight importance
+Input: Forgetting data Df, Model gθ
+Output: Weight importance ¯Ω
+1: ¯Ω ← {0}
+2: Ω ← weight importances(Df, gθ) (Eq. 2)
+3: for l in range L do
+4:
+Get importance of l-th layer Ωl ← Ω
+5:
+Normalize Ωl ←
+Ωl − Min(Ωl)
+Max(Ωl) − Min(Ωl)
+6:
+Update ¯Ωl ← {1 − Ωl}
+7: end for
+8: return ¯Ω
+storage. To tackle this limitation, we define regularization-based unlearning for which the goal is to
+achieve the goal above without explicit use of Dr:
+LRegUL(Df; θ) = LUL(Df; θ) + R(Df, gθ)
+(5)
+Here, R(·) is the regularization term used to overcome forgetting of knowledge on the remaining
+data Dr. In the following subsections, we introduce two novel regularization methods designed to
+overcome representation- and weight-level forgetting during the unlearning process.
+Regularization using adversarial examples.
+The motivation of using adversarial examples stems
+from the work of Ilyas et al. (2019), which showed that perturbations added to x to generate an ad-
+versarial example x′ contain class-specific features of the attack target label ¯y ̸= y. Based on this
+finding, we utilize generated adversarial examples as part of regularization R(·) to preserve class-
+specific knowledge previously learned by the model, overcoming forgetting during unlearning at the
+representation-level. Let Df be a set of Nf images: {(x(i)
+f , y(i)
+f )}Nf
+i=1. Prior to the unlearning pro-
+cess, we generate adversarial examples x′
+f using the targeted PGD attack with a randomly selected
+attack target label ¯y ̸= yf. We generate Nadv adversarial examples per input xf. Then, we have
+¯Df = {(x′(k)
+f
+, ¯y(k)
+f )}
+¯
+Nf
+k=1 where ¯Nf = Nf × Nadv. During unlearning, we add LCE( ¯Df; θ) as a
+regularization term with adversarial examples:
+LAdv
+UL (Df; θ) = LUL(Df; θ) + RAdv(Df, gθ)
+= LUL(Df; θ) + LCE( ¯Df; θ)
+(6)
+An intuitive illustration of this approach in the representation-level is shown in Figure 1. The gen-
+erated adversarial examples ¯Df mimic the remaining dataset Dr, providing information of the pre-
+trained decision boundary within the representation space. As a result, by adding LCE( ¯Df; θ) as a
+regularizer to the unlearning process, the model can learn a new decision boundary that minimizes
+LUL (in Eq. 3 and 4) while simultaneously attempting to keep the decision boundary of the original
+model. The pseudocode for generating adversarial examples is in Algorithm 1.
+Regularization with weight importance.
+We also propose a regularization using weight impor-
+tance to overcome forgetting at the weight-level. As depicted in Figure 1, our approach is to maintain
+the weights that were less important for Df prediction as much as possible, while allowing changes
+in weights that are considered important for correctly predicting Df. That is, it is to prevent the
+weight-level forgetting by penalizing weights that were less important when predicting Df.
+For this, we calculate the weight importance with MAS before unlearning given gθ and Df, and
+normalize the measured importances Ωl within each l-th layer to lie within [0, 1]. Note that this
+normalized importance Ωl assigns large values to weights important for Df. Therefore, we define
+¯Ωl = 1 − Ωl as the weight importance for the regularization used for unlearning, so that more
+important weights are updated more. The objective including weight importance regularization in
+5
+
+Preprint. Under review.
+Table 2: Evaluation results before and after unlearning k instances from ResNet-50 pretrained on
+respective image classification datasets. While using negative gradients only loses significant infor-
+mation on Dr, our proposed methods ADV and ADV+IMP retain predictive performance on Dr as
+well as Dtest, while completely forgetting instances in Df.
+CIFAR-10
+CIFAR-100
+ImageNet-1K
+k = 4
+k = 16
+k = 64
+k = 128
+k = 4
+k = 16
+k = 64
+k = 128
+k = 4
+k = 16
+k = 64
+k = 128
+Df (↓)
+BEFORE
+100.0
+100.0
+99.38
+99.53
+100.0
+100.0
+100.0
+100.0
+91.66
+87.50
+84.90
+86.72
+NEGGRAD
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+ADV
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+ADV+IMP
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+Dr (↑)
+BEFORE
+99.60
+99.60
+99.60
+99.60
+99.98
+99.98
+99.98
+99.98
+87.42
+87.42
+87.42
+87.42
+NEGGRAD
+38.44
+15.79
+9.22
+7.11
+99.71
+66.97
+26.20
+11.64
+83.34
+61.18
+40.50
+30.16
+ADV
+79.40
+69.70
+66.97
+53.49
+83.90
+89.18
+81.07
+76.28
+74.13
+81.09
+76.02
+69.01
+ADV+IMP
+82.95
+85.75
+72.77
+54.51
+83.89
+89.91
+89.48
+82.86
+74.16
+81.77
+79.36
+75.33
+Dtest (↑)
+BEFORE
+92.59
+92.59
+92.59
+92.59
+77.10
+77.10
+77.10
+77.10
+76.01
+76.01
+76.01
+76.01
+NEGGRAD
+36.56
+15.87
+9.28
+7.11
+74.54
+48.07
+21.11
+10.19
+72.53
+53.30
+35.61
+26.73
+ADV
+74.34
+65.14
+62.23
+49.47
+60.00
+63.17
+57.43
+53.89
+62.12
+70.42
+65.89
+59.73
+ADV+IMP
+77.53
+79.65
+67.08
+50.82
+60.50
+63.69
+62.83
+58.44
+65.15
+70.97
+68.72
+65.09
+addition to regularization via adversarial examples can be written as:
+LAdv+Imp
+UL
+(Df; θ) = LAdv
+UL (Df; θ) + RImp(Df, gθ)
+= LAdv
+UL (Df; θ) +
+�
+i
+¯Ωi(θi − ˜θi)2
+(7)
+where i is the index of each weight and ˜θ is the initial weight of the pre-trained classifier before
+unlearning. The pseudocode of measuring weight importance is shown in Algorithm 2. Throughout
+various experiments, we observe that applying the regularization using adversarial examples is al-
+ready effective to overcome the forgetting for knowledge of Dr, and the additional regularization
+with weight importance further enhances performance even further, especially in more harder sce-
+narios such as continual unlearning. The pseudocode of the overall unlearning pipeline is shown in
+the supplementary material.
+4
+EXPERIMENTS
+In this section, we evaluate our proposed instance-wise unlearning methods in various image clas-
+sification benchmarks. We first describe our experimental setup, including datasets, baselines and
+experimental details. We then show that our methods effectively preserves knowledge of remaining
+data while unlearning instances that should be forgotten in both single-task and continual unlearning
+scenarios. Lastly, we offer qualitative analyses on three parts: prediction patterns, decision boundary
+and layer-wise representations in unlearning.
+4.1
+SETUP
+Datasets and baselines.
+We evaluate our unlearning methods on three different image classifi-
+cation datasets: CIFAR-10, CIFAR-100 (Krizhevsky et al., 2009), and ImageNet-1K (Deng et al.,
+2009). Also, we use the ResNet-50 (He et al., 2016) as a base model. The experimental results of
+various base models are available in the appendix. The compared methods are as follows: BEFORE,
+the pre-trained model before unlearning; NEGGRAD (Golatkar et al., 2020), fine-tuning on Df using
+negative gradients (i.e. LMS
+UL); CORRECT, fine-tuning using LCor
+UL ; ADV is our proposed method using
+adversarial examples (i.e. LAdv
+UL ); ADV+IMP, our unlearning method using both adversarial examples
+and the weight importance regularization (i.e. LAdv+Imp
+UL
+).
+Experimental details.
+For each dataset, we randomly pick k ∈ {4, 16, 64, 128} images from the
+entire training dataset as the unlearning data Df and consider the remaining as Dr. For the unlearn-
+ing, we use a SGD optimizer with a learning rate of 1e-3, weight decay of 1e-5, and momentum
+of 0.9 across all experiments. We take early stopping when the model attains zero accuracy from
+the unlearning data Df. For generating adversarial examples from Df, we use L2-PGD targeted
+attack (Madry et al., 2017) with a learning rate of 1e-1, attack iterations of 100 and ϵ = 0.4. It gen-
+erates 20 adversarial examples for CIFAR-10 and 200 examples for CIFAR-100 and ImageNet-1K.
+For the weight importance regularization, we set regularization strength λ = 1 in Eq. 5.
+6
+
+Preprint. Under review.
+Table 3: Results analogous to Table 6, but with unlearning via relabeling each image in Df to an
+arbitrarily chosen class. We see a similar trend where CORRECT loses significant information on
+Dr, while our proposed methods retain predictive performance on Dr as well as Dtest.
+CIFAR-10
+CIFAR-100
+ImageNet-1K
+k = 4
+k = 16
+k = 64
+k = 128
+k = 4
+k = 16
+k = 64
+k = 128
+k = 4
+k = 16
+k = 64
+k = 128
+Df (↑)
+BEFORE
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+CORRECT
+100.0
+100.0
+100.0
+100.0
+100.0
+100.0
+100.0
+99.84
+100.0
+100.0
+100.0
+100.0
+ADV
+95.0
+100.0
+99.375
+98.28
+90.0
+100.0
+100.0
+98.28
+100.0
+100.0
+87.5
+71.32
+ADV+IMP
+90.0
+100.0
+53.75
+50.16
+80.0
+86.25
+20.63
+15.16
+100.0
+100.0
+8.59
+4.30
+Dr (↑)
+BEFORE
+99.60
+99.60
+99.60
+99.60
+99.98
+99.98
+99.98
+99.98
+87.42
+87.42
+87.42
+87.42
+CORRECT
+28.39
+11.75
+12.33
+9.71
+96.14
+74.84
+31.79
+18.64
+84.34
+82.94
+76.21
+68.03
+ADV
+81.43
+85.53
+83.36
+81.06
+69.55
+92.94
+94.64
+96.32
+70.05
+83.09
+84.75
+84.54
+ADV+IMP
+83.43
+91.15
+94.76
+90.57
+68.77
+90.73
+96.68
+96.44
+74.18
+83.34
+83.27
+80.15
+Dtest (↑)
+BEFORE
+92.59
+92.59
+92.59
+92.59
+77.10
+77.10
+77.10
+77.10
+76.01
+76.01
+76.01
+76.01
+CORRECT
+27.62
+11.79
+12.16
+9.80
+69.82
+53.11
+24.37
+14.64
+73.26
+71.90
+65.68
+58.25
+ADV
+76.35
+79.15
+76.95
+74.61
+51.23
+65.62
+66.79
+68.56
+64.81
+72.02
+73.41
+73.32
+ADV+IMP
+78.08
+84.24
+86.92
+82.82
+50.60
+64.28
+69.15
+68.60
+64.94
+72.20
+71.82
+68.92
+4.2
+MAIN RESULTS
+Results on various datasets.
+Table 6 shows evaluation results before and after unlearning k in-
+stances from ResNet-50 models pre-trained on each of three different datasets. With respect to
+accuracies on Df, we find that ResNet-50 can completely forget up to k = 128 instances with
+consistently zero post-unlearning accuracies. On CIFAR-10, using negative gradients only results
+in significant loss of accuracy on the remaining data (i.e. Dr and Dtest), performing worse than
+random-choice when the number of forgetting instances is as large as 128. Meanwhile, adding regu-
+larization with adversarial examples boosts the accuracy by more than 40% depending on the number
+of instances to forget. Incorporating weight importances from MAS provides further improvement.
+Results from CIFAR-100 and ImageNet-1K show a similar trend except when k = 4, where adding
+our regularization approaches deteriorates performance. This well aligns with our intuition as the
+model can easily misclassify a small number of examples by tweaking a small number of model
+parameters, hence forgetting Df without losing much information on Dr and Dtest despite lack
+of regularization. The benefit of using adversarial examples is also small when k is small as the
+diversity amongst images in Dadv is limited by the number of instances to forget.
+Table 7 shows results analogous to Table 6, but with the goal of relabeling data points in Df to
+arbitrarily chosen labels rather than misclassifying. We find that a similar trend, where ADV attains
+significantly less forgetting in Dr and Dtest compared to CORRECT, while succesfully relabeling all
+points in most cases. While ADV+IMP show even less forgetting, it loses accuracy in relabeling Df,
+showing that regularization via weight importance focuses too much on retaining previous knowl-
+edge rather than adapting to corrections provided in Df. An intuitive explanation on why this occurs
+particularly in relabeling is that while misclassifying can be done easily by driving the input to its
+closest decision boundary, relabeling can be difficult if the new class is far from the original class in
+the representation space. The difficulty rises even more when the size of Df is large, in which case
+more parameters in the network are discouraged from being updated during unlearning.
+Correcting natural adversarial examples.
+Leveraging the ImageNet-A (Hendrycks et al., 2021)
+dataset consisting of natural images that are misclassified with high-confidence by strong classifiers,
+we test whether our method can make corrections on these adversarial examples, while preserving
+knowledge from the original training data. For this experiment, we consider Df to consist k adver-
+sarial images from ImageNet-A, and adjust a ResNet-50 model pre-trained on ImageNet-1K to cor-
+rectly classify Df via our unlearning framework. Table 4 shows the results for k = {16, 32, 64, 128}.
+We find that correcting predictions of a small number of images (e.g. k = 16), finetuning the model
+na¨ıvely with cross-entropy only attains the best accuracy in both Dr and Dtest. When correcting
+larger number of images, however, the absence of regularization terms results in larger forgetting in
+Dr compared to ADV and ADV+IMP, with a performance gap that consistently increases with the
+number of adversarial images. Another takeaway is that regularization via weight importance does
+not help in this scenario, even showing a significant drop in Df accuracy when a large number of
+adversarial images are introduced. This implies that using weight importances imposes too strong
+a regularzation that correcting predictions for Df itself becomes non-trivial. We conjecture that the
+aggregation of important parameters for predictions in Df cover a large proportion of the network
+with large k, and that careful search for the Pareto optimal between accuracies on Df and on Dr is
+required.
+7
+
+Preprint. Under review.
+Table 4: Correcting adversarial images from
+ImageNet-A. ADV achieves the least forgetting,
+while ADV+IMP fails to correct large number of
+predictions due to strong regularization.
+ImageNet-A
+k = 16
+k = 32
+k = 64
+k = 128
+Df (↑)
+BEFORE
+0.0
+0.0
+0.0
+0.0
+CORRECT
+100.0
+100.0
+100.0
+100.0
+ADV
+100.0
+100.0
+95.31
+83.44
+ADV+IMP
+100.0
+100.0
+10.94
+9.38
+Dr (↑)
+BEFORE
+87.46
+87.46
+87.46
+87.46
+CORRECT
+84.41
+83.29
+80.79
+77.38
+ADV
+81.75
+83.80
+83.74
+83.44
+ADV+IMP
+81.82
+83.73
+83.53
+82.86
+Dtest (↑)
+BEFORE
+76.15
+76.15
+76.15
+76.15
+CORRECT
+73.21
+72.04
+69.91
+66.73
+ADV
+70.89
+72.58
+72.68
+72.36
+ADV+IMP
+70.98
+72.51
+72.39
+71.68
+Table 5: Unlearning instances continually by in-
+crements of kCL = 8 images per step. Our meth-
+ods outperform NEGGRAD in the continual un-
+learning scenario as well.
+CIFAR-100 (kCL = 8)
+k = 8
+k = 16
+k = 64
+k = 128
+Df (↓)
+BEFORE
+100.0
+100.0
+100.0
+100.0
+NEGGRAD
+0.0
+0.0
+0.0
+0.52
+ADV
+0.0
+0.0
+1.04
+0.0
+ADV+IMP
+0.0
+0.0
+0.0
+1.04
+Dr (↑)
+BEFORE
+99.98
+99.98
+99.98
+99.98
+NEGGRAD
+80.58
+31.85
+6.60
+1.89
+ADV
+80.33
+70.54
+59.67
+38.16
+ADV+IMP
+81.46
+72.78
+62.30
+47.14
+Dtest (↑)
+BEFORE
+77.10
+77.10
+77.10
+77.10
+NEGGRAD
+58.20
+24.48
+5.73
+1.22
+ADV
+57.56
+50.43
+43.48
+30.10
+ADV+IMP
+58.33
+51.97
+45.09
+36.17
+Continual unlearning.
+In real-world scenarios, it is likely that data removal requests come as
+a stream, rather than all at once. Ultimately, despite continual unlearning requests, we need the
+unlearning method that can delete the requested data while maintaining performance for the rest data.
+Thus, we consider the setting of deleting k = {8, 16, 64, 128} data by repeating the procedure of
+continually unlearning Df in small fragments of size kCL = 8. Table 5 shows the results of continual
+unlearning in the model trained with ResNet-50 on CIFAR-100. We observe that NEGGRAD suffers
+from large forgetting as the iteration of unlearning procedure increases. On the other hand, our
+proposed method shows significantly less forgetting while effectively deleting for Df even after
+multiple iterations of unlearning.
+4.3
+QUALITATIVE ANALYSIS
+Through further analysis, we gather insight on the following questions: Q1. Is there any particular
+pattern in how the model unlearns a set of instances (i.e. does the model use any particular label as
+a retainer for deleted data)? Q2. How does the model isolate out instances in Df from its previous
+decision boundary? Q3. How do layer-wise representations of data points in Df and Dr change
+before and after unlearning? For interpretable visualizations, we perform the following analysis on
+a ResNet-18 model pre-trained on CIFAR-10.
+(a) NEGGRAD
+(b) ADV
+(c) ADV+IMP
+Figure 2: Confusion matrices showing average
+pairwise frequencies of pre- (Y-axis) and post-
+unlearning (X-axis) prediction labels from Df. A
+hue closer to blue indicates higher frequency. Our
+unlearning framework does not produce any dis-
+cernible correlation in misclassification.
+A1. Our method shows no pattern in mis-
+classification.
+We first check whether the un-
+learned model classifies all instances in Df to
+a particular set of labels. The model exhibit-
+ing no correlation between true labels and new
+misclassified labels is crucial with respect to
+data privacy, as it indicates that the unlearn-
+ing process avoids the so-called Streisand ef-
+fect where data instances being forgotten unin-
+tentionally becomes more noticeable (Golatkar
+et al., 2020). Figure 2 shows the confusion ma-
+trices of (pre-unlearning label, post-unlearning
+label) pairs from Df for k = 512. We find no
+distinguishable pattern when unlearning with our methods as well as NegGrad, which shows that no
+specific label is used as a retainer, which adds another layer of security against adversaries in search
+of unlearned data points.
+A2. Our method effectively preserves the decision boundary.
+We check whether the adversar-
+ial examples generated from forgetting data help in preserving the decision boundary in the feature
+space. Figure 3 shows t-SNE (Van der Maaten & Hinton, 2008) visualizations of final-layer acti-
+vations from examples in Dr and Df before and after unlearning. We find that unlearning through
+only negative gradient significantly distorts the previous decision boundary, leading to poor predic-
+8
+
+NegGrad
+0.5
+9
+80
+0.4
+7
+6
+labels
+0.3
+5
+4
+0.2
+m
+2
+0.1
+1
+0.0
+0
+2
+7
+Predicted labelsOurs (Adv)
+0.5
+9
+8
+0.4
+7
+6
+labels
+0.3
+5
+4
+0.2
+m
+2
+0.1
+1
+1
+0.0
+0
+2
+7
+Predicted labelsOurs（Adv+Imp）
+0.5
+9
+8
+0.4
+7
+6
+labels
+0.3
+5
+4
+0.2
+m
+2
+0.1
+1
+1
+1
+0.0
+0
+2
+3
+4
+>
+9
+Predicted labelsPreprint. Under review.
+(a) BEFORE
+(b) NEGGRAD
+(c) ADV
+(d) ADV+IMP
+Figure 3: t-SNE plots of CIFAR-10 datapoints in Df (triangles) and Dr (dots) before and after
+unlearning. Colors indicate true labels for all plots. Regularization with adversarial examples and
+weight importance effectively preserves the decision boundary while migrating instances in Df
+towards the class boundary to induce misclassification.
+tive performance across Dr. However, when we incorporate adversarial samples from instances in
+Df, the decision boundary is well-preserved with unlearned examples being inferred as boundary
+cases in-between multiple classes. Even for examples that lie far from the decision boundary be-
+fore unlearning, our method successfully relocates the corresponding representations towards the
+decision boundary, while keeping each class cluster intact.
+(a) NEGGRAD
+(b) ADV
+(c) ADV+IMP
+Figure 4: Layer-wise CKA correlations on Df
+(top row) and Dr (bottom row) between repre-
+sentations before (X-axis) and after (Y-axis) un-
+learning. Brighter color indicates higher CKA cor-
+relation. NEGGRAD results in large forgetting of
+high-level features in not only Df, but also Dr.
+Our approaches, on the other hand, selectively for-
+get high-level features only in Df.
+A3. Our method unlearns data by forgetting
+high-level features.
+Lastly, we compare the
+representations at each layer of the model be-
+fore and after unlearning to identify where the
+intended forgetting occurs. For this analysis, we
+leverage CKA (Kornblith et al., 2019) which
+measures correlations between representations
+given two distinct models. Figure 4 shows the
+CKA correlation heatmaps between the origi-
+nal ResNet-18 model pre-trained on CIFAR-10
+and the same model after unlearning. Results
+show that for examples in Df, representations
+are no longer aligned starting from the 10-th
+layer while the representations before that layer
+still resemble those from the original model.
+This indicates that the model forgets examples
+by forgetting high-level features, while simi-
+larly recognizing low-level features in images
+as the original model. This insight is consis-
+tent with previous observations in the contin-
+ual learning literature that more forgettable ex-
+amples exhibit peculiarities in high-level fea-
+tures (Toneva et al., 2018).
+5
+CONCLUDING REMARKS
+We propose an instance-wise unlearning framework that deletes information from a pre-trained
+model given a set of data instances with mixed labels. Rather than undoing the influence of given
+instances during the pre-training, we aim for a stronger form of unlearning via intended misclas-
+sification. We develop two regularization techniques that reduce forgetting on the remaining data,
+one utilizing adversarial examples of deleting instances and another leveraging weight importances
+to focus updates to parameters responsible for propagating information we wish to forget. Both ap-
+proaches are agnostic to the choice of architecture, and requires access only to the pre-trained model
+and instances requested for deletion. Experiments on various image classification datasets showed
+that our methods effectively mitigates forgetting on remaining data, while completely misclassify-
+ing deletion data. Further qualitative analyses show that our unlearning framework does not show
+any pattern in misclassification (i.e. the Streisand effect), preserves the decision boundary with the
+help of adversarial examples, and unlearns by forgetting high-level features of deleting data. These
+9
+
+Residual Data (D_r)
+1.0
+16
+14 
+0.8
+Case 3: -CE(D_f) + CE(adv) + Reg(importance)
+12
+0.6
+10
+0.4
+4 -
+0.2
+2
+Fo
+0.0
+0
+2
+4
+6
+8
+10
+12 
+14
+16
+Before UnlearningForgettingData(D_f)
+1.0
+16
+14 -
+F0.8
+12
+0.6
+10
+Case 1: - CE(D_f)
+8
+ 0.4
+6
+4 -
+0.2
+2 
+0
+0.0
+2
+4
+6
+8
+10
+12
+14
+16
+Before UnlearningForgettingData(D_f)
+1.0
+16
+14 
+0.8
+12
+Case 2: -CE(D_f) + CE(adv)
+0.6
+10
+8
+0.4
+6
+4 -
+0.2
+2
+0
+0.0
+2
+4
+6
+8
+10
+12
+14
+16
+Before UnlearningForgettingData(D_f)
+1.0
+16
+14 
+ 0.8
+Case 3: -CE(D_f) + CE(adv) + Reg(importance)
+12
+0.6
+10
+8
+0.4
+6
+4
+0.2
+2
+0.0
+2
+4
+6
+8
+10
+12
+14
+16
+Before UnlearningResidual Data (D_r)
+1.0
+16
+14 
+0.8
+12
+ 0.6
+Case 1: - CE(D_f)
+10
+ 0.4
+9
+4 -
+0.2
+2 
+0
+0.0
+0
+2
+4
+6
+8
+10
+12
+14 
+16
+Before UnlearningResidual Data (D_r)
+1.0
+16
+14 
+0.8
+12
+Case 2: -CE(D_f) + CE(adv)
+0.6
+10
+8
+0.4
+6
+4 -
+0.2
+2
+0.0
+0
+2
+4
+6
+8
+10
+12 
+14 
+16
+Before UnlearningPreprint. Under review.
+observations shed light towards future work evaluating the utility our approach as a defense mech-
+anism against membership inference attacks that predict whether a data point was included in the
+training set by using posterior confidence (Shokri et al., 2017; Salem et al., 2018; Yeom et al., 2018;
+Sablayrolles et al., 2019) or its distance to nearby decision boundaries (Choquette-Choo et al., 2021;
+Li & Zhang, 2021). Removing harmful information that lead to socially unfair and biased predic-
+tions based upon sensitive traits such as race, gender, and religion (Mehrabi et al., 2021) is another
+potential contribution from this work.
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+A
+APPENDIX
+A.1
+PSEUDO CODE OF OVERALL UNLEARNING PROCESS
+Algorithm 3 The pseudo code of overall unlearning process the case of using LMS
+UL.
+1: UNLEARNACC = 100
+2: MAXEP = 100
+3: EP = 0
+4: ¯Dr ← Generate adversarial examples with Algorithm 1
+5: ¯Ω ← Measure weight importance with Algorithm 2
+6: ˜θ ← θ
+7: while UNLEARNACC ̸= 0 do
+8:
+Minimize Eqn (6) and (7)
+9:
+UNLEARNACC = GetAccuracy(Df, gθ)
+10:
+if EP > MAXEP then
+11:
+break
+12:
+EP += 1
+13:
+end if
+14: end while
+15: return ˆθ
+Algorithm 4 The pseudo code of overall unlearning process the case of using LCor
+UL .
+1: UNLEARNACC = 0
+2: MAXEP = 100
+3: EP = 0
+4: ¯Dr ← Generate adversarial examples with Algorithm 1
+5: ¯Ω ← Measure weight importance with Algorithm 2
+6: ˜θ ← θ
+7: while UNLEARNACC ̸= 100 do
+8:
+Minimize Eqn (6) and (7)
+9:
+UNLEARNACC = GetAccuracy(Df, gθ)
+10:
+if EP > MAXEP then
+11:
+break
+12:
+EP += 1
+13:
+end if
+14: end while
+15: return ˆθ
+A.2
+ADDITIONAL RESULTS ON VARIOUS MODELS
+Results on various models.
+Figure 5 shows unlearning results on CIFAR-100, but with different
+model architectures. We find that our methods effectively preserve knowledge outside the forgetting
+data, resulting in up to 40% boost in accuracy. NegGrad again outperforms our methods when k = 4,
+but soon breaks down when unlearning more instances. Interestingly, SqueezeNet and MobileNetv2
+suffer from larger forgetting in Dr and Dtest than ResNet-50, possibly due to the width being nar-
+rower as previously investigated by Mirzadeh et al. (2022). ViT also suffers from large forgetting, an
+observation consistent with previous work which showed that ViT suffers more catastrophic forget-
+ting compared to other CNN-based methods in continual learning due to Transformer architectures
+requiring large amounts of data. We also evaluate the results of unlearning on ImageNet-1K with
+varying k in Figure 6. Our proposed methods prevent forgetting knowledge about the rest data Dr
+better than NegGrad in all cases where k is greater than 8. At the same time, the methods effectively
+delete information about Df.
+A.3
+SUPPLEMENTARY MATERIALS FOR REBUTTAL
+14
+
+Preprint. Under review.
+Table 6: Evaluation results before and after unlearning k instances from ResNet-50 pretrained on
+respective image classification datasets. While using negative gradients only loses significant infor-
+mation on Dr, our proposed methods ADV and ADV+IMP retain predictive performance on Dr as
+well as Dtest, while completely forgetting instances in Df.
+CIFAR-10
+CIFAR-100
+k = 4
+k = 16
+k = 64
+k = 128
+k = 4
+k = 16
+k = 64
+k = 128
+Df (↓)
+BEFORE
+100.0
+100.0
+99.38
+99.53
+100.0
+100.0
+100.0
+100.0
+ORACLE
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+NEGGRAD
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+ADV
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+ADV+IMP
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+Dr (↑)
+BEFORE
+99.60
+99.60
+99.60
+99.60
+99.98
+99.98
+99.98
+99.9
+ORACLE
+93.43
+98.74
+99.72
+98.97
+99.68
+99.96
+96.17
+96.74
+NEGGRAD
+38.44
+15.79
+9.22
+7.11
+99.71
+66.97
+26.20
+11.64
+ADV
+79.40
+69.70
+66.97
+53.49
+83.90
+89.18
+81.07
+76.28
+ADV+IMP
+82.95
+85.75
+72.77
+54.51
+83.89
+89.91
+89.48
+82.86
+Dtest (↑)
+BEFORE
+92.59
+92.59
+92.59
+92.59
+77.10
+77.10
+77.10
+77.10
+ORACLE
+86.28
+90.21
+91.01
+89.44
+77.49
+64.41
+67.06
+66.88
+NEGGRAD
+36.56
+15.87
+9.28
+7.11
+74.54
+48.07
+21.11
+10.19
+ADV
+74.34
+65.14
+62.23
+49.47
+60.00
+63.17
+57.43
+53.89
+ADV+IMP
+77.53
+79.65
+67.08
+50.82
+60.50
+63.69
+62.83
+58.44
+Table 7: Results analogous to Table 6, but with unlearning via relabeling each image in Df to an
+arbitrarily chosen class. We see a similar trend where CORRECT loses significant information on
+Dr, while our proposed methods retain predictive performance on Dr as well as Dtest.
+CIFAR-10
+CIFAR-100
+ImageNet-1K
+k = 4
+k = 16
+k = 64
+k = 128
+k = 4
+k = 16
+k = 64
+k = 128
+k = 4
+k = 16
+k = 64
+k = 128
+Df (↑)
+BEFORE
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+ORACLE
+100.0
+100.0
+100.0
+100.0
+100.0
+100.0
+100.0
+100.0
+CORRECT
+100.0
+100.0
+100.0
+100.0
+100.0
+100.0
+100.0
+99.84
+100.0
+100.0
+100.0
+100.0
+ADV
+95.0
+100.0
+99.375
+98.28
+90.0
+100.0
+100.0
+98.28
+100.0
+100.0
+87.5
+71.32
+ADV+IMP
+90.0
+100.0
+53.75
+50.16
+80.0
+86.25
+20.63
+15.16
+100.0
+100.0
+8.59
+4.30
+Dr (↑)
+BEFORE
+99.60
+99.60
+99.60
+99.60
+99.98
+99.98
+99.98
+99.98
+87.42
+87.42
+87.42
+87.42
+ORACLE
+94.90
+99.73
+99.94
+99.90
+97.94
+99.90
+99.97
+99.79
+CORRECT
+28.39
+11.75
+12.33
+9.71
+96.14
+74.84
+31.79
+18.64
+84.34
+82.94
+76.21
+68.03
+ADV
+81.43
+85.53
+83.36
+81.06
+69.55
+92.94
+94.64
+96.32
+70.05
+83.09
+84.75
+84.54
+ADV+IMP
+83.43
+91.15
+94.76
+90.57
+68.77
+90.73
+96.68
+96.44
+74.18
+83.34
+83.27
+80.15
+Dtest (↑)
+BEFORE
+92.59
+92.59
+92.59
+92.59
+77.10
+77.10
+77.10
+77.10
+76.01
+76.01
+76.01
+76.01
+ORACLE
+87.33
+91.65
+91.99
+91.57
+71.56
+74.05
+74.93
+74.15
+CORRECT
+27.62
+11.79
+12.16
+9.80
+69.82
+53.11
+24.37
+14.64
+73.26
+71.90
+65.68
+58.25
+ADV
+76.35
+79.15
+76.95
+74.61
+51.23
+65.62
+66.79
+68.56
+64.81
+72.02
+73.41
+73.32
+ADV+IMP
+78.08
+84.24
+86.92
+82.82
+50.60
+64.28
+69.15
+68.60
+64.94
+72.20
+71.82
+68.92
+15
+
+Preprint. Under review.
+(a) MobileNetv2 (Sandler et al., 2018)
+(b) SqueezeNet (Iandola et al., 2016)
+(c) ViT (Dosovitskiy et al., 2020)
+Figure 5: Experimental results before and after unlearning varying k instances from various models
+on CIFAR-100.
+16
+
+100
+Original
+NegGrad
+Ours (Adv)
+80
+Ours (Adv+Imp)
+Acc.
+60
+40
+20
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+Original
+NegGrad
+Ours (Adv)
+80
+Ours (Adv+Imp)
+Acc.
+60
+40
+20
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+ Acc.
+Original
+60
+NegGrad
+Ours (Adv)
+40
+Ours (Adv+Imp)
+20
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearnina dataset (Df)100
+80
+Dr Acc.
+60
+40
+Original
+20
+NegGrad
+Ours (Adv)
+Ours (Adv+Imp)
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+Df Acc.
+60
+.★-Original
+-. NegGrad
+Ours (Adv)
+40
+Ours (Adv+Imp)
+20
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+Dr Acc.
+60
+40
+Original
+20
+NegGrad
+Ours (Adv)
+Ours (Adv+Imp)
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+Df Acc.
+60
+★:Original
+NegGrad
+Ours (Adv)
+40
+Ours (Adv+Imp)
+20
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+Dr Acc.
+Original
+60
+NegGrad
+Ours (Adv)
+40
+Ours (Adv+Imp)
+20
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+Dr Acc.
+Original
+60
+NegGrad
+Ours (Adv)
+40
+Ours (Adv+Imp)
+20
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)Preprint. Under review.
+(a) MobileNet v2
+(b) ResNet34
+(c) DenseNet121
+Figure 6: Experimental results before and after unlearning varying k instances from various models
+on ImageNet-1K.
+(a) Analysis for entropy-accuracy
+(b) Analysis for a forgotten label
+Figure 7: Experimental analysis with CIFAR-10 dataset using ResNet-18. We randomly select single
+image (k = 1) for unlearning and unlearn it with NegGrad. All experiments are conducted with 100
+seeds. Each class number denotes a specific label, such as {airplane : 0, automobile : 1, bird : 2, cat
+: 3, deer : 4, dog : 5, frog : 6, horse : 7, ship : 8, truck : 9}.
+17
+
+100
+Original
+NegGrad
+Ours (Adv)
+80
+Ours (Adv+Imp)
+Acc.
+60
+40
+20
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+Acc.
+60
+40
+Original
+20
+NegGrad
+Ours (Adv)
+Ours (Adv+Imp)
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+Acc.
+60
+40
+Original
+20
+NegGrad
+Ours (Adv)
+Ours (Adv+Imp)
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+r Acc.
+60
+D
+40
+Original
+20
+NegGrad
+Ours (Adv)
+Ours (Adv+Imp)
+0
+L
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+Df Acc.
+60
+★:Original
+NegGrad
+Ours (Adv)
+40
+Ours (Adv+Imp)
+20
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+r Acc.
+60
+D
+40
+Original
+20
+NegGrad
+Ours (Adv)
+Ours (Adv+Imp)
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+Df Acc.
+60
+★:Original
+NegGrad
+Ours (Adv)
+40
+Ours (Adv+Imp)
+20
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+Dr Acc.
+60
+40
+Original
+20
+NegGrad
+Ours (Adv)
+Ours (Adv+Imp)
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)100
+80
+Df Acc.
+60
+Original
+NegGrad
+Ours (Adv)
+40
+Ours (Adv+Imp)
+20
+0
+1
+2
+4
+8
+16
+32
+64
+128
+256
+Number of unlearning dataset (Df)

8NE1T4oBgHgl3EQfngRF/content/tmp_files/2301.03309v1.pdf.txt

@@ -0,0 +1,1059 @@
+Mapping Charge-Transfer Excitations in
+Bacteriochlorophyll Dimers from First Principles
+Zohreh Hashemi1, Matthias Knodt1, Mario R. G. Marques1, Linn
+Leppert1,2
+1)Institute of Physics, University of Bayreuth, Bayreuth 95440, Germany,
+2)MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The
+Netherlands
+E-mail: l.leppert@utwente.nl
+Abstract.
+Photoinduced charge-transfer excitations are key to understand the primary
+processes of natural photosynthesis and for designing photovoltaic and photocatalytic devices.
+In this paper, we use Bacteriochlorophyll dimers extracted from the light harvesting apparatus
+and reaction center of a photosynthetic purple bacterium as model systems to study such
+excitations using first-principles numerical simulation methods. We distinguish four different
+regimes of intermolecular coupling, ranging from very weakly coupled to strongly coupled,
+and identify the factors that determine the energy and character of charge-transfer excitations
+in each case.
+We also construct an artificial dimer to systematically study the effects of
+intermolecular distance and orientation on charge-transfer excitations, as well as the impact of
+molecular vibrations on these excitations. Our results provide design rules for tailoring charge-
+transfer excitations in Bacteriochloropylls and related photoactive molecules, and highlight
+the importance of including charge-transfer excitations in accurate models of the excited-state
+structure and dynamics of Bacteriochlorophyll aggregates.
+arXiv:2301.03309v1  [physics.chem-ph]  9 Jan 2023
+
+2
+1. Introduction
+Photoinduced charge-transfer excitations are of central importance to the primary processes
+of natural photosynthesis and for photovoltaic and photocatalytic applications [1, 2].
+In
+organic semiconductors, charge-transfer excitations are believed to be important intermediates
+between excited states localized on donor molecules and charge-separated electron-hole states
+on acceptor and donor units, respectively, even though the exact mechanism of charge-
+separation is debated [3–12].
+In photosynthesis, the efficient conversion of solar energy
+into chemical energy is achieved by structurally complex aggregates of Bacteriochlorophylls
+(BCL), Chlorophylls, and other pigment molecules embedded in transmembrane proteins
+that modulate their structure and function. These pigment-protein complexes form light-
+harvesting complexes and reaction centers that are responsible for photon absorption,
+excitation-energy transfer, and charge-separation. Their main operating principles are well-
+understood due to a wealth of crystallographic and spectroscopic studies complemented by
+numerical modelling using semi-empirical and first-principles approaches [13–22].
+Figure 1. Crystal structure of BCL aggregates in the reaction center (RC) and light-harvesting
+II (LHII) complex of the purple bacteria Rhodobacter sphaeroides and Rhodoblastus
+acidophilus, respectively. Dimers of BCLs are highlighted in color using (a) pink for the
+special pair PA – PB, (b) orange for the A branch dimer PA – BA, (c) red for a dimer from
+the B800 and blue for a dimer from the B850 ring of the LHII complex. Hydrogen atoms are
+omitted for clarity.
+In purple bacteria, charge separation occurs in reaction centers (RCs) comprising a
+hexameric aggregate of four BCLs and two Bacteriopheophytins, tightly surrounded by
+several protein chains [23–25]. The primary four BCL molecules of this reaction center are
+shown in Figure 1a, highlighting the so-called special pair (SP), a strongly-coupled dimer of
+BCLs called PA – PB in the following. Charge separation in the bacterial RC is initiated by a
+series of energy- and charge-transfer excitations that involve the SP and proceed along the A
+branch, the photoactive of the two pseudo-symmetric branches the RC consists of [19,26–28].
+In Figure 1b, we have highlighted the A-branch dimer PA – BA that has been speculated to be
+involved in the primary charge-separation step, although this assignment is debated in the
+literature [29–32]. Excitation energy reaches the RC through a cascade of excitation-energy
+
+c
+B
+B
+A
+B850
+B
+B
+A
+B8003
+transfer processes that are initiated in the light harvesting II (LHII) complex, consisting of
+two rings of BCL molecules dubbed B850 and B800, respectively, and shown in Figure 1c.
+Neighboring BCLs in the B800 ring are only weakly coupled and excitation-energy transfer
+is well-described by Förster dipole-dipole coupling [33]. In the B850 ring, neighboring BCL
+molecules are closer and intermediate between the weakly coupled B800 and the strongly
+coupled special pair BCLs.
+The excited states that are believed to be responsible for excitation energy transfer
+in and between the light-harvesting complexes and the RC, are commonly thought of as
+Frenkel-like excitons that are spatially relatively localized on one or two BCL molecules [34].
+Semi-empirical models based on Frenkel-excitons Hamiltonians have played an important
+role in modelling the excitation-energy and charge-transfer dynamics in large photosynthetic
+pigment-protein complexes [35–37]. However, for a reliable and predictive representation
+of the electronic coupling between adjacent pigments, charge-transfer excitations need to
+be included in these model Hamiltonians [36, 38–40], calling for accurate first-principles
+calculations of such excitations.
+For computationally efficient first-principles methods such as time-dependent density
+functional theory (TDDFT), charge-transfer excitations were long considered a major
+challenge due to their inherently nonlocal nature, i.e., the spatial separation of the occupied
+and virtual orbitals contributing to these excitations [41].
+TDDFT with optimally-tuned
+range-separated hybrid functionals is a viable solution to this problem, and has been used
+to predict excited states of molecular systems and solids with great success [42–48]. In
+these exchange-correlation functionals, the presence of long-range exact exchange leads to
+asymptotically correct potentials. Additionally, a parameter controlling the range-separation
+of exact and semilocal exchange can be used to tune the energies of the highest occupied and
+the lowest unoccupied orbitals to correspond to the negative of the ionization potentials and
+the electron affinity, respectively, within the conceptual framework of generalized Kohn-Sham
+[49]. Both conditions are crucial for accurately capturing charge-transfer excitations within
+linear-response TDDFT [50] and have been extended to solvated molecular systems [46, 51]
+and extended solids [48].
+An alternative approach for calculating charge-transfer excitations of molecules and
+solids is the GW+Bethe-Salpeter Equation (GW+BSE) approach [52,53]. While this method
+was initially primarily applied to solids, recent years have witnessed a multitude of studies
+that have demonstrated the accuracy and predictive power of the GW+BSE method for small
+molecules [54–56] and larger molecular complexes [57–61].
+In particular, we [62] and
+others [61] benchmarked the accuracy of the GW+BSE approach against experiment and
+wavefunction-based methods and found excellent agreement for the Qy and Qx excitations
+of a range of BCL and Chlorophyll molecules.
+We showed that both eigenvalue self-
+consistent GW calculations and one-shot G0W0 calculations where the zeroth-order single-
+particle Green’s function G0 and screened Coulomb interaction W0 were constructed from a
+DFT eigensystem obtained with an optimally-tuned range-separated hybrid functional lead
+to the best results. TDDFT with an optimally-tuned hybrid-functional performed slightly
+worse and tended to overestimate the energy of the Qy excitations, in agreement with previous
+
+4
+studies [57].
+In this article, we report a systematic first-principles study of charge-transfer excitations
+in BCL dimers - the smallest structural units in which excitations with intermolecular charge-
+transfer character can be observed. These BCL dimers, extracted from the LHII complex and
+RC of purple bacteria, constitute our model systems. Our goal is to elucidate the factors that
+determine the energy and character of these excitations, in particular their mixing with the
+coupled Qy and Qx excitations of the dimers. We treat these dimers as representative of four
+different regimes of intermolecular coupling resulting in distinct charge-transfer properties:
+1. The B800 dimer is very weakly coupled with Qy and Qx excitations resembling those
+of the monomeric units and high-energy charge-transfer excitations due to vanishing orbital
+overlap. 2. The A-branch dimer is more strongly coupled and exhibits one charge-transfer
+excitation corresponding to electron transfer from PA to BA. We use the notation P+
+A B−
+A to
+indicate the direction of charge-transfer in the following.
+This charge-transfer excitation
+is ∼0.4 eV higher in energy than the coupled Qx excitations. 3. The B850 dimer is even
+more strongly coupled. The lowest-energy charge-transfer excitation mixes with the coupled
+Qx excitations and another charge-transfer state appears at higher energies. 4. Finally, the
+special pair SP is the most strongly coupled case with three charge-transfer excitations mixing
+with the coupled Qx excitations. Additionally, we construct an artificial BCL dimer and
+systematically study the effects of intermolecular distance and orientation on charge-transfer
+excitations. We also estimate the effect of molecular vibrations on charge-transfer excitations.
+We do this by calculating the vibrational normal modes of a dimeric system and determining
+the renormalization of excitation energies for structures distorted along normal modes. This
+allows us to identify vibrational modes with pronounced effects on charge-transfer excitations.
+Finally, we comment on differences and similarities between TDDFT with an optimally-tuned
+range separated hybrid functional and the GW+BSE approach.
+2. Computational Methods
+2.1. First-Principles Methods and Computational Details
+For all calculations reported in this article, we used TDDFT as implemented in TURBOMOLE
+version 7.5 [63] and the GW+BSE approach as implemented in MOLGW version 3.0 [64].
+Briefly, in the linear-response formulation of both methods the excitation energies Ωn can be
+obtained by solving the matrix eigenvalue equation CZ = Ω2
+nZ, where C is
+Cijσ,klτ = (εiσ −εjσ)2δi jδ jlδστ +2�εiσ −εjσ
+√εkτ −εlτKi jσ,klτ
+(1)
+and the indices i,k refer to occupied, j,l to virtual orbitals and σ,τ to spin-indices.
+Differences between TDDFT and the GW+BSE approach enter Equation 1 in two distinct
+ways: 1. Through the differences between virtual and occupied orbital energies εiσ − εjσ
+which are obtained from a (generalized) Kohn-Sham calculation in TDDFT and from the GW
+approach in GW+BSE. 2. Through the kernel matrix element Ki jσ,klτ, which depends on
+the exchange-correlation kernel fxc,σ - the functional derivative of the exchange-correlation
+
+5
+potential - in TDDFT, and on the screened Coulomb interaction W, typically evaluated in the
+random phase approximation and at zero frequency, in the BSE approach [65–68].
+Here we use the optimally-tuned range-separated hybrid functional ωPBE for our
+TDDFT calculations.
+We use a range-separation parameter ω=0.171 a−1
+0 , which we
+determined previously for a single BCL a molecule [62].
+The optimal-tuning procedure
+follows the recipe by Stein et al. and ensures that the HOMO eigenvalue corresponds to
+the ionization potential and the LUMO eigenvalue corresponds to the electron affinity of the
+molecule [69]. We do not perform a new tuning procedure for the dimers for general reasons:
+Using the same ω for each dimer allows us to compare the electronic and excited state
+structure of these systems on the same footing. Furthermore, optimal tuning of conjugated
+systems of increasing size leads to artificially low values of ω and, thus, a dominance
+of semilocal exchange at long range, which deteriorates the description of charge-transfer
+excitations [46,70].
+For our GW+BSE calculations we use a "one-shot" G0W0 approach in which we
+construct the zeroth-order single-particle Green’s function G0 and the screened Coulomb
+interaction W0 from DFT eigenvalues and eigenfunctions calculated using the same ωPBE
+as described above. This approach leads to excellent agreement with experimental excitation
+energies and reference values from wavefunction-based methods for a range of BCL and
+Chlorophyll molecules [62]. Range-separated hybrid functionals have been shown to lead
+to accurate charge-transfer excitations for larger molecular complexes as well [57,71]. In all
+calculations we used a def2-TZVP basis set, and the frozen core and resolution-of-the-identity
+approximations (with the DeMon auxiliary basis set [72]). We did not apply the Tamm-
+Dancoff approximation in any of the results reported in this paper. In our G0W0 calculations,
+we used the optimized virtual subspace method by Bruneval with an aug-cc-pVDZ basis set
+for the reduced virtual orbital subspace [73]. With these settings, our excitation energies
+are converged to within 40 meV. Further details on our convergence tests can be found in
+Section 2.2 and in the Supplemental Material (SM).
+For evaluating the character of the excited states, we calculated their transition densities.
+Since the transition density vanishes for charge-transfer states, we calculated the difference
+density ∆ni = ni − n0 between the excited (ni) and the ground-state density (n0) for every
+excitation i. The excited-state density ni is calculated as the diagonal part of the excited
+state density matrix γii(r,r′) = N
+� Ψi(r,r2,r3,...,rn)Ψi(r′,r2,r3,...,rn)dr2...drn, where N is
+the number of electrons and Ψi is the generalized Kohn-Sham excited-state wavefunction, that
+consists of a sum of Slater determinants of generalized Kohn-Sham orbitals with coefficients
+obtained from TDDFT [74]. To quantify the magnitude of charge transfer we integrated over
+subsystem difference densities. For this purpose, we subdivided the volume containing the
+difference densities of the dimer into subsystem volumes, each containing one pigment. Our
+aim is to assign each grid point of the difference-density grid to its closest pigment molecule.
+For achieving this, we used the distances between grid points and each molecule’s atomic
+coordinates (including hydrogen atoms), as previously done in Ref. [75].
+Finally, to obtain a mode-resolved picture of the effect of thermally-activated vibrations
+(Section 3.3), we relaxed a dimer structure using the B3LYP approximation for the exchange-
+
+6
+correlation functional and def2-TZVP basis set, and evaluated its normal modes and
+frequencies. Using the harmonic approximation, we can relate the amplitude of these normal
+modes with the thermal energy of a molecule. Thus, we distorted the dimer structure along
+its lowest-frequency normal modes at a temperature of 300 K. In this manner, we generated
+60 distortions of the dimer, that we then studied using TDDFT calculations using the ωPBE
+functional. All these calculations were performed using the tools provided in the TURBOMOLE
+package.
+2.2. Convergence of G0W0+BSE calculations
+We carefully tested that our GW+BSE results are converged. Due to the large size of a
+BCL dimer, featuring more than 300 electrons, the calculation of the GW self-energy which
+requires summation over virtual states is computationally demanding. We therefore used the
+optimized virtual subspace method implemented in the MOLGW code, in which a reduced
+virtual orbital subspace represented by a comparably small basis set is used to evaluate the
+GW self-energy [73].
+Figure 2. Convergence of GW (a) HOMO-LUMO gap and (c) energy of the first excited
+state of BCL a monomer as a function of the number of basis functions. Blue data points
+correspond to calculations in which the same basis set is used for the occupied orbitals and the
+virtual subspace. Red points correspond to calculations using the optimized virtual subspace
+method. Lines are fits to these data points. Convergence of the HOMO-LUMO gap and energy
+of the first excited state is shown in panel (b) and (d) for the B850 dimer, respectively. Here,
+green corresponds to using the same basis set for the occupied orbitals and the virtual subspace
+and pink to calculations using the optimized virtual subspace method.
+
+(c)
++
+SCF basis = GW basis
+1.65
+SCF basis = GW basis
+SCF basis = Def2-TZVP
+SCF basis = Def2-TZVP
+4.20
+1.60
+4.15
+.55
+4.10
+1.50
+4.05
+Monomer
+Monomer
+1.45
+4.00
+120014001600
+800
+1000 1200 1400 1600
+800
+1000
+N
+(d)
+N
+b
+basis
+basis
+SCF basis = GW basis
+.65
+4.00
+SCF basis = GW basis
+ excitation (eV)
+SCF basis = Def2-TZVP
+SCF basis = Def2-TZVP
+3.95
+1.60
+3.90
+1.55
+3.85
+1.50
+3.80
+>1.45
+3.75
+3.70
+Dimer
+1.40
+Dimer
+3.65
+1500
+2000
+2500
+3000
+3500
+1500
+2500
+3500
+2000
+3000
+N
+N
+basis
+basis7
+We start by testing the convergence of the HOMO-LUMO gap, and the Qy and Qx
+excitations of a BCL a monomer with respect to basis set size without the optimized virtual
+subspace method (Table S1). In agreement with our previous results [62], we find that the
+def2-TZVP basis set deviates by less than 10 meV from the considerably larger aug-cc-pVTZ
+basis. We proceeded by calculating the convergence of the Qy and Qx excitations of the BCL
+monomer as a function of the number of virtual orbitals Nvirt included in the evaluation of
+the GW self-energy using the def2-TZVP basis (Figure S1). We find that for Nvirt = 500
+both excitations are converged to within 80 meV from the limit of infinite Nvirt. Based on
+these findings we continued by evaluating the effect of using a smaller basis set for the
+virtual subspace [73].
+The results for the HOMO-LUMO gap and the Qy excitation are
+plotted in Figure 2a and c, and show that the optimized virtual subspace method leads to
+an underestimation of the HOMO-LUMO gap and the Qy excitation energy as compared to
+the conventional method in which the same basis set is used for all orbitals. We find that using
+the aug-ccpVDZ basis for the optimized virtual subspace in conjunction with Nvirt = 500 leads
+to a fortuitous error cancellation and results in a HOMO-LUMO gap and Qy and Qx excitation
+energies that are within less than 50 meV of the results obtained with the conventional method
+and Nvirt → ∞ (Figure S2).
+For the dimer, we therefore chose Nvirt = 1000 and the same strategy for determining the
+optimized virtual subspace. We find very similar results for the convergence of the HOMO-
+LUMO gap and the first bright coupled Qy excitation shown in Figure 2b and d. All GW+BSE
+results reported in this paper are therefore based on calculations using the def2-TZVP basis
+set for the occupied orbitals and the aug-ccpVDZ basis for the optimized virtual subspace.
+2.3. Construction of the Model Systems
+We constructed our model systems from the x-ray crystallographic structures of the purple
+bacteria Rhodobacter sphaeroides (structure ID 1M3X in the Protein Data Base) [76] and
+Rhodoblastus acidophilus (structure ID 1NKZ) [77].
+In all structures, we replaced the
+phytyl tail with hydrogen. Hydrogen atoms not resolved in the experimental crystal structure
+were added using AVOGADRO and their positions were optimized while keeping the rest of
+the structure fixed. These geometry optimizations were performed using TURBOMOLE and
+the B3LYP exchange-correlation functional. The reaction center dimers PA – PB and PA –
+BA (Figures 1a and b) were constructed using structure 1M3X while the B800 and B850
+ring dimers (Figure 1c) were extracted from 1NKZ.
+These molecules correspond to ID
+numbers BCL307 and BCL309 for the B800, and BCL302 and BCL303 for the B850 ring.
+We additionally constructed an artificial dimer consisting of two exactly equivalent BCL a
+molecules (using molecule PA) that we initially oriented in the same way as the special pair
+dimer PA – PB by aligning their transition dipole moments (as calculated with TDDFT) with
+those of PA and PB, respectively. We are providing all relevant structure files necessary to
+reproduce the results of this article in the SM.
+
+8
+3. Discussion and Results
+3.1. Charge-Transfer Excitations in RC and LHII Dimers
+We start by comparing the excitation spectrum of the four dimeric systems shown in Figure 1a-
+c using TDDFT and GW+BSE. The energies and oscillator strengths of the first 15 excitations
+of each system can be found in Table S3 and S4.
+The spectra are shown in Figure 3a
+and b, respectively, and allow for several observations.
+First, we find that TDDFT and
+GW+BSE predict qualitatively very similar spectra. The most striking difference appears for
+the B800 dimer, for which the coupled Qy excitations calculated with TDDFT are ∼0.3 eV
+higher in energy than with GW+BSE while all other excitations are at similar energies. This
+observation is consistent with our results for single BCL a molecules for which TDDFT with
+optimally-tuned ωPBE consistently overestimates the Qy excitation energy by ∼0.3 eV [62]
+and therefore leads to an underestimation of the Qy – Qx energy difference as compared
+to experiment.
+Interestingly, this overestimation as compared to GW+BSE, while still
+present, is less pronounced for the other three dimers and seems to decrease with increasing
+intermolecular coupling.
+Figure 3. Excitation spectrum of B800, A-branch, B850, and SP dimers using (a) TDDFT
+with ωPBE and (b) the G0W0@ωPBE+BSE approach. Arrows mark dark excitations without
+(D) and with (CT) charge-transfer character. The shaded areas are calculated by folding the
+excitation energies with Gaussian functions with a width of 0.08 eV as a guide to the eye.
+Second, we find several dark excitations for all four systems, predicted at very similar
+energies with TDDFT and the GW+BSE approach. We analyze the charge-transfer character
+
+(b)
+(a)
+TD-OPBE
+G.W.@oPBE/BSE
+0.8
+0.8
+Str
+B800
+Str
+B800
+lator
+Oscillator
+0.6
+0.6
+Oscil
+0.4
+0.4
+D.D
+2.3
+0
+0.2
+0.2
+0
+0
+0.8
+0.8
+Str
+Str
+A-branch
+A-branch
+ 0.6
+lat
+0.4
+0.4
+Osci
+0.2
+0.2
+0
+0.8
+B850
+B850
+0.6
+Oscill
+0.4
+CT
+CT
+0.2
+0.2
+0
+0.8
+Str
+ 0.8
+SP
+SP
+S
+≥0.6
+lato
+OsC
+0.2
+0.2
+1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
+1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
+Excitation energy (eV)
+Excitation energy (eV)9
+of these excitations by calculating their difference densities and integrating over subsystem
+difference densities as described in Section 2.1. The energy and character of these dark
+excitations considerably differs for our four dimers. For the B800 dimer, we find three dark
+excitations, E5, E6, and E7, ∼0.7 eV above the coupled Qx excitations which are almost
+degenerate. The difference densities (Figure S3 and Table 1) do not indicate any charge-
+transfer character for these excitations - their charge distribution is primarily localized on
+only one BCL in each excitation, and looks similar to those of the monomeric system. Charge-
+transfer excitations can be found at around 3.0 eV, consistent with the large distance of 20 Å
+between the B800 molecules, measured as the distance between their centers of masses.
+dimer
+molecule label
+charge distribution
+E3
+E4
+E5
+E6
+E7
+B800
+B307
+0
+0
+0
+0
+0
+B309
+0
+0
+0
+0
+0
+A-branch
+PA
+0
+0
+-0.97
+0
+0
+BA
+0
+0
+0.97
+0
+0
+B850
+B302
+0
+-0.78
+-0.11
+0.91
+0
+B303
+0
+0.78
+0.11
+-0.91
+0
+SP
+PA
+-0.69
+0
+0
+0.83
+-0.76
+PB
+0.69
+0
+0
+-0.83
+0.76
+Table 1. Difference density integrated over subsystem volumes. The first two excitations,
+i.e., E1 and E2, are not included since their difference densities integrate to zero in all studied
+systems.
+The molecules PA and BA of the A-branch dimer are ∼13 Å apart, leading to stronger
+intermolecular coupling and the appearance of a charge-transfer state in the energy range
+considered here. Figure 3 shows that for this system the coupled Qy and Qx excitations
+are split and the first dark excitation is ∼0.3 eV higher in energy than the second coupled
+Qx excitation. Contrary to the B800 dimer, this dark excitation has clear charge-transfer
+character (Table 1) and corresponds to P+
+A B−
+A . The character of the two following dark states
+is unchanged as compared to B800 apart from a redshift.
+In the B850 dimer with ∼ 11Å distance, the stronger intermolecular coupling leads to
+a further redshift of the dark excitations. We find that a dark state mixes with the coupled
+Qx excitations leading to charge-transfer character in E4 and E5. Another charge-transfer
+excitation in which charge is moved in the other direction is found ∼0.3 eV higher in energy.
+The excitation spectrum of the special pair dimer SP is yet different. Due to the strong
+intermolecular coupling of the two molecules which are only 9 Å apart, three charge-transfer
+excitations appear at relatively low energies.
+The first one is lower in energy than the
+first coupled Qx excitation and corresponds to P+
+A P−
+B , whereas the other two are above the
+coupled Qx excitations and correspond to P−
+A P+
+B and P+
+A P−
+B , respectively. Note that due to
+
+10
+the overestimation of the coupled Qy excitations by TDDFT, GW+BSE predicts the energy
+gap between the coupled Qy excitations and CT1 to be twice as large as TDDFT. Nonetheless,
+since the qualitative features of all four excitation spectra and the charge-transfer character
+of all excitations is similar, we use TDDFT for all further calculations and report GW+BSE
+results in the SM.
+3.2. Charge-Transfer Excitations in Artificial Dimer
+The dimeric systems extracted from the RC and LHII crystal structures discussed in
+Section 3.1, differ in their distance, relative orientation, and the structural details of the two
+molecular subunits comprising the dimer. To disentangle these effects, we therefore proceeded
+by performing TDDFT calculations for an artificial dimeric system constructed as discussed
+in Section 2. The structural parameters that define the distance and relative orientations of
+this dimer are shown in Figure 4. We measure the distance between the molecules r as the
+distance between their centers of masses R1 and R2, i.e., r = |r| = |R1 − R2|. Their relative
+orientation is defined by three angles α, β, and γ. The first angle, α, is a rotation around the
+normal vector of the plane spanned by the Qy and Qx transition dipole moments of a single
+molecule, i.e., it is approximately perpendicular to the porphyrin-ring plane. The second
+rotation axis, associated with β, corresponds to r = R1 −R2. The third rotation, γ, is around
+the axis given by the cross product of r and the normal vector of the Qy – Qx plane. For our
+further discussion, we also distinguish between the four functional groups FG1, FG2, FG3,
+and FG4, highlighted in Figure 4.
+Figure 4. Structure of artificial dimer based on two identical PA molecules. We highlight four
+functional groups FG1 (in green), FG2 (in red), FG3 (in pink), and FG4 (in orange). Hydrogen
+atoms are omitted for clarity.
+We start by investigating the effect of changing the distance r between the molecules PA1
+and PA2, fixing the relative orientation of the molecules such that it corresponds to the one
+found in the special pair dimer SP. Figure 5a shows the excitation spectra of dimers separated
+by 9, 11, and 13 Å, corresponding to the center-of-mass difference found in the special pair
+SP, the B850 dimer, and the A-branch dimer of Section 3.1, respectively. Note that distances
+smaller than 9 Å are not possible for the artificial dimer due to overlap between the FG3
+functional groups. Decreasing the center-of-mass difference leads to a redshift and splitting of
+the coupled Qy excitations accompanied by a redistribution of oscillator strength between the
+
+P
+P
+A2
+FG
+PaintX lite11
+two excitations, in accordance with expectations from Kasha’s exciton theory [78]. The effect
+on the coupled Qx excitations cannot be discussed without also considering the higher-energy
+charge-transfer excitations. The latter are redshifted when going from 13 Å to 11 Å, and mix
+with the coupled Qx excitations at 9 Å, similar to the situation in the special pair dimer SP.
+The corresponding charge distributions based on subsystem integrals of difference densities
+are shown in Table 2 and demonstrate that for the system at r = 9 Å , all excitations in the
+energy-range of the coupled Qx excitations and the higher energy dark states exhibit charge-
+transfer character. We classify E4, which is in the energy range of the coupled Qx excitations
+and corresponds to transfer of half an electron from PA1 to PA2 as a partial charge-transfer state
+(PCT) in Figure 5a. Our results are qualitatively similar when using the GW+BSE approach,
+as shown in Figure S5 and consistent with our discussion in Section 3.1.
+Figure 5.
+(a) Absorption spectra of artificial dimer with r = 9 Å (blue), r = 11 Å (red),
+and r = 13 Å (green). Arrows mark excitations with charge-transfer character. The shaded
+areas are calculated by folding the excitation energies with Gaussian functions with a width of
+0.08 eV as a guide to the eye. (b) The excitation energy of the first two charge-transfer (CT1
+and CT2) excitations and the first four dark states (D1-D4) as a function of r. The color scale
+represents the charge-transfer character of each excitation based on the absolute value of the
+integrated subsystem difference densities. (c) ∆R (see main text) as a function of the rotation
+angle α (top), β (middle), and γ (bottom). Blue lines are periodic fits and serve as a guide to
+the eye. The color scale corresponds to the change in energy ∆E of CT1 as compared to the
+unrotated reference structure.
+These trends are even more apparent in Figure 5b, where we plot the energy of all
+dark excitations as a function of distance and indicate their charge-transfer character in color.
+In the energy range considered here, there are four dark excitations without charge-transfer
+character which are essentially independent of distance and are only redshifted and acquire
+substantial charge-transfer character at relatively small r.
+The two charge-transfer states
+exhibit a significant distance dependence and are red-shifted by almost 1 eV with decreasing
+r but lose some of their charge-transfer character at the smallest distance where they start
+mixing with the coupled Qx excitations.
+For investigating the effect of the relative orientation of the two molecules, we fixed
+the intermolecular distance at 13 Å. Shorter distances were not possible due to overlap of
+functional groups for some orientations. Since rotations around the angles α, β, and γ do not
+commute, we treat them separately from each other, i.e., we first consider rotations around
+
+(a)
+(b)
+(c)
+0.9
+1.0
+2
+0.1
+α
+■ CT,
+●CT.
+^D1
+←D2
+13 A
+1
+0.9
+0.0
+0.8
+3.0
+11A
+0
+0.8
+-1
+-0.1
+0.7
+９A
+一
+-2 
+Excitation energy (eV)
+2.8
+-0.2
+0.7
+-3 
+2
+0.1
+0.6
+2.6
+0.5
+0
+0.5
+△R
+-1
+0.4
+-2
+2.4
+0.4
+-0.2
+-3 
+0.3
+0.3
+0.1
+V
+2.2
+0.2
+0.2
+0.0
+0
+PCT
+CT
+-1 
+-0.1
+0.1
+0.1
+2.0
+-2 
+-0.2
+-3
+0.0
+0.0
+1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7
+8
+10
+12
+14
+18
+20
+22
+0
+50
+100
+150
+ 200
+250
+¥300
+350
+16
+Angle
+Excitation energy (eV)
+r (A)12
+r (Å)
+molecule
+charge distribution
+E4
+E5
+E6
+E7
+E8
+9
+PA1
+-0.48
+0.21
+0.28
+-0.62
+-0.63
+PA2
+0.48
+-0.21
+-0.28
+0.62
+0.63
+11
+PA1
+0
+-0.96
+0.96
+0
+0
+PA2
+0
+0.96
+-0.96
+0
+0
+13
+PA1
+0
+0
+0
+-0.99
+0.99
+PA2
+0
+0
+0
+0.99
+-0.99
+Table 2.
+Charge distribution on each molecule in the artificial dimer upon excitation as
+calculated by integration over subsystem difference densities. The first two excitations, i.e.,
+E1 and E2, are not included since their subsystem difference densities integrate to zero.
+α for fixed β and γ, then rotations around β for fixed α and γ, and finally rotations around
+γ for fixed α and β. For each structure, we determine the smallest intermolecular distance
+between every two individual atoms in PA1 and PA2, R. The difference between R in the
+reference (unrotated) structure from each rotated structure, ∆R = Rre f −Rrot, as a function of
+rotation angle, is shown in Figure 5c. Since charge-transfer excitations CT1 and CT2 follow
+similar trends, we only show the change in energy of CT1 upon rotation in Figure 5c. Negative
+(positive) values of ∆ECT1 = ECT1
+ref −ECT1
+rot correspond to a redshift (blue-shift) of the excitation
+energy.
+Rotations around α and β correspond to orientations with smaller R than in the reference
+structure. Consequently, we observe increased intermolecular coupling and hence a redshift
+of the charge-transfer state by up to ∼0.2 eV. For the structure for which we observe the largest
+effect (corresponding to a β rotation of 120 degrees), it is primarily the relative orientation and
+distance of carbon chains determining the intermolecular coupling (Figure S7a). For many of
+the other structures that show pronounced redshifts, we find that the functional groups of
+the two BCLs highlighted in Figure 4 are in close spatial proximity (see Figure S7b for an
+example). In contrast, the rotation around the angle γ results primarily in structures with
+positive ∆R and a blueshift of the charge-transfer excitation by up to ∼0.1 eV. We note that in
+the majority of structures rotated around γ, the functional groups FG1, FG2, and FG4 are
+far apart from the second BCL. However, for some structures, overlap between FG2 and
+the second BCL molecule led to unrealistic structures that were excluded from Figure 5c.
+Overall, the γ rotation primarily leads to geometries with weaker intermolecular coupling and
+an overall blueshift in energy of the charge-transfer excitation.
+3.3. Vibrational Renormalization of Charge-Transfer Excitations
+Excitations of different spatial localization and character are known to be affected in different
+ways by molecular vibrations [79]. Our goal here is to provide a mode-resolved picture
+of excitation energy renormalization in a BCL dimer due to thermally-activated vibrations,
+
+13
+following earlier work by Hele et al. [80]. For this purpose we started from the crystal
+structure of the special pair dimer SP and performed a full geometry optimization using
+the def2-TZVP basis set and B3LYP exchange-correlation functional. In the absence of the
+protein environment and other co-factors, no external force fixes PA and PB in the parallel
+configuration they have in vivo.
+Consequently, the relaxed structure differs considerably
+from SP, and is more akin to the A-branch dimer. Since our aim is to provide a qualitative
+picture, we proceed with this structure which is dynamically stable, i.e., without imaginary
+normal modes. We note, however, that the excitation spectrum of the relaxed dimer, shown
+in Figure 6a, differs from the spectra discussed so far. In particular, the spectrum displays a
+charge-transfer state CT1 at ∼1.6 eV (see also Table S8). This state mixes with the coupled
+Qy excitations and corresponds to the transfer of 0.78 of an electron from PA to PB (see Table
+S9). A second charge-transfer state CT2 mixes with the coupled Qx excitations, while the
+third one, CT3, is energetically well-separated from the Q-band excitations at ∼2.7 eV.
+Figure 6. (a) Absorption spectrum of relaxed dimer. Arrows mark the first three charge-
+transfer excitations, (b) Excitation energy renormalization ∆E as a function of normal mode
+frequency for CT1, CT2, and CT3. Negative (positive) values of ∆E correspond to a redshift
+(blueshift), (c) Visualization of the first two normal modes which correspond to intermolecular
+rotations (see main text).
+We calculate the vibrational normal modes of the relaxed dimer using the same basis
+set and exchange-correlation functional but with a very fine grid for the quadrature of the
+exchange-correlation energy. We then distort the structure along the 60 first vibrational normal
+modes with a distortion amplitude corresponding to a temperature of 300 K. The excitation
+spectrum of each distorted structure is then calculated with TDDFT as before, i.e., with
+ωPBE with ω = 0.171 a−1
+0 . We define the excitation energy renormalization of excitation
+n as ∆En = En
+ref − En
+dis. Here we focus on how molecular vibrations affect charge-transfer
+excitations, but note that ∆E for the coupled Qy and Qx excitations can also be substantial as
+shown in Figure S8.
+The excitation energy renormalization of the charge-transfer excitations CT1, CT2, and
+CT3 is shown in Figure 6b. High-frequency modes correspond to intramolecular vibrations
+such as C-C and C-H stretch modes, which are not thermally activated and only have a small
+effect on the energy of the three charge-transfer states. In contrast, low-frequency modes
+correspond to intermolecular vibrations that change the orbital overlap between neighboring
+molecules and thus have a more substantial impact. In particular, we find that the two lowest-
+
+(a)
+(b)
+(c)
+CT
+0.5
+0.10
+CT
+(2)
+0.05
+O
+30-0000
+-0.05
+0.10
+CT
+-0.15
+0.1
+0 (1)
+-0.20
+80
+1.6
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+20
+40
+60
+100
+140
+0
+160
+Excitation energy (eV)
+0 (cm-l)14
+frequency modes lead to substantial changes of all three charge-transfer states. Both modes
+correspond to a rotational motion of the porphyrin planes of the BCL molecules with respect to
+each other as indicated in Figure 6c. The first modes leads to a redshift of all three excitations
+which is with ∼0.2 eV most pronounced for CT1, the second one leads to a smaller blueshift
+of CT1 and CT3 and a slight redshift of CT2. These results qualitatively agree with our results
+in Section 3.2, suggesting that thermally-activated vibrational modes can significantly affect
+the energy of charge-transfer excitations affecting their charge-transfer character and mixing
+with other delocalized and localized excitations of the system.
+4. Summary and Conclusions
+In summary, we have presented a systematic first-principles study of charge-transfer
+excitations in BCL dimers. Our model systems are inspired by molecular aggregates found
+in the LHII complex and RC of purple bacteria and cover a wide range of intermolecular
+coupling strengths, and consequently, excited-state structures. Charge-transfer excitations
+can be found in a wide range of energies, primarily depending on intermolecular distance
+and orientation. BCL molecules have a complex three-dimensional structure with several
+functional groups, a long phytyl tail, and other carbon chains protruding out of the porphyrin
+plane. In vivo, i.e., within the evolutionary-optimized protein networks of the photosynthetic
+apparatus, the protein environment determines the distance, orientation, and structural
+details of these aggregates.
+Furthermore, the protein environment indirectly affects the
+excited state structure and dynamics of BCL aggregates through dielectric screening and
+electrostatic effects [75,81,82,82–89]. Therefore our results can not directly be used to infer
+charge-transfer mechanisms in photosynthetic systems Nonetheless, they provide an intuitive
+understanding and design rules for tailoring charge-transfer excitations in BCLs and similar
+photoactive molecules. Furthermore, they explicitly confirm the importance of charge-transfer
+excitations for a correct description of the Q-band excitations of BCL aggregates [40]. We
+hope that our results inspire future calculations of the excited-state structure and dynamics of
+pigment-protein complexes and chromophore aggregates based on model Hamiltonians, that
+include charge-transfer excitations.
+Furthermore, we have compared our results based on TDDFT with the optimally-
+tuned ωPBE functional to calculations using the GW+BSE approach.
+While charge-
+transfer excitations appear at very similar energies with both approaches, coupled Qy
+excitations are systematically overestimated by TDDFT as compared to the GW+BSE
+approach.
+Previous studies suggest that Qy excitation energies from GW+BSE are in
+better agreement with wavefunction-based methods and experiment than TDDFT with ωPBE
+[61, 62]. However, accurate benchmarks for larger molecular aggregates are missing and
+we therefore do not think that a clear recommendation for using GW+BSE instead of
+TDDFT is warranted. Nonetheless, with advances in code implementation [90–92] and in the
+combination of GW+BSE with discrete and polarizable continuum models [93,94] and other
+QM/MM methods [95], GW+BSE calculations of large molecular aggregates are becoming
+computationally feasible, demonstrated in a recent study by Förster et al. [61]. Further study
+
+15
+of the accuracy and predictive power of TDDFT, with exchange-correlation functionals that
+capture the nonlocal nature of charge-transfer excitations for such aggregates is necessary.
+Supplementary Material
+Additional convergence data, excitation energies, difference densities and transition densities
+not shown in the main text, and structure files.
+Acknowledgements
+This work was supported by the Bavarian State Ministry of Science and the Arts through the
+Elite Network Bavaria (ENB) and through computational resources provided by the Bavarian
+Polymer Institute (BPI).
+References
+[1] Wahadoszamen M, Margalit I, Ara A M, van Grondelle R and Noy D 2014 Nat. Comm. 5 5287 URL
+https://www.nature.com/articles/ncomms6287
+[2] Zoppi L and Baldridge K K 2018 Int. J. Quant. Chem. 118 e25413 URL https://onlinelibrary.
+wiley.com/doi/abs/10.1002/qua.25413
+[3] Muntwiler M, Yang Q, Tisdale W A and Zhu X Y 2008 Phys. Rev. Lett. 101 196403 URL https:
+//link.aps.org/doi/10.1103/PhysRevLett.101.196403
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+Impact, Attention, Influence:
+Early Assessment of Autonomous Driving Datasets
+Daniel Bogdoll†‡*, Jonas Hendl‡*, Felix Schreyer†, Nishanth Gowda†, Michael F¨arber‡ and J. Marius Z¨ollner†‡
+†FZI Research Center for Information Technology, Germany
+bogdoll@fzi.de
+‡Karlsruhe Institute of Technology, Germany
+Abstract—Autonomous Driving (AD), the area of robotics with
+the greatest potential impact on society, has gained a lot of
+momentum in the last decade. As a result of this, the number
+of datasets in AD has increased rapidly. Creators and users of
+datasets can benefit from a better understanding of developments
+in the field. While scientometric analysis has been conducted in
+other fields, it rarely revolves around datasets. Thus, the impact,
+attention, and influence of datasets on autonomous driving
+remains a rarely investigated field. In this work, we provide a
+scientometric analysis for over 200 datasets in AD. We perform
+a rigorous evaluation of relations between available metadata
+and citation counts based on linear regression. Subsequently, we
+propose an Influence Score to assess a dataset already early on
+without the need for a track-record of citations, which is only
+available with a certain delay.
+Index Terms—Robotics, Autonomous Driving, Datasets, Influ-
+ence, Impact, Attention, Scientometrics, Bibliometrics
+I. INTRODUCTION
+Autonomous driving technology does not only affect ur-
+ban transportation [1] and delivery of goods [2], but also
+farming [3] or warehouse logistics [4]. With the progress
+of deep learning and this growing interest in AD in many
+fields of robotic, the number of related datasets is consistently
+increasing. The datasets have also increased in size and many
+have become increasingly specialized [5]. The most extensive
+collection of datasets known to us, ad-datasets, currently
+lists 231 datasets in the domain [6]. However, not all of
+them are being equally used in the robotics community, the
+distribution of their citations is heavily skewed. As part of
+the more impactful works, well known datasets for the core
+tasks perception and prediction dominate [7]–[9]. As part of
+the long tail, many datasets for niche research areas exist [10]–
+[12]. Well known datasets tend to bring many advantages with
+them: They enable comparison between works, have higher
+quality, advanced tooling, and often community knowledge
+and support is available. The increasing number of datasets,
+which are potentially interesting but lack reputation, leads to a
+lot of untapped potential: Many researchers are hesitant to use
+such datasets and stick to old, but established ones instead [6].
+This is why we asked ourselves the question: Given a novel
+dataset without a multi-year track record of citations, is there
+a way to estimate its future development? Datasets with a high
+potential might be more appealing already in their early days.
+* These authors contributed equally
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+2022
+year
+5
+10
+15
+20
+25
+30
+35
+40
+publications
+0
+2000
+4000
+6000
+8000
+10000
+citations
+Fig. 1.
+Course of published datasets and citations of the accompanying
+publications in the domain of AD. This growing number of datasets, initially
+without reputation, holds a great deal of untapped potential as researchers
+struggle to use new datasets for their research. Datasets as listed on ad-
+datasets [6], citation counts from Semantic Scholar [13].
+Research Gap. To date, citations are mostly used to as-
+sess datasets, which are not available early on. Thus, new
+datasets can have a hard time gaining traction, which results
+in untapped potential. It is yet not well understood if and
+how metadata of datasets relate to future impact or how they
+can be utilized to assess datasets early on. To the best of our
+knowledge, such an analysis has not yet been performed.
+Contribution. In order to analyze the field of dataset, we
+first assembled the largest collection of datasets with enriched
+metadata available, including over 200 datasets with metadata
+from three different sources. We then applied linear regression
+to evaluate factors which relate to the future impact of datasets,
+measured in citations. Finally, we propose the Influence Score
+(IS), which is a mean to assess datasets early on without the
+need of a multi-year track record of citations. The IS can be
+used to assess any datasets at any given year, which also allows
+for later analysis. Our work aims to help researchers from
+the robotics community to better understand and assess the
+performance of datasets. This can lead to the design of better
+and thus more influential datasets as well as an actionable
+analysis of new datasets to assess their potential. All data used
+in this work is as of January 04, 2023. All code is available
+on GitHub.
+arXiv:2301.02200v1  [cs.DL]  5 Jan 2023
+
+II. RELATED WORK
+Here, we give an introduction to the scientometrics, biblio-
+metrics, and altmetrics, followed by dataset analysis.
+A. Scientometrics, Bibliometrics, and Altmetrics
+Scientometrics, Bibliometrics, and Altmetrics are highly
+intertwined fields that focus on the analysis of science and
+its processes as a whole, written works of science, and online
+communication of science, respectively [14].
+Scientometrics. Ravenscroft et al. [15] examined the impact
+of research by comparing citation-based metrics, such as
+citation count or h-index [16], with altmetrics and impact other
+than citations, e.g., societal and economic impact. However,
+they found no strong relationship between the fields. Hicks
+et al. [17] suggest using multiple factors to portray multiple
+aspects.
+Leydesdorff et al. [18] claim that citations are equated to im-
+pact and evaluate the relationship between impact and research
+quality. They found that short-term citations signify the invest-
+ment in a current discourse, while long-term citations signify
+acceptance as reliable scientific knowledge. However, some
+researchers question if or to what extent citations measure
+scientific impact and point to issues, e.g., inconsistent reasons
+for citations [19]. Problems include the cumulative advantages
+already successful papers experience [20], self-citations, which
+men do more often [21], negative citations, and citing out
+of reasons that do not reflect actual use or relevance [22].
+Valenzuela et al. [23] presented a method to identify four types
+of citations: ”Related work, Comparison, Using the work,
+Extending the work” [23], which is used by Semantic Scholar
+to determine “Highly Influential Citations” [24]. However, it
+shows a high correlation with citations.
+The field of trend detection analyzes large corpora of works
+to detect upcoming patterns [25]–[28]. Lopez Belmonte et
+al. [29] analyzed publications in Machine Learning and Big
+Data and found exponential growth of publications. They
+compared the popularity of keywords and the h-index.
+Bibliometrics. Citations can be aggregated on different
+levels, e.g., for the papers of one author as the h-index does,
+or on the journal level, like the journal impact factor (JIF),
+which is the two-year average ratio of citations to articles
+published. The JIF is ill-suited for evaluating individual papers
+by means of the journal it was published in [30]. This is due
+to the heavy skewness of the distribution of citation counts
+within journals [31]. The Hirsch-index, usually referred to as
+the h-index, combines the productivity of an author with the
+impact of their individual papers. Using the h-index increases
+robustness compared to simply counting the total number of
+citations, as few highly cited papers have little effect on the
+h-index. In addition, there have been efforts to recommend
+papers and citations [32], [33], predict future citation counts
+of papers [34]–[37] and the impact of scientists [38]. Such
+approaches remain challenging and are often domain-specific.
+Bornmann and Marx [39] have proposed to expand the
+bibliometric analysis by not only considering citations but also
+references. Following this idea, reference analysis has been
+used to identify influential references [40].
+Altmetrics. Online interactions with papers are referred to
+as altmetrics and are usually available earlier than citations,
+which gives altmetrics an advantage over bibliometrics [41].
+Bornmann and Marx [42] examined if Altmetrics can be used
+to predict paper quality which was measured through peer as-
+sessments and found that both tweets and readers do, with the
+latter having a stronger relationship. Lamb et al. [43] showed
+that the Altmetric Attention Score is a predictor of the citations
+of a paper in ecology and conservation. Zavrel et al. [44]
+clustered papers released at the International Conference on
+Machine Learning (ICML) in 2022 and calculated a score
+for their impact. They used Twitter mentions, citations, the
+authors’ average h-index, and an award for outstanding papers
+rewarded by the conference itself. They claim to do a “sim-
+ple combination of these four scores to calculate an impact
+score” [44] but do not reveal the formula. F¨arber analyzed
+GitHub repositories of papers, mostly from the field of AI, and
+found a power-law distribution of stars and forks [45]. While
+Haustein et al. claim that ”Altmetrics measures scientific
+impact based on online references and activity” [46], many
+disagree with equating altmetrics with impact. For example,
+Sugimoto states that ”attention is not impact” and calls online
+interaction with scientific works ”attention” [47]. Altmetrics
+might reflect broader or societal impact [41].
+B. Dataset Analysis
+Bogdoll et al. [5] gathered metadata of over 200 datasets
+in the field of autonomous driving. Similarly, F¨arber and
+Lamprecht released the data set knowledge graph, which is
+a collection of over 2,000 datasets with added metadata [48].
+D’Ulizia et al. [49] analyzed the metadata of datasets for fake
+news detection. Utamachant and Anutariya [50] analyzed the
+datasets of Thailand’s national open data portal, but relied
+on domain experts to assess impact. Nguyen and Weller
+proposed FAIRnets, a service to search for neural networks
+and their related datasets [51] published on GitHub. They build
+upon the Findable, Accessible, Interoperable, Reusable (FAIR)
+principles [52], which ”put specific emphasis on enhancing
+the ability of machines to automatically find and use the
+data” [52]. Khan et al. [53] analyzed datasets from the Global
+Biodiversity Information Facility (GBIF) which publishes
+datasets with a DOI and indexes datasets in biodiversity.
+They promote data standards and the reuse of datasets [54]
+as well as accompanying publications, which they call ”data
+papers”, that describe a dataset thoroughly [55]. Khan et
+al. [53] report a strong correlation between dataset download
+numbers and citation counts, and suggest that downloads and
+citations signify a similar kind of impact. They also find
+correlations between altmetrics and citations. Moreover, they
+question whether every citation means the usage of a dataset
+and point to differences in citation behavior. F¨arber et al.
+proposed an approach to find methods and datasets which
+authors actually used when citing the related paper [56].
+However, unrealistically few dataset usages were identified.
+
+AD-Datasets
+Altmetric
+Semantic Scholar
+List of Datasets
+with Enriched
+Metadata
+Regression Analysis
+of Citations
+and Metadata
+Influence
+Score
+Fig. 2. Overview: First, we collect data from various sources and combine them to a single list of datasets. Based on this, we perform a regression analysis
+to determine which metadata correlate with future prediction counts. Based on the metadata, we compute our Influence Score (IS).
+III. REGRESSION ANALYSIS
+Here, we first introduce our taxonomy of terms related to the
+assessment of datasets. Subsequently, we introduce our data
+sources. Based on these, we describe the regression analysis of
+citations and metadata. In Section IV, we present the resulting
+Influence Score. Figure 2 gives an overview over this process.
+A. Taxonomy
+As became clear in Section II, no common language for
+specific aspects in the domain has evolved yet. Thus, we
+introduce a taxonomy to clearly describe different aspects with
+respect to the development of a dataset or paper. As general
+terms for this, we utilize success, progress, performance, or
+potential. For concrete aspects, we establish the following
+terms, where each one can be applied to any single paper:
+Impact: We use the number of citations to measure the sci-
+entific impact of a paper, which is common in Scientometrics,
+but not without criticism [19].
+Attention: The online reception, such as tweets and
+Wikipedia articles mentioning a paper, represents the attention
+by researchers and the public.
+Influence: We refer to the resulting score of our proposed
+method, which combines a multitude of aspects, as the influ-
+ence, or IS, of a dataset. We deem this term appropriate for
+any method that goes beyond purely impact-based assessment.
+B. Data Sources and Selection
+We used three sources for our data: ad-datasets.com [6],
+the
+Semantic
+Scholar
+Academic
+Graph
+API
+[13],
+and
+altmetric.com [57]. Based on the DOI and arXiv-Id from
+ad-datasets, we automatically extracted the metadata of papers
+from Semantic Scholar and altmetric.com. Based on these
+papers, all of which describe datasets, we performed data
+exploration, regression, and the computation of the IS.
+AD-Datasets: This web tool offers an overview of over
+200 data sets in AD
+[5]. It includes a detailed breakdown
+of most dataset entries by 20 different meta categories,
+provided by the authors and the research community. This
+way, relations between datasets, accompanying papers and
+further metadata are available. The underlying data is stored
+in the JSON format and can be accessed accordingly. In this
+work, we utilize the nframes and nsensors metadata, which
+indicate the size of datasets in different dimensions, which is
+a potential aspect of the relevance of a dataset.
+Altmetric: We used the API by altmetric.com [57], which
+provides insight into online attention and readership. These
+properties are provided by the following categories:
+Attention Score: The aascurr aggregates different sources
+into a single score [58]. It is a weighted count of different
+online sources. For example, the weight for a reference on
+Wikipedia is 3, while Twitter and Reddit mentions are both
+weighted with 0.25. Unfortunately, the history of this score is
+only provided for the most recent year.
+Attention Score after three months: The aas3m is the
+percentile of the papers’ Attention Score three months after
+publication. The percentile is calculated in comparison to
+papers that have been released at a similar time.
+Readers: The number of people nreaders that have saved
+a paper in their reference management software. Reading
+a paper is less significant than citing it, but the number of
+readers might imply interest in a paper early on. The number
+of readers is provided individually for multiple reference
+management services, which we sum into a single count for
+online readers. Altmetric.com cannot verify the number of
+readers, thus, it is not included in the attention score. This
+is a relevant attribute, as it decouples the metrics. However,
+there are no historic data available.
+Semantic Scholar: For every accompanying paper of a
+dataset, we pulled data from Semantic Scholar. Sometimes,
+multiple datasets are described in the same paper, which will
+lead to the same information for those datasets. We extracted
+the following nested data:
+• List of referenced papers, including for each a list of all
+citing papers and the year of citation.
+• List of authors and their respective publications, including
+for each publication a list of citing papers and the year
+of citation.
+• List of citing papers, including for each a list of citing
+papers and the year of citation.
+Wherever possible, we collected associated timestamps,
+including the publication year apub of each paper. The first
+
+two categories, while dynamic, are directly available. We use
+the citations of references as a measure of the impact of
+references. Having impactful references might indicate that a
+paper is covering popular topics within AD or that the authors
+are knowledgeable in the field.
+The performance of authors can be estimated by evaluating
+their paper count and how many citations they have received,
+which becomes only meaningful over time. As discussed
+earlier, not every citation means usage of a dataset. While it
+would have been interesting to take into account, in which
+section a paper has been cited, this data was not available for
+most papers. Based on the ncit3 citations from the previous
+three years, citations of works that cited a dataset signify the
+value created by working with the dataset, which is why we
+included those.
+A critical aspect of the collected data is that oftentimes, no
+historic information was available. Also, oftentimes, data was
+not available due to limitations, e.g., Altmetric is incompatible
+with DOIs from IEEE publications, which are common in the
+fields of robotics, autonomous driving, and machine learning.
+Similarly, Ravenscroft et al. [15] expressed concerns about
+Altmetric, as they were unable to find 40 % of the papers
+they analyzed, all from the field of computer science.
+C. Data Aggregation
+To further utilize the raw data we collected, we aggregated
+some of it with the aim to assemble a finite list of features
+that describe a dataset. We aggregated some of our data
+sources using the concept of the h-index formula, as it is
+widely known, transparent, and easy to reproduce. In order
+to analyze smaller timespans, we deviated from the typical 5-
+year duration and calculated multiple 3-year indexes ourselves.
+For authors, we applied the h3-index for each individual.
+We then aggregated the h-indices of all authors of a paper via
+the arithmetic mean in autµh3. Respectively, we applied the
+h-index formula to references and citations. For the references
+of a paper, the refh3 is calculated identically to the way it
+is utilized for authors. Just like an author has papers with
+citations, a paper has references with citations. A high h-index
+for references would signify that several of the referenced
+papers gained lots of attraction. We also applied the h3-index
+formula to the citations and their citations to get the h3-index
+of citations cith3, following Schubert et al. [59]. The final list
+of all extracted and calculated features can be found in Table I.
+D. Cluster Analysis and Regression Setup
+We evaluated our computed features with respect to their
+ability to predict future citations. Therefor, we performed
+linear regression. For this, we first computed clusters of
+the datasets to determine a meaningful time horizon. Subse-
+quently, we defined our regression setup.
+Cluster Analysis: To show that there are meaningful vari-
+ations between clusters, we looked at the impact of papers
+for up to 2 years after publication in a journal or conference
+1
+0
+1
+2
+years after publication
+0
+50
+100
+150
+200
+250
+300
+350
+400
+ncits
+Fig. 3. Development of the number of citations for publication-clusters over
+a dynamic 3-year window. Papers are clustered based on similar performance.
+Semantic Scholar also tracks citations of pre-prints, which leads to citations
+prior to the publication date of the final work.
+proceedings. As visualized in Fig. 3, clear clusters are visi-
+ble, where line-thickness indicates cluster size. For k-means
+clustering, we used six clusters based on the elbow plot,
+which shows which additional cluster provides a non-marginal
+reduction of the total variation within clusters. The growth of
+citation counts behaves exponentially for the top performing
+works. A clear differentiation between all clusters becomes
+apparent already after one year, which we chose as the time
+horizon for the regression. This allowed us to include more
+recent papers, which would have been excluded otherwise due
+to their missing track record of citations.
+Regression Setup: As independent variables, we included
+the features nsensors, apub, refh3, autµh3, ncit3, and aas3m,
+as shown in Table I, in order to estimate the citation count after
+one year. Preliminary data exploration suggested a curvilinear
+relationship between aas3m and the number of citations.
+Therefore, a quadratic term was added. All predictors were
+standardized by subtracting the mean and dividing by the
+standard deviation prior to the analysis. The feature ncit3 was
+log(x+1)-transformed to ensure a normal distribution of the
+residuals, which are the error terms of the regression.
+For the regression, we were able to utilize 111 datasets, as
+values for all included features were available, and they had
+been released at least one year prior. Residuals and collinearity,
+the ability to linearly predict one independent variable with
+other independent variables, were checked. The collinearity
+was quantified through the variance inflation factor of each
+regressor which all were lower than three. We performed the
+Breusch-Pagan and White test for heteroskedasticity, which is
+the inconsistency of the variance of residuals at different levels
+of the dependent variable. Both tests indicated that we do not
+have sufficient evidence for the presence of heteroskedasticity.
+Still, robust standard errors were used to ensure the standard
+errors are calculated correctly in the presence of heteroskedas-
+ticity which at worst leads to standard errors being estimated
+larger.
+
+Feature
+Description
+Availability
+Standardized
+Log(x+1)
+Influence Score
+Source
+nframes
+Number of frames in the dataset
+At publication
+–
+–
+✓
+AD-Datasets [6]
+nsensors
+Number of sensor types
+At publication
+✓
+✗
+✓
+AD-Datasets [6]
+apub
+Year of publication
+At publication
+✓
+✗
+✗
+Semantic Scholar [13]
+refh3
+3 year h-index of references
+At publication
+✓
+✗
+✓
+Semantic Scholar [13]
+autµh3
+Mean 3 year h-index of authors papers
+At publication
+✓
+✗
+✓
+Semantic Scholar [13]
+ncit3
+Number of citations within past 3 years
+Anytime after publication
+✓
+✓
+✓
+Semantic Scholar [13]
+cith3
+3 year h-index of citations
+>3 years after publication
+–
+–
+✓
+Semantic Scholar [13]
+aascurr
+Altmetric Attention Score
+Anytime after publication
+–
+–
+✓
+Altmetric [57]
+aas3m
+Altmetric Attention Score at 3 mos
+After 3 months
+✓
+✗
+✗
+Altmetric [57]
+nreaders
+Number of readers
+Anytime after publication
+–
+–
+✓
+Altmetric [57]
+TABLE I
+OVERVIEW OF METADATA USED FOR THE REGRESSION ANALYSIS AND THE INFLUENCE SCORE.
+We chose not to include nframes for the regression because
+numerous of the datasets did not contain this meta-information.
+However, we examined a model in which the feature was
+included, which did not lead to new findings.
+E. Regression Analysis
+With the explained regression setup, we were now interested
+in finding statistically significant predictor variables for the
+citation count at the end of the year after publication.
+The aas3m and aas2
+3m were positively related to the number
+of citations and both relationships were significant at <0.0001.
+Both coefficients were positive. All other features were not
+significantly related to the number of citations. The results are
+reported in Table II.
+coef
+std err
+z
+P>|z|
+[0.025
+0.975]
+refh3
+0.0987
+0.103
+0.96
+0.337
+-0.103
+0.3
+autµh3
+0.071
+0.086
+0.824
+0.41
+-0.098
+0.24
+apub
+0.0281
+0.084
+0.337
+0.736
+-0.136
+0.192
+nsensors
+0.1383
+0.108
+1.287
+0.198
+-0.072
+0.349
+aas3m
+0.803
+0.116
+6.895
+0
+0.575
+1.031
+aas2
+3m
+0.3375
+0.072
+4.72
+0
+0.197
+0.478
+intercept
+2.6402
+0.117
+22.48
+0
+2.41
+2.87
+TABLE II
+REGRESSION FOR CITATIONS AFTER ONE YEAR. REGRESSION
+COEFFICIENTS AND 95% CONFIDENCE INTERVAL ARE REPRESENTED ON
+THE LOG SCALE.
+Since only one feature showed a relationship with the
+number of citations, we do not consider a stable prediction
+of citations possible with the available data. In order to still
+perform an early evaluation of datasets, in the following we
+present our Influence Score (IS).
+IV. INFLUENCE SCORE
+We propose the Influence Score (IS), which includes a
+variety of features that are available early on. These are
+weighted dynamically in order to receive an indication of the
+relative performance of any given dataset at any given time.
+The calculation compares each data set with all existing ones
+from the domain, so that relative differences and trends are
+immediately recognizable.
+Percentiles are used to allow relative scoring within the
+surrounding group of datasets. The data sets roughly follow a
+normal distribution in their IS scores. As shown in Table I, we
+utilize eight different features for the IS: nframes, nsensors,
+refh3, autµh3, ncit3, cith3, aascurr and nreaders. This way,
+we consider more than just the citations, but do not exclude
+them: If early citations are already available, they become a
+meaningful part of the score, as the relation to other datasets
+of the peer group is relevant. This way, citation velocity is
+included. The IS is defined as follows:
+IS(paper) = 1/n ∗
+n
+�
+i=0
+percentile(featurei)
+(1)
+where:
+i = Feature Index
+n = Number of available features
+Only features, which are available, are dynamically included
+in the IS. As we used percentiles of each feature to facil-
+itate the understanding of the features, common issues are
+mitigated. E.g., typical feature values change over time: For
+example, with the growth of AD, the ncit value of a paper
+today is likely higher than a decade ago, which becomes
+clearly visible in Figure 1. Furthermore, commonly observed
+values for features might differ between different fields. This
+helps people who are not familiar with AD or the features to
+easily assess if the score a dataset achieved is high or low.
+A. Qualitative Demonstration
+To showcase the IS, we compare exemplary the development
+of the five most and least cited papers with a latest publication
+in 2019, by their IS and visualize the results in Fig. 4.
+It becomes clearly visible, that the two groups are easily
+distinguishable by their IS, but also that differences within
+the groups are visible.
+The individual features show different pictures: For refh3,
+also papers with only a few citations can have meaningful
+references in their works. ncitt3 and cith3 only confirm what
+was known by our data selection, as we selected the datasets
+by citation count. autµh3 shows, how successful datasets can
+also boost personal careers, as some authors became professors
+and remained active in their field. nsensors and nframes show
+
+2014
+2016
+2018
+2020
+2022
+years
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+IS
+2014
+2016
+2018
+2020
+2022
+years
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+percentile refh3
+2014
+2016
+2018
+2020
+2022
+years
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+percentile ncit3
+2014
+2016
+2018
+2020
+2022
+years
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+percentile cith3
+2014
+2016
+2018
+2020
+2022
+years
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+percentile aut h3
+2014
+2016
+2018
+2020
+2022
+years
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+percentile nsensors
+2014
+2016
+2018
+2020
+2022
+years
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+percentile nframes
+KITTI
+nuImages
+Cars
+Synthia
+Waymo Open Perception
+Daimler Stereo Pedestrian Detection Benchmark
+TRoM
+DriveSeg (Semi-auto)
+DriveSeg (MANUAL)
+WZ-traffic
+Fig. 4. Influence Score and individual features for different datasets. We show exemplary results for the five best and worst performing datasets of all time,
+measured by citations, with a latest release in 2019 for historical data. We also show six individual features of the IS, where historical data was available.
+rather static results, with a trend towards larger datasets being
+more successful.
+B. Quantitative Demonstration
+In order to show the quantitative performance of the IS, we
+showcase all datasets released in 2022 in a detailed overview
+in Table III. Such a pre-filtering process is useful in order
+to explore novel datasets. Here, it becomes clear that the
+IS captures a wide variety of different aspects of a dataset.
+Of particular interest is the fact that even low-performing
+data sets can lead in certain features. Thus, if a researcher
+is interested in certain aspects of a dataset, they can simply
+focus on the features they are interested in and omit the others,
+which enables less-known datasets to be discovered and used.
+Figure 5 shows an overview of the IS distributions.
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+IS
+0
+2
+4
+6
+8
+10
+number of datasets from 2022
+Fig. 5. Distribution of the Influence Score (IS) of all datasets from 2022.
+V. CONCLUSION
+In this paper, we addressed the lack of knowledge with
+respect to the scientific impact, attention, and influence of
+datasets in robotics. Our focus was on an early assessment
+of datasets, given a flood of new datasets published every
+year. We analyzed impact measured by citations and evalu-
+ated relations of metadata and features which we extracted
+from multiple online sources. Our regression analysis showed
+no strong relation between future citations and our selected
+features. Subsequently, we presented our developed Influence
+Score (IS). This score utilizes a set of eight features to assess
+any dataset also early on. This is based on an analysis within
+the peer group of all datasets, which allows for the early
+detection of relative trends.
+Our work contributes to a better understanding of datasets,
+which enables researchers to find and assess published
+datasets in the domain of autonomous driving without the
+need of waiting for a track record of citations.
+Limitations and Outlook: For our work, we evaluated the
+paper accompanying the dataset assuming that the paper is a
+good representation of the dataset. When measuring scientific
+impact through citations, we think this holds because the
+paper is actually the cited scientific work. However, not every
+citation might be meaningful, positive, or indicate the usage
+of a dataset. Ideally, large language models could evaluate
+if a dataset is actually used, if cited. Khan et al. [53], who
+analyzed datasets in biodiversity, suggested that the correlation
+between the number of downloads and citations signifies
+that these two measures are comparable representations of
+impact. However, in the domain of AD, download numbers
+are typically not available, but this might change. As some
+datasets are presented in the same paper, a further decoupling
+of accompanying papers and the respective datasets would be
+helpful. We found, that the quality and availability of metadata
+in AD provided by the creators of datasets varies strongly.
+Thus, standards should be established [90]. While we focussed
+on dataset and paper specific features for this work, we are
+also interested in the venue or journal of publication, which
+can be considered as an additional feature in future work.
+
+IS
+ncit3
+cith3
+refh3
+autµh3
+nframes
+nsensors
+aascurr
+nreaders
+Waymo Block-NeRF [60]
+0.82
+0.7
+0.53
+0.95
+0.87
+–
+–
+1.0
+0.89
+SHIFT [61]
+0.62
+0.25
+0.2
+0.99
+0.88
+0.94
+0.77
+0.65
+0.42
+Street Hazards [62]
+0.62
+0.67
+0.58
+0.83
+0.92
+0.16
+0.25
+0.73
+0.45
+KITTI-360-APS [63]
+0.48
+0.23
+0.06
+0.68
+0.67
+0.56
+0.25
+0.97
+0.21
+ScribbleKITTI [64]
+0.45
+0.2
+0.15
+0.83
+0.97
+0.32
+0.25
+0.63
+0.02
+BDD100K-APS [63]
+0.42
+0.23
+0.06
+0.68
+0.67
+0.1
+0.25
+0.97
+0.21
+Ithaca365 [65]
+0.4
+0.15
+0.15
+0.68
+0.22
+0.82
+0.58
+0.53
+0.26
+CODA [66]
+0.39
+0.2
+0.15
+0.75
+0.36
+–
+0.25
+0.53
+0.33
+Rope3D [67]
+0.37
+0.15
+0.15
+0.83
+0.42
+0.53
+0.58
+0.25
+0.25
+Comma2k19 LD [68]
+0.37
+0.12
+0.15
+0.71
+0.56
+–
+–
+0.64
+0.02
+RoadSaW [69]
+0.29
+0.09
+0.06
+0.27
+0.19
+0.83
+0.25
+–
+–
+K-Radar [70]
+0.26
+0.09
+0.06
+0.47
+0.11
+0.44
+0.93
+0.4
+0.26
+CARLA-WildLife [71]
+0.26
+0.03
+0.06
+0.92
+0.12
+–
+0.25
+0.34
+0.08
+AugKITTI [72]
+0.25
+0.03
+0.06
+0.88
+0.16
+–
+–
+0.25
+0.1
+WildDash 2 [73]
+0.24
+0.17
+0.2
+0.52
+0.08
+–
+0.25
+–
+–
+MONA [74]
+0.23
+0.03
+0.06
+0.36
+0.48
+–
+0.25
+–
+–
+Street Obstacle Sequences [71]
+0.23
+0.03
+0.06
+0.92
+0.12
+0.07
+0.25
+0.34
+0.08
+HDBD [75]
+0.22
+0.03
+0.06
+0.16
+0.64
+–
+0.58
+–
+–
+GLARE [76]
+0.22
+0.03
+0.06
+0.47
+0.29
+–
+–
+0.42
+0.06
+Boreas [77]
+0.22
+0.29
+0.25
+0.14
+0.2
+–
+–
+–
+–
+Autonomous Platform Inertial [78]
+0.21
+0.2
+0.25
+0.36
+0.03
+–
+–
+–
+–
+aiMotive [79]
+0.2
+0.03
+0.06
+0.33
+0.04
+0.41
+0.93
+0.44
+0.13
+CarlaScenes [80]
+0.2
+0.03
+0.06
+0.52
+0.2
+–
+0.77
+–
+–
+LUMPI [81]
+0.19
+0.09
+0.06
+0.02
+0.07
+0.74
+0.58
+–
+–
+A9 [82]
+0.19
+0.25
+0.25
+0.14
+0.1
+–
+0.58
+0.29
+0.12
+Amodal Cityscapes [83]
+0.19
+0.09
+0.06
+0.27
+0.43
+0.11
+0.25
+0.29
+0.08
+R-U-MAAD [84]
+0.16
+0.03
+0.06
+0.33
+0.35
+–
+0.25
+0.15
+0.06
+TJ4DRadSet [85]
+0.15
+0.12
+0.15
+0.14
+0.05
+–
+–
+0.42
+0.02
+OpenMPD [86]
+0.14
+0.18
+0.15
+0.02
+0.07
+0.27
+0.58
+–
+–
+I see you [87]
+0.12
+0.03
+0.06
+0.09
+0.01
+–
+–
+0.44
+0.08
+SceNDD [88]
+0.11
+0.09
+0.06
+0.16
+0.19
+–
+–
+0.15
+0.02
+exiD [89]
+0.11
+0.12
+0.06
+0.02
+0.24
+–
+0.25
+–
+–
+TABLE III
+INFLUENCE SCORE AND FEATURES FOR DATASETS RELEASED IN 2022. SORTED BY IS, TOP 3 FEATURES BOLD.
+VI. ACKNOWLEDGMENT
+This work results partly from the KIGLIS project supported
+by the German Federal Ministry of Education and Research
+(BMBF), grant number 16KIS1231. We want to thank both
+Altmetric and Semantic Scholar, who have provided us with
+the necessary API accesses for this work.
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FdE1T4oBgHgl3EQf-wYy/content/tmp_files/2301.03572v1.pdf.txt

@@ -0,0 +1,1607 @@
+Non-oscillating Early Dark Energy and
+Quintessence from α-Attractors
+Lucy Brissenden, Konstantinos Dimopoulos and Samuel S´anchez
+L´opez
+Consortium for Fundamental Physics, Physics Department,
+Lancaster University, Lancaster LA1 4YB, United Kingdom.
+E-mail: l.brissenden@lancaster.ac.uk, k.dimopoulos1@lancaster.ac.uk,
+s.sanchezlopez@lancaster.ac.uk
+Abstract.
+Early dark energy (EDE) is one of the most promising possibilities in order to
+resolve the Hubble tension: the discrepancy between early and late-Universe measurements
+of the Hubble constant. In this paper we propose a model of a scalar field which can explain
+both EDE and late Dark Energy (DE) in a joined manner without additional fine-tuning.
+The field features kinetic poles as with α-attractors. Our model provides an injection of EDE
+near matter-radiation equality, and redshifts away shortly after via free-fall, later refreezing to
+become late-time DE at the present day. Using reasonable estimates of the current constraints
+on EDE from the literature, we find that the parameter space is narrow but viable. As such
+our model is readily falsifiable. In contrast to other work in EDE, our model is non-oscillatory,
+which causes its decay to be faster than that of the usual oscillatory EDE, thereby achieving
+better agreement with observations.
+arXiv:2301.03572v1  [astro-ph.CO]  9 Jan 2023
+
+Contents
+1
+Introduction
+1
+1.1
+The Hubble tension
+2
+1.2
+Early Dark Energy
+2
+1.3
+α-attractors
+3
+1.4
+Quintessence
+4
+2
+The Model
+5
+2.1
+Lagrangian and Field Equations
+5
+2.2
+Shape of Potential and Expected Behaviour
+5
+2.3
+Asymptotic forms of the scalar potential
+5
+2.3.1
+Expected Field Behaviour
+7
+2.4
+Tuning requirements
+8
+3
+Numerical Simulation
+9
+4
+Results and analysis
+11
+4.1
+Parameter Space
+11
+4.2
+Field Behaviour
+13
+5
+Initial Conditions
+14
+6
+Conclusions
+18
+A Quintessential Inflation
+19
+1
+Introduction
+In the last few decades cosmological observations of the early and late Universe have con-
+verged into a broad understanding of the history of our Universe from the very first seconds
+of its existence until today. Thus, cosmology has developed a standard model called the
+concordance model, or in short ΛCDM.
+However, the latest data might imply that the celebrated ΛCDM model is not that
+robust after all. In particular, there is a 5-σ discrepancy between the measurements of the
+current expansion rate, the Hubble constant H0, as inferred by early Universe observations
+compared with late Universe observations. This Hubble tension has undermined our confi-
+dence in ΛCDM and as such it is investigated intensely at present.
+In this work we study a toy model that can simultaneously solve the Hubble tension
+and explain the current accelerated expansion with no more tuning that in ΛCDM. Our
+model introduces a scalar field which plays both the role of early dark energy (EDE) and
+quintessence. In contrast to most other works in the literature which consider scalar fields
+as EDE, ours is not an oscillating scalar field.
+We use natural units with c = ¯h = 1, the reduced Planck mass mP = 1/
+√
+8πG =
+2.43 × 1018GeV and consider a positive signature metric (−1, +1, +1, +1) throughout the
+present work.
+– 1 –
+
+1.1
+The Hubble tension
+Measurements in observational cosmology can broadly be classified into two groups. These
+are measurements of quantities which depend only on the early-time history of our Universe
+(such as the cosmic microwave background (CMB) radiation at redshift z ≃ 1100, or Baryon
+Acoustic Oscillations (BAO)) and measurements of quantities which depend on present-day
+observations (the primary example of this is the cosmic distance ladder, which measures the
+redshift of observable astrophysical objects such as Cepheid stars and type-1a supernovae,
+at redshift z = O(1)).
+The value of the Hubble constant H0 can in principle be inferred from both early and
+late-time measurements. However, it has been found that while early-time measurements are
+in good agreement with each other, they disagree with current late-time data. Latest analysis
+of the CMB temperature anisotropies’ data gives the value inferred from Planck satellite [1],
+H0 = 67.44 ± 0.58 km s−1Mpc−1,
+(1.1)
+and a distance scale measurement using Cepheid-SN 1a data from the SH0ES collaboration
+[2] as
+H0 = 73.04 ± 1.04 km s−1Mpc−1.
+(1.2)
+This is a 5σ tension which includes estimates of all systematic errors and which the SH0ES
+team conclude has “no indication of arising from measurement uncertainties or analysis varia-
+tions considered to date”. It is becoming increasingly apparent with successive measurements
+that this tension is likely to have a theoretical resolution [3, 4], which can have many possible
+sources [5, 6].
+1.2
+Early Dark Energy
+One proposed class of solutions to the Hubble tension is models of Early Dark Energy (EDE),
+whose early works include references [7–10], followed by many others, e.g. see Refs. [5, 11–32].
+These involve an injection of energy in the dark energy sector at around the time of matter-
+radiation equality, which then dilutes or otherwise decays away faster than the background
+energy density, such that it becomes negligible before it can be detected in the CMB. As
+briefly reviewed below, such models result in a slight change in the expansion history of the
+Universe, bumping up the value of the Hubble parameter at the present day.
+It has previously been concluded [3, 5, 6] that EDE models are most likely to source
+a theoretical resolution to the Hubble tension. One reason for this is that EDE can effect
+substantial modifications to H0 without significant effect on other cosmological parameters
+which are tightly constrained by observations.1 In particular, EDE models can be incorpo-
+rated into existing scalar-field models of inflation and late-time dark energy; one example of
+the latter is the model detailed in this work.
+However, precisely because EDE models exist so close in time to existing observational
+data, they have significant constraints; the primary consideration being that EDE must be
+subdominant at all times and must decay away fast enough to be essentially negligible at
+the time of last scattering translating to a redshift rate that is faster than radiation [8]. So
+far, in previous works in EDE, this has been achieved by considering first or second-order
+phase transitions (e.g. [23], [29]). These abrupt events might have undesirable side-effects
+1Models which modify other cosmological parameters are often unable to reconcile their changes with
+current observational constraints on said parameters (see Ref. [5] for a comprehensive review).
+– 2 –
+
+such as inhomogeneities from bubble collisions or topological defects. Other proposed models
+[5, 7, 8, 23–30] typically feature oscillatory behaviour to achieve the rapid decay rate necessary
+for EDE to be negligible at last scattering. As with the original proposal in Ref. [7], the
+EDE field is taken to oscillate around its Vacuum Expectation Value (VEV) in a potential
+minimum which is tuned to be of order higher than quartic. As a result, its energy density
+decays on average as ∝ a−n, with 4 < n < 6. In contrast, in our model, the EDE scalar field
+experiences a period of kinetic domination, where the field is in non-oscillatory free-fall and
+its density decreases as ∝ a−6, exactly rather than approximately.
+Before continuing, we briefly explain how EDE manages to increase the value of H0
+as from CMB observations. Measurements of the CMB temperature anisotropies provide
+very tight constraints on the cosmological parameters. One would therefore think that this
+severely limits models which alter the Universe content and dynamics at this time. However,
+there are certain classes of models for which this is not the case. These are models that affect
+both the Hubble parameter and rs, the comoving sound horizon2 (in this case during the
+drag epoch, shortly after recombination), given by
+rs =
+� ∞
+zd
+cs(z)
+H(z)dz,
+(1.3)
+where cs(z) is the sound speed and H(z) is the Hubble parameter, both as a function of
+redshift.
+An additional amount of dark energy in the Universe increases the total density, which in
+turn increases the Hubble parameter because of the Friedmann equation ρ ∝ H2. Therefore,
+EDE considers such a brief increase at or before decoupling, which lowers the value of the
+sound horizon because it increases H(z) in Eq. (1.3). However, there is a way to avoid this
+being evident in and therefore disproved by current CMB measurements. This is because
+BAO and CMB measurements do not constrain the value of the sound horizon directly.
+For example, BAO measurements do not constrain the sound horizon alone, but the com-
+bination H(z)rs [33]. The observations of the Planck satellite measure the quantity θ∗ ≡ r∗
+D∗
+[34], the angular scale of the sound horizon; given by ratio of the comoving sound horizon to
+the angular diameter distance at which we observe fluctuations. Both of these measurements
+entail an assumption of ΛCDM cosmology and can be shown to be equally constrained by
+other models, provided that they make only small modifications which simultaneously lower
+the value of rs and increase H0.
+EDE may have a significant drawback, however, in that it does not alleviate the σ8
+tension (associated with matter clustering) and may in fact exacerbate it [3, 35]. As with
+many others, our model does not attempt to solve this problem.
+1.3
+α-attractors
+Our model unifies EDE with late DE in the context of α-attractors. An earlier attempt
+for such unification in the same theoretical context can be seen in Ref. [30]. However, this
+proposal is also of oscillatory EDE.
+α-attractors [36–44], which appear naturally in conformal field theory or supergravity
+theories, are a class of models whose inflationary predictions continuously interpolate between
+those of chaotic inflation [45] and those of Starobinsky [46] and Higgs inflation [47].
+In
+2This is the characteristic scale of BAO, typically approximately proportional to the value of the cosmo-
+logical horizon at that point by rs =
+1
+√
+3rH assuming spatial flatness.
+– 3 –
+
+supergravity, introducing curvature to the internal field-space manifold can give rise to a
+non-trivial K¨ahler metric, which results in kinetic poles for some of the scalar fields of the
+theory. The free parameter α is inversely proportional to said curvature. It is also worth
+clarifying what is meant by the word “attractor”. It is not only used in the usual sense (i.e.,
+field trajectories during inflation flowing to a unique one, regardless of the initial conditions),
+but also to refer to the fact that the inflationary predictions are largely insensitive of the
+specific characteristics of the model under consideration. Such an attractor behaviour is seen
+for sufficiently large curvature (small α) in the internal field-space manifold.
+In practical terms, the scalar field has a non-canonical kinetic term, featuring two poles,
+which the field cannot transverse. To aid our intuition, the field can be canonically normalised
+via a field redefinition, such that the finite poles for the non-canonical field are transposed
+to infinity for the canonical one. As a result, the scalar potential is “stretched” near the
+poles, resulting in two plateau regions, which are useful for modelling inflation, see e.g. Refs.
+[48–53] or quintessence [54], or both, in the context of quintessential inflation [54–56].
+Following the standard recipe, we introduce two poles at ϕ = ±
+√
+6α mP by considering
+the Lagrangian
+L =
+− 1
+2(∂ϕ)2
+(1 −
+ϕ2
+6α m2
+P )2 − V (ϕ) ,
+(1.4)
+where ϕ is the non-canonical scalar field and we use the short-hand notation (∂ϕ)2 ≡
+gµν∂µϕ ∂νϕ. We then redefine the non-canonical field in terms of the canonical scalar field φ
+as
+dφ =
+dϕ
+1 −
+ϕ2
+6αm2
+P
+⇒
+ϕ = mP
+√
+6α tanh
+�
+φ
+√
+6α mP
+�
+.
+(1.5)
+It is obvious that the poles ϕ = ±
+√
+6α are transposed to infinity.
+In terms of the canonical field, the Lagrangian now reads
+L = −1
+2(∂φ)2 − V (φ).
+(1.6)
+1.4
+Quintessence
+“Early” Dark Energy is so named in order to make it distinct from “late” Dark Dnergy,
+which is the original source of the name (and often just called Dark Energy (DE)). In cos-
+mological terms the latter is just beginning to dominate the Universe at present, making up
+approximately 70% of the Universe’s energy density [57]. This is the mysterious unknown
+substance that is responsible for the current accelerating expansion of the Universe and has
+equation-of-state (barotropic) parameter of w = −1.03 ± 0.03 [1].
+Late DE that is due to an (as-yet-undiscovered) scalar field is called quintessence [58],
+so-named because it is the “fifth element” making up the content of the Universe 3. In this
+case, the Planck-satellite bound on the barotropic parameter of DE is −1 ≤ w < −0.95 [1].
+Quintessence is distinct from other explanations for DE because a scalar field has a variable
+barotropic parameter and can therefore exhibit completely different behaviour in different
+periods of the Universe’s history. In order to get it to look like late-time DE, a scalar field
+should be dominated by its potential density, making its barotropic parameter sufficiently
+3After baryonic matter, dark matter, photons and neutrinos.
+– 4 –
+
+close to −1. It is useful to consider the CPL parametrization, which is obtained by Taylor
+expanding w(z) near the present as [59, 60]
+w(z) = w0 + wa
+z
+z + 1 ,
+(1.7)
+where wa ≡ −(dw/da)0. The Planck satellite observations impose the bounds [1]
+−1 ≤ w < −0.95
+wa = −0.29+0.32
+−0.26 .
+(1.8)
+2
+The Model
+2.1
+Lagrangian and Field Equations
+Consider a potential of the form
+V (ϕ) = VX exp
+�
+−λeκϕ/mP
+�
+,
+with VΛ ≡ exp
+�
+−λeκ
+√
+6α�
+VX ,
+(2.1)
+where α, κ, λ are dimensionless model parameters, VX is a constant energy density scale and
+ϕ is the non-canonical scalar field with kinetic poles given by the typical alpha attractors form
+(see [40]) with Lagrangian density given by Eq. (1.4).4 In the above, VΛ is the vacuum density
+at present. To assist our intuition, we switch to the canonically normalised (canonical) scalar
+field φ, using the transformation in Eq. (1.5). In terms of the canonical scalar field, the
+Lagrangian density is then given by Eq. (1.6), where the scalar potential is
+V (φ) = exp
+�
+λeκ
+√
+6α�
+VΛ exp
+�
+−λeκ
+√
+6α tanh(φ/
+√
+6α mP)�
+.
+(2.2)
+As usual, the Klein-Gordon equation of motion for the homogeneous canonical field is
+¨φ + 3H ˙φ + V ′(φ) = 0 ,
+(2.3)
+where the dot and prime denote derivatives with respect to the cosmic time and the scalar
+field respectively, and we assumed that the field was homogenised by inflation, when the
+latter overcame the horizon problem.
+2.2
+Shape of Potential and Expected Behaviour
+Henceforth we will discuss the behaviour of the field in terms of the variation, i.e. movement
+in field space, of the canonical field.
+2.3
+Asymptotic forms of the scalar potential
+We are interested in two limits for the potential above: φ → 0 (ϕ → 0) and φ → +∞ (ϕ →
+√
+6α mP ). The first limit would correspond to matter-radiation equality. In this limit, the
+potential is
+4The model parameter is VX and not VΛ, the latter being generated by VX and the remaining model
+parameters as shown in Eq. (2.1).
+– 5 –
+
+Veq ≃ exp
+�
+λ(eκ
+√
+6α − 1)
+�
+VΛ exp(−κλ φeq/mP) ,
+(2.4)
+where the subscript ‘eq’ denotes the time of matter-radiation equality when the field un-
+freezes. It is assumed that the field was originally frozen there. We discuss and justify this
+assumption in Sec. 5. After unfreezing, it is considered that the field has not varied much,
+for the above approximation to hold, i.e.
+0 ≲ φeq ≪
+√
+6αmP .
+(2.5)
+This is a reasonable assumption given that the field begins shortly before matter-radiation
+equality frozen at the origin, unfreezing at some point during this time 5.
+At large φ (φ → ∞), the non-canonical field is near the kinetic pole (ϕ →
+√
+6α mP).
+Then the potential in this limit is
+V0 ≃ VΛ
+�
+1 + 2κλeκ
+√
+6α√
+6α exp
+�
+−
+2φ0
+√
+6α mP
+��
+,
+(2.6)
+which, even for sub-Planckian total field excursion in φ, should be a good approximation for
+sufficiently small α. The subscript ‘0’ denotes the present time.
+The above approximations describe well the scalar potential near equality and the
+present time, as shown in Fig. 1. As we exlain below, in between these regions, the scalar
+field free-falls and becomes oblivious of the scalar potential as the term V ′(φ) in its equation
+of motion (2.3) becomes negligible.
+Canonical Potential
+Approximation at Low Field Values
+Approximation at High Field Values
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+-120
+-119
+-118
+-117
+-116
+-115
+ϕ
+mP √(6 α)
+log V(ϕ)
+mP
+4
+VΛ
+mP
+4 = 10-120.068
+α =0.0002
+κ=200
+λ=0.01
+Figure 1: Graph of the canonical potential and its two approximations for small and large
+field values, given in Eqs. (2.4) and (2.6) respectively.
+These approximations are useful
+because they are simple exponential potentials with known attractors, so we know the type
+of behaviour the field should exhibit when each approximation is valid. It can be readily seen
+that, after leaving the origin the field jumps off a potential plateau and is free-falling as a
+result.
+5There is no suggestion in the EDE literature [5, 7, 8, 23–30] that the field has to unfreeze at any particular
+time, as long as it does not grow to larger than the allowed fraction and its energy density is essentially
+negligible by the time of decoupling.
+– 6 –
+
+2.3.1
+Expected Field Behaviour
+Here we explain the rationale behind the mechanism envisaged. We make a number of crude
+approximations, which enable us to follow the evolution of the scalar field, but which need
+to be carefully examined numerically. We do so in the next section.
+First, we consider that originally the field is frozen at zero (for reasons explained in
+Sec. 5). Its energy density is such that it remains frozen there until equality, when it thaws
+following the appropriate exponential attractor, since Veq in Eq. (2.4) is approximately ex-
+ponential [61]. Assuming that this is the subdominant attractor requires that the strength
+of the exponential is [62, 63]
+Z ≡ κλ >
+√
+3 .
+(2.7)
+The subdominant exponential attractor dictates that the energy density of the rolling scalar
+field mimics the dominant background energy density. Thus, the density parameter of the
+field is constant, given by the value [61–63]
+Ωeq
+φ ≃ 3
+Z2 =
+3
+(κλ)2 < 1
+(2.8)
+This provides an estimate of the moment when the originally frozen scalar field, unfreezes and
+begins rolling down its potential. Unfreezing happens when Ωφ (which is growing while the
+field is frozen, because the background density decreases with the expansion of the Universe)
+obtains the above value.
+However, after unfreezing, the field soon experiences the full exp(exp) steeper than
+exponential potential so, it does not follow the subdominant attractor any more but it free-
+falls,6 such that its density scales as ρφ ≃ 1
+2 ˙φ2 ∝ a−6, until it refreezes at a larger value φF .
+This value is estimated as follows.
+In free-fall, the slope term in the equation of motion (2.3) of the field is negligible, so
+that the equation is reduced to ¨φ + 3H ˙φ ≃ 0, where H = 2/3t after equality. The solution is
+φ(t) = φeq + C
+teq
+�
+1 − teq
+t
+�
+,
+(2.9)
+where C is an integration constant.
+From the above, it is straightforward to find that
+˙φ = Ct−2. Thus, the density parameter at equality is
+Ωeq
+φ = ρφ
+ρ
+����
+eq
+=
+1
+2C2t−4
+eq
+4
+3( mP teq)2 = 3
+8
+C2
+(mP teq)2
+⇒ C =
+�
+8
+3Ωeq
+φ mP teq =
+√
+8
+κλ mP teq ,
+(2.10)
+where we used Eq. (2.8), ρφ ≃ 1
+2 ˙φ2 and that ρ = 1/6πGt2 = 4
+3(mP /t)2. Thus, the field freezes
+at the value
+φ0 = φeq + C/teq = φeq +
+√
+8
+κλ mP ,
+(2.11)
+where we considered that teq ≪ tfreeze < t0 .
+Using that teq ∼ 104 y and t0 ∼ 1010 y, we can estimate
+Veq
+V0
+≃
+Ωeq
+φ ρeq
+0.7 ρ0
+≃
+30
+7(κλ)2
+� t0
+teq
+�2
+≃
+3
+7(κλ)2 × 1013 .
+(2.12)
+6i.e. its energy density is dominated by its kinetic energy density only.
+– 7 –
+
+Now, from Eqs. (2.4) and (2.6) we find
+Veq
+V0
+≃
+eλ(eκ
+√
+6α−1) exp(−κλ φeq/mP )
+1 + 2κλ eκ
+√
+6α√
+6α exp
+�
+−2φ0/
+√
+6α mP
+� .
+(2.13)
+In view of Eqs. (2.5) and (2.11), the above can be written as
+Veq
+V0
+≃
+eλ(eκ
+√
+6α−1)
+1 + 2κλ eκ
+√
+6α√
+6α e−2
+√
+8/κλ
+√
+6α .
+(2.14)
+Taking Ωeq
+φ ≃ 0.1 as required by EDE, Eq. (2.8) suggests
+κλ ≃
+√
+30 .
+(2.15)
+Combining this with Eq. (2.12) we obtain
+e
+√
+30
+κ (eκ
+√
+6α−1) ∼ 1012/7 ,
+(2.16)
+where we have ignored the 2nd term in the denominator of the right-hand-side of Eq. (2.14).
+From the above we see that, κ is large when α is small. Taking, as an example, α = 0.01
+we obtain κ ≃ 18 and λ ≃ 0.30 (from Eq. (2.15)). With these values, the second term in the
+denominator of the right-hand-side of Eq. (2.14), which was ignored above, amounts to the
+value 3.2. This forces a correction to the ratio Veq/V0 of order unity, which means that the
+order-of-magnitude estimate in Eq. (2.16) is not affected.
+Using the selected values, Eq. (2.11) suggests that the total excursion of the field is
+∆φ = φ0 − φeq =
+√
+8
+κλ mP ≃ 0.5 mP ,
+(2.17)
+i.e. it is sub-Planckian. In the approximation of Eq. (2.4), we see that the argument of the
+exponential becomes κλ∆φ/mP ≃ 2.7 > 1, where we used Eq. (2.15). This means that the
+approximation breaks down and the exp(exp) potential is felt as considered, as depicted also
+in Fig. 1.
+For small α the eventual exponential potential in Eq. (2.6) is steep, which suggests that
+field rushes towards the minimum at infinity and the barotropic parameter is w ≈ −1 because
+the potential is dominated by the constant VΛ.
+2.4
+Tuning requirements
+Our model addresses in a single shot two cosmological problems: firstly, the Hubble tension
+between inferences of H0 using early and late-time data; and secondly, the reason for the
+late-time accelerated expansion of the Universe; late DE. However, it is subject to some
+tuning. Namely, the two free parameters κ and λ, the intrinsic field-space curvature dictated
+by α, and the scale of the potential introduced by VΛ.
+As we have seen κ and λ seem to take natural values, not too far from order unity.
+Regarding α we only need that it is small enough to lead to rapid decrease of the exponential
+contribution in the scalar potential in Eq. (2.6), leaving the constant VΛ to dominate at
+present. We show in the next section that α ∼ 10−4 is sufficient for this task. This leaves
+VΛ itself. The required tuning of this parameter is given by VΛ =
+� HPlanck
+0
+HSH0ES
+0
+�2
+V Planck
+Λ
+, where
+– 8 –
+
+V Planck
+Λ
+is given by the Planck 2018 [1] estimate of ρ0, the density today, multiplied by ΩΛ,
+the estimate of the density parameter of dark energy today, i.e.
+V Planck
+Λ
+= ΩΛρ0.
+Since
+� HPlanck
+0
+HSH0ES
+0
+�2
+≃ ( 67.44
+73.04)2 = 0.8525 we see that the required fine-tuning of our VΛ is not different
+from the fine-tuning introduced in ΛCDM, but, in contrast to ΛCDM, our proposal addresses
+two cosmological problems; not only late DE but also the Hubble tension.7
+3
+Numerical Simulation
+In order to numerically solve the dynamics of the system, it is enough to solve for the scale
+factor a(t), the field φ(t) and the background fluid densities ρm(t) and ρr(t), as every other
+quantity depends on these.
+They are governed by the Friedmann equations, the Klein-
+Gordon equation and the continuity equations respectively. Of course, the Klein-Gordon
+equation is a second order ODE, while the continuity equations are first order so that we
+need the initial value and velocity of φ and just the initial value of ρm and ρr as initial
+conditions. As described above, the field starts frozen and unfreezes around matter-radiation
+equality. Effectively, this means using φini = 0 and ˙φini = 0 as initial conditions, a few e-
+folds before matter-radiation equality, while the initial radiation and matter energy densities
+are chosen to satisfy the bounds obtained by Planck [1] at matter-radiation equality, i.e.,
+ρm(teq) = ρr(teq) = 1.27 × 10−110m4
+P.
+For convenience, we rewrite the equations in terms of the logarithmic energy densities
+˜ρm(t) = ln (ρm(t)/m4
+P) and ˜ρr(t) = ln (ρr(t)/m4
+P). Plugging the first Friedmann equation in
+the Klein-Gordon equation, gives
+¨φ(t) +
+�
+3ρ(t)
+mP
+˙φ(t) + dV
+dφ = 0,
+(3.1)
+˙˜ρm(t) +
+�
+3ρ(t)
+mP
+= 0,
+(3.2)
+˙˜ρr(t) + 4
+3
+�
+3ρ(t)
+mP
+= 0,
+(3.3)
+where 3m2
+PH2(t) = ρ(t) = [ exp(˜ρm(t))+exp(˜ρr(t))]m4
+P+ρφ(t) and ρφ(t) = K(φ(t))+V (φ(t))
+where K(φ(t)) = 1
+2( ˙φ(t))2 and V (φ(t)) is given by Eq. (2.2).
+As mentioned above, we assume the field to be frozen at an ESP, such that it could have
+been the inflaton or a spectator field at earlier times. The time of unfreezing is then controlled
+only by the parameters of the model’s potential.8 The densities of matter and radiation are
+scaled back to find some initial conditions at some arbitrary redshift, zini = 104, before
+equality.
+The differential solver records three “events” during solving: matter-radiation equality,
+triggered by the obvious condition; decoupling, triggered by the total energy density taking
+the correct value; and the present day, triggered by the field making up the correct fraction of
+the total energy density (as estimated by the Planck satellite [1]). These values are saved to
+an association so that they can later be searched to identify points which fulfill the necessary
+7In our simulations we use VΛ = 10−120.068 m4
+P as assumed also in Fig. 1.
+8 Although we could use an estimate for the initial time, it turns out that it makes no difference to the
+numerical results or the behaviour of the field and simply offsets the differential equations.
+– 9 –
+
+Initial Densities
+Calculation
+Value
+Matter
+ρm = 3ΩPlanck
+m,0
+m2
+P (HSH0ES
+0
+)2
+3.84 × 10−121m4
+P
+Radiation
+π2
+30g∗(T Planck
+CMB, 0)4
+9.56 × 10−125m4
+P
+Table 1: Table of present-day densities, where the present matter density parameter is
+ΩPlanck
+m,0
+= 0.3111, T Planck
+CMB, 0 = 2.7255 K and the effective relativistic degrees of freedom of
+radiation are g∗ = 3.36, calculated by taking the photon and neutrino contribution into
+account (see section 5 of [64]).
+Variable
+Initial Value
+Source
+Redshift
+zinitial = 104
+chosen to be shortly before
+matter-radiation equality
+Time
+tini = 0.1m−1
+P
+chosen to be close to zero
+(see footnote 8)
+Field Value
+φ(tini) = 0
+simplified initial conditions
+Rate of change of Field
+Value
+˙φ(tini) = 0
+simplified initial conditions
+Density of Matter
+ρm(tini) = 3.84 × 10−109m4
+P
+ρm(t0)Planck(zini + 1)3
+Density of Radiation
+ρr(tini) = 1.24 × 10−108m4
+P
+ρr(t0)Planck(zini + 1)4
+E-folds elapsed
+Nini = 0
+chosen for convenience
+Table 2: Table detailing the initial conditions for the differential equations.
+constraints, in order to find a viable parameter space. Once the final event is recorded, the
+solver is terminated.
+Event
+Criteria
+Justification
+Matter-Radiation
+Equality
+ρm(teq) = ρr(teq)
+Theoretical Definition
+Last Scattering
+ρm(tls) = 4.98 × 10−112 m4
+P
+Extrapolation
+from
+ΛCDM
+initial conditions (see Table 2)
+using Planck results ρm(zeq)
+with zeq = 1089.80 [1]
+Present Day
+Ωφ = 0.6889
+Planck data [1]
+Table 3: Table of events recorded during the numerical solving of equations and how.
+If a field point does not meet the conditions for the final event (i.e. the present day),
+this indicates that the field began the simulation as the dominant component and will never
+reach the correct energy density. The point is thrown away. Finally, reasonable observational
+and theoretical constraints to the parameter space are applied to the data collected, which
+are outlined in Table 4.
+– 10 –
+
+Parameter
+to
+be
+constrained
+Source
+Description
+Constraint
+Density
+parameter
+of
+the field at equality
+EDE
+literature
+[25]
+Upper limit governed by the
+maximum value that does
+not impede structure forma-
+tion; lower limit is so that
+EDE actually has an effect
+0.015 ≤ Ωeq
+φ < 0.107
+Density parameter
+of the field at
+Last Scattering
+EDE
+literature
+[8]
+This is the upper limit that
+ensures EDE cannot cur-
+rently be detected in the
+CMB
+Ωls
+φ < 0.015
+Density parameters of
+the field at Last Scatter-
+ing and Equality
+Theoretical
+Achieves desired behaviour
+of the field
+Ωeq
+φ > Ωls
+φ
+Density
+parameter
+of
+the field today
+Planck 2018
+[1]
+Observational constraint
+0.6833 ≤ Ω0
+φ ≤ 0.6945
+Barotropic parameter of
+the field today
+Planck 2018
+Observational constraint
+−1 ≤ w0
+φ ≤ −0.95
+Running of the
+barotropic parameter
+today
+Planck 2018
+[1]
+Observational constraint
+−0.55 ≤ wa
+φ ≤ 0.03
+Hubble constant
+SH0ES
+2021 [2]
+Observational constraint
+72.00≤
+H0
+km s−1 Mpc−1 ≤74.08
+Total Field Excursion
+Theoretical
+From analytical estimates,
+the total excursion of the
+field should ideally be sub-
+Planckian
+φ0 − φeq < mP
+Table 4: Table describing and justifying constraints used to identify the viable parameter
+space. In the above, wa
+φ = − dwφ
+da
+���
+0, c.f. Eq. (1.8).
+4
+Results and analysis
+4.1
+Parameter Space
+As evident from Figs. 2, 3 and 4, we find that κ ∼ 102 and λ ∼ 10−3, which are rather
+reasonable values. In particular, the value of κ suggests that the mass-scale which suppresses
+the non-canonical field ϕ in the original potential in Eq. (2.1) is near the scale of grand
+unification ∼ 10−2 mP. Regarding the curvature of field space we find α ∼ 10−4, which again
+is not unreasonable.
+The viable parameter space suggests that κλ >
+√
+3, which contradicts our assumption
+in Eq. (2.7). This implies that, unlike the analytics in Sec. 2.3.1, the field does not adopt
+the subdominant exponential scaling attractor but the slow-roll exponential attractor, which
+leads to domination [61, 63].
+As the field thaws and starts following this attractor, the
+approximation in Eq. (2.4) breaks down as the field experiences the full exp(exp) potential,
+which is steeper that exponential (see Fig. 1). Consequently, instead of becoming dominant
+the field free-falls. This contradiction with our discussion in Sec. 2.3.1 is not very important.
+– 11 –
+
+0.0000
+0.0001
+0.0002
+0.0003
+0.0004
+0.0005
+0.0006
+0.0007
+0
+100
+200
+300
+400
+500
+600
+700
+α
+κ
+VΛ
+mP
+4
+= 10-120.068,
+0 < λ < 0.027
+Figure 2:
+Parameter space slice in the κ − α plane with 0 < λ < 0.027 and VΛ =
+10−120.068m4
+P.
+The blue dotted line is the boundary of the region that produces non-
+inflationary results (see below), while the orange region is constituted by the success-
+ful points, i.e., those for which the constraints detailed in Table 4 are satisfied.
+Note
+that the region bounded in blue is not equal to the range of the scan, which goes from
+0 ≤ κ ≤ 700, 0 ≤ α ≤ 0.00071. This is because points with potential larger than a certain
+starting value result in the field beginning the simulation dominant, which means that the
+Universe goes into inflation which cannot terminate and will never meet the numerical end
+condition for the present day. These points are very close to the viable parameter space for
+these two parameters and therefore must be thrown away.
+The existence of the scaling attractor provided an easy analytic estimate for the moment
+when the field unfreezes. It turns out that, because the scaling attractor has been substituted
+by the slow-roll attractor, the field unfreezes because its potential energy density becomes
+comparable to the total energy density, going straight into free-fall. In is much harder to
+analytically estimate when exactly this takes place, but the eventual result (free-fall) is the
+same.
+The redshift of matter-radiation equality occurs earlier than usual at zeq ≃ 4000. How-
+ever, equality occurs well before last scattering, zeq > zls and its redshift is only indirectly
+inferred by observations. In contrast, the redshift of last scattering is where we would expect
+it at zls ≃ 1087. Theoretical constraints suggest zls ≃ 1090 [65], and the observations of the
+Planck satellite suggest zls = 1089.80 ± 0.21 [1].
+– 12 –
+
+0.0000
+0.0001
+0.0002
+0.0003
+0.0004
+0.0005
+0.0006
+0.0007
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+α
+λ
+VΛ
+mP
+4
+= 10-120.068,
+0 < κ < 700
+Figure 3: Parameter space slice in the λ−α plane with 0 < κ < 700 and VΛ = 10−120.068m4
+P.
+The orange region is constituted by the successful points, i.e., those for which the constraints
+detailed in Table 4 are satisfied.
+4.2
+Field Behaviour
+The field behaves as expected, with the mild modification of the attractor solution at un-
+freezing (slow-roll instead of scaling), which leads to free-fall. The evolution is depicted in
+Figs. 5, 6, 7 and 8 for the example point at α = 0.0005, κ = 145, λ = 0.008125, and Vλ
+tuned to the SH0ES cosmological constant [2]. The observables obtained in this case (i.e. the
+values of H0, w0 and wa) are shown in Table 5. The behaviour of the Hubble parameter is a
+function of redshift as can be seen in Fig. 7.
+As mentioned in Table 4, the maximum allowed value of the EDE density parameter
+at equality is just over 0.1.
+However, it is possible that this is too lenient a constraint
+because unlike the models for which this constraint was developed, our model has a true
+free-fall period, which means it redshifts away exactly as a−6 rather than below this rate
+as in oscillatory behaviour (see Figs. 5 and 8). A full MCMC analysis may provide a more
+accurate constraint for non-oscillatory models.
+At present, the exponential contribution to the potential density in Eq. (2.6) is largely
+subdominant to VΛ, so the contribution of the scalar field to the total density budget is
+almost constant, as in ΛCDM. Its barotropic parameter is, therefore, wφ ≈ −1 (see Fig. 5).
+Technically, it is not exactly -1 but its running is negligible, with the viable parameter space
+for wa fitting easily within the constraint in Eq. (1.8) by some ten orders of magnitude (see
+Table 5).
+– 13 –
+
+0
+100
+200
+300
+400
+500
+600
+700
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+κ
+λ
+VΛ
+mP
+4
+= 10-120.068,
+0 < α < 0.00071
+Figure 4: Parameter space slice in the λ − κ plane with 0 < α < 0.00071 and VΛ =
+10−120.068m4
+P. The orange region is constituted by the successful points, i.e., those for which
+the constraints detailed in Table 4 are satisfied.
+5
+Initial Conditions
+Our model accounts for both EDE and late-time dark energy in a non-oscillatory manner
+(in contrast to Ref. [30]). The field is frozen at early times, thawing just before matter-
+radiation equality when its density grows to nearly 0.1 of the total value (see Fig. 6), as set
+by constraints in Ref. [25]. A steep exp(− exp) potential then forces the field into free-fall,
+causing its energy density to dilute away as ρφ ∝ a−6. After this, the field hits the asymptote
+of the exponential decay and refreezes, becoming dominant at present (see Fig. 8).
+Thus, we achieve DE-like behaviour at the present day by ensuring that the field re-
+freezes after its period of free-fall, therefore remaining at a constant energy density equal to
+the value of the potential density at that point. Although this constant potential density is
+initially negligible, the expansion of the Universe causes the density of matter to decrease.
+Because the field refreezes at a potential density that is comparable to the density of matter
+at present, the field starts to become dominant at the present day. Once it begins to domi-
+nate the Universe, the field thaws again, but the density of the Universe is dominated by a
+constant contribution VΛ, as with ΛCDM.
+The obvious question is why our scalar field finds itself frozen at the origin in the first
+place. One compelling explanation is the following. We assume that the origin is an enhanced
+symmetry point (ESP) such that, at very early times, an interaction of ϕ with some other
+scalar field χ traps the rolling of ϕ at zero. The idea follows the scenario explored in Ref. [66].
+– 14 –
+
+wϕ
+wm+r
+wUniverse
+0
+2
+4
+6
+8
+-1.0
+-0.5
+0.0
+0.5
+1.0
+3671
+1350
+496
+182
+66
+24
+8
+2
+N
+z
+VΛ
+mP
+4 = 10-120.068
+α =0.0005
+κ=145
+λ=0.008125
+Figure 5: Barotropic parameter of the scalar field (dotted green), of the background perfect
+fluid (full blue) and of the sum of both components (full black), for α = 0.0005, κ = 145, λ =
+0.008125, and VΛ = 10−120.068m4
+P.
+Density parameter of field Ωϕ
+0
+2
+4
+6
+8
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+3671
+1350
+496
+182
+66
+24
+8
+2
+N
+z
+VΛ
+mP
+4 = 10-120.068
+α =0.0005
+κ=145
+λ=0.008125
+Figure 6: The density parameter of the scalar field, for α = 0.0005, κ = 145, λ = 0.008125,
+and VΛ = 10−120.068m4
+P, as a function of the redshift (top) and e-folds (bottom) elapsed since
+the beginning of the simulation.
+In this scenario, the scalar potential includes the interaction
+∆V = 1
+2g2ϕ2χ2 ,
+(5.1)
+– 15 –
+
+HϕCDM
+HCDM only
+HΛCDM
+8.0
+8.2
+8.4
+8.6
+8.8
+9.0
+9.2
+50
+100
+150
+200
+250
+2.35 2.03 1.74 1.48 1.24 1.03 0.84 0.66 0.50 0.36 0.23 0.11 0.01
+N
+z
+VΛ
+mP
+4 = 10-120.068
+α =0.0005
+κ=145
+λ=0.008125
+Figure 7: The Hubble parameter (in units of km s−1Mpc−1) of a Universe with the modelled
+scalar field (green), a classical ΛCDM simulation (black), and one with only matter and
+radiation (blue), as a function of the redshift (top) and the e-folds (bottom) elapsed since
+the beginning of the simulation.
+log[ρm/mP
+4]
+log[ρr/mP
+4]
+log[ρϕ/mP
+4]
+log[(ρm+ρr)/mP
+4]
+0
+2
+4
+6
+8
+-125
+-120
+-115
+-110
+-105
+3671
+1350
+496
+182
+66
+24
+8
+2
+N
+z
+VΛ
+mP
+4 = 10-120.068
+α =0.0005
+κ=145
+λ=0.008125
+Figure 8: The logarithmic densities of matter (dot-dashed red), radiation (dotted orange),
+the sum of both (solid blue) and the scalar field (dashed green), as a function of the redshift
+(top) and the e-folds (bottom) elapsed since the beginning of the simulation. The horizontal
+full line represents the (SH0ES) energy density of the Universe at present.
+where the coupling g < 1 parametrises the strength of the interaction.
+– 16 –
+
+Constraint
+Field Value
+0.015 ≤ Ωeq
+φ < 0.107
+0.05178
+Ωls
+φ < 0.015
+0.001722
+Ωeq
+φ > Ωls
+φ
+YES
+0.6833 ≤ Ω0
+φ ≤ 0.6945
+0.6889
+−1 ≤ w0
+φ ≤ −0.95
+-1.000
+−0.55 ≤ wa
+φ ≡ − dwφ
+da
+���
+0 ≤ 0.03
+−4.850 × 10−11
+72.00 ≤
+H0
+km s−1 Mpc−1 ≤ 74.08
+73.27
+κλ
+1.178
+(φ0 − φeq)/mP < 1
+0.4274
+Table 5: Table giving the constraints and their corresponding values for an example point,
+α = 0.0005, κ = 145, λ = 0.008125, and VΛ tuned to the SH0ES cosmological con-
+stant, in the viable parameter space.
+The Hubble constant obtained in this example is
+H0 = 73.27 km/s Mpc.
+We assume that initially ϕ is rolling down its steep potential.9 Then, the interaction
+in Eq. (5.1) provides a modulated effective mass-squared m2
+eff = g2ϕ2 to the scalar field χ.
+When ϕ crosses the origin, this effective mass becomes momentarily zero. If the variation of
+the ϕ field (i.e. the speed | ˙ϕ| in field space) is large enough, then there is a window around
+the origin when | ˙meff| ≫ m2
+eff (because, | ˙ϕ| ≫ ϕ2 ≃ 0). This violates adiabaticity and leads
+to copious production of χ-particles [66].10
+As the field moves past the ESP, the produced χ particles become heavy, which takes
+more energy from the ϕ field, producing an effective potential incline in the direction the
+ϕ field is moving. Indeed, the particle production generates an additional linear potential
+∼ g|ϕ|nχ [66], where nχ is the number density of the produced χ-particles. This number
+density is constant because the duration of the effect is much smaller than a Hubble time,
+so that we can ignore dilution from the Universe expansion. The rolling ϕ field climbs up
+the linear potential until its kinetic energy density is depleted. Then the field momentarily
+stops and afterwards reverses its motion (variation) back to the origin. When crossing the
+origin again, there is another bout of χ-particle production, which increases nχ and makes the
+linear potential steeper to climb. This time, ϕ variation halts at a value closer to the origin.
+Then, the field reverses its motion and rushes through the origin again. Another outburst of
+χ-particle production steepens the linear potential further. The process continues until the
+9For away from the origin, the scalar potential V (ϕ) does not have to be of the form in Eq. (2.1). In fact,
+it is conceivable that ϕ might play the role of the inflaton field too (see Appendix).
+10Near the origin, when ϕ ≃ 0, the ϕ-field is approximately canonically normalised, as suggested by Eq. (1.5),
+so the considerations of Ref. [66] are readily applicable.
+– 17 –
+
+ϕ-field is trapped at the origin [63, 66].
+The trapping of a rolling scalar field at an ESP can take place only if the χ-particles do
+not decay before trapping occurs. If they did, the nχ would decrease and the potential g|ϕ|nχ
+would not be able to halt the motion (variation) of the ϕ-field. The end result of this process is
+that all the kinetic energy density of the rolling ϕ has been given to the χ-particles. Now, since
+ϕ is trapped at the origin, the effective mass of the χ-particles is zero, which means that they
+are relativistic matter, with density scaling as ρχ ∝ a−4. As far as ϕ is concerned, it is trapped
+at the origin and its density is only ρϕ = V (ϕ = 0) = e−λVX = constant (cf. Eq. (2.1)).
+After some time, it may be assumed that the χ-particles do eventually decay into the
+standard model particles, which comprise the thermal bath of the hot Big Bang. The con-
+fining potential, which is proportional to nχ, disappears but, we expect the ϕ-field to remain
+frozen at the origin because the scalar potential V (ϕ) in Eq. (2.1) is flat enough there. As we
+have discussed, the ϕ-field unfreezes again in matter-radiation equality. The above scenario
+is depicted in Fig. 9
+For simplicity, we have considered that, apart from the obvious violation of adiabacity at
+the ESP, the χ direction is otherwise approximately flat and the χ-field has a negligible bare
+mass compared to the ϕ field. It would be more realistic to consider a non-zero bare mass for
+the χ-particles, which when they become non-relativistic (much later than the trapping of
+ϕ) can safely decay to the thermal bath of the hot Big Bang, reheating thereby the Universe,
+e.g. in a manner not dissimilar to Ref. [67].
+The above scenario is one possible explanation of the initial condition considered and
+not directly relevant to the scope of this work - numerical simulations simply assume that
+the field begins frozen at the origin. Other possibilities to explain our initial condition exist,
+for example considering a thermal correction of the form δV ∝ T 2ϕ2, which would make the
+origin an effective minimum of the potential at high temperatures and drive the ϕ-field there.
+6
+Conclusions
+In conclusion, we have studied in detail a non-oscillatory model of unified early and late dark
+energy, which resolves the Hubble tension and simultaneously explains the observed current
+accelerated expansion with no more fine tuning than ΛCDM. Our model considers a single
+scalar field in the context of α-attractors, as in Ref. [30], but in our case the field is not
+oscillating; instead after equality, it free-falls with energy density decreasing as a−6, faster
+than most early dark energy (EDE) proposals and the fastest possible.
+In our proposed scenario, the scalar field lies originally frozen at the origin, until it
+thaws near the time of equal matter-radiation densities, when it becomes EDE. Afterwards
+it free-falls until it refreezes at a lower potential energy density value, which provides the
+vacuum density of ΛCDM. We showed that the total excursion of the field in configuration
+space is sub-Planckian, which implies that our potential is stable under radiative corrections.
+One explanation of our initial conditions is that the origin is an enhanced symmetry
+point (ESP). Our scalar field is originally kinetically dominated until it is trapped at the ESP
+when crossing it.11 As we discuss in Appendix A, the scalar field could even be the inflaton,
+which after inflation rolls down its runaway potential until it becomes trapped at the ESP.
+Our potential in Eq. (2.1) really serves to demonstrate that a model unifying EDE with
+ΛCDM can be achieved with a suitably steep runaway potential. With the parameters of
+our model assuming rather natural values, thereby not introducing fine-tuning additional to
+11A thermal correction to the scalar potential can have a similar effect.
+– 18 –
+
+ln ρ
+ln a
+ρφ
+ρr + ρm
+today
+equality
+ESP
+e−λVX
+VΛ
+Figure 9: Schematic log-log plot depicting the evolution of the density of the scalar field
+ρφ (solid blue line) and the density of radiation and matter ρr + ρm (dashed red line) in
+the case when the decay of the kinetic energy density of the trapped scalar field generates
+the thermal bath of the hot Big Bang (as in Ref. [REF]). Originally the φ-field is rushing
+towards the minimum of the potential, dominated by its kinetic density, so that ρφ ∝ a−6
+(free-fall). When it crosses the enhanced symmetry point (ESP) its interaction to the χ-
+field (cf. Eq. (5.1)) traps the rolling φ-field at the ESP while all its kinetic energy is given
+to χ-particles, which soon decay into the radiation and matter of the hot Big Bang (the
+decay is assumed to be quick, just after trapping). Afterwards, the φ-field stays frozen, with
+energy density V (φ = 0) = e−λVX (cf. Eq. (2.1)) until much later, when its potential density
+is comparable to the background. Then it unfreezes before dominating, acting as early dark
+energy at the time near matter-radiation equality, and subsequently free-falls to its value φ0,
+with potential density approximately VΛ = constant. The field stays there until the present
+when it dominates the Universe and becomes late dark energy.
+that of ΛCDM, we show that this is indeed possible with a simple design. The challenge lies
+in constructing a concrete theoretical framework for such a potential.
+Acknowledgements: LB is supported by STFC. KD is supported (in part) by the Lancaster-
+Manchester-Sheffield Consortium for Fundamental Physics under STFC grant: ST/T001038/1.
+SSL is supported by the FST of Lancaster University.
+A
+Quintessential Inflation
+Is it possible that our scalar field can not only be early and late dark energy, but also be the
+inflaton field, responsible for accelerated expansion in the early Universe?
+– 19 –
+
+The α-attractors construction leads to two flat regions in the scalar potential of the
+canonical field, as the kinetic poles of the non-caninical field are displaced to infinity. This
+idea has been employed in the construction of quintessential inflation models in Refs. [54–56],
+where the low-energy plateau was the quintessential tail, responsible for quintessence and the
+high-energy plateau was responsible for inflation.
+However, if we inspect the potential in Eq. (2.1) at the poles ϕ = ±
+√
+6α mP, we find that
+the potential for the positive pole is V (ϕ+) = VΛ as expected, while for the negative pole we
+have V (ϕ−) = VΛ exp
+�
+2λ sinh
+�
+κ
+√
+6α
+��
+. For the values of the parameters obtained (κ ∼ 102,
+λ ∼ 10−3 and α ∼ 10−4) it is easy to check that V (ϕ−) is unsuitable for the inflationary
+plateau. Thus, our model needs to be modified to lead to quintessential inflation.
+The first modification is a shift in field space such that our new field is
+˜ϕ = ϕ + Φ ,
+(A.1)
+where Φ is a constant. The α-attractors construction applies now on the new field ˜ϕ for
+which the Lagrangian density is given by the expression in Eq. (1.4) with the substitution
+ϕ → ˜ϕ. The poles of our new field lie at ˜ϕ± = ±
+√
+6˜α mP, where ˜α is the new α-attractors
+parameter.
+We want all our results to remain unaffected, which means that, for the positive pole,
+Eq. (A.1) suggests
+ϕ+ =
+√
+6α mP = ˜ϕ+ − Φ =
+√
+6˜α mP − Φ ⇒ ˜α = 1
+6
+� Φ
+mP
++
+√
+6α
+�2
+.
+(A.2)
+The above, however, is not enough. It turns out we need to modify the scalar potential
+as well.
+This modification must be such that near the positive pole the scalar potential
+reduces to the one in Eq. (2.1). A simple proposal is
+V ( ˜ϕ) = VX exp{−2λ sinh[κ( ˜ϕ − Φ)/mP]} ,
+(A.3)
+which indeed reduces to Eq. (2.1) when κ( ˜ϕ − Φ) = κϕ > mP Note that κ
+√
+6α > 1 is implied
+from the requirement that near the positive pole we have κ
+√
+6α mP = κϕ+ > mP. The ESP
+discussed in Sec. 5 is now located at ˜ϕ = Φ, such that Eq. (5.1) is now ∆V = 1
+2g2( ˜ϕ − Φ)2χ2.12
+We are interested in investigating the inflationary plateau. This is generated for the
+canonical field near the negative pole ˜ϕ− = −
+√
+6˜α mP, where the scalar potential of the
+canonical field “flattens out” [40].
+Assuming that Φ >
+√
+6α mP, we have that ˜ϕ− − Φ = −2Φ −
+√
+6α mP ≃ −2Φ, where we
+used Eq. (A.2). Hence, for the potential energy density of the inflationary plateau we obtain
+Vinf = V ( ˜ϕ−) ≃ VX exp[−2λ sinh(−2κΦ/mP)]
+≃ exp
+�
+λ eκ
+√
+6α�
+VΛ exp[λ exp(2κΦ/mP)]
+= exp
+�
+λ(eκ
+√
+6α + e2κΦ/mP)
+�
+VΛ ≃ VΛ exp
+�
+λ e2κΦ/mP
+�
+,
+(A.4)
+where we used Eq. (2.1) and that in −2 sinh(−x) ≃ ex, when x ≫ 1.
+12Near the ESP the potential does not approximate Eq. (2.1). However, we assume that, after unfreezing,
+the field rolls away fast from the ESP, such that soon the exp(exp) form of the potential becomes valid and
+the evolution is the one discussed in the main text of our paper.
+– 20 –
+
+With α-attractors, the inflationary predictions are ns = 1 − 2/N and r = 12˜α/N2 [40],
+where ns is the spectral index of the scalar curvature perturbation and r is the ratio of
+the spectrum of the tensor curvature perturbation to the spectrum of the scalar curvature
+perturbation, with N being the number of inflationary efolds remaining after the cosmo-
+logical scales exit the horizon.
+Typically, N = 60 − 65 for quintessential inflation, which
+means that ns = 0.967 − 0.969, in excellent agreement with the observations [68]. For the
+tensor-to-scalar ratio the observations provide the bound r < 0.036 [69], which suggests
+˜α < 0.003 N2 = 10.8 − 12.7.
+The COBE constraint requires Vinf ∼ 10−10 m4
+P. Using that VΛ ∼ 10−120 m4
+P, Eq. (A.4),
+suggests that κΦ/mP = 1
+2 ln(110 ln 10/λ). Hence. the conditions Φ >
+√
+6α mP and κ
+√
+6α > 1
+suggest
+1 < κ
+√
+6α < κΦ/mP = 1
+2 ln(110 ln 10/λ) .
+(A.5)
+Our findings in Sec. 4 are marginally in agreement with the above requirements.
+For
+example, taking α = 0.0006 and κ = 100 we find κ
+√
+6α = 6 and then Eq. (A.5) suggests
+λ < 1.556 × 10−3.
+We also find Φ/mP >
+√
+6α = 0.06, which is rather reasonable.
+Then,
+Eq. (A.2) implies ˜α > 12α = 7.2 × 10−3, which comfortably satisfies the observational con-
+straint on r. In fact, taking N ≃ 60, we find r = 12˜α/N2 > α/25 = 2.4 × 10−5.
+The above should be taken with a pinch of salt because the approximations employed
+are rather crude. However, they seem to suggest that our augmented model in Eq. (A.3)
+may lead to successful quintessential inflation while also resolving the Hubble tension, with no
+more fine-tuning than that of ΛCDM. A full numerical investigation is needed to confirm this.
+References
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+1
+A Submillimeter-Wave FMCW Pulse-Doppler Radar
+to Characterize the Dynamics of Particle Clouds
+Tomas Bryllert, Member, IEEE, Marlene Bonmann, and Jan Stake, Senior Member, IEEE
+Abstract—This work presents a 340-GHz frequency-modulated
+continuous-wave (FMCW) pulse-Doppler radar. The radar sys-
+tem is based on a transceiver module with about one milli-
+Watt output power and more than 30-GHz bandwidth. The
+front-end optics consists of an off-axis parabola fed by a horn
+antenna from the transceiver unit, resulting in a collimated radar
+beam. The digital radar waveform generation allows for coherent
+and arbitrary FMCW pulse waveforms. The performance in
+terms of sensitivity and resolution (range/cross-range/velocity) is
+demonstrated, and the system’s ability to detect and map single
+particles (0.1–10 mm diameter), as well as clouds of particles, at
+a 5-m distance, is presented. A range resolution of ∼1 cm and
+a cross-range resolution of a few centimeters (3-dB beam-width)
+allow for the characterization of the dynamics of particle clouds
+with a measurement voxel size of a few cubic centimeters. The
+monitoring of particle dynamics is of interest in several industrial
+applications, such as in the manufacturing of pharmaceuticals
+and the control/analysis of fluidized bed combustion reactors.
+Index Terms—FMCW, pulse-Doppler, radar, remote sensing,
+sensors, submillimeter waves, terahertz systems, transceivers
+I. INTRODUCTION
+F
+OR many industrial applications, such as in the manu-
+facturing of pharmaceuticals [1], or energy conversion
+using fluidized bed reactors [2], the industrial process involves
+particles or powders dispersed in a process reactor. It is neces-
+sary to monitor the particle dynamics to maintain the process
+quality and to gain insights regarding the process. Therefore,
+measuring the particle concentration and the local particle
+velocities at a high update rate and high spatial resolution
+is desirable. Ideally, these quantities should be measured ex
+vivo without inserting any physical probes into the reactors
+so that the introduction of measurement sensors does not
+alter the processes. In particular, this is required in harsh
+process environments [3]. Frequency-modulated continuous-
+wave (FMCW) range-Doppler radar operating at center fre-
+quencies (fc) within the submillimeter wave range [4] of
+the electromagnetic spectrum offers a realistic opportunity to
+provide the desired information.
+Compared to other contactless measurement methods using
+visible or infrared light [5], [6], the submillimeter wavelength
+range allows more penetration depth into dense particle clouds
+[7] and is less sensitive to contaminations on the reactor
+access windows. The radar technique also allows for Doppler
+Manuscript received January 1st, 2023. This work was supported in part
+by the Swedish Foundation for Strategic Research (SSF) under the contract
+ITM17-0265.
+Tomas Bryllert, Marlene Bonmann, and Jan Stake are with the Tera-
+hertz and Millimetre Wave Laboratory, Chalmers University of Technol-
+ogy, SE-412 96 Gothenburg, Sweden. (e-mail: tomas.bryllert@chalmers.se;
+marbonm@chalmers.se; jan.stake@chalmers.se)
+processing, which reveals information about the velocities
+of the particles [8]. Compared with more traditional radar
+techniques in the microwave and millimeter wave region [9],
+there are a few properties that favor submillimeter waves [10]:
+• Short wavelengths (λ) result in higher sensitivity for
+detecting smaller particles since the radar cross-section
+of particles in the Rayleigh regime scales as λ−4;
+• Wide bandwidth and, thereby, a higher range resolution.
+For example, a 30-GHz bandwidth results in a theoretical
+range resolution of 5 mm;
+• The cross-range resolution for a fixed antenna size,
+typically limited by the access window size in an ac-
+tual application, improves with high frequency since the
+diffraction-limited resolution scales with λ.
+Several FMCW radars for high-resolution, 3D imaging have
+been presented with center frequencies above 300 GHz [11]–
+[14]. These systems use ranging to produce 3D static images
+and are not using pulse-Doppler processing [15]. FMCW
+radars using MMIC-transceivers based on SiGe technology
+have been demonstrated in the millimeter wave region [16],
+including promising performance up to 480 GHz [17]. Still,
+submillimeter-wave transceivers, with a high dynamic range at
+room temperature, require diode technology [18], [19]. Cooper
+et al. [10] reported a FMCW range-Doppler radar system at
+660 GHz, demonstrating the range-Doppler concept’s feasibil-
+ity at submillimeter wave frequencies, but with few details.
+This work presents the implementation of a FMCW pulse-
+Doppler radar based on a 340-GHz transceiver module with
+30-GHz bandwidth [20]. A digital waveform generator con-
+trols the system. The transceiver module provides an accept-
+able trade-off between performance and hardware complexity,
+resulting in a relatively compact tripod-mounted radar design,
+as shown in Fig. 1. The form factor allows easy implementa-
+tion in industrial scenarios. The performance of the transceiver
+modules and their application in a 3D imaging radar was pre-
+sented in [13]. Here the implementation of the coherent pulse
+generation and signal processing to realize range-Doppler
+radar operation are explained, together with the resulting radar
+system’s noise- and resolution performance. Furthermore, the
+ability of the radar to detect single particles with diameters
+ranging from 100 µm – 500 µm is demonstrated. The accuracy
+of the velocity measurements is validated by comparing the
+measured range-Doppler profile of a falling metal sphere with
+known weight and diameter to the standard free-fall model.
+The results demonstrate that the performance of the radar
+system is highly suitable for the suggested industrial scenarios.
+arXiv:2301.00558v1  [physics.ins-det]  2 Jan 2023
+
+2
+Fig. 1. Photograph of the radar system. The front-end optics and electronics
+are mounted on a base plate together with analog and digital baseband
+circuitry.
+II. METHOD
+A. Radar electronics and optics
+Fig. 2 shows a schematic block diagram of the 340-GHz
+FMCW range-Doppler radar. The system architecture is a
+frequency up-converted, frequency multiplied FMCW radar.
+A few hardware details deserve to be highlighted: The digital
+waveform generator is an FPGA-controlled arbitrary waveform
+card with 4 Gb of useful memory and a maximum sampling
+rate of >6 Gs/s of which 4 Gs/s is used in the current work.
+The card can write >100 ms of 1-GHz bandwidth waveform
+data directly from memory. This means that, in a coher-
+ent pulse-Doppler processing interval (CPI), typically much
+shorter than 100 ms, an arbitrary pulse train of FMCW pulses
+can be transmitted – and then repeated. Multiple FMCW
+waveforms can therefore be interleaved, addressing different
+parts of the system bandwidth (323 – 357 GHz) within a
+coherent processing interval. This capability can be used to
+extract frequency-resolved (spectroscopic) information from
+the scene in an efficient way. This feature is not used in the
+presented performance demonstrations. The baseband chirp,
+typically 1-GHz bandwidth, generated by the digital hardware,
+is centered at 1 GHz. This signal is up-converted to X-band
+using frequency mixing and a 9.6-GHz local oscillator (LO)
+and is then passed on to the transceiver unit. The transceiver
+unit multiplies the X-band chirp by a factor of 32 for a
+total final bandwidth of 32 GHz and transmits the signal, now
+centered at ∼340 GHz. The radar echoes are received back in
+the transceiver and are mixed on the outgoing signal straight
+down to the baseband using a balanced configuration [21].
+The front-end 340-GHz Schottky diode circuit is designed
+to operate as a frequency multiplier (x2) and sub-harmonic
+mixer - thereby simultaneously operating as a transmitter and
+receiver. The transceiver’s LO chain consists of an InGaAs
+pHEMT active frequency multiplier MMIC (x8) developed by
+Gotmic AB and a 170-GHz Schottky diode frequency doubler.
+The GaAs Schottky barrier diode circuits were fabricated
+in the Nanofabrication Laboratory at Chalmers university of
+technology. Originally, the complete transceiver module was
+developed for a 16-channel, high frame-rate, imaging radar
+[13] by Wasa Millimeter Wave AB, and is described in detail
+in [20].
+At the output of the transceiver unit, a circular horn from
+Custom Microwave Inc is used as a feed antenna for the optical
+system. This feedhorn illuminates a 4” off-axis parabolic
+mirror from Edmund Optics with an effective focal length of
+6”. The optical system results in a collimated radar beam.
+The digital hardware on the receiver side consists of an
+eight-channel, 250-Ms/s digitizer from National Instruments
+(1 channel is used). The digitizer is controlled by an FPGA
+which gives deterministic timing control. The digitizer card
+(PXIe format) integrates with a PC controller via a PXIe
+bus allowing for real-time signal processing and display. The
+waveform card, the analog-to-digital converter (ADC), and
+the local oscillator run from a common 10-MHz reference
+resulting in a fully coherent system.
+B. Radar signal processing
+Typical radar parameters used in the experiments presented
+in this work are:
+• Pulse bandwidth = 32 GHz;
+• Pulse time = 41 µs;
+• Pulse repetition interval (PRI) = 102.4 µs or 51.2 µs;
+• Number of pulses coherently processed (nP RI) = 128;
+• Target distance 4 – 6 m.
+Fig. 3 shows a block diagram of the signal processing.
+The data matrix format that is coherently processed is of
+the form: (nr of samples per pulse, ns) × (nP RI). After
+down-conversion in the transceiver, the received baseband (IF)
+signal is in the frequency range of 21 – 31 MHz, which is
+digitized. The data is digitally filtered with a finite impulse
+response bandpass filter (FIR BPF), converted to IQ format
+with the help of the Hilbert transform, down-converted to
+complex baseband, and decimated by a factor of 16 to 1.5 ×
+Nyquist limited sampling (15.625 Ms/s IQ), with: n′
+s = 640.
+In reality, several samples at the beginning and the end of
+each waveform are discarded (due to low-frequency ringing),
+leaving 590 samples instead of 640. This also reduces the
+used bandwidth from 32 GHz to 29.5 GHz. Both the pulse
+compression in range and the Doppler processing can be
+done using Fourier transforms in FMCW pulse-Doppler radar,
+which means that the signal processing can be done with a 2D
+fast Fourier transform (FFT) over the coherent data matrix –
+with appropriate windowing functions and digital filters. The
+output displayed for the radar user is the logarithm of the
+squared amplitude of the radar signal in a range-Doppler map.
+
+45 cm
+30 cm3
+Fig. 2. Schematic block diagram of the 340-GHz FMCW pulse-Doppler radar.
+Fig. 3. Schematic block diagram of the digital signal processing steps.
+C. Radar characterization and evaluation
+To demonstrate the performance of the radar system in
+terms of the noise floor, range and velocity resolution, and
+small particle detection, the following measurements were
+conducted: noise floor measurements, range and Doppler reso-
+lution, detection of small particles, and velocity measurements
+of a free-falling metal sphere. To study the origin of the
+noise floor in zero-Doppler and at finite Doppler frequency,
+the noise floor was measured without a target under four
+different conditions: First, with ADC only; second, with ADC
+together with IF amplifiers; third, ADC with IF amplifiers
+and a 10.1 GHz continuous wave (CW) signal driving the
+transceiver, fourth, ADC with IF amplifier and a chirp signal
+driving the transceivers. Additionally, the noise floor as a
+function of target strength was measured. Different radar cross
+sections (RCSs) were achieved by placing a corner reflection
+at different positions in the radar beam.
+Increasing the number of pulses per CPI, with other radar
+parameters fixed, the S/N for a target should increase linearly
+with the number of pulses (integration time) if the target
+and the radar system remain coherent and if the noise is
+uncorrelated with the radar signal. To verify this, a radar
+measurement on a static, corner reflector target was performed
+with nP RI= 16, 32, 64, 128, 256, and 512 per CPI.
+Three metal beads with a 2-mm diameter were glued onto
+a string and positioned at a 5 m distance to demonstrate the
+radar system’s range resolution. The target with three beads
+on a string was positioned so that all beads were illuminated
+by the radar beam and angled so that the beads were separated
+in range by approximately 3 cm. Another radar measurement
+was performed to display the velocity resolution while gently
+tapping the string to make it vibrate.
+To investigate the radar system’s ability to detect small
+particles, the radar beam is folded with a flat metallic mirror
+to be directed vertically upwards. A transparent plastic box
+was placed directly above the folding mirror to collect the
+particles. This way different test materials could be dropped
+straight into the radar beam. This experiment used 2-mm and
+10-mm diameter metal beads, 500-µm diameter quartz sand,
+and 100-µm spherical glass beads.
+The velocity measurement of the radar system was validated
+by comparing the measured velocity of a free-falling metal
+sphere of known diameter (1.27 cm) and weight (m = 8.44 g)
+with an analytical free-fall model. Letting the metal bead
+drop towards the radar it moves vertically under gravity and
+quadratic air resistance. Solving Newton’s second law of
+motion, the velocity (v) and position (x) with time (t) are
+then described by
+v = vt tanh (t/τ)
+(1a)
+x = x0 − vtτ ln (cosh (t/τ))
+(1b)
+with the terminal velocity vt =
+�
+(2mg/(Aρaircd)) and the
+characteristic time τ = vt/g, where g is the gravity of Earth, m
+is the mass of the metal bead, ρair is the air density at normal
+temperature pressure, A is the metal beads cross-section, cd
+is the drag coefficient (here 0.47 for a sphere [22]), and x0 is
+the initial position.
+
+Ref.10MHz
+LO
+Mixer
+BPF
+RFAmplifier
+TxRx
+9.6 GHz
+10.2-11 GHz
+RF in
+x32
+IF out
+326.4-352GHz
+Software for frequency
+DAC
+0.6-1.4 GHz
+>6Gs/swaveform
+LPF
+control
+generator
+Software for data
+IF Amplifier
+BPF
+processinganddata
+ADC
+display
+250 Ms/s digitizerns
+Hilbert Down conversion
+Range
+Data decimation Windowing
+FIR BPF
+LPF
+n.
+processing
+transform
+n
+_npRI
+FFT
+Coherent
+data matrix
+Doppler
+Windowing
+processing
+Display
+10log1o(I Ampl. 12)
+npRI
+FFT4
+Fig. 4. Noise floor in the range-Doppler map. (a) General view of the noise
+floor with the cuts that are presented in (b-d) indicated. (b) Constant range
+cut. (c) Constant velocity cut at zero-Doppler. (d) Constant velocity cut at
+finite Doppler.
+III. RESULTS
+A. Noise performance
+Fig. 4 shows the noise floor at different hardware settings
+and at different cuts through the range-Doppler map as indi-
+cated in Fig 4(a). No target is used in these measurements
+which have the purpose of demonstrating the origins of the
+noise floor for the radar.
+Ideally, the noise floor in the whole range-Doppler map
+should be set by thermal noise, deteriorated by the loss and
+noise figure of the front-end electronics, and scaled by the IF
+amplification. The transceiver unit trades noise performance
+for simplicity though. Using the same balanced pair of Schot-
+tky diode circuits for the final stage frequency multiplication
+and subharmonic homodyne down-conversion to baseband
+[20], the transceiver unit can be made quite compact – at
+the cost of excess noise. The excess noise comes in two
+shapes – through a conversion loss in the subharmonic mixing
+that is worse than would be the case in a dedicated mixer
+and through excess amplitude modulated noise from the LO
+(FMCW chirp) that mix into the IF side of the transceiver
+(despite the balanced configuration). In addition, Fig. 4(c)
+shows that excess noise is generated in zero-Doppler from
+driving the RF hardware with short (40 µs) chirps with high
+bandwidth. The cost of the excess noise is acceptable, though,
+since S/N is generally sufficient in the application scenarios
+that are evaluated.
+Fig. 5 shows how the noise floor for zero-Doppler and for
+finite Doppler is affected by the strength (RCS) of a static
+target. The noise floor is calculated as the mean when aver-
+aging over relevant range bins within the IF filter bandwidth
+(excluding the range-bin with the target response). The noise
+floor in zero-Doppler is not random noise but the result of
+sidelobes and amplitude/phase modulation of the waveform,
+as well as multiple reflections in the RF hardware. These
+effects are not seen at a finite Doppler frequency since the
+sidelobe/modulation/reflection pattern is identical from pulse
+Fig. 5. The noise floor in zero-Doppler and at a finite Doppler frequency as
+a function of target strength (RCS).
+Fig. 6.
+S/N as a function of the number of pulses used in the coherent
+processing. The target was a static corner cube.
+to pulse and therefore, only appear in the zero-Doppler bin. At
+strong target returns, the noise floor increases at finite Doppler
+frequencies, but then as a general increase of the noise floor
+in the whole range-Doppler plane - indicating that this noise
+increase originates in the actual noise of the RF carrier.
+Fig. 6 shows the radar signal of a static target and the
+noise floor versus the number of pulses per CPI. The S/N,
+when comparing the target signal with the Doppler noise
+floor increased linearly as expected. Thus verifying that the
+target and the radar system remain coherent and the noise is
+uncorrelated with the radar signal. As discussed above, the
+noise in zero-Doppler (the static noise floor) originates from
+the radar signal, meaning no improvement in S/N in zero-
+Doppler is seen at longer integration times.
+B. Range resolution, small particle detection, and velocity
+measurement
+Fig. 7 shows that the three metal beads with 2-mm diameter
+are clearly separated in the radar measurement with the signal
+
+-30
+-40
+(a)
+(b)
+6
+60
+-40
+Radar signal (dB)
+5.5
+(p)
+-80
+(w)
+-50
+Radar signal (
+5
+-100
+ADCsonly
+4.5
+(b)
+-60
+-120
+ADCs +IFamps.
+4
+ADCs + IF amps. + chirp
+-70
+-140
+ADCs + IF amps. + 1GHz cw
+3.5
+-80
+-160
+-4
+-2
+0
+2
+4
+-4
+-2 
+0
+2
+4
+Velocity (m/s)
+Velocity (m/s)
+40
+-40
+(d)
+60
+-60
+(dB)
+(dB)
+Radar signal (
+-80
+-80
+ signal
+100
+100
+ADCs only
+ADCs only
+Radar
+120
+ADCs+IFampS.
+120
+-ADCs+IFamps
+ADCs + IF amps. + chirp
+ADCs + IF amps.+ chirp
+-140
+-140
+ADCs +IF amps.+ 1GHz cw
+ADCs +IFamps.+1GHz cw
+-160
+-160
+3.5
+4
+4.5
+5
+5.5
+6
+3.5
+4
+4.5
+5
+5.5
+6
+Range (m)
+Range (m)-30
+Noise level (dB)
+40
+Static.noise.floor.
+50
+-60
+Doppler...noise...floor
+-70
+-40
+-20
+0
+20
+40
+Target signal (dB)0
+-10
+Target
+Radar signal (dB)
+20
+30
+Static noise floor
+40
+50
+Doppler noise floor
+-60
+-70
+16
+32
+64
+128
+256
+512
+Number of PRl5
+Fig. 7. Range and velocity resolution. (a-c) Shows a radar measurement of
+three beads with 2-mm diameter, demonstrating that the beads are resolved
+in range when positioned 3 cm apart in the range direction. The S/N is
+approximately 20 dB. (d) The string is vibrating, moving the beads in different
+directions and resulting in small Doppler shifts.
+peaks measured to be 3 cm and 3.1 cm apart and are visible
+with a S/N of approximately 20 dB. When lightly tapping the
+string, the beads are also separated in Doppler due to the fine
+Doppler resolution of 0.04 m/s per Doppler bin.
+Figures 8(a-b) show photographs of the materials used for
+testing the radar system’s ability to detect small particles. For
+each material, Figures 8(c-f) show the corresponding range-
+Doppler maps integrated over several CPI. This way, one
+can clearly see the acceleration of the 2-mm diameter metal
+bead and the 10-mm diameter metal sphere toward the radar.
+Each detection corresponds to a separate CPI, or “frame”,
+of the radar with a frame rate of 6.2 frames/s. For 500-µm
+diameter sand grains and 100-µm diameter glass spheres, the
+integrated particle stream over several pinches of particles is
+clearly visible. Fig. 8(e) shows clear detection of single sand
+grains. At a 4.3 m distance, the sand grains hit the plastic
+box and bounce to a stop. The deflection from the plastic
+box appears as positive Doppler velocity. In conclusion, all
+tested materials could be detected with significant S/N at a 5-m
+distance, proving the radar instrument’s suitability to monitor
+particle clouds’ dynamics.
+Fig. 8(c) includes the predicted trajectory for the 10-mm
+diameter metal sphere from the free-fall model (1), which
+indicates that the measurement agrees very well with the
+theory, thus supporting the velocity measurement of the radar.
+IV. CONCLUSIONS
+We
+have
+presented
+a
+340-GHz
+frequency-modulated
+continuous-wave pulse-Doppler radar. The performance of the
+radar is described and shown to follow what is expected
+from theoretical predictions. The instrument’s sensitivity and
+Fig. 8.
+Measurement of falling objects at a 5-m distance. (a-b) Show the
+photographs of the tested materials. The time-integrated range-Doppler image
+of (c) a 10-mm diameter falling metal bead, (d) a 2-mm diameter metal bead,
+(e) a few pinches of 500 µm sand grains, and (f) a few pinches of 100 µm
+glass spheres.(c) Shows the predicted trajectory from the free-fall model (1).
+resolution, both in the spatial domain and in Doppler velocity,
+are adequate to map the dynamics of particle clouds. This
+is demonstrated by performing radar measurements on free-
+falling particles with grain sizes down to 100-µm diameter.
+The mapping of particle clouds is relevant in many industrial
+applications, such as in the manufacturing of pharmaceuticals
+or energy conversion using fluidized bed reactors. Future work
+will demonstrate the radar technique in these applications.
+ACKNOWLEDGMENT
+The authors would like to thank Mats Myremark for ma-
+chining mechanical parts for the measurement setup; Vladimir
+Drakinskiy for his help with the fabrication of the front-
+end terahertz circuits; Divya Jayasankar for valuable feedback
+on the manuscript and help with LATEX. The devices were
+fabricated and measured in the Nanofabrication Laboratory
+and Kollberg Laboratory, respectively, at Chalmers University
+of Technology, Gothenburg, Sweden.
+REFERENCES
+[1] P. Bawuah and J. A. Zeitler, “Advances in terahertz time-domain spec-
+troscopy of pharmaceutical solids: A review,” TrAC Trends in Analytical
+Chemistry, vol. 139, p. 116272, 2021, doi: 10.1016/j.trac.2021.116272.
+
+-30
+5.4
+-30
+(a)
+(b)
+-40
+5.35
+-40
+5.5
+Range (m)
+beads
+(m)
+50
+5
+5.3
+-60
+-60
+4.5
+5.25
+-70
+-70
+-80
+5.2
+0.4-0.2
+0
+80
+-2
+0
+2
+0.2
+Velocity (m/s)
+Velocity(m/s)
+5.4
+-30
+0
+(c)
+d
+20
+5.35
+-40
+40
+(w)
+50
+Range
+50
+5.2
+5.35.4
+5.3
+-60
+5.25
+-70
+-80
+4
+4.5
+5
+5.5
+6
+5.2
+-0.2
+0.2
+80
+-0.4
+Range (m)
+Velocity (m/s)(a)
+(b)
+10 mm
+~0.5mm
+~0.1mm
+2mm
+5.8
+-30
+5.8
+-30
+(c)
+(d)
+5.4
+-40
+-40
+Radar signal (dB)
+5.4
+Radar signal (dB)
+Range (m)
+Range (m)
+-50
+50
+5
+5
+free f
+ta
+-60
+60
+4.6
+4.6
+-70
+-70
+4.2
+-80
+4.2
+-80
+-4
+-2
+0
+2
+-4
+-2
+0
+2
+Velocity(m/s)
+Velocity(m/s)
+5.8
+-30
+5.8
+-30
+(e)
+(f)
+-40
+-40
+5.4
+Radar signal (dB)
+5.4
+Radar signal (dB)
+Range (m)
+ngle 
+grains
+Range (m)
+-50
+50
+5
+lastic box
+5
+-60
+-60
+4.6
+4.6
+-70
+-70
+4.2
+-80
+4.2
+-80
+-4
+-2
+0
+2
+-4
+-2
+0
+2
+Velocity(m/s)
+Velocity (m/s)6
+TABLE I
+COMPARISON OF SUBMILLIMETER-WAVE RADARS
+Center frequency
+Bandwidth
+Output power
+Comment
+Technology
+Reference
+(GHz)
+(GHz)
+(mW)
+350
+19
+4
+FMCW
+Schottky diode
+[11]
+675
+30
+0.5
+FMCW pulse-Doppler
+Schottky diode
+[10]
+340
+29
+0.6
+FMCW
+Schottky diode
+[23]
+332
+16
+0.2
+FMCW, MIMO
+Schottky diode
+[12]
+340
+30
+1
+FMCW
+Schottky diode
+[13]
+383
+80
+8
+FMCW
+mHEMT
+[24]
+480
+55
+0.06
+FMCW
+SiGe
+[17]
+340
+30
+1
+FMCW pulse-Doppler
+Schottky diode
+This work
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+200–240-GHz FMCW radar transceiver module,” IEEE Transactions on
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+Tomas Bryllert was born in V¨axj¨o, Sweden, in
+1974. He received an M.Sc. degree in physics and a
+Ph.D. in semiconductor physics from Lund Univer-
+sity, Lund, Sweden, in 2000 and 2005, respectively.
+In 2006, he joined the Microwave Electronics
+Laboratory, Chalmers University of Technology,
+G¨oteborg, Sweden. From 2007 to 2009, he was
+with the Submillimeter Wave Advanced Technology
+(SWAT) group, Jet Propulsion Laboratory, California
+Institute of Technology, Pasadena, CA, USA. He is
+currently with the Terahertz and Millimetre Wave
+Laboratory at Chalmers University of Technology, G¨oteborg, Sweden. He is
+also the co-founder and Chief Executive Officer of Wasa Millimeter Wave AB,
+a company that develops and fabricates millimeter wave products. Dr. Bryllert
+also works part-time in the new concepts team at Saab AB. His research
+interests include submillimeter wave electronic circuits and their applications
+in imaging and radar systems.
+
+7
+Marlene Bonmann was born in Karlsruhe, Ger-
+many, in 1988. She received an M.Sc. degree in
+physics and astronomy and a Ph.D. in Microtechnol-
+ogy and Nanoscience from the Chalmers University
+of Technology, Gothenburg, Sweden, in 2014 and
+2020, respectively.
+She is currently with the Terahertz and Millimetre
+Wave Laboratory at the Chalmers University of
+Technology.
+Jan Stake (S’95–M’00–SM’06) was born in Ud-
+devalla, Sweden, in 1971. He received an M.Sc.
+degree in electrical engineering and a Ph.D. in
+microwave electronics from the Chalmers University
+of Technology, Gothenburg, Sweden, in 1994 and
+1999, respectively.
+In 1997, he was a Research Assistant at the
+University of Virginia, Charlottesville, VA, USA.
+From 1999 to 2001, he was a Research Fellow
+with the Millimetre Wave Group at the Rutherford
+Appleton Laboratory, Didcot, UK. He then joined
+Saab Combitech Systems AB, Link¨oping, Sweden, as a Senior RF/microwave
+Engineer, until 2003. From 2000 to 2006, he held different academic positions
+with the Chalmers University of Technology and from 2003 to 2006, he was
+also the Head of the Nanofabrication Laboratory, Department of Microtech-
+nology and Nanoscience (MC2). In 2007, he was a Visiting Professor with the
+Sub-millimetre Wave Advanced Technology (SWAT) Group at Caltech/JPL,
+Pasadena, CA, USA. In 2020, he was a Visiting Professor at TU Delft.
+He is currently a Professor and the Head of the Terahertz and Millimetre
+Wave Laboratory at the Chalmers University of Technology. He is also
+the co-founder of Wasa Millimeter Wave AB, Gothenburg, Sweden. His
+research interests include graphene electronics, high-frequency semiconductor
+devices, terahertz electronics, submillimeter wave measurement techniques,
+and terahertz systems.
+Prof. Stake served as the Editor-in-Chief for the IEEE Transactions on
+Terahertz Science and Technology between 2016 and 2018 and as Topical
+Editor between 2012 and 2015.
+
+HAGLOFS

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+arXiv:2301.00334v1  [quant-ph]  1 Jan 2023
+Complete Genuine Multipartite Entanglement Monotone
+Yu Guo∗
+Institute of Quantum Information Science, Shanxi Datong University, Datong, Shanxi 037009, China
+A complete characterization and quantification of entanglement, particularly the multipartite
+entanglement, remains an unfinished long-term goal in quantum information theory. As long as
+the multipartite system is concerned, the relation between the entanglement contained in different
+partitions or different subsystems need to take into account. The complete multipartite entanglement
+measure and the complete monogamy relation is a framework that just deals with such a issue. In this
+paper, we put forward conditions to justify whether the multipartite entanglement monotone (MEM)
+and genuine multipartite entanglement monotone (GMEM) are complete, completely monogamous,
+and tightly complete monogamous according to the feature of the reduced function. Especially,
+we proposed a class of complete MEMs and a class of complete GMEMs via the maximal reduced
+function for the first time.
+By comparison, it is shown that, for the tripartite case, this class
+of GMEMs is better than the one defined from the minimal bipartite entanglement in literature
+under the framework of complete MEM and complete monogamy relation. In addition, the relation
+between monogamy, complete monogamy, and the tightly complete monogamy are revealed in light
+of different kinds of MEMs and GMEMs.
+PACS numbers: 03.67.Mn, 03.65.Db, 03.65.Ud.
+I.
+INTRODUCTION
+Entanglement, as one of the most puzzling features in
+quantum mechanics, has been widely used as an essen-
+tial resource for quantum communication [1–3], quantum
+cryptography [4, 5], and quantum computing [6, 7], etc.
+The utility of an entangled state for these applications
+is often directly related to the degree or type of entan-
+glement contained in it.
+Therefore, efficiently quanti-
+fying and characterizing multipartite entanglement is of
+paramount importance.
+Especially, the genuine multi-
+partite entanglement, as one of the important types of
+entanglement, offers significant advantages in quantum
+tasks compared with bipartite entanglement [8].
+The phenomenon becomes much more complex for
+multipartite entanglement, particularly the genuinely
+multipartite entanglement, entanglement shared between
+all of the particles.
+Over the years, many multipar-
+tite entanglement measures have been proposed, such as
+the “residual tangle” which reports the genuine three-
+qubit entanglement [9], the genuinely multipartite con-
+currence [10], the k-ME concurrence [11], the m con-
+currence [12], the generalization of negativity [13], the
+SL-invariant multipartite measure of entanglement [14–
+19], and the α-entanglement entropy [20], concurrence
+triangle [21], concentratable entanglement [22], geomet-
+ric mean of bipartite concurrence [23], concurrence tri-
+angle induced genuine multipartite entanglement mea-
+sure [24], and a general way of constructing genuine mul-
+tipartite entanglement monotone is proposed in Ref. [25].
+In Ref. [26], we proposed a framework of complete mul-
+tipartite entanglement monotone from which the entan-
+glement between any partitions or subsystems with the
+coarsening relation could be compared with each other.
+∗ guoyu3@aliyun.com
+In the context of describing multipartite entanglement,
+another fundamental task is to understand how entan-
+glement is distributed over many parties since it reveals
+fundamental insights into the nature of quantum correla-
+tions [8] and has profound applications in both quantum
+communication [27, 28] and other area of physics [29–
+33].
+This characteristic trait of distribution is known
+as the monogamy law of entanglement [27, 34], which
+means that the more entangled two parties are, the
+less correlated they can be with other parties.
+Quan-
+titatively, the monogamy of entanglement is described
+by an inequality [9, 29, 34–36] or equality [26, 37, 38],
+involving a bipartite entanglement monotone or multi-
+partite entanglement monotone (MEM). Consequently,
+considerable research has been undertaken in this direc-
+tion [9, 26, 29, 34–39].
+Very recently, we discussed when the genuine multi-
+partite entanglement measure is complete [40] with the
+same spirit as in Ref. [26]. Under such a sense, the hierar-
+chy structure of the entanglement in the system is clear.
+Moreover, whether the multipartite entanglement mea-
+sure is proper or not can be justified together with the
+framework of complete monogamy relation for the mul-
+tipartite system established in Ref. [26]. The framework
+of complete monogamy relation is based on the complete
+multipartite entanglement measure [26, 40, 41].
+With
+this postulates, the distribution of entanglement appears
+more explicitly.
+Multipartite entanglement measure is always defined
+via the bipartite entanglement measure. Let SX be the
+set of all density matrices acting on the state space HX.
+Recall that, a function E : SAB → R+ is called a mea-
+sure of entanglement [42, 43] if (1) E(σAB) = 0 for any
+separable density matrix σAB ∈ SAB, and (2) E be-
+haves monotonically decreasing under local operations
+and classical communication (LOCC). Moreover, convex
+measures of entanglement that do not increase on average
+
+2
+under LOCC are called entanglement monotones [42, 44].
+By replacing SAB with SA1A2···An, it is just the mul-
+tipartite entanglement measure/monotone, and denoted
+by E(n).
+Any bipartite entanglement monotone corre-
+sponds to a concave function on the reduced state when
+it is evaluated for the pure states [44]. For any entangle-
+ment measure E, if
+h
+�
+ρA�
+= E
+�
+|ψ⟩⟨ψ|AB�
+(1)
+is concave, i.e.
+h[λρ1 + (1 − λ)ρ2] ≥ λh(ρ1) + (1 −
+λ)h(ρ2) for any states ρ1, ρ2, and any 0 ≤ λ ≤ 1,
+then the convex roof extension of E, i.e., EF
+�
+ρAB�
+≡
+min �n
+j=1 pjE
+�
+|ψj⟩⟨ψj|AB�
+, is an entanglement mono-
+tone, where the minimum is taken over all pure state
+decompositions of ρAB = �n
+j=1 pj|ψj⟩⟨ψj|AB. We call h
+the reduced function of E and HA the reduced subsystem
+throughout this paper.
+An n-partite pure state |ψ⟩ ∈ HA1A2···An is called
+biseparable if it can be written as |ψ⟩
+=
+|ψ⟩X ⊗
+|ψ⟩Y
+for some bipartition of A1A2 · · · An (for exam-
+ple, A1A3|A2A4 is a bipartition of A1A2A3A4). An n-
+partite mixed state ρ is biseparable if it can be writ-
+ten as a convex combination of biseparable pure states
+ρ = �
+i pi|ψi⟩⟨ψi|, wherein the contained {|ψi⟩} can be
+biseparable with respect to different bipartitions (i.e., a
+mixed biseparable state does not need to be separable
+with respect to any particular bipartition). If ρ is not
+biseparable, then it is called genuinely entangled. A mul-
+tipartite entanglement measure E(n) is called a genuine
+multipartite entanglement measure if (i) E(n)(σ) = 0 for
+any biseparable state σ, (ii) E(n)(ρ) > 0 for any genuine
+entangled state, and (iii) it is convex [10].
+A genuine
+multipartite entanglement measure is called a genuine
+multipartite entanglement monotone (GMEM) if it does
+not increase on average under LOCC.
+In Refs. [25, 26, 40], we present MEMs and GMEMs
+that are defined by the sum of the reduced function on
+pure states and then extended to mixed states via the
+convex-roof structure. The aim of this paper is to give
+a condition that can justify when the MEMs and the
+GMEMs defined in this way is complete and completely
+monogamous. Moreover, we give another way of defining
+MEMs and the GMEMs from the maximal reduced func-
+tion and then discuss when these quantities are complete
+and completely monogamous.
+The remainder of this paper is organized as follows.
+In Sec.
+II, we introduce some preliminaries.
+Sec.
+III
+discusses the properties of the reduced functions of the
+entanglement monotones so far in literature.
+Sec.
+IV
+is divided into two subsections.
+Subsec.
+A discusses
+the MEM defined by the sum of reduced functions, and
+in Subsec. B, we give the MEMs defined by the max-
+imal reduced function.
+Both of theses two MEMs are
+explored under the framework of the complete measure
+and the complete monogamy relation.
+In Sec.
+V, we
+consider three kinds of GMEMs which are defined by the
+sum of reduced functions, the maximal reduced function,
+and the minimal reduced function, respectively, under
+the framework the complete measure and the complete
+monogamy relation. We present a conclusion in Sec. VI.
+II.
+NOTATIONS AND PRELIMINARIES
+The framework of the complete entanglement mea-
+sure/monotone is closely related to the coarser relation
+of multipartite partition. We first introduce three kinds
+of coarser relation in Subsec. A, from which we then re-
+view the complete MEM, complete GMEM, monogamy
+relation and complete monogamy relation, respectively,
+in the latter three subsections.
+A.
+Coarser relation of multipartite partition
+Let X1|X2| · · · |Xk and Y1|Y2| · · · |Yl be two partitions
+of A1A2 · · · An or subsystem of A1A2 · · · An (for instance,
+partition AB|C|DE is a 3-partition of the 5-particle sys-
+tem ABCDE with X1 = AB, X2 = C and X3 = DE).
+We denote by [40]
+X1|X2| · · · |Xk ≻a Y1|Y2| · · · |Yl,
+(2)
+X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl,
+(3)
+X1|X2| · · · |Xk ≻c Y1|Y2| · · · |Yl
+(4)
+if Y1|Y2| · · · |Yl can be obtained from X1|X2| · · · |Xk by
+(a) discarding some subsystem(s) of X1|X2| · · · |Xk,
+(b) combining some subsystems of X1|X2| · · · |Xk,
+(c) discarding some subsystem(s) of some subsystem(s)
+Xk provided that Xk = Ak(1)Ak(2) · · · Ak(f(k)) with
+f(k) ⩾ 2,
+respectively. For example, A|B|C|D ≻a A|B|D ≻a B|D,
+A|B|C|D ≻b AC|B|D ≻b AC|BD, A|BC ≻c A|B.
+Furthermore, if X1|X2| · · · |Xk ≻ Y1|Y2| · · · |Yl, we de-
+note by Ξ(X1|X2| · · · |Xk −Y1|Y2| · · · |Yl) the set of all the
+partitions that are coarser than X1|X2| · · · |Xk and either
+exclude any subsystem of Y1|Y2| · · · |Yl or include some
+but not all subsystems of Y1|Y2| · · · |Yl [40]. For exam-
+ple, Ξ(A|B|CD|E − A|B) = {CD|E, A|CD|E, B|CD|E,
+A|CD, B|CD, B|C|E, B|D|E, A|D|E, A|C|E, A|E,
+B|E, A|C, A|D, B|C, B|D, C|E, D|E}.
+B.
+Complete MEM
+A multipartite entanglement measure E(n) is called a
+unified multipartite entanglement measure if it satisfies
+the unification condition [26]:
+(i) (additivity):
+E(n)(A1A2 · · · Ak ⊗ Ak+1 · · · An)
+= E(k)(A1A2 · · · Ak) + E(n−k)(Ak+1 · · · An),
+(5)
+
+3
+holds for all ρA1A2···An
+∈ SA1A2···An, hereafter
+E(n)(X) refers to E(n)(ρX);
+(ii) (permutation invariance):
+E(n)(A1A2 · · · An)
+=
+E(n)(Aπ(1)Aπ(2) · · · Aπ(n)),
+for all ρA1A2···An
+∈
+SA1A2···An and any permutation π;
+(iii) (coarsening monotone):
+E(k)(X1|X2| · · · |Xk) ⩾ E(l)(Y1|Y2| · · · |Yl)
+(6)
+holds
+for
+all
+ρA1A2···An
+∈
+SA1A2···An
+when-
+ever
+X1|X2| · · · |Xk
+≻a
+Y1|Y2| · · · |Yl,
+where
+X1|X2| · · · |Xk and Y1|Y2| · · · |Yl are two partitions
+of A1A2 · · · An or subsystem of A1A2 · · · An, the
+vertical bar indicates the split across which the en-
+tanglement is measured..
+E(n) is called a complete multipartite entanglement mea-
+sure if it satisfies both the conditions above and the hi-
+erarchy condition [26]:
+(iv) (tight coarsening monotone):
+Eq. (6) holds for
+all ρ ∈ SA1A2···An whenever X1|X2| · · · |Xk ≻b
+Y1|Y2| · · · |Yl.
+C.
+Complete GMEM
+Let E(n)
+g
+be a genuine multipartite entanglement mea-
+sure. It is defined to be a unified genuine multipartite
+entanglement measure if it satisfies the unification con-
+dition [40], i.e.,
+(i) (permutation invariance):
+E(n)
+g
+(A1A2 · · · An)
+=
+E(n)
+g
+(Aπ(1)Aπ(2) · · · Aπ(n)),
+for all ρA1A2···An
+∈
+SA1A2···An
+g
+and any permutation π;
+(ii) (coarsening monotone):
+E(k)
+g (X1|X2| · · · |Xk) > E(l)
+g (Y1|Y2| · · · |Yl)
+(7)
+holds for all ρA1A2···An
+∈ SA1A2···An
+g
+whenever
+X1|X2| · · · |Xk ≻a Y1|Y2| · · · |Yl.
+A unified GMEM E(n)
+g
+is call a complete genuine multi-
+partite entanglement measure if E(n)
+g
+admits the hierar-
+chy condition [40], i.e.,
+(iii) (tight coarsening monotone):
+E(k)
+g (X1|X2| · · · |Xk) ≥ E(l)
+g (Y1|Y2| · · · |Yl)
+(8)
+holds
+for
+all
+ρ
+∈
+SA1A2···An
+g
+whenever
+X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl.
+D.
+Monogamy Relation
+For an bipartite entanglement measure E, E is said to
+be monogamous if [9, 39]
+E(A|BC) ⩾ E(AB) + E(AC).
+(9)
+However, Equation (9) is not valid for many entangle-
+ment measures [9, 35, 37] but some power function of
+Q admits the monogamy relation (i.e., Eα(A|BC) ⩾
+Eα(AB) + Eα(AC) for some α > 0). In Ref. [37], we
+improved the definition of monogamy as: A bipartite
+measure of entanglement E is monogamous if for any
+ρ ∈ SABC that satisfies the disentangling condition, i.e.,
+E(ρA|BC) = E(ρAB),
+(10)
+we have that E(ρAC) = 0, where ρAB = TrCρABC.
+With respect to this definition, a continuous measure E
+is monogamous according to this definition if and only if
+there exists 0 < α < ∞ such that
+Eα(ρA|BC) ⩾ Eα(ρAB) + Eα(ρAC)
+(11)
+for all ρ acting on the state space HABC with fixed
+dim HABC = d < ∞ (see Theorem 1 in Ref. [37]).
+In Ref. [26], in order to characterize the distribution
+of entanglement in a “complete” sense, the term “com-
+plete monogamy” of the unified multipartite entangle-
+ment measure is proposed.
+For a unified multipartite
+entanglement measure E(n), it is said to be completely
+monogamous if for any ρ ∈ SA1A2···An that satisfies [26]
+E(k)(X1|X2| · · · |Xk) = E(l)(Y1|Y2| · · · |Yl)
+(12)
+with X1|X2| · · · |Xk ≻a Y1|Y2| · · · |Yl we have that
+E(∗)
+g (Γ) = 0
+(13)
+holds for all Γ ∈ Ξ(X1|X2| · · · |Xk − Y1|Y2| · · · |Yl), here-
+after the superscript (∗) is associated with the partition
+Γ, e.g., if Γ is a n-partite partition, then (∗) = (n). For
+example, E(3) is completely monogamous if for any ρABC
+that admits E(3)(ABC) = E(2)(AB) we get E(2)(AC) =
+E(2)(BC) = 0. Let E(n) be a complete multipartite en-
+tanglement measure. E(n) is defined to be tightly com-
+plete monogamous if for any ρ ∈ SA1A2···An that satis-
+fies [26]
+E(k)(X1|X2| · · · |Xk) = E(l)(Y1|Y2| · · · |Yl)
+(14)
+with X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl we have that
+E(∗)
+g (Γ) = 0
+(15)
+holds for all Γ ∈ Ξ(X1|X2| · · · |Xk − Y1|Y2| · · · |Yl). For
+instance, E(3) is tightly complete monogamous if for any
+ρABC that admits E(3)(ABC) = E(2)(A|BC) we have
+E(2)(BC) = 0.
+Let E(n)
+g
+be a genuine multipartite entanglement mea-
+sure. We denote by SA1A2···Am
+g
+the set of all genuine en-
+tangled states in SA1A2···Am. E(n)
+g
+is completely monog-
+amous if it obeys Eq. (7) [40]. A complete genuine mul-
+tipartite entanglement measure E(n)
+g
+is tightly complete
+
+4
+monogamous if it satisfies the genuine disentangling con-
+dition, i.e., either for any ρ ∈ SA1A2···Am
+g
+that satis-
+fies [40]
+E(k)
+g (X1|X2| · �� · |Xk) = E(l)
+g (Y1|Y2| · · · |Yl)
+(16)
+with X1|X2| · · · |Xk ≻b Y1|Y2| · · · |Yl we have that
+E(∗)
+g (Γ) = 0
+(17)
+holds for all Γ ∈ Ξ(X1|X2| · · · |Xk − Y1|Y2| · · · |Yl), or
+E(k)
+g (X1|X2| · · · |Xk) > E(l)
+g (Y1|Y2| · · · |Yl)
+(18)
+holds for any ρ ∈ SA1A2···Am
+g
+.
+In Ref. [26], we showed that the tightly complete
+monogamy is stronger than the complete monogamy for
+the complete MEMs that defined by the convex-roof ex-
+tension. One can easily find that it is also true for any
+complete GMEM defined by the convex-roof extension.
+III.
+STRICT CONCAVITY AND
+SUBADDITIVITY OF THE REDUCED
+FUNCTION
+Any entanglement monotone, when evaluated on pure
+states, is uniquely determined by its reduced function
+and vice versa. Therefore, the feature of the entangle-
+ment monotone defined via the convex-roof extension
+rests with the quality of its reduced function. In Ref. [38],
+we proved that the bipartite entanglement monotone is
+monogamous whenever its reduced function is strictly
+concave. In ths Section, we review all the reduced func-
+tions of the entanglement monotones in literature so far
+and then discuss the subadditivity of theses functions. As
+what we will show in the next two Sections, the subaddi-
+tivity is affinitive with the completeness of the measures
+for some kind of MEM/GMEM.
+A.
+Strict concavity
+The reduced functions of the entanglement of for-
+mation Ef [45, 46], tangle τ [47], concurrence C [48–
+50], negativity N [51], the Tsallis q-entropy of entangle-
+ment Eq [52], and the R´enyi α-entropy of entanglement
+Eα [44, 53] are
+h(ρ) = S(ρ),
+hτ(ρ) = h2
+C(ρ) = 2(1 − Trρ2),
+hN(ρ) = 1
+2[(Tr√ρ)2 − 1],
+hq(ρ) = 1 − Trρq
+q − 1
+,
+q > 0,
+hα(ρ) = (1 − α)−1 ln(Trρα),
+0 < α < 1,
+respectively, where S is the von Neumann entropy. It has
+been shown that h, hτ, hC, hN, hq, and hα are not only
+concave but also strictly concave [38, 44, 54] (where the
+strict concavity of hN is proved very recently in Ref. [55]).
+The reduced functions of the entanglement monotones
+induced by the fidelity-based distances EF, EF ′, and
+EAF are [56]
+hF(ρ) = 1 − Trρ3,
+hF ′(ρ) = 1 −
+�
+Trρ2�2 ,
+hAF(ρ) = 1 −
+�
+Trρ3,
+respectively. They are strictly concave [40].
+In Ref. [55], four kinds of partial norm of entangle-
+ment are investigated: the partial-norm of entanglement
+E2, the minimal partial-norm of entanglement Emin, the
+reinforced minimal partial-norm of entanglement Emin′,
+and the partial negativity ˆN. The reduced functions of
+E2, Emin, E′
+min, and ˆN are
+h2(ρ) = 1 − ∥ρ∥,
+hmin(ρ) = ∥ρ∥min,
+hmin′(ρ) = r(ρ)∥ρ∥min,
+ˆh(ρ) =
+�
+δ1δ2,
+where r(ρ) denotes the rank of ρ, ∥ · ∥ is the operator
+norm, i.e., ∥X∥ = sup|ψ⟩ ∥A|ψ⟩∥,
+∥ρ∥min =
+�
+λ2
+min,
+λmin < 1,
+0,
+λmin = 1,
+and δ1, δ2 are the two largest eigenvalues of ρ. All of them
+are concave but not strictly concave (ˆh is only strictly
+concave on qubit states), and these entanglement mono-
+tones are not monogamous [55].
+B.
+Subadditivity
+We summarize the subadditivity of the reduced func-
+tions in literature as following:
+(i) S is additive and subadditive [54], i.e.,
+S(ρ ⊗ σ) = S(ρ) + S(σ)
+(19)
+and
+S(ρAB) ≤ S(ρA) + S(ρB),
+(20)
+respectively.
+(ii) Sq is subadditive iff q > 1, but not additive, and
+for 0 < q < 1, Sq is neither subadditive nor super-
+additive [57] (superadditivity refers to Sq(ρAB) ⩾
+Sq(ρA) + Sq(ρB)). In addition,
+Sq(ρA ⊗ ρB) = Sq(ρA) + Sq(ρB)
+(21)
+iff ρA or ρB is pure [57].
+
+5
+TABLE I. Comparing of the properties of the reduced func-
+tions.
+C, SC, SA, and A signify the function is concave,
+strictly concave, subadditive, and additive, respectively.
+E
+h
+C
+SC
+SA
+A
+Ef
+S
+✓
+✓
+✓
+✓
+C
+�
+2(1 − Trρ2)
+✓
+✓
+✓
+×
+τ
+2(1 − Trρ2)
+✓
+✓
+✓
+×
+Eq
+1−Trρq
+q−1
+✓(q > 0) ✓(q > 1) ✓(q > 1) ×
+Eα
+ln(Trρα)
+1−α
+, α ∈ (0, 1)
+✓
+✓
+×
+✓
+NF
+(Tr√ρ)2−1
+2
+✓
+✓
+×
+×
+EF
+1 − Trρ3
+✓
+✓
+✓
+×
+EF′
+1 − (Trρ2)2
+✓
+✓
+✓a
+×
+EAF
+1 −
+�
+Trρ3
+✓
+✓
+✓a
+×
+E2
+1 − ∥ρ∥
+✓
+×
+✓
+×
+Emin
+∥ρ∥min
+✓
+×
+×
+×
+Emin′
+r(ρ)∥ρ∥min
+✓
+×
+×
+×
+ˆ
+N
+√
+δ1δ2
+✓b
+×
+✓a
+×
+a We conjecture that they are subadditive.
+b We conjecture that it is concave.
+(iii) hα is additive but not subadditive [58, 59].
+(iv) hτ is subadditive [60], i.e.,
+1 + Trρ2
+AB ≥ Trρ2
+A + Trρ2
+B.
+(22)
+In particular, the equality holds iff ρA or ρB is
+pure [26].
+(v) hN is neither subadditive nor supperadditive [26].
+Item (iv) implies hC is subadditive and the equality holds
+iff ρA or ρB is pure. hF is subadditive since it coincides
+with Sq/2 (q = 3). We conjecture that hF ′ and hAF are
+subadditive.
+Proposition 1. h2 is subadditive, i.e.,
+1 + ∥ρAB∥ ⩾ ∥ρA∥ + ∥ρB∥
+(23)
+holds for any ρAB ∈ SAB.
+In particular, the equality
+holds iff ρA or ρB is a pure state.
+Proof. Note that partial trace is a quantum channel and
+any quantum channel can be regarded as a operator on
+the space of the trace-class operators. The norm of quan-
+tum channel in such a sense is 1. Therefore ∥ρAB∥ ⩾
+∥ρA,B∥. Moreover, if 1 + ∥ρAB∥ = ∥ρA∥ + ∥ρB∥, then
+∥ρA∥ = 1 or ∥ρB∥ = 1, which completes the proof.
+Let
+ρAB = 1
+2|ψ⟩⟨ψ| + 1
+2|φ⟩⟨φ|
+with |ψ⟩ =
+�
+4
+5|00⟩+
+�
+1
+5|11⟩ and |φ⟩ =
+�
+4
+5|22⟩+
+�
+1
+5|33⟩.
+It is clear that ∥ρAB∥min = 1
+2 > ∥ρA∥min + ∥ρB∥min =
+1/10 + 1/10 = 1/5.
+That is, ∥ · ∥min is not subaddi-
+tive. Clearly, hmin′ is also not subadditive. According
+to Proposition 1, hmin and hmin′ are subadditive on the
+states that satisfies r(ρAB) = r(ρA) = r(ρB) = 2. One
+can easily verifies that h2, hmin, hmin′, and ˆh are not
+additive.
+We conjecture that ˆh is subadditive, i.e.,
+ˆh(ρAB) ⩽ ˆh(ρA) + ˆh(ρB)
+(24)
+holds for any ρAB ∈ SAB. In what follows, we always
+assume that hF ′, hAF, and ˆh are subadditive, and that
+ˆh is concave.
+The reduced functions of parametrized entanglement
+monotones in Ref. [61] and Ref. [62] are
+hq′(ρ) = 1 − Trρq,
+q > 1,
+and
+hα′(ρ) = Trρα − 1,
+0 < α < 1,
+respectively. Obviously, the properties of these two func-
+tions above are the same as that of hq, although they are
+different from Eq [61, 62]. We summarize the properties
+of theses reduced functions in Table I for more conve-
+nience.
+IV.
+COMPLETE MEM
+A.
+Complete MEM from sum of the reduced
+functions
+In Ref. [26], we put forward several complete MEMs
+defined by the sum of the reduced functions on all the
+single subsystems. In fact, this scenario is valid for all en-
+tanglement monotones. Let |ψ⟩A1A2···An be a pure state
+in HA1A2···An and h be a non-negative concave function
+on SX. We define
+E(n)(|ψ⟩A1A2···An) = 1
+2
+�
+i
+h(ρAi)
+(25)
+and then extend it to mixed states by the convex-roof
+structure.
+We denote E(n) by E(n)
+f
+, C(n), τ (n), E(n)
+q
+,
+E(n)
+α , N (n)
+F , E(n)
+F , E(n)
+F ′ , E(n)
+AF, E(n)
+2
+, E(n)
+min, E(n)
+min′, and ˆN (n)
+whenever h = S, hC, hτ, hq, hα, hN, hF, hF ′, hAF, h2,
+hmin, hmin′, and ˆh, respectively. Here, E(n)
+f
+, C(n), τ (n),
+E(n)
+q
+, E(n)
+α , and N (n)
+F
+have been discussed in Ref. [26]
+for the first time. The coefficient “1/2” is fixed by the
+unification condition when E(n) is regarded as a unified
+MEM. One need note here that E(n)
+F , E(n)
+F ′ , and E(n)
+AF are
+different from E(n)
+F,F , E(n)
+F ′,F , and E(n)
+AF,F respectively in
+Ref. [56].
+Theorem 1. Let E(n) be a non-negative function defined
+as in Eq. (25). Then the following statements hold true.
+
+6
+(i) E(n) is a unified MEM and is completely monoga-
+mous;
+(ii) E(n) is a complete MEM iff h is subadditive;
+(iii) E(n) is tightly complete monogamous iff h is sub-
+additive with
+h(ρAB) = h(ρA) + h(ρB) ⇒ ρAB is separable. (26)
+Proof. We only need to discuss the case of n = 3 with no
+loss of generality.
+(i) For any |ψ⟩ABC ∈ HABC, we let E(2)(ρAB) =
+�
+i piE(2)(|ψi⟩) = 1
+2
+�
+i pi[h(ρA
+i ) + h(ρB
+i )]. Then
+E(3)(|ψ⟩ABC) = 1
+2
+�
+h(ρA) + h(ρB) + h(ρC)
+�
+≥ 1
+2
+�
+h(ρA) + h(ρB)
+�
+≥ 1
+2
+�
+i
+pi[h(ρA
+i ) + h(ρB
+i )]
+= E(2)(ρAB).
+That is, E(3) satisfies Eq. (6) for pure states and it is
+completely monogamous on pure states. For any mixed
+state ρABC, we let E(3)(ρABC) = �
+j qjE(3)(|ψj⟩) and
+E(2)(ρAB
+j
+) = �
+i pi(j)E(2)(|ψi(j)⟩) = 1
+2
+�
+i pi(j)[h(ρA
+i(j)) +
+h(ρB
+i(j))]. Then
+E(3)(ρABC) = 1
+2
+�
+j
+qj
+�
+h(ρA
+j ) + h(ρB
+j ) + h(ρC
+j )
+�
+≥ 1
+2
+�
+j
+qj
+�
+hj(ρA) + hj(ρB)
+�
+≥ 1
+2
+�
+i,j
+qjpi(j)[h(ρA
+i(j)) + h(ρB
+i(j))] ≥ E(2)(ρAB),
+i.e., it is a unified MEM. If E(3)(ρABC) = E(2)(ρAB),
+it yields h(ρC
+j ) = 0 for any j, and thus |ψj⟩ABC =
+|ψj⟩AB|ψj⟩C. Therefore it is completely monogamous.
+(ii) If E(3) is a complete MEM, then E(3)(|ψ⟩ABC) ≥
+E(2)(|ψ⟩A|BC) for any |ψ⟩ABC, which implies h(ρBC) ≤
+h(ρB) + h(ρC). That is, h is subadditive since |ψ⟩ABC
+is arbitrarily given.
+Conversely, if h is subadditive,
+then E(3)(|ψ⟩ABC) ≥ E(2)(|ψ⟩A|BC) for any pure state
+|ψ⟩ABC. For any mixed state ρABC, we let E(3)(ρABC) =
+�
+j qjE(3)(|ψj⟩). Then
+E(3)(ρABC) = 1
+2
+�
+j
+qj
+�
+h(ρA
+j ) + h(ρB
+j ) + h(ρC
+j )
+�
+≥ 1
+2
+�
+j
+qj
+�
+hj(ρA) + hj(ρBC)
+�
+≥ E(2)(ρA|BC),
+i.e., it is a complete MEM.
+(iii) It can be easily checked using the argument anal-
+ogous to that of (ii) together with the fact that, if E(n)
+is tightly complete monogamous, it is automatically a
+complete MEM.
+TABLE II. Comparing of E(n) with different different reduced
+functions, and E (n).
+CM and TCM signify the measure is
+completely monogamous and tightly completel monogamous,
+respectively.
+MEM
+Unified
+Complete
+CM
+TCM
+E(n)
+f
+✓
+✓
+✓
+✓
+C(n)
+✓
+✓
+✓
+✓
+�� (n)
+✓
+✓
+✓
+✓
+E(n)
+q
+✓
+✓
+✓
+✓a
+E(n)
+α
+✓
+×
+✓
+×
+N (n)
+F
+✓
+×
+✓
+×
+E(n)
+F
+✓
+✓
+✓
+✓a
+E(n)
+F′
+✓
+✓b
+✓
+✓a
+E(n)
+AF
+✓
+✓b
+✓
+✓a
+E(n)
+2
+✓
+✓
+✓
+✓
+E(n)
+min
+✓
+×
+✓
+×
+E(n)
+min′
+✓
+×
+✓
+×
+ˆ
+N (n)
+✓
+✓b
+✓
+✓a
+E (n) (n ≥ 4)
+✓
+✓
+✓
+✓
+a It is tightly complete monogamous under the assumption that
+h is subadditive and Eq. (26) holds.
+b It is complete under the assumption that h is subadditive.
+By Theorem 1, we can conclude: (i) E(n)
+f
+, C(n), τ (n),
+E(n)
+q
+, E(n)
+α , N (n)
+F , E(n)
+F , E(n)
+F ′ , E(n)
+AF, E(n)
+2
+, E(n)
+min, E(n)
+min′,
+and ˆN (n) are unified MEMs and are completely monog-
+amous; (ii) E(n)
+f
+, C(n), τ (n), E(n)
+q
+, E(n)
+F , E(n)
+F ′ , E(n)
+AF,
+E(n)
+2
+, and ˆN (n) are complete MEMs; (iii) E(n)
+α , N (n)
+F ,
+E(n)
+min, and E(n)
+min′ are not complete MEMs since the asso-
+ciated reduced functions are not subadditive which vio-
+late the hierarchy condition for some states. (iv) E(n)
+f
+,
+C(n), τ (n), and E(n)
+2
+are tightly complete monogamous.
+However E(n)
+2
+, E(n)
+min, E(n)
+min′, and ˆN (n) are not monoga-
+mous.Together with Theorem in Ref. [38], we obtain that,
+for these MEMs, both monogamy and tightly complete
+monogamy are stronger than the complete monogamy
+under the frame work of the complete MEM, and that
+monogamy is stronger than both complete monogamy
+and tightly complete monogamy (e.g., E(n)
+2
+).
+In particular, if h is subadditive with h(ρAB)
+=
+h(ρA) + h(ρB) implies ρAB = ρA ⊗ ρB, then E(n) is
+tightly complete monogamous. S, hτ, hC, and h2 be-
+long to such situations. We also conjecture that hq, hF,
+hF ′, hAF, and ˆh belong to such situations as well. That
+is, we conjecture that E(n)
+q
+, E(n)
+F , E(n)
+F ′ , E(n)
+AF, and ˆN (n)
+are tightly complete monogamous.
+In Ref. [25], we put forward several multipartite en-
+tanglement measures which are defined by the sum of all
+bipartite entanglement. Let |ψ⟩A1A2···An be a pure state
+in HA1A2···An and h be a non-negative concave function
+
+7
+on SX. We define [25]
+E(n)(|ψ⟩A1A2···An)
+=
+
+
+
+
+
+1
+2
+�
+i1≤···≤is,s<n/2
+h(ρAi1Ai2 ···Ais ),
+if n is odd,
+1
+2
+�
+i1≤···≤is<n,s≤n/2
+h(ρAi1Ai2 ···Ais ),
+if n is even,(27)
+for pure states and for mixed states by the convex-roof
+structure.
+Note that E(n) is just E12···n(2) in Ref. [25]
+provided that the corresponding bipartite entanglement
+measure is an entanglement monotone. Clearly,
+E(n) ≤ E(n),
+(28)
+and in general, E(n) < E(n) whenever n ≥ 4. Indeed, we
+can easily show that E(n)(|ψ⟩) < E(n)(|ψ⟩) iff |ψ⟩ is not
+fully separable, i.e., |ψ⟩ ̸= |ψ⟩A1|ψ⟩A2 ⊗ · · · ⊗ |ψ⟩An. E(3)
+coincides with E(3) but E(n) is different from E(n) when-
+ever n ≥ 4. The following Proposition is straightforward
+by the definition of E(n).
+Proposition 2. Let E(n) be a non-negative function de-
+fined as in Eq. (27), n ≥ 4. Then E(n) is a complete
+MEM and it is completely monogamous and tightly com-
+plete monogamous.
+Then, when n ≥ 4, all these MEMs E(n) with the
+reduced functions we mentioned above are complete
+MEMs, and are not only completely monogamous but
+also tightly complete monogamous. We compare all the-
+ses MEMs in Table II for convenience.
+B.
+Complete MEM from the maximal reduced
+function
+Let |ψ⟩A1A2···An be a pure state in HA1A2···An and h
+be a non-negative concave function. We define
+E′(n)(|ψ⟩A1A2···An) = max
+i
+h(ρAi)
+(29)
+and then extend it to mixed states by the convex-roof
+structure. By definition, E′(n) ≤ E(n) if h is subadditive.
+Theorem 2. Let E′(n) be a MEM defined as in Eq. (29).
+Then (i) E′(3) is a complete MEM but not tightly com-
+plete monogamous, and if h is strictly concave, E′(3) is
+completely monogamous, and (ii) E′(n) is not complete
+whenever n ≥ 4.
+Proof. (i) It is clear that the unification condition and
+the hierarchy condition are valid for E′(3), thus E′(3) is
+a complete MEM. Let E′(3)(|ψ⟩ABC) = E′(2)(|ψ⟩A|BC),
+then ρBC is not necessarily separable. Thus, E′(3) is not
+tightly complete monogamous. If h is strictly concave
+and E′(3)(|ψ⟩ABC) = E′(2)(ρAB), then E′(2)(|ψ⟩A|BC) =
+E′(2)(|ψ⟩B|AC) = E′(2)(ρAB).
+Therefore, |ψ⟩ABC =
+|ψ⟩AB|ψ⟩C by Theorem in Ref. [38].
+That is, E′(3) is
+completely monogamous.
+(ii) Let
+|W4⟩ = 1
+2 (|1000⟩ + |0100⟩ + |0010⟩ + |0001⟩), (30)
+we have E′(4)(|W4⟩) < E′(2)(|W4⟩AB|CD) since
+ρA = ρB = ρC = ρD =
+�
+3/4
+0
+0
+1/4
+�
+and the bipartite reduced state is maximal mixed two
+qubit state. That is, it violates the hierarchy condition.
+This complete the proof.
+We denote the corresponding E′(n) in the previous sub-
+section by E′(n)
+f , C′(n), τ ′(n), E′(n)
+q
+, E′(n)
+α , N ′(n)
+F , E′(n)
+F ,
+E′(n)
+F ′ , E′(n)
+AF, E′(n)
+2 , E′(n)
+min, E′(n)
+min′, and
+ˆ
+N ′(n), respec-
+tively. Then, by Theorem 2, all of them are complete
+MEMs but not tightly complete monogamous by The-
+orem 2 for the case of n = 3, and E′(3)
+f , C′(3), τ ′(3),
+E′(3)
+q , E′(3)
+α , N ′(3)
+F , E′(3)
+F , E′(3)
+F ′ , and E′(3)
+AF are completely
+monogamous, E′(n)
+f , C′(n), τ ′(n), E′(n)
+q
+, E′(n)
+α , N ′(n)
+F ,
+E′(n)
+F , E′(n)
+F ′ , E′(n)
+AF, E′(n)
+2 , E′(n)
+min, E′(n)
+min′, and ˆ
+N ′(n) are
+not complete MEMs whenever n ≥ 4.
+If h is not strictly concave, then E′(3) is not completely
+monogamous. For example, we take
+|ψ⟩ABC = |ψ⟩AB1|ψ⟩B2C,
+(31)
+where B1B2 means HB has a subspace isomorphic to
+HB1 ⊗ HB2 and up to local unitary on system B1B2. We
+assume
+E′(3)
+min(|ψ⟩ABC) = E′(2)
+min(|ψ⟩A|BC),
+ˆ
+N ′(3)(|ψ⟩ABC) = ˆ
+N ′(2)(|ψ⟩A|BC),
+then
+E′(3)
+min(|ψ⟩ABC) = E′(2)
+min(|ψ⟩AB1) = E′(2)
+min(ρAB),
+ˆ
+N ′(3)(|ψ⟩ABC) = ˆ
+N ′(2)(|ψ⟩AB1) = ˆ
+N ′(2)(ρAB),
+and ρBC is entangled. In addition, we take
+|φ⟩ABC =
+1
+√
+3|000⟩ + 1
+√
+3|101⟩ + 1
+√
+3|110⟩.
+(32)
+It is straightforward that
+E′(3)
+2 (|φ⟩ABC)
+= E′(2)
+2 (|φ⟩A|BC) = E′(2)
+2 (|φ⟩AB|C) = E′(2)
+2 (|φ⟩B|AC)
+= E′(2)
+2 (ρAB) = E′(2)
+2 (ρAC) = E′(2)
+2 (ρBC)
+= 1/3.
+Namely, E′(3)
+2 , E′(3)
+min, E′(3)
+min′, and
+ˆ
+N ′(3) are not com-
+pletely monogamous. Namely, for these four complete
+MEMs, monogamy coincides with complete monogamy,
+tightly complete monogamy seems stronger than both
+monogamy and complete monogamy.
+
+8
+TABLE III. Comparing of E′(3) with different reduced func-
+tions, E′(4) (n ≥ 4), and E ′(4) (n ≥ 4).
+MEM
+Unified
+Complete
+CM
+TCM
+E′(3)
+f
+✓
+✓
+✓
+×
+C′(3)
+✓
+✓
+✓
+×
+τ ′(3)
+✓
+✓
+✓
+×
+E′(3)
+q
+✓
+✓
+✓
+×
+E′(3)
+α
+✓
+✓
+✓
+×
+N ′(3)
+F
+✓
+✓
+✓
+×
+E′(3)
+F
+✓
+✓
+✓
+×
+E′(3)
+F′
+✓
+✓
+✓
+×
+E′(3)
+AF
+✓
+✓
+✓
+×
+E′(3)
+2
+✓
+✓
+×
+×
+E′(3)
+min
+✓
+✓
+×
+×
+E′(3)
+min′
+✓
+✓
+×
+×
+ˆ
+N ′(3)
+✓
+✓
+×
+×
+E′(4) (n ≥ 4)
+?
+×
+?
+×
+E ′(n) (n ≥ 4)
+?
+×
+?
+×
+It is worthy mentioning here that E′(n) may not
+a unified MEM if n ≥ 4 since it may occur that
+E′(k)(X1|X2| · · · |Xk)
+<
+E′(l)(Y1|Y2| · · · |Yl) for some
+state ρ
+∈
+SA1A2···An
+whenever X1|X2| · · · |Xk
+≻a
+Y1|Y2| · · · |Yl.
+Let |ψ⟩A1A2···An be a pure state in HA1A2···An and h
+be a non-negative concave function on SX. We define
+E′(n)(|ψ⟩A1A2···An) =
+max
+i1≤···≤is,s≤n/2 h(ρAi1Ai2 ···Ais) (33)
+for pure states and for mixed states by the convex-roof
+structure. By definition,
+E′(n) ≤ E′(n),
+(34)
+E′(3) coincides with E′(3), E′(n) satisfies the hierarchy
+condition, but it may violate the unification condition.
+We give a comparison for theses MEMs in Table III for
+more clarity.
+The case E′(n) < E′(n) occurs whenever n ≥ 4. It is
+clear that E′(4)(|W4⟩) < E′(4)(|W4⟩) for any E′(4) and
+E′(4) with the reduced functions we considered in Sec.
+III. In addition, for the state
+|ϕ⟩ =
+√
+5
+4 |0000⟩ +
+√
+5
+4 |1111⟩ + 1
+4|0100⟩ +
+√
+5
+4 |1010⟩, (35)
+we have E′(4) < E′(4) for any E′(4) and E′(4) men-
+tioned above except for h = hmin and h = ˆh since
+ρA = ρB = ρC = ρD and the eigenvalues of ρA is
+{5/8, 3/8} and the eigenvalues of the bipartite reduced
+state is {3/8, 5/16, 5/16}.
+V.
+COMPLETE GMEM
+A.
+Complete GMEM from sum of the reduced
+functions
+In Ref. [40], we discussed the completeness of GMEMs
+defined by sum of all reduced functions of the single
+subsystems with the reduced functions corresponding to
+Ef, C, τ, Eq, and Eα.
+We consider here the general
+case for any given bipartite entanglement monotone. Let
+|ψ⟩A1A2···An be a pure state in HA1A2···An and h be a
+non-negative concave function on SX. We define
+E(n)
+g
+(|ψ⟩A1A2···An)
+=
+
+
+
+1
+2
+�
+i h(ρAi),
+h(ρAi) > 0 for any i,
+0,
+h(ρAi) = 0 for some i,
+(36)
+and then extend it to mixed states by the convex-
+roof structure. By Proposition 1 and Proposition 4 in
+Ref. [40], together with Theorem 1, we have the follow-
+ing statement.
+Proposition 3. Let E(n)
+g
+be a non-negative function de-
+fined as in Eq. (36). Then the following statements hold
+true.
+(i) E(n)
+g
+is a unified GMEM and is completely monog-
+amous;
+(ii) E(n)
+g
+is a complete GMEM iff h is subadditive;
+(iii) E(n)
+g
+is tightly complete monogamous iff h is sub-
+additive with Eq. (26) holds.
+We denote E(n)
+g
+in the previous Section by E(n)
+g,f , C(n)
+g
+,
+τ (n)
+g
+, E(n)
+g,q , E(n)
+g,α, N (n)
+g,F , E(n)
+g,F, E(n)
+g,F ′, E(n)
+g,AF, E(n)
+g,2 , E(n)
+g,min,
+E(n)
+g,min′, and ˆN (n)
+g
+, respectively.
+By Proposition 3, we
+can conclude: (i) All theses GMEMs are unified GMEMs
+and are completely monogamous; (ii) E(n)
+g,f , C(n)
+g
+, τ (n)
+g
+,
+E(n)
+g,q , E(n)
+g,F, E(n)
+g,F ′, E(n)
+g,AF, E(n)
+g,2 , and ˆN (n)
+g
+are complete
+GMEMs; (iii) E(n)
+g,α, N (n)
+g,F , E(n)
+g,min, and E(n)
+g,min′ are not
+complete GMEMs since the associated reduced functions
+are not subadditive and thus they violate the hierarchy
+condition for some states. (iv) E(n)
+g,f , C(n)
+g
+, τ (n)
+g
+, and E(n)
+g,2
+are tightly complete monogamous. Therefore, for these
+GMEMs, tightly complete monogamy are stronger than
+the complete monogamy under the frame work of the
+complete GMEM.
+By the assumption, we conjecture that E(n)
+g,q , E(n)
+g,F,
+E(n)
+g,F ′, E(n)
+g,AF, and ˆN (n)
+g
+are tightly complete monoga-
+mous.
+That is, E(n)
+g
+is complete, completely monoga-
+mous, tightly complete monogamous, if and only if E(n)
+is complete, completely monogamous, tightly complete
+monogamous, respectively.
+A similar quantity, εg−12···n(2), is also put forward in
+Ref. [25]. Let |ψ⟩A1A2···An be a pure state in HA1A2···An
+
+9
+TABLE IV. Comparing of E(n)
+g
+with different reduced func-
+tions and E (n)
+g
+(n ≥ 4).
+GMEM
+Unified
+Complete
+CM
+TCM
+E(n)
+g,f
+✓
+✓
+✓
+✓
+C(n)
+g
+✓
+✓
+✓
+✓
+τ (n)
+g
+✓
+✓
+✓
+✓
+E(n)
+g,q
+✓
+✓
+✓
+✓a
+E(n)
+g,α
+✓
+×
+✓
+×
+N (n)
+g,F
+✓
+×
+✓
+×
+E(n)
+g,F
+✓
+✓
+✓
+✓a
+E(n)
+g,F′
+✓
+✓b
+✓
+✓a
+E(n)
+g,AF
+✓
+✓b
+✓
+✓a
+E(n)
+g,2
+✓
+✓
+✓
+✓
+E(n)
+g,min
+✓
+×
+✓
+×
+E(n)
+g,min′
+✓
+×
+✓
+×
+ˆ
+N (n)
+g
+✓
+✓b
+✓
+✓a
+E (n)
+g
+(n ≥ 4)
+✓
+✓
+✓
+✓
+a Assume that h is subadditive and Eq. (26) holds.
+b Assume that h is subadditive.
+and h be a non-negative concave function on SX. We
+define
+E(n)
+g
+(|ψ⟩A1A2···An)
+=
+�
+E(n)(|ψ⟩A1A2···An),
+h(ρAi) > 0 for any i,
+0,
+h(ρAi) = 0 for some i, (37)
+and then extend it to mixed states by the convex-roof
+structure. Notice here that E(n)
+g
+is slightly different than
+εg−12···n(2) in which the factor “1/2” is ignored.
+Clearly,
+E(n)
+g
+≤ E(n)
+g
+,
+(38)
+and E(3)
+g
+coincides with E(3)
+g
+but E(n)
+g
+is different from
+E(n)
+g
+whenever n ≥ 4. E(n)
+g
+is just Eg−12···n(2) in Ref. [25]
+if the corresponding bipartite entanglement measure is
+an entanglement monotone. The following Proposition
+can be easily checked.
+Proposition 4. Let E(n) be a non-negative function de-
+fined as in Eq. (37), n ≥ 4. Then E(n) is a complete
+MEM and it is completely monogamous and tightly com-
+plete monogamous.
+That is, for the case of n ≥ 4, all these MEMs E(n)
+g
+with
+the reduced functions we discussed in Sec. III are com-
+plete GMEMs, and are not only completely monogamous
+but also tightly complete monogamous. For convenience,
+we list all these MEMs in Table IV. In addition, it is ob-
+vious that E(n)
+g
+< E(n)
+g
+whenever n ≥ 4 for any E(4)
+g
+and
+E(4)
+g
+mentioned above.
+B.
+Complete GMEM from the maximal reduced
+function
+Let |ψ⟩A1A2···An be a pure state in HA1A2···An and h
+be a non-negative concave function on the set of density
+matrices. We define
+E(n)
+g′ (|ψ⟩A1A2···An)
+=
+�
+max
+i
+h(ρAi),
+if h(ρAi) > 0 for any i,
+0,
+if h(ρAi) = 0 for some i,
+(39)
+and then extend it to mixed states by the convex-roof
+structure. From Theorem 2, we have the following Propo-
+sition.
+Proposition 5. Let E(n)
+g′
+be a GMEM defined as in
+Eq. (39). Then (i) E(3)
+g′
+is a complete GMEM but not
+tightly complete monogamous, and if h is strictly con-
+cave, E(3)
+g′
+is completely monogamous, and (ii) E(n)
+g′
+is
+not complete whenever n ≥ 4.
+We denote E(n)
+g′
+the corresponding GMEMs men-
+tioned in the previous Subsection by E(n)
+g′,f, C(n)
+g′ , τ (n)
+g′ ,
+E(n)
+g′,q, E(n)
+g′,α, N (n)
+g′,F , E(n)
+g′,F, E(n)
+g′,F ′, E(n)
+g′,AF, E(n)
+g′,2, E(n)
+g′,min,
+E(n)
+g′,min′, and ˆN (n)
+g′ , respectively. Then all these GMEMS
+are complete GMEMs but not tightly complete monoga-
+mous for the case of n = 3, E(3)
+g′,f, C(3)
+g′ , τ (3)
+g′ , E(3)
+g′,q, E(3)
+g′,α,
+N (3)
+g′,F , E(3)
+g′,F, E(3)
+g′,F ′, and E(3)
+g′,AF, are completely monog-
+amous, all of these GMEMs are not complete GMEMs
+whenever n ≥ 4.
+One need note here that, when h is not strictly con-
+cave, E(n)
+g′
+is not a unified GMEM since it may hap-
+pen that E(k)
+g′ (X1|X2| · · · |Xk) = E(l)
+g′ (Y1|Y2| · · · |Yl) for
+some ρA1A2···An ∈ SA1A2···An
+g
+with X1|X2| · · · |Xk ≻a
+Y1|Y2| · · · |Yl, namely, it violates Eq. (7).
+In addition,
+E(n)
+g′
+also violates Eq. (17) or Eq. (18). For example, we
+take the state in Eq. (31) with both |ψ⟩AB1 and |ψ⟩B2C
+are entangled. We assume
+E(3)
+g′,min(|ψ⟩ABC) = E(2)
+g′,min(|ψ⟩A|BC),
+ˆN (3)
+g′ (|ψ⟩ABC) = ˆN (2)
+g′ (|ψ⟩A|BC),
+then
+E(3)
+g′,min(|ψ⟩ABC) = E(2)
+g′,min(|ψ⟩AB1) = E(2)
+g′,min(ρAB),
+ˆN (3)
+g′ (|ψ⟩ABC) = ˆN (2)
+g′ (|ψ⟩AB1) = ˆN (2)
+g′ (ρAB),
+and ρBC is entangled.
+In addition, for the sate in
+Eq. (32), we have
+E(3)
+g′,2(|φ⟩ABC)
+= E(2)
+g′,2(|φ⟩A|BC) = E(2)
+g′,2(|φ⟩AB|C) = E(2)
+g′,2(|φ⟩B|AC)
+= E(2)
+g′,2(ρAB) = E(2)
+g′,2(ρAC) = E(2)
+g′,2(ρBC)
+= 1
+3.
+
+10
+TABLE V. Comparing of E(3)
+g′
+with different reduced func-
+tions, E(4)
+g′ , and E (n)
+g′ .
+GMEM
+Unified
+Complete
+CM
+TCM
+E(3)
+g′,f
+✓
+✓
+✓
+×
+C(3)
+g′
+✓
+✓
+✓
+×
+τ (3)
+g′
+✓
+✓
+✓
+×
+E(3)
+g′,q
+✓
+✓
+✓
+×
+E(3)
+g′,α
+✓
+✓
+✓
+×
+N (3)
+g′,F
+✓
+✓
+✓
+×
+E(3)
+g′,F
+✓
+✓
+✓
+×
+E(3)
+g′,F′
+✓
+✓
+✓
+×
+E(3)
+g′,AF
+✓
+✓
+✓
+×
+E(3)
+g′,2
+×
+×
+×
+×
+E(3)
+g′,min
+×
+×
+×
+×
+E(3)
+g′,min′
+×
+×
+×
+×
+ˆ
+N (3)
+g′
+×
+×
+×
+×
+E(n)
+g′
+(n ≥ 4)
+?
+×
+?
+×
+E (n)
+g′
+(n ≥ 4)
+?
+×
+?
+×
+That is, whenever h is strictly concave, E(n)
+g′
+is complete,
+completely monogamous, tightly complete monogamous,
+if and only if E′(n) is complete, completely monogamous,
+tightly complete monogamous, respectively.
+For the states that admit the form
+|η⟩ABC = |η⟩AB1|η⟩B2C
+(40)
+where B1B2 refers to HB has a subspace isomorphic to
+HB(x)
+1
+⊗HB(x)
+2
+such that up to local unitary on system B,
+we have E(3)
+g′ (|η⟩ABC) = E(2)
+g′ (|η⟩B|AC) whenever h(ρ ⊗
+σ) ≥ h(ρ) and h(ρ ⊗ σ) ≥ h(σ) for any ρ and σ, and ρAC
+is a product state. We therefore have the following fact.
+Proposition 6. If h is strictly concave and h(ρ ⊗ σ) ≥
+h(ρ) and h(ρ⊗σ) ≥ h(σ) for any ρ and σ, E(3)
+g′ defined as
+in Eq. (39) is tightly complete monogamous on the states
+that admit the form (40).
+In fact, we always have h(ρ⊗σ) ≥ h(ρ) and h(ρ⊗σ) ≥
+h(ρ) if h ∈ {S, hC, hτ, hq, hα, hN, hF, hF ′, hAF}. So
+E(n)
+g′,f, C(n)
+g′ , τ (n)
+g′ , E(n)
+g′,q, E(n)
+g′,α, N (n)
+g′,F , E(n)
+g′,F, E(n)
+g′,F ′, and
+E(n)
+g′,AF are tightly complete monogamous on the states
+with the form as in Eq. (40). Proposition 6 is also valid
+when we replacing E(3)
+g′
+with E′(3).
+Let |ψ⟩A1A2···An be a pure state in HA1A2···An and h
+be a non-negative concave function on SX. We define
+E(n)
+g′ (|ψ⟩A1A2···An)
+=
+�
+E′(n)(|ψ⟩A1A2···An),
+if h(ρAi) > 0 for any i,
+0,
+if h(ρAi) = 0 for some i,(41)
+for pure states and for mixed states by the convex-roof
+structure. By definition,
+E(n)
+g′
+≤ E(n)
+g′ ,
+(42)
+E(3)
+g′
+coincides with E′(3)
+g , and E(n)
+g′
+satisfies the hierarchy
+condition, but it violates the unification condition if n ≥
+4. It is easy to see that all these GMEMs E(n)
+g′
+with the
+reduced function we discussed are not complete GMEMs
+whenever n ≥ 4. We give comparison for these GMEMs
+in Table V.
+For the case of n ≥ 4, it is possible that E(n)
+g′
+< E(n)
+g′ .
+For example, for |W4⟩ and the state in Eq. (35) we have
+E(4)
+g′ < E(4)
+g′ for any E(4)
+g′ and E(4)
+g′ mentioned above except
+for h = hmin and h = ˆh.
+C.
+GMEM from the minimal reduced function
+With h is a non-negative concave function on the set
+of density matrices, when we define
+E(n)
+g′′ (|ψ⟩A1A2···An) = min
+i
+h(ρAi),
+(43)
+and then extend it to mixed states by the convex-roof
+structure, it is a GMEM. Moreover, we can define
+E(n)
+g′′ (|ψ⟩A1A2···An) =
+min
+i1≤···≤is,s≤n/2 h(ρAi1 Ai2···Ais ), (44)
+and then extend it to mixed states by the convex-roof
+structure, it is also a GMEM. For example, GMC, de-
+noted by Cgme [31], is defined as in Eq. (44).
+Recall
+that,
+Cgme(|ψ⟩) := min
+γi∈γ
+�
+2
+�
+1 − Tr(ρAγi )2�
+for pure state |ψ⟩ ∈ HA1A2···Am, where γ = {γi} repre-
+sents the set of all possible bipartitions of A1A2 · · · Am,
+and via the convex-roof extension for mixed states.
+We denote E(n)
+g′′ the corresponding GMEMs mentioned
+in the previous Subsection by E(n)
+g′′,f, C(n)
+g′′ , τ (n)
+g′′ , E(n)
+g′′,q,
+E(n)
+g′′,α, N (n)
+g′′,F , E(n)
+g′′,F, E(n)
+g′′,F ′, E(n)
+g′′,AF, E(n)
+g′′,2, E(n)
+g′′,min,
+E(n)
+g′′,min′, and
+ˆN (n)
+g′′ , respectively, and denote E(n)
+g′′
+by
+E(n)
+g′′,f, C(n)
+g′′
+(or Cgme), ˆτ (n)
+g′′ , E(n)
+g′′,q, E(n)
+g′′,α, N (n)
+g′′,F , E(n)
+g′′,F,
+E(n)
+g′′,F ′, E(n)
+g′′,AF, E(n)
+g′′,2, E(n)
+g′′,min, E(n)
+g′′,min′, and ˆ
+N (n)
+g′′ , respec-
+tively.
+By definition,
+E(n)
+g′′ ≤ E(n)
+g′′ ≤ E(n)
+g′
+≤ E(n)
+g
+(45)
+for any h, and E(3)
+g′′ = E(3)
+g′′ . If n ≥ 4, there does exist
+state such that E(n)
+g′′ < E(n)
+g′′ . For example, we take
+|ψ⟩ABCD = |ψ⟩AB1|ψ⟩B2C1|ψ⟩C2D,
+
+11
+0
+0.2
+0.4
+0.6
+0.8
+1
+t
+0
+0.5
+1
+1.5
+E
+(a) Cg
+0
+0.2
+0.4
+0.6
+0.8
+1
+t
+0
+0.5
+1
+E
+(b) Eg,F′
+0
+0.2
+0.4
+0.6
+0.8
+1
+t
+0
+0.2
+0.4
+0.6
+0.8
+E
+(c) Eg,2
+FIG. 1. (color online). Comparing (a) C(3)
+g
+and C(3)
+g′
+for |Ψ⟩.
+(b) E(3)
+g,F′ and E(3)
+g′,F′, and (c) Comparing E(3)
+g,2 and E(3)
+g��,2 for
+|Ψ⟩. E(3)
+g′ = E(3)
+g′′ in such a case.
+where X1X2 refers to HX has a subspace isomorphic
+to HX(x)
+1
+⊗ HX(x)
+2
+such that up to local unitary on sys-
+tem X. If h(ρB2) < h(ρA) and h(ρB2) < h(ρD), then
+E(4)
+g′′ (|ψ⟩ABCD) = h(ρB2) < E(4)
+g′′ (|ψ⟩ABCD). In addition,
+for the state in Eq. (35),
+E(4)
+g′′,min = 5
+16 < E(4)
+g′′,min = 3
+8,
+ˆ
+N (4)
+g′′ =
+√
+15
+8
+√
+2 < ˆN (4)
+g′′ =
+√
+15
+8 .
+Cgme is not a complete GMEM since it does not satisfy
+the hierarchy condition (8) [40]: Let
+|ξ⟩ =
+√
+5
+4 |0000⟩ + 1
+4|1111⟩ +
+√
+5
+4 |0100⟩ +
+√
+5
+4 |1010⟩, (46)
+then
+Cgme(|ξ⟩) = C(|ξ⟩ABC|D) =
+√
+15
+8
+< C(|ξ⟩AB|CD) =
+√
+65
+8 .
+In general, E(n)
+g′′ and E(n)
+g′′ do not obey the unification
+condition (7) and the hierarchy condition (8).
+For in-
+stance, for the state as in Eq. (31), we have
+E(3)
+g′′,min(|ψ⟩ABC) = E(2)
+g′′,min(|ψ⟩B|AC),
+ˆN (3)
+g′′ (|ψ⟩ABC) = ˆN (2)
+g′′ (|ψ⟩B|AC),
+and
+E(3)
+g′′,min(|ψ⟩ABC) < E(2)
+g′′,min(|ψ⟩A|BC)
+= E(2)
+g′′,min(|ψ⟩AB1) = E(2)
+g′′,min(ρAB),
+E(3)
+g′′,min(|ψ⟩ABC) < E(2)
+g′′,min(|ψ⟩C|AB)
+= E(2)
+g′′,min(|ψ⟩B2C) = E(2)
+g′′,min(ρBC),
+ˆ
+N (3)
+g′′ (|ψ⟩ABC) < ˆN (2)
+g′′ (|ψ⟩A|BC)
+= ˆ
+N (2)
+g′′ (|ψ⟩AB1) = ˆN (2)
+g′′ (ρAB),
+ˆ
+N (3)
+g′′ (|ψ⟩ABC) < ˆN (2)
+g′′ (|ψ⟩AB|C)
+= ˆ
+N (2)
+g′′ (|ψ⟩B2C) = ˆN (2)
+g′′ (ρBC).
+In addition,
+C(ρBD) ≈ 0.839 > Cgme(|ξ⟩)
+for the pure state |ψ⟩ in Eq. (46). Let
+|ζ⟩ABC = λ0|000⟩ + λ2|101⟩ + λ3|110⟩
+(47)
+with λ0 ≥ λ2 ≥ λ3 > 0. If we take λ0 =
+√
+5
+√
+12, λ2 =
+1
+√
+3, and λ3 = 1
+2 in Eq. (47), then E(3)
+g′′,2(|ζ⟩ABC) = 1/4,
+but E(2)
+g′′,2(|ζ⟩A|BC) = 5/12, E(2)
+g′′,2(|ζ⟩AB|C) = 1/3. In
+general, for the state
+|ω⟩ABC = λ0|000⟩ + λ2|101⟩ + λ3|110⟩ + λ4|111⟩
+with λ0λ4 > 0, max{λ2, λ3} > 0 and min{λ2, λ3} = 0,
+then (i) ρAC and ρBC are separable while ρAB is entan-
+gled whenever λ3 > 0, and (ii) ρAB and ρBC are separable
+while ρAC is entangled whenever λ2 > 0. From this we
+can arrive at (i) if λ4 is small enough, then
+Cgme(|ω⟩ABC) = C(|ω⟩AB|C) < C(|ω⟩A|BC),
+C(|ω⟩AB|C) < C(|ω⟩B|AC),
+Cgme(|ω⟩ABC) < C(ρAB),
+
+12
+and (ii) if λ4 is small enough, then
+Cgme(|ω⟩ABC) = C(|ω⟩B|AC) < C(|ω⟩C|AB),
+C(|ω⟩B|AC) < C(|ω⟩A|BC),
+Cgme(|ω⟩ABC) < C(ρAC).
+For example, when taking λ2
+0 = 7/9, λ3 = λ4 = 1/3, we
+get
+Cgme(|ω⟩ABC) ≈ 0.5879,
+C(|ω⟩A|BC) = C(|ω⟩B|AC) ≈ 0.8315,
+C(ρAB) ≈ 0.8090;
+when taking λ2
+0 = 7/9, λ2 = λ4 = 1/3, we get
+Cgme(|ω⟩ABC) ≈ 0.5879,
+C(|ω⟩A|BC) = C(|ω⟩C|AB) ≈ 0.8315,
+C(ρAC) ≈ 0.8090.
+For the generalized GHZ state
+|GHZ⟩ = λ0|0⟩⊗n + λ1|1⟩⊗n + · · · λd−1|d − 1⟩⊗n, (48)
+E(n)
+g′′
+and E(n)
+g′′
+are complete monogamous and tightly
+complete monogamous. For this state, E(n)
+g′′
+= E(n)
+g′
+=
+E(n)
+g′′ = E(n)
+g′ , and nE(n)
+g′′ = nE(n)
+g′
+= 2E(n)
+g
+. Moreover, for
+such a state, all the entanglement are shared between all
+of the particles. We thus regard this state as the maxi-
+mal genuinely entangled state, and it reaches the maxi-
+mal value whenever λ0 = λ1 = · · · = λd−1 = 1/
+√
+d for
+the multi-qudit case.
+Comparing E(3)
+g′′ with E(3)
+g′
+and E(3)
+g , E(3)
+g′
+seems the
+best one since (i) it is complete and completely monoga-
+mous whenever the reduced function is strictly concave,
+(ii) it can be easily calculated, and (iii) it is monogamous
+iff it is completely monogamous. For the case of n ≥ 4,
+E(n), E(n), E(n)
+g
+, and E(n)
+g
+seems better the other cases as
+a MEM/GMEM as these measures admit the postulates
+of a complete MEM/GMEM.
+At last, we calculate these GMEMs for the following
+examples,
+|Ψ⟩ =
+√
+t|000⟩ +
+√
+1 − t|111⟩,
+|Φ⟩ = √p|100⟩ + √q|010⟩ +
+�
+1 − p − q|001⟩.
+For the GHZ class state |Ψ⟩, Eg′ coincides with Eg′′ and
+Eg′ is equivalent to Eg (see Fig. 1 for detail). For |Φ⟩,
+Eg, Eg′ and Eg′′ reflect roughly the same tendency (see
+Fig. 2 for detail).
+VI.
+CONCLUSION
+We developed a grained scenario of investigating the
+MEM and GMEM based on its reduced functions and
+then explored these measures in light of the framework of
+the complete MEM and complete monogamy relation re-
+spectively. We provided criteria that can verify whether
+(a) Cg
+(b) Eg,F′
+(c) Eg,2
+FIG. 2. (color online). Comparing (a) C(3)
+g , C(3)
+g′
+and C(3)
+g′′ ,
+(b) E(3)
+g,F′, E(3)
+g′,F′ and E(3)
+g′′,F′, (c) E(3)
+g,2, E(3)
+g′,2 and E(3)
+g′′,2 for |Φ⟩
+with p ≥ q ≥ 1 − p − q > 0, respectively.
+a MEM/GMEM is good or not.
+By comparision, for
+tripartite case, the MEM and GMEM via the maximal
+reduced function seems finer than that of the minimal
+reduced function as it not only can be easily calculated
+but also is complete and completely monogamous. And
+for the n-partite case with n ≥ 4, the MEM and GMEM
+via the sum of the reduced function sound better than
+the other one in the framework of complete MEM and
+complete monogamy relation.
+In addition, our findings show that, whether the re-
+duced function is strictly concave and whether it is
+subadditive is of crucial important.
+We can also con-
+
+9,2
+E
+q/.2
+E
+gll,20.4
+0.6
+p0.5
+0
+0
+0.2
+q
+0.4
+0.8
+0.6
+1a.F
+E
+g'.F
+E
+gll,F0.4
+0.6
+p0.5
+0
+0
+0.2
+b
+0.4
+0.8
+0.6
+1.9
+gr0.4
+0.6
+p0.5
+0
+0
+0.2
+b
+0.4
+0.8
+0.6
+113
+clude that the monogamy is stronger than the complete
+monogamy in general, they are equivalent to each other
+for some case such as the MEM and GMEM via the max-
+imal reduced function for the tripartite case, and the
+tightly complete monogamy is stronger than the com-
+plete monogamy in general. We also find that, in the
+framework of complete MEM, the hierarchy condition is
+stronger than the unification condition in general but it
+is not true for some case such as the MEM and GMEM
+via the maximal bipartite entanglement.
+ACKNOWLEDGMENTS
+This work is supported by the National Natural Sci-
+ence Foundation of China under Grant No. 11971277, the
+Fund Program for the Scientific Activities of Selected Re-
+turned Overseas Professionals in Shanxi Province under
+Grant No. 20220031, and the Scientific Innovation Foun-
+dation of the Higher Education Institutions of Shanxi
+Province under Grant No. 2019KJ034.
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+

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+ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE
+YONGQIAO WANG AND TAKASHI NISHIMURA
+Abstract. In this paper, on envelopes created by circle families in the plane, answers to all four basic
+problems (existence problem, representation problem, problem on the number of envelopes, problem on
+relationships of definitions) are given.
+1. Introduction
+Throughout this paper, I is an open interval and all functions, mappings are of class C∞ unless
+otherwise stated.
+Envelopes of planar regular curve families have fascinated many pioneers since the dawn of differential
+analysis (for instance, see [3]). In most typical cases, straight line families have been studied. In [6], by
+solving four basic problems on envelopes created by straight line families in the plane (existence problem,
+representation problem, uniqueness problem and equivalence problem of definitions), the second author
+constructs a general theory for envelopes created by straight line families in the plane. On the other
+hand, circle families in the plane are non-negligible families because the envelopes of them have already
+had an important application, namely, an application to Seismic Survey. Following 7.14(9) of [1], a brief
+explanation of Seismic Survey is given as follows. In the Eucledian plane R2, consider the “ground level
+curve” C parametrized by γ : I → R2. Suppose that there is a stratum of granite below the top layer of
+sandstone and that the dividing curve, denoted by M, is parametrized by �f : I → R2. Seismic Survey is
+the following method to obtain an approximation of �f as precisely as possible. Take one fixed point A of
+C and consider an explosion at A. Assume that the sound waves travel in straight lines and are reflected
+from M, arriving back at points γ(t) of C where their times of arrival are exactly recorded by sensors
+located along C (see Figure 1). It is known that there exists a curve W parametrized by f : I → R2
+Figure 1. Reflection of sound waves.
+with well-defined normals such that each broken line of a reflected ray starting at A and finishing on C
+2010 Mathematics Subject Classification. 57R45, 58C25.
+Key words and phrases. Circle family, Envelope, Frontal, Creative, Creator.
+1
+arXiv:2301.04478v1  [math.DG]  11 Jan 2023
+
+A
+(ti)(t2)(t3)(t4)(ts)
+C
+M2
+Y. WANG AND T. NISHIMURA
+can be replaced by a straight line which is normal to W and of the same total length. The curve W is
+called the orthotomic of M relative to A and conversely the curve M is called the anti-orthotomic of W
+relative to A. Then, an envelope created by the circle family
+�
+(x, y) ∈ R2 �� ||(x, y) − γ(t)|| = ||f(t) − γ(t)||
+�
+t∈I
+recovers W (see Figure 2). After obtaining the parametrization f of W, the parametrization �f of M
+Figure 2. An envelope created by the circle family.
+can be easily obtained by using the anti-orthotomic technique developed in [5]. Therefore, in order to
+investigate the parametrization of W as precisely as possible, construction of general theory on envelopes
+created by circle families is very important.
+In this paper, we construct a general theory on envelopes created by circle families in the plane. For a
+point P of R2 and a positive number λ, the circle C(P,λ) centered at P with radius λ is naturally defined
+as follows, where the dot in the center stands for the standard scalar product.
+C(P,λ) =
+�
+(x, y) ∈ R2 �� ((x, y) − P) · ((x, y) − P) = λ2�
+.
+For a curve γ : I → R2 and a positive function λ : I → R+, the circle family C(γ,λ) is naturally defined as
+follows. Here, R+ stands for the set consisting of positive real numbers.
+C(γ,λ) =
+�
+C(γ(t),λ(t))
+�
+t∈I .
+It is reasonable to assume that at each point γ(t) the normal vector to the curve γ is well-defined. Thus,
+we easily reach the following definition.
+Definition 1. A curve γ : I → R2 is called a frontal if there exists a mapping ν : I → S1 such that the
+following identity holds for each t ∈ I, where S1 is the unit circle in R2.
+dγ
+dt (t) · ν(t) = 0.
+For a frontal γ, the mapping ν : I → S1 given above is called the Gauss mapping of γ.
+By definition, a frontal is a solution of the first order linear differential equation defined by Gauss mapping
+ν. Thus, for a fixed mapping ν : I → S1 the set consisting of frontals with a given Gauss mapping
+ν : I → S1 is a linear space. For frontals, [4] is recommended as an excellent reference. Hereafter in this
+paper, the curve γ : I → R2 for a circle family C(γ,λ) is assumed to be a frontal.
+In this paper, the following is adopted as the definition of an envelope created by a circle family.
+Definition 2. Let C(γ,λ) be a circle family. A mapping f : I → R2 is called an envelope created by C(γ,λ)
+if there exists a mapping �ν : I → S1 such that the following two hold for any t ∈ I.
+(1)
+df
+dt(t) · �ν(t) = 0.
+
+A
+(ti) (t2) (t3) (t4) (ts)
+f(ts)
+-f(t4)
+f(t2)f(t3)
+f(ti)ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE
+3
+(2) f(t) ∈ C(γ(t),λ(t)).
+By definition, as same as an envelope created by a hyperplane family (see [6]), an envelope created by
+a circle family is a solution of a first order linear differential equation with one constraint condition.
+Moreover, again by definition, an envelope created by a circle family is a frontal with Gauss mapping
+�ν : I → S1. On the other hand, since there is one constraint condition, again as same as an envelope
+created by a hyperplane family, the set of envelopes created by a given circle family is in general not a
+linear space.
+Problem 1.
+(1) Given a circle family C(γ,λ), find a necessary and sufficient codition for the family
+to create an envelope in terms of γ, ν and λ.
+(2) Suppose that a circle family C(γ,λ) creates an envelope.
+Then, find a parametrization of the
+envelope in terms of γ, ν and λ.
+(3) Suppose that a circle family C(γ,λ) creates an envelope. Then, find a criterion for the number of
+distinct envelopes created by C(γ,λ) in terms of γ, ν and λ.
+Note 1.
+(1) (1) of Problem 1 is a problem to seek the integrability conditions. There are various cases, for instance
+the concentric circle family {{(x, y) ∈ R2 | x2 + y2 = t2}}t∈R+ does not create an envelope while the
+parallel-translated circle family {{(x, y) ∈ R2 | (x − t)2 + y2 = 1}}t∈R does create two envelopes. Thus,
+(1) of Problem 1 is significant.
+(2) The following Example 1 shows that the apparently well-known method to obtain the envelope seems
+to be useless in this case. Thus, (2) of Problem 1 is important and the positive answer to it is much
+desired.
+(3) The following Example 2 shows that there are at least three cases: the case having a unique envelope,
+the case having exactly two envelopes and the case having uncountably many envelopes. Thus, (3) of
+Problem 1 is meaningful and interesting.
+Example 1. Let γ : R → R2 be the mapping defined by γ(t) =
+�
+t3, t6�
+. Set ν(t) =
+1
+√
+4t6+1
+�
+−2t3, 1
+�
+. It
+is clear that the mapping γ is a frontal with Gauss mapping ν : R → S1. Let λ : R → R+ be the constant
+function defined by λ(t) = 1.
+Then, it seems that the circle family C(γ,λ) creates envelopes.
+Thus,
+we can expect that the created envelopes can be obtained by the well-known method. Set F(x, y, t) =
+�
+x − t3�2 +
+�
+y − t6�2 − 1. Then, we have the following.
+D
+=
+�
+(x, y) ∈ R2
+���� ∃t such that F(x, y, t) = ∂F
+∂t (x, y, t) = 0
+�
+=
+�
+(x, y) ∈ R2 ��� ∃t such that
+�
+x − t3�2 +
+�
+y − t6�2 − 1 = 0, −6t2 �
+x − t3�
+− 12t5 �
+y − t6�
+= 0
+�
+=
+�
+(x, y) ∈ R2 ��� ∃t such that
+�
+x − t3�2 +
+�
+y − t6�2 − 1 = 0, t2 ��
+x − t3�
++ 2t3 �
+y − t6��
+= 0
+�
+=
+�
+(x, y) ∈ R2 �� x2 + y2 = 1
+� � �
+(x, y) ∈ R2 ���
+�
+x − t3�2 +
+�
+y − t6�2 − 1 = 0, x = t3 − 2t3 �
+y − t6��
+=
+�
+(x, y) ∈ R2 �� x2 + y2 = 1
+� � �
+(x, y) ∈ R2 ���
+�
+−2t3 �
+y − t6��2 +
+�
+y − t6�2 = 1, x = t3 �
+1 − 2y + 2t6��
+=
+�
+(x, y) ∈ R2 �� x2 + y2 = 1
+� � ��
+t3 ∓
+2t3
+√
+4t6 + 1, t6 ±
+1
+√
+4t6 + 1
+�
+∈ R2
+���� t ∈ R
+�
+.
+In Example 3 of Section 3, it turns out that the set D calculated here is actually larger than the set of
+envelopes created by C(γ,λ), namely the unit circle
+�
+(x, y) ∈ R2 �� x2 + y2 = 1
+�
+is redundant. Therefore,
+unfortunately, the apparently well-known method to obtain the envelopes does not work well in this case.
+The circle family C(γ,λ) and the candidate of its envelope are depicted in Figure 3.
+Example 2.
+(1) Let γ : R+ → R2 be the mapping defined by γ(t) = (0, 1 + t). Then, it is clear that
+γ is a frontal. Let λ : R+ → R+ be the positive function defined by λ(t) = 1+t. Then, it is easily
+seen that the origin (0, 0) of the plane R2 itself is a created envelope by the circle family C(γ,λ)
+and that there are no other envelopes created by C(γ,λ). Hence, the number of created envelopes
+is one in this case.
+(2) The parallel-translated circle family {{(x, y) ∈ R2 | (x − t)2 + y2 = 1}}t∈R creates exactly two
+envelopes.
+(3) Let γ : R → R2 be the constant mapping defined by γ(t) = (0, 0). Then, it is clear that γ is a
+frontal. Let λ : R → R+ be the constant function defined by λ(t) = 1. Then, for any function
+
+4
+Y. WANG AND T. NISHIMURA
+Figure 3. The circle family C(γ,λ) and the candidate of its envelope.
+θ : R → R, the mapping f : R → R2 defined by f(t) = (cos θ(t), sin θ(t)) is an envelope created
+by the circle family C(γ,λ). Hence, there are uncountably many created envelopes in this case.
+In order to solve Problem 1, we prepare several terminologies that can be derived from a frontal
+γ : I → R2 with Gauss mapping ν : I → S1 and a positive function λ : I → R+. For a frontal γ : I → R2
+with Gauss mapping ν : I → S1, following [2], we set µ(t) = J(ν(t)), where J is the anti-clockwise
+rotation by π/2. Then we have a moving frame {µ(t), ν(t)}t∈I along the frontal γ. Set
+ℓ(t) = dν
+dt (t) · µ(t),
+β(t) = dγ
+dt (t) · µ(t).
+The pair of functions (ℓ, β) is called the curvature of the frontal γ with Gauss mapping ν. We want to
+focus the ratio of dλ
+dt (t) and β(t). The following definition is the key of this paper.
+Definition 3. Let γ : I → R2, λ : I → R+ be a frontal with Gauss mapping ν : I → S1 and a positive
+function respectively.
+Then, the circle family C(γ,λ) is said to be creative if there exists a mapping
+�ν : I → S1 such that the following identity holds for any t ∈ I.
+dλ
+dt (t) = −β(t) (�ν(t) · µ(t)) .
+Set cos θ(t) = −�ν(t) · µ(t). Then, the creative condition is equivalent to say that there exists a function
+θ : I → R satisfying the following identity for any t ∈ I.
+dλ
+dt (t) = β(t) cos θ(t).
+By definition, any family of concentric circles with expanding radius is not creative, and it is clear that
+such the circle family does not create an envelope. Under the above preparation, Problem 1 is solved as
+follows.
+Theorem 1. Let γ : I → R2 be a frontal with Gauss mapping ν : I → S1 and let λ : I → R+ be a
+positive function. Then, the following three holds.
+(1) The circle family C(γ,λ) creates an envelope if and only if C(γ,λ) is creative.
+(2) Suppose that the circle family C(γ,λ) creates an envelope f : I → R2. Then, the created envelope
+f is represented as follows.
+f(t) = γ(t) + λ(t)�ν(t).
+where �ν : I → S1 is the mapping defined in Definition 3.
+
+ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE
+5
+(3) Suppose that the circle family C(γ,λ) creates an envelope. Then, the number of envelopes created
+by C(γ,λ) is characterized as follows.
+(3-i) The circle family C(γ,λ) creates a uinique envelope if and only if the set consisting of t ∈ I
+satisfying β(t) ̸= 0 and dλ
+dt (t) = ±β(t) is dense in I.
+(3-ii) There are exactly two distinct envelopes created by C(γ,λ) if and only if the set of t ∈ I
+satisfying β(t) ̸= 0 is dense in I and there exists at least one t0 ∈ I such that the strict
+inequality | dλ
+dt (t0)| < |β(t0)| holds.
+(3-∞) There are uncountably many distinct envelopes created by C(γ,λ) if and only if the set of t ∈ I
+satisfying β(t) ̸= 0 is not dense in I.
+By the assertion (2) of Theorem 1, it is reasonable to call �ν the creator for an envelope f created by
+C(γ,λ).
+This paper is organized as follows. Theorem 1 is proved in Section 2. In Section 3, several examples to
+which Theorem 1 is effectively applicable are given. Finally, in Section 4, relations of several definitions
+of an envelope created by a circle family are investigated.
+2. Proof of Theorem 1
+2.1. Proof of the assertion (1) of Theorem 1. Suppose that C(γ,λ) is creative. By definition, there
+exists a mapping �ν : I → S1 such that the equality dλ
+dt (t) = −β(t) (�ν(t) · µ(t)) holds for any t ∈ I. Set
+f(t) = γ(t) + λ(t)�ν(t).
+Then, since (f(t) − γ(t)) · (f(t) − γ(t)) = λ2(t), it follows f(t) ∈ C(γ(t),λ(t)). Morever, since
+df
+dt (t) = dγ
+dt (t) + dλ
+dt (t)�ν(t) + λ(t)d�ν
+dt (t),
+we have the following.
+df
+dt (t) · (f(t) − γ(t))
+=
+�dγ
+dt (t) + dλ
+dt (t)�ν(t) + λ(t)d�ν
+dt (t)
+�
+· (λ(t)�ν(t))
+=
+dγ
+dt (t) · (λ(t)�ν(t)) + dλ
+dt (t)λ(t)
+=
+(β(t)µ(t)) · (λ(t)�ν(t)) + (−β(t) (�ν(t) · µ(t))) λ(t)
+=
+β(t)λ(t) (µ(t) · �ν(t)) − β(t)λ(t) (�ν(t) · µ(t))
+=
+0.
+Hence, f is an envelope created by the circle family C(γ,λ).
+Conversely, suppose that the circle family C(γ,λ) creates an envelope f : I → R. Then, by definition, it
+follows that f(t) ∈ C(γ(t),λ(t)) and df
+dt(t) · (f(t) − γ(t)) = 0. The condition f(t) ∈ C(γ(t),λ(t)) implies that
+there exists a mapping �ν : I → S1 such that the following equality holds for any t ∈ I.
+f(t) = γ(t) + λ(t)�ν(t).
+Then, since
+df
+dt (t) = dγ
+dt (t) + dλ
+dt (t)�ν(t) + λ(t)d�ν
+dt (t),
+we have the following.
+0
+=
+df
+dt (t) · (f(t) − γ(t))
+=
+�dγ
+dt (t) + dλ
+dt (t)�ν(t) + λ(t)d�ν
+dt (t)
+�
+· (λ(t)�ν(t))
+=
+(β(t)µ(t)) · (λ(t)�ν(t)) + dλ
+dt (t)λ(t)
+=
+λ(t)
+�
+β(t) (µ(t) · �ν(t)) + dλ
+dt (t)
+�
+.
+
+6
+Y. WANG AND T. NISHIMURA
+Since λ(t) is positive for any t ∈ I, it follows
+β(t) (µ(t) · �ν(t)) + dλ
+dt (t) = 0.
+Therefore, the circle family C(γ,λ) is creative.
+2
+2.2. Proof of the assertion (2) of Theorem 1. The proof of the assertion (1) given in Subsection 2.1
+proves the assertion (2) as well.
+2
+2.3. Proof of the assertion (3) of Theorem 1.
+2.3.1. Proof of (3-i). Suppose that the circle family C(γ,λ) creates a unique envelope. Then, for any t ∈ I
+the unit vector �ν(t) satisfying
+dλ
+dt (t) = −β(t) (�ν(t) · µ(t))
+must be uniquely determined. Hence, under considering continuity of two functions dλ
+dt and β, it follows
+that the set consisting of t ∈ I satisfying dλ
+dt (t) = ±β(t) ̸= 0 must be dense in I.
+Conversely, suppose that the set consisting of t ∈ I satisfying dλ
+dt (t) = ±β(t) ̸= 0 is dense in I. Then,
+under considering continuity of the function t �→ �ν(t) · µ(t), it follows that �ν(t) · µ(t) = ±1 for any t ∈ I.
+Thus, the created envelope f(t) = γ(t) + λ(t)�ν(t) must be unique.
+2
+2.3.2. Proof of (3-ii). Suppose that there are exactly two distinct envelopes created by C(γ,λ). Then, by
+the equality dλ
+dt (t) = −β(t) (�ν(t) · µ(t)) , the set consisting of t ∈ I satisfying β(t) ̸= 0 must be dense in
+I. Suppose moreover that the set of t ∈ I satisfying the equality dλ
+dt (t) = ±β(t) holds for any t ∈ I.
+Then, it follows that the set consisting of t ∈ I satisfying dλ
+dt (t) = ±β(t) ̸= 0 is dense in I. Then, by the
+assertion (3-i), the given circle family must create a unique envelope. This contradicts the assumption
+that there are exactly two distinct envelopes. Hence, there must exist at least one t0 ∈ I such that the
+strict inequality | dλ
+dt (t0)| < |β(t0)| holds.
+Conversely, suppose that the set of t ∈ I satisfying β(t) ̸= 0 is dense in I and there exists at least
+one t0 ∈ I such that the strict inequality | dλ
+dt (t0)| < |β(t0)| holds. Then, it follows that there must exist
+an open interval �I in I such that the absolute value |�ν(t) · µ(t)| = | cos θ(t)| is less than 1 for any t ∈ �I.
+Thus, it follows θ(t) ̸= −θ(t) for any t ∈ �I. Hence, for any t ∈ �I, there exist exactly two distinct unit
+vectors �ν+(t), �ν−(t) corresponding �ν+(t) · µ(t) = − cos θ(t) and �ν−(t) · µ(t) = − cos (−θ(t)) respectively.
+Therefore, the circle family must create exactly two distinct envelopes.
+2
+2.3.3. Proof of (3-∞). Suppose that there are uncountably many distinct envelopes created by C(γ,λ).
+Suppose moreover that the set of t ∈ I such that β(t) ̸= 0 is dense in I. Then, from (3-i) and (3-ii),
+it follows that the circle family C(γ,λ) must create a unique envelope or two distinct envelopes. This
+contradicts the assumption that there are uncountably many distinct envelopes created by C(γ,λ). Hence,
+the set of t ∈ I such that β(t) ̸= 0 is never dense in I.
+Conversely, suppose that the set of t ∈ I such that β(t) ̸= 0 is not dense in I. This assumption implies
+that there exists an open interval �I in I such that β(t) = 0 for any t ∈ �I. On the other hand, since C(γ,λ)
+creates an envelope f0, the equality
+dλ
+dt (t) = −β(t) (�ν(t) · µ(t))
+holds for any t ∈ I. Thus, there are no restrictions for the value �ν(t) · µ(t) for any t ∈ �I. Take one
+point t0 of �I and denote the �ν for the envelope f0 by �ν0. Then, by using the standard technique on
+bump functions, we may construct uncountably many distinct creators �νa : I → S1 (a ∈ A) such that
+the following (a), (b), (c) and (d) hold, where A is a set consisting uncountably many elements such that
+0 ̸∈ A.
+(a) The equality dλ
+dt (t) = −β(t) (�νa(t) · µ(t)) holds for any t ∈ I and any a ∈ A.
+(b) For any t ∈ I − �I and any a ∈ A, the equality �νa(t) = �ν0(t) holds.
+(c) For any a ∈ A, the property �νa(t0) ̸= �ν0(t0) holds.
+(d) For any wo distinct a1, a2 ∈ A, the property �νa1(t0) ̸= �νa2(t0) holds.
+Therefore, the circle family C(γ,λ) creates uncountably many distinct envelopes.
+2
+
+ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE
+7
+3. Examples
+Example 3. We examine Example 1 by applying Theorem 1. In Example 1, γ : R → R2 is given by
+γ(t) =
+�
+t3, t6�
+. Thus, we can say that ν : R → S1 and µ : R → S1 are given by ν(t) =
+1
+√
+4t6+1
+�
+−2t3, 1
+�
+and µ(t) =
+1
+√
+4t6+1
+�
+−1, −2t3�
+respectively. Moreover, the radius function λ : R → R is the constant
+function defined by λ(t) = 1. Thus,
+dλ
+dt (t) = 0.
+By calculation, we have
+β(t) = dγ
+dt (t) · µ(t) = −3t2(1 + 4t6)
+√
+4t6 + 1
+.
+Therefore, the unit vector �ν(t) ∈ S1 satisfying
+dλ
+dt (t) = −β(t) (�ν(t) · µ(t))
+exsists and it must have the form
+�ν(t) = ±ν(t) =
+±1
+√
+4t6 + 1
+�
+−2t3, 1
+�
+.
+Hence, by (1) of Theorem 1, the circle family C(γ,λ) creates an envelope f : R → R2. By (2) of Theorem
+1, f is parametrized as follows.
+f(t)
+=
+γ(t) + λ(t)�ν(t)
+=
+�
+t3, t6�
+±
+1
+√
+4t6 + 1
+�
+−2t3, 1
+�
+=
+�
+t3 ∓
+2t3
+√
+4t6 + 1
+, t6 ±
+1
+√
+4t6 + 1
+�
+.
+Finally, by (3-ii) of Theorem 1, the number of distinct envelopes created by the circle family C(γ,λ) is
+exactly two.
+Therefore, Theorem 1 reveals that the set D calculated in Example 1 is certainly the union of the unit
+circle and the set of two envelopes of C(γ,λ).
+Example 4. We examine (1) of Example 2 by applying Theorem 1. In (1) of Example 2, γ : R+ → R2
+is given by γ(t) = (0, 1 + t). Thus, if we define the unit vector ν(t) = (1, 0), ν : R+ → S1 gives the Gauss
+mapping of γ. By definition, µ(t) = (0, 1) and thus we have β(t) = dγ
+dt (t) · µ(t) = 1. On the other hand,
+the radius function λ : R+ → R+ has the form λ(t) = 1 + t in this example. Thus, the created condition
+dλ
+dt (t) = −β(t) (�ν(t) · µ(t))
+becomes simply
+(∗)
+1 = − (�ν(t) · (0, 1))
+in this case. If we take �ν(t) = (0, −1), then the above equality holds for any t ∈ R+. Thus, by (1) of
+Theorem 1, the circle family C(γ,λ) creates an envelope. By (2) of Theorem 1, the parametrization of the
+created envelope is
+f(t) = γ(t) = λ(t)�ν(t) = (0, 1 + t) + (1 + t) (0, −1) = (0, 0) .
+Finally, notice that for any t ∈ R+ the creative condition (*) in this case holds if and only if �ν(t) =
+(0, −1) = −µ(t). Thus, by (3-i) of Theorem 1, the origin (0, 0) is the unique envelope created by C(γ,λ).
+Example 5. Theorem 1 can be applied also to (2) of Example 2 as follows. In this example, γ(t) = (t, 0)
+and λ(t) = 1. Thus, we may take ν(t) = (0, −1), µ(t) = (1, 0). We have β(t) = dγ
+dt (t) · µ(t) = 1. Since the
+radius function λ is a constant function, the created condition
+dλ
+dt (t) = −β(t) (�ν(t) · µ(t))
+becomes simply
+0 = − (�ν(t) · (0, 1))
+
+8
+Y. WANG AND T. NISHIMURA
+in this case. Thus, for any t ∈ R, the created condition is satisfied if and only if �ν(t) = ±(1, 0). Hence, by
+(1) of Theorem 1, the circle family C(γ,λ) creates an envelope. By (2) of Theorem 1, the parametrization
+of the created envelope is
+f(t) = γ(t) = λ(t)�ν(t) = (t, 0) ± (0, −1) = (t, ∓1) .
+Finally, by (3-ii) of Theorem 1, the number of envelope created by C(γ,λ) is exactly two.
+Example 6. Theorem 1 can be applied even to (3) of Example 2 as follows. In this example, γ(t) = (0, 0)
+and λ(t) = 1. Thus, every mapping ν : R → S1 can be taken as Gauss mapping of γ. In particular, γ is
+a frontal. We have β(t) = dγ
+dt (t) · µ(t) = 0. Since the radius function λ is a constant function λ(t) = 1,
+the created condition
+dλ
+dt (t) = −β(t) (�ν(t) · µ(t))
+becomes simply
+0 = 0
+in this case. Thus, for any �ν : R → S1, the created condition is satisfied. Hence, by (1) of Theorem
+1, the circle family C(γ,λ) creates an envelope. By (2) of Theorem 1, the parametrization of the created
+envelope is
+f(t) = γ(t) = λ(t)�ν(t) = (0, 0) + �ν(t) = �ν(t).
+Finally, by (3-∞) of Theorem 1, there are uncountably many distinct envelope created by C(γ,λ).
+Example 7. Let γ : R+ → R2 be the mapping defined by γ(t) = (t, 0) and let λ : R+ → R+ be the
+positive function defined by λ(t) = t2.
+The circle family C(γ,λ) and the candidate of its envelope is
+depicted in Figure 4. Defining the mapping ν : R+ → S1 by ν(t) = (0, −1) clarifies that the mapping γ
+Figure 4. The circle family C(γ,λ) and the candidate of its envelope.
+is a frontal. Then, µ(t) = J(ν(t)) = (1, 0) and β(t) = dγ
+dt (t) · µ(t) = (1, 0) · (1, 0) = 1. We want to seek a
+mapping �ν : R+ → S1 satisfying
+dλ
+dt (t) = −β(t) (�ν(t) · µ(t)) ,
+namely, a mapping �ν : R+ → S1 satisfying
+2t = −((�ν(t) · (1, 0))).
+Since �ν(t) ∈ S1, from the above expression, it follows that such �ν(t) does not exist if 1
+2 < t. Thus, the
+circle family C(γ,λ) is not creative and it creates no envelopes by (1) of Theorem 1.
+
+0.5
+0.5
+1.0
+0.5ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE
+9
+Example 8. This example is almost the same as Example 7. The difference from Example 7 is only the
+parameter space. In Example 8, the parameter space I is
+�
+0, 1
+2
+�
+. That is to say, in this example, R+ in
+Example 7 is replaced by
+�
+0, 1
+2
+�
+and all other settings in Example 7 remain without change.
+Then, from calculations in Example 7, it follows that the given circle family C(γ,λ) is creative. Thus,
+by (1) of Theorem 1, C(γ,λ) creates an envelope. It is easily seen that the expression of �ν(t) must be as
+follows.
+�ν(t) =
+�
+−2t, ±
+�
+1 − 4t2
+�
+.
+Therefore, by (2) of Theorem 1, an envelope f created by C(γ,λ) is parametrized as follows.
+f(t)
+=
+γ(t) + λ(t)�ν(t)
+=
+(t, 0) + t2 �
+−2t, ±
+�
+1 − 4t2
+�
+=
+�
+t − 2t3, ±t2�
+1 − 4t2
+�
+.
+Finally, by (3-ii) of Theorem 1, it follows that the number of distinct envelopes created by the circle
+family C(γ,λ) is exactly two.
+Example 9. Let γ : R → R2 be the mapping defined by γ(t) = (t3, t2) and let λ : R → R+ be the
+constant function defined by λ(t) = 1.
+The circle family C(γ,λ) and the candidate of its envelope is
+depicted in Figure 5. It is easily seen that the mapping ν : R → S1 defined by ν(t) =
+1
+√
+4+9t2 (2, −3t)
+Figure 5. The circle family C(γ,λ) and the candidate of its envelope.
+gives the Gauss mapping for γ. Thus, γ is a frontal. By definition, the mapping µ : R → S1 has the form
+µ(t) =
+1
+√
+4+9t2 (3t, 2). By calculation, we have
+β(t) = dγ
+dt (t) · µ(t) = t
+�
+4 + 9t2.
+Since the radius function λ is constant, it follows dλ
+dt (t) = 0. Thus, for any t ∈ R, the unit vector �ν(t)
+satisfying
+dλ
+dt (t) = −β(t) (�ν(t) · µ(t)) ,
+always exists. Namely we have
+�ν(t) = ±ν(t) =
+±1
+√
+4 + 9t2 (2, −3t) .
+
+4 F
+.4
+-2
+2
+4
+210
+Y. WANG AND T. NISHIMURA
+Thus, by (1) of Theorem 1, C(γ,λ) creates an envelope, and the created envelope f : R → R2 has the
+following form by (2) of Theorem 1.
+f(t) = γ(t) + λ(t)�ν(t) =
+�
+t3, t2�
+±
+1
+√
+4 + 9t2 (2, −3t) =
+�
+t3 ±
+2
+√
+4 + 9t2 , t2 ∓
+3t
+√
+4 + 9t2
+�
+.
+Finally, by (3-ii) of Theorem 1, there are no other envelopes created by C(γ,λ).
+4. Alternative definitions
+In Definition 2 of Section 1, the definition of envelope created by the circle family is given. In [1], the
+set consisting of the images of envelopes defined in Definition 2 is called E2 envelope (denoted by E2)
+and two alternative definitions (called E1 envelope and D envelope) are given as follows.
+Definition 4 (E1 envelope [1]). Let γ : I → R2, λ : I → R+ be a frontal and a positive function
+respectively. Let t0 be a parameter of I and fix it. Assume that
+lim
+ε→0 C(γ(t0),λ(t0)) ∩ C(γ(t0+ε),λ(t0+ε))
+is not the empty set and denote the set by I(t0). Take one point e1(t0) = (x(t0), y(t0)) of I(t0). Then, the
+set consisting of the images of smooth mappings e1 : I → R2, if exists, is called an E1 envelope created
+by the circle family C(γ,λ) and is denoted by E1.
+Definition 5 (D envelope [1]). Let γ : I → R2, λ : I → R+ be a frontal and a positive function
+respectively. Set
+F(x, y, t) = ||(x, y) − γ(t)||2 − (λ(t))2 .
+Then, the following set is called the D envelope created by the circle family C(γ,λ) and is denoted by D.
+�
+(x, y) ∈ R2 | ∃t ∈ I such that F(x, y, t) = ∂F
+∂t (x, y, t) = 0
+�
+.
+Concerning the relationships among E1, E2 and D for a given circle family C(γ,λ), the following is
+known.
+Fact 1 ([1]). E1 ⊂ D and E2 ⊂ D.
+In this section, we study more precise relationships among E1, E2 and D.
+4.1. The relationship between E1 and E2. We first establish the relationship between E1 and E2 as
+follows.
+Theorem 2. E1 = E2.
+Proof. We first show E1 ⊂ E2. Let t0 be a parameter of I and let {ti}i=1,2,... be a sequence of I conversing
+to t0. Take a point (x(t0), y(t0)) of E1. Then, we may assume that a point (x(ti), y(ti)) is taken from
+the intersection of two circles C(γ(ti), λ(ti)) ∩ C(γ(t0), λ(t0)) and satisfies
+lim
+ti→t0(x(ti), y(ti)) = (x(t0), y(t0)).
+Then, we have the following.
+||(x(ti), y(ti)) − γ(ti)||2
+=
+(λ(ti))2
+(1)
+||(x(ti), y(ti)) − γ(t0)||2
+=
+(λ(t0))2 .
+(2)
+For j = 0, 1, 2, . . ., set γ(tj) = (γx(tj), γy(tj)). Subtracting (2) from (1) yields the following.
+−2 (x(ti) (γx(ti) − γx(t0)) + y(ti) (γy(ti) − γy(t0))) + (γx(ti))2 − (γx(t0))2 + (γy(ti))2 − (γy(t0))2
+=
+(λ(ti))2 − (λ(t0))2 .
+Since limi→∞ ti = t0 and limti→t0(x(ti), y(ti)) = (x(t0), y(t0)), this equality implies
+−2
+�
+x(t0)dγx
+dt (t0) + y(t0)dγy
+dt (t0)
+�
++ 2
+�
+γx(t0)dγx
+dt (t0) + γy(t0)dγy
+dt (t0)
+�
+= 2λ(t0)dλ
+dt (t0).
+Hence we have
+−
+1
+λ(t0) (x(t0) − γx(t0), y(t0) − γy(t0)) ·
+�dγx
+dt (t0), dγy
+dt (t0)
+�
+= dλ
+dt (t0).
+
+ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE
+11
+Notice that the vector
+1
+λ(t0) (x(t0) − γx(t0), y(t0) − γy(t0)) =
+1
+λ(t0) ((x(t0), y(t0)) − γ(t0)) is a unit vector
+and
+�
+dγx
+dt (t0), dγy
+dt (t0)
+�
+= β(t0)µ(t0). Thus the creative condtion is satisfied at t = t0. Therefore, by the
+proof of (1) of Theorem 1, the point (x(t0), y(t0)) must belong to E2.
+Conversely, suppose that the circle family C(γ,λ) creates an E2 envelope f : I → R2. By (2) of Theorem
+1, f has the following representation.
+f(t) = γ(t) + λ(t)�ν(t).
+For a point P ∈ R2 and a unit vector v ∈ S1, the straight line L(P, v) is naturally defined as follows.
+L(P,v) =
+�
+(x, y) ∈ R2 | ((x, y) − P) · v = 0
+�
+.
+Then, since
+df
+dt (t)·�ν(t) =
+�dγ
+dt (t) + dλ
+dt (t) · �ν(t) + λ(t)d�ν
+dt (t)
+�
+·�ν(t) = dγ
+dt (t)·�ν(t)+dλ
+dt (t) = β(t) (µ(t) · �ν(t))+dλ
+dt (t) = 0,
+f is an E2 envelope created by the straight line family
+L(f,�ν) =
+�
+L(f(t),�ν(t))
+�
+t∈R .
+Take one parameter t0 ∈ I and let {ti}i=1,2,... ⊂ I be a sequence converging to t0. Since for the straight
+line family L(f,�ν) the image of E2 envelope is the same as E1 emvelope (see (c) of Theorem 1 in [6]), for
+any sufficiently large i ∈ N there exists a point
+(x(ti), y(ti)) ∈ L(f(t0),�ν(t0)) ∩ L(f(ti),�ν(ti))
+such that limi→∞ (x(ti), y(ti)) = f(t0). Hence for any sufficiently large i ∈ N there must exist a point
+(�x(ti), �y(ti)) ∈ C(γ(t0),λ(t0)) ∩ C(γ(ti),λ(ti))
+such that limi→∞ (�x(ti), �y(ti)) = f(t0) (see Figure 6). Therefore, the point f(t0) ∈ R2 belongs to E1.
+Figure
+6. Existence
+of
+(�x(ti), �y(ti))
+∈
+C(γ(t0),λ(t0)) ∩ C(γ(ti),λ(ti))
+satisfying
+limi→∞ (�x(ti), �y(ti)) = f(t0).
+Since f is an arbitrary envelop created by C(γ,λ) and t0 is an arbitrary parameter in I, it follows that
+E2 ⊂ E1.
+□
+
+L((ti),入ti)
+(α(ti),y(ti))
+f(to)
+L((to),入(to)
+f(ti)
+(α(ti), y(ti))
+C((to),入(to))
+((ti),入(ti))12
+Y. WANG AND T. NISHIMURA
+4.2. A relationship between E2 and D. In this subsection, we prove the following theorem which
+asserts that D = E2 if and only if γ : I → R2 is non-singular, and D contains not only E2 but also the
+circle C(γ(t),λ(t)) at a singular point t of γ when γ is singular.
+Theorem 3. Let γ : I → R2, λ : I → R+ be a frontal and a positive function respectively. Suppose that
+the circle family C(γ,λ) is creative. Then, the following hold.
+D = E2 ∪
+�
+� �
+t∈Σ(γ)
+C(γ(t),λ(t))
+�
+� .
+Here, Σ(γ) stands for the set consisting of singular points of γ : I → R2.
+Proof. Recall that
+D =
+�
+(x, y) ∈ R2 | ∃t ∈ I such that F(x, y, t) = ∂F
+∂t (x, y, t) = 0
+�
+.
+Let (x0, y0) be a point of D. Since F(x, y, t) = ||(x, y) − γ(t)||2 − |λ(t)|2, it follows the following (a) and
+(b).
+(a) There exists a t ∈ I such that ((x0, y0) − γ(t)) · ((x0, y0) − γ(t)) − (λ(t))2 = 0.
+(b)
+d(((x0,y0)−γ(t))·((x0,y0)−γ(t))−(λ(t))2)
+dt
+= 0.
+The condition (a) implies that there exists a t ∈ I and a unit vector ν1(t) ∈ S1 at the t ∈ I such that
+(x0, y0) = γ(t) − λ(t)ν1(t).
+The condition (b) implies that there exists a t ∈ I such that
+dγ
+dt (t) · ((x0, y0) − γ(t)) − dλ
+dt (t)λ(t) = 0.
+Since dγ
+dt (t) = β(t)µ(t), just by substituting, we have that there exists a t ∈ I and a unit vector ν1(t) ∈ S1
+at the t ∈ I satisfying
+λ(t)
+�
+β(t) (µ(t) · ν1(t)) + dλ
+dt (t)
+�
+= 0.
+Since λ(t) > 0 for any t ∈ I, it follows that there exists a t ∈ I and a unit vector ν1(t) ∈ S1 at the t ∈ I
+satisfying
+dλ
+dt (t) = −β(t) (µ(t) · ν1(t)) .
+On the other hand, since C(γ,λ) is creative, there must exist a smooth unit vector field �ν : I → S1
+along γ : I → R2 such that
+dλ
+dt (t) = −β(t) (µ(t) · �ν(t))
+for any t ∈ I. Suppose that the parameter t ∈ I is a regular point of γ. Then, β(t) ̸= 0 at the t ∈ I.
+Thus, at the t ∈ I, the unit vector ν1(t) must be �ν(t). Therefore, by the proof of (1) of Theorem 1, at
+the regular point t ∈ I of γ, it follows
+D = E2.
+Suppose that the parameter t ∈ I is a singular point of γ. Then, β(t) = 0 at the t ∈ I. Thus, for any
+unit vector v ∈ S1, the following holds at the t ∈ I.
+dλ
+dt (t) = −β(t) (µ(t) · v) .
+Hence, at the singular point t ∈ I, we may choose any unit vector v ∈ S1 as the unit vector ν1(x).
+Therefore, by the proof of (1) of Theorem 1, at the singular point t ∈ I of γ, it follows
+D = E2 ∪ C(γ(t),λ(t)).
+□
+
+ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE
+13
+Acknowledgement
+The first author is supported by the National Natural Science Foundation of China (Grant No.
+12001079), Fundamental Research Funds for the Central Universities (Grant No. 3132023205) and China
+Scholarship Council.
+References
+[1] J. W. Bruce and P. J. Giblin, Curves and Singularities (second edition), Cambridge University Press, Cambridge, 1992.
+https://doi.org/10.1017/CBO9781139172615
+[2] T. Fukunaga and M. Takahashi, Existence and uniqueness for Legendre curves, J. Geom., 104 (2013), 297–307.
+https://doi.org/10.1007/s00022-013-0162-6
+[3] E. Hairer and G. Wanner, Analysis by Its History, Undergraduate Texts in Mathematics, Springer New York, NY,
+2008. https://doi.org/10.1007/978-0-387-77036-9
+[4] G. Ishikawa, Singularities of frontals, Adv. Stud. Pure Math., 78, 55–106, Math. Soc. Japan, Tokyo, 2018.
+https://doi.org/10.2969/aspm/07810055
+[5] S. Janeczko and T. Nishimura, Anti-orthotomics of frontals and their applications, J. Math. Anal. Appl., 487 (2020),
+124019. https://doi.org/10.1016/j.jmaa.2020.124019
+[6] T. Nishimura, Hyperplane families creating envelopes, Nonlinearity, 35 (2022), 2588. https://doi.org/10.1088/1361-
+6544/ac61a0
+School of Science, Dalian Maritime University, Dalian 116026, P.R. China
+Email address: wangyq@dlmu.edu.cn
+Research Institute of Environment and Information Sciences, Yokohama National University, Yokohama
+240-8501, Japan
+Email address: nishimura-takashi-yx@ynu.ac.jp
+

JdE3T4oBgHgl3EQfXQrW/content/tmp_files/load_file.txt

@@ -0,0 +1,466 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf,len=465
+page_content='ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE  YONGQIAO WANG AND TAKASHI NISHIMURA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In this paper, on envelopes created by circle families in the plane, answers to all four basic  problems (existence problem, representation problem, problem on the number of envelopes, problem on  relationships of definitions) are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Introduction  Throughout this paper, I is an open interval and all functions, mappings are of class C∞ unless  otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Envelopes of planar regular curve families have fascinated many pioneers since the dawn of differential  analysis (for instance, see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In most typical cases, straight line families have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In [6], by  solving four basic problems on envelopes created by straight line families in the plane (existence problem,  representation problem, uniqueness problem and equivalence problem of definitions), the second author  constructs a general theory for envelopes created by straight line families in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' On the other  hand, circle families in the plane are non-negligible families because the envelopes of them have already  had an important application, namely, an application to Seismic Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Following 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='14(9) of [1], a brief  explanation of Seismic Survey is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In the Eucledian plane R2, consider the “ground level  curve” C parametrized by γ : I → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Suppose that there is a stratum of granite below the top layer of  sandstone and that the dividing curve, denoted by M, is parametrized by �f : I → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Seismic Survey is  the following method to obtain an approximation of �f as precisely as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Take one fixed point A of  C and consider an explosion at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Assume that the sound waves travel in straight lines and are reflected  from M, arriving back at points γ(t) of C where their times of arrival are exactly recorded by sensors  located along C (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' It is known that there exists a curve W parametrized by f : I → R2  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Reflection of sound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  with well-defined normals such that each broken line of a reflected ray starting at A and finishing on C  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' 57R45, 58C25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Circle family, Envelope, Frontal, Creative, Creator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='04478v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='DG]  11 Jan 2023  A  (ti)(t2)(t3)(t4)(ts)  C  M2  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' NISHIMURA  can be replaced by a straight line which is normal to W and of the same total length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' The curve W is  called the orthotomic of M relative to A and conversely the curve M is called the anti-orthotomic of W  relative to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, an envelope created by the circle family  �  (x, y) ∈ R2 �� ||(x, y) − γ(t)|| = ||f(t) − γ(t)||  �  t∈I  recovers W (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' After obtaining the parametrization f of W, the parametrization �f of M  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' An envelope created by the circle family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  can be easily obtained by using the anti-orthotomic technique developed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Therefore, in order to  investigate the parametrization of W as precisely as possible, construction of general theory on envelopes  created by circle families is very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  In this paper, we construct a general theory on envelopes created by circle families in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' For a  point P of R2 and a positive number λ, the circle C(P,λ) centered at P with radius λ is naturally defined  as follows, where the dot in the center stands for the standard scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  C(P,λ) =  �  (x, y) ∈ R2 �� ((x, y) − P) · ((x, y) − P) = λ2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  For a curve γ : I → R2 and a positive function λ : I → R+, the circle family C(γ,λ) is naturally defined as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Here, R+ stands for the set consisting of positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  C(γ,λ) =  �  C(γ(t),λ(t))  �  t∈I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  It is reasonable to assume that at each point γ(t) the normal vector to the curve γ is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus,  we easily reach the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' A curve γ : I → R2 is called a frontal if there exists a mapping ν : I → S1 such that the  following identity holds for each t ∈ I, where S1 is the unit circle in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  dγ  dt (t) · ν(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  For a frontal γ, the mapping ν : I → S1 given above is called the Gauss mapping of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  By definition, a frontal is a solution of the first order linear differential equation defined by Gauss mapping  ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, for a fixed mapping ν : I → S1 the set consisting of frontals with a given Gauss mapping  ν : I → S1 is a linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' For frontals, [4] is recommended as an excellent reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Hereafter in this  paper, the curve γ : I → R2 for a circle family C(γ,λ) is assumed to be a frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  In this paper, the following is adopted as the definition of an envelope created by a circle family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let C(γ,λ) be a circle family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' A mapping f : I → R2 is called an envelope created by C(γ,λ)  if there exists a mapping �ν : I → S1 such that the following two hold for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (1)  df  dt(t) · �ν(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  A  (ti) (t2) (t3) (t4) (ts)  f(ts)  f(t4)  f(t2)f(t3)  f(ti)ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE  3  (2) f(t) ∈ C(γ(t),λ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  By definition, as same as an envelope created by a hyperplane family (see [6]), an envelope created by  a circle family is a solution of a first order linear differential equation with one constraint condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Moreover, again by definition, an envelope created by a circle family is a frontal with Gauss mapping  �ν : I → S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' On the other hand, since there is one constraint condition, again as same as an envelope  created by a hyperplane family, the set of envelopes created by a given circle family is in general not a  linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (1) Given a circle family C(γ,λ), find a necessary and sufficient codition for the family  to create an envelope in terms of γ, ν and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (2) Suppose that a circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Then, find a parametrization of the  envelope in terms of γ, ν and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (3) Suppose that a circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, find a criterion for the number of  distinct envelopes created by C(γ,λ) in terms of γ, ν and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Note 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (1) (1) of Problem 1 is a problem to seek the integrability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' There are various cases, for instance  the concentric circle family {{(x, y) ∈ R2 | x2 + y2 = t2}}t∈R+ does not create an envelope while the  parallel-translated circle family {{(x, y) ∈ R2 | (x − t)2 + y2 = 1}}t∈R does create two envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus,  (1) of Problem 1 is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (2) The following Example 1 shows that the apparently well-known method to obtain the envelope seems  to be useless in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, (2) of Problem 1 is important and the positive answer to it is much  desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (3) The following Example 2 shows that there are at least three cases: the case having a unique envelope,  the case having exactly two envelopes and the case having uncountably many envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, (3) of  Problem 1 is meaningful and interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let γ : R → R2 be the mapping defined by γ(t) =  �  t3, t6�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Set ν(t) =  1  √  4t6+1  �  −2t3, 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' It  is clear that the mapping γ is a frontal with Gauss mapping ν : R → S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let λ : R → R+ be the constant  function defined by λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Then, it seems that the circle family C(γ,λ) creates envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Thus,  we can expect that the created envelopes can be obtained by the well-known method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Set F(x, y, t) =  �  x − t3�2 +  �  y − t6�2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  D  =  �  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' y) ∈ R2  ���� ∃t such that F(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' t) = ∂F  ∂t (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' t) = 0  �  =  �  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' y) ∈ R2 ��� ∃t such that  �  x − t3�2 +  �  y − t6�2 − 1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' −6t2 �  x − t3�  − 12t5 �  y − t6�  = 0  �  =  �  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' y) ∈ R2 ��� ∃t such that  �  x − t3�2 +  �  y − t6�2 − 1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' t2 ��  x − t3�  + 2t3 �  y − t6��  = 0  �  =  �  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' y) ∈ R2 �� x2 + y2 = 1  � � �  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' y) ∈ R2 ���  �  x − t3�2 +  �  y − t6�2 − 1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' x = t3 − 2t3 �  y − t6��  =  �  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' y) ∈ R2 �� x2 + y2 = 1  � � �  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' y) ∈ R2 ���  �  −2t3 �  y − t6��2 +  �  y − t6�2 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' x = t3 �  1 − 2y + 2t6��  =  �  (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' y) ∈ R2 �� x2 + y2 = 1  � � ��  t3 ∓  2t3  √  4t6 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' t6 ±  1  √  4t6 + 1  �  ∈ R2  ���� t ∈ R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  In Example 3 of Section 3, it turns out that the set D calculated here is actually larger than the set of  envelopes created by C(γ,λ), namely the unit circle  �  (x, y) ∈ R2 �� x2 + y2 = 1  �  is redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Therefore,  unfortunately, the apparently well-known method to obtain the envelopes does not work well in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  The circle family C(γ,λ) and the candidate of its envelope are depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (1) Let γ : R+ → R2 be the mapping defined by γ(t) = (0, 1 + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, it is clear that  γ is a frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let λ : R+ → R+ be the positive function defined by λ(t) = 1+t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, it is easily  seen that the origin (0, 0) of the plane R2 itself is a created envelope by the circle family C(γ,λ)  and that there are no other envelopes created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Hence, the number of created envelopes  is one in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (2) The parallel-translated circle family {{(x, y) ∈ R2 | (x − t)2 + y2 = 1}}t∈R creates exactly two  envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (3) Let γ : R → R2 be the constant mapping defined by γ(t) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, it is clear that γ is a  frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let λ : R → R+ be the constant function defined by λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, for any function  4  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' NISHIMURA  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' The circle family C(γ,λ) and the candidate of its envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  θ : R → R, the mapping f : R → R2 defined by f(t) = (cos θ(t), sin θ(t)) is an envelope created  by the circle family C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Hence, there are uncountably many created envelopes in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  In order to solve Problem 1, we prepare several terminologies that can be derived from a frontal  γ : I → R2 with Gauss mapping ν : I → S1 and a positive function λ : I → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' For a frontal γ : I → R2  with Gauss mapping ν : I → S1, following [2], we set µ(t) = J(ν(t)), where J is the anti-clockwise  rotation by π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then we have a moving frame {µ(t), ν(t)}t∈I along the frontal γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Set  ℓ(t) = dν  dt (t) · µ(t),  β(t) = dγ  dt (t) · µ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  The pair of functions (ℓ, β) is called the curvature of the frontal γ with Gauss mapping ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' We want to  focus the ratio of dλ  dt (t) and β(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' The following definition is the key of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let γ : I → R2, λ : I → R+ be a frontal with Gauss mapping ν : I → S1 and a positive  function respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Then, the circle family C(γ,λ) is said to be creative if there exists a mapping  �ν : I → S1 such that the following identity holds for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  dλ  dt (t) = −β(t) (�ν(t) · µ(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Set cos θ(t) = −�ν(t) · µ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, the creative condition is equivalent to say that there exists a function  θ : I → R satisfying the following identity for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  dλ  dt (t) = β(t) cos θ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  By definition, any family of concentric circles with expanding radius is not creative, and it is clear that  such the circle family does not create an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Under the above preparation, Problem 1 is solved as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let γ : I → R2 be a frontal with Gauss mapping ν : I → S1 and let λ : I → R+ be a  positive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, the following three holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (1) The circle family C(γ,λ) creates an envelope if and only if C(γ,λ) is creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (2) Suppose that the circle family C(γ,λ) creates an envelope f : I → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, the created envelope  f is represented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  f(t) = γ(t) + λ(t)�ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  where �ν : I → S1 is the mapping de���ned in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE  5  (3) Suppose that the circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, the number of envelopes created  by C(γ,λ) is characterized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (3-i) The circle family C(γ,λ) creates a uinique envelope if and only if the set consisting of t ∈ I  satisfying β(t) ̸= 0 and dλ  dt (t) = ±β(t) is dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (3-ii) There are exactly two distinct envelopes created by C(γ,λ) if and only if the set of t ∈ I  satisfying β(t) ̸= 0 is dense in I and there exists at least one t0 ∈ I such that the strict  inequality | dλ  dt (t0)| < |β(t0)| holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (3-∞) There are uncountably many distinct envelopes created by C(γ,λ) if and only if the set of t ∈ I  satisfying β(t) ̸= 0 is not dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  By the assertion (2) of Theorem 1, it is reasonable to call �ν the creator for an envelope f created by  C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Theorem 1 is proved in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In Section 3, several examples to  which Theorem 1 is effectively applicable are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Finally, in Section 4, relations of several definitions  of an envelope created by a circle family are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Proof of Theorem 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Proof of the assertion (1) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Suppose that C(γ,λ) is creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' By definition, there  exists a mapping �ν : I → S1 such that the equality dλ  dt (t) = −β(t) (�ν(t) · µ(t)) holds for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Set  f(t) = γ(t) + λ(t)�ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Then, since (f(t) − γ(t)) · (f(t) − γ(t)) = λ2(t), it follows f(t) ∈ C(γ(t),λ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Morever, since  df  dt (t) = dγ  dt (t) + dλ  dt (t)�ν(t) + λ(t)d�ν  dt (t),  we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  df  dt (t) · (f(t) − γ(t))  =  �dγ  dt (t) + dλ  dt (t)�ν(t) + λ(t)d�ν  dt (t)  �  (λ(t)�ν(t))  =  dγ  dt (t) · (λ(t)�ν(t)) + dλ  dt (t)λ(t)  =  (β(t)µ(t)) · (λ(t)�ν(t)) + (−β(t) (�ν(t) · µ(t))) λ(t)  =  β(t)λ(t) (µ(t) · �ν(t)) − β(t)λ(t) (�ν(t) · µ(t))  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Hence, f is an envelope created by the circle family C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Conversely, suppose that the circle family C(γ,λ) creates an envelope f : I → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, by definition, it  follows that f(t) ∈ C(γ(t),λ(t)) and df  dt(t) · (f(t) − γ(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' The condition f(t) ∈ C(γ(t),λ(t)) implies that  there exists a mapping �ν : I → S1 such that the following equality holds for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  f(t) = γ(t) + λ(t)�ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Then, since  df  dt (t) = dγ  dt (t) + dλ  dt (t)�ν(t) + λ(t)d�ν  dt (t),  we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  0  =  df  dt (t) · (f(t) − γ(t))  =  �dγ  dt (t) + dλ  dt (t)�ν(t) + λ(t)d�ν  dt (t)  �  (λ(t)�ν(t))  =  (β(t)µ(t)) · (λ(t)�ν(t)) + dλ  dt (t)λ(t)  =  λ(t)  �  β(t) (µ(t) · �ν(t)) + dλ  dt (t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  6  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' NISHIMURA  Since λ(t) is positive for any t ∈ I, it follows  β(t) (µ(t) · �ν(t)) + dλ  dt (t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Therefore, the circle family C(γ,λ) is creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Proof of the assertion (2) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' The proof of the assertion (1) given in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='1  proves the assertion (2) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Proof of the assertion (3) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Proof of (3-i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Suppose that the circle family C(γ,λ) creates a unique envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, for any t ∈ I  the unit vector �ν(t) satisfying  dλ  dt (t) = −β(t) (�ν(t) · µ(t))  must be uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Hence, under considering continuity of two functions dλ  dt and β, it follows  that the set consisting of t ∈ I satisfying dλ  dt (t) = ±β(t) ̸= 0 must be dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Conversely, suppose that the set consisting of t ∈ I satisfying dλ  dt (t) = ±β(t) ̸= 0 is dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then,  under considering continuity of the function t �→ �ν(t) · µ(t), it follows that �ν(t) · µ(t) = ±1 for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Thus, the created envelope f(t) = γ(t) + λ(t)�ν(t) must be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Proof of (3-ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Suppose that there are exactly two distinct envelopes created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, by  the equality dλ  dt (t) = −β(t) (�ν(t) · µ(t)) , the set consisting of t ∈ I satisfying β(t) ̸= 0 must be dense in  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Suppose moreover that the set of t ∈ I satisfying the equality dλ  dt (t) = ±β(t) holds for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Then, it follows that the set consisting of t ∈ I satisfying dλ  dt (t) = ±β(t) ̸= 0 is dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, by the  assertion (3-i), the given circle family must create a unique envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' This contradicts the assumption  that there are exactly two distinct envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Hence, there must exist at least one t0 ∈ I such that the  strict inequality | dλ  dt (t0)| < |β(t0)| holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Conversely, suppose that the set of t ∈ I satisfying β(t) ̸= 0 is dense in I and there exists at least  one t0 ∈ I such that the strict inequality | dλ  dt (t0)| < |β(t0)| holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, it follows that there must exist  an open interval �I in I such that the absolute value |�ν(t) · µ(t)| = | cos θ(t)| is less than 1 for any t ∈ �I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Thus, it follows θ(t) ̸= −θ(t) for any t ∈ �I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Hence, for any t ∈ �I, there exist exactly two distinct unit  vectors �ν+(t), �ν−(t) corresponding �ν+(t) · µ(t) = − cos θ(t) and �ν−(t) · µ(t) = − cos (−θ(t)) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Therefore, the circle family must create exactly two distinct envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Proof of (3-∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Suppose that there are uncountably many distinct envelopes created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Suppose moreover that the set of t ∈ I such that β(t) ̸= 0 is dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, from (3-i) and (3-ii),  it follows that the circle family C(γ,λ) must create a unique envelope or two distinct envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' This  contradicts the assumption that there are uncountably many distinct envelopes created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Hence,  the set of t ∈ I such that β(t) ̸= 0 is never dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Conversely, suppose that the set of t ∈ I such that β(t) ̸= 0 is not dense in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' This assumption implies  that there exists an open interval �I in I such that β(t) = 0 for any t ∈ �I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' On the other hand, since C(γ,λ)  creates an envelope f0, the equality  dλ  dt (t) = −β(t) (�ν(t) · µ(t))  holds for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, there are no restrictions for the value �ν(t) · µ(t) for any t ∈ �I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Take one  point t0 of �I and denote the �ν for the envelope f0 by �ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, by using the standard technique on  bump functions, we may construct uncountably many distinct creators �νa : I → S1 (a ∈ A) such that  the following (a), (b), (c) and (d) hold, where A is a set consisting uncountably many elements such that  0 ̸∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (a) The equality dλ  dt (t) = −β(t) (�νa(t) · µ(t)) holds for any t ∈ I and any a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (b) For any t ∈ I − �I and any a ∈ A, the equality �νa(t) = �ν0(t) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (c) For any a ∈ A, the property �νa(t0) ̸= �ν0(t0) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (d) For any wo distinct a1, a2 ∈ A, the property �νa1(t0) ̸= �νa2(t0) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Therefore, the circle family C(γ,λ) creates uncountably many distinct envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  2  ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Examples  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' We examine Example 1 by applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In Example 1, γ : R → R2 is given by  γ(t) =  �  t3, t6�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, we can say that ν : R → S1 and µ : R → S1 are given by ν(t) =  1  √  4t6+1  �  −2t3, 1  �  and µ(t) =  1  √  4t6+1  �  −1, −2t3�  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Moreover, the radius function λ : R → R is the constant  function defined by λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus,  dλ  dt (t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  By calculation, we have  β(t) = dγ  dt (t) · µ(t) = −3t2(1 + 4t6)  √  4t6 + 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Therefore, the unit vector �ν(t) ∈ S1 satisfying  dλ  dt (t) = −β(t) (�ν(t) · µ(t))  exsists and it must have the form  �ν(t) = ±ν(t) =  ±1  √  4t6 + 1  �  −2t3, 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Hence, by (1) of Theorem 1, the circle family C(γ,λ) creates an envelope f : R → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' By (2) of Theorem  1, f is parametrized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  f(t)  =  γ(t) + λ(t)�ν(t)  =  �  t3, t6�  ±  1  √  4t6 + 1  �  −2t3, 1  �  =  �  t3 ∓  2t3  √  4t6 + 1  , t6 ±  1  √  4t6 + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Finally, by (3-ii) of Theorem 1, the number of distinct envelopes created by the circle family C(γ,λ) is  exactly two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Therefore, Theorem 1 reveals that the set D calculated in Example 1 is certainly the union of the unit  circle and the set of two envelopes of C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' We examine (1) of Example 2 by applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In (1) of Example 2, γ : R+ → R2  is given by γ(t) = (0, 1 + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, if we define the unit vector ν(t) = (1, 0), ν : R+ → S1 gives the Gauss  mapping of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' By definition, µ(t) = (0, 1) and thus we have β(t) = dγ  dt (t) · µ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' On the other hand,  the radius function λ : R+ → R+ has the form λ(t) = 1 + t in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, the created condition  dλ  dt (t) = −β(t) (�ν(t) · µ(t))  becomes simply  (∗)  1 = − (�ν(t) · (0, 1))  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' If we take �ν(t) = (0, −1), then the above equality holds for any t ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, by (1) of  Theorem 1, the circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' By (2) of Theorem 1, the parametrization of the  created envelope is  f(t) = γ(t) = λ(t)�ν(t) = (0, 1 + t) + (1 + t) (0, −1) = (0, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Finally, notice that for any t ∈ R+ the creative condition (*) in this case holds if and only if �ν(t) =  (0, −1) = −µ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, by (3-i) of Theorem 1, the origin (0, 0) is the unique envelope created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Theorem 1 can be applied also to (2) of Example 2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In this example, γ(t) = (t, 0)  and λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, we may take ν(t) = (0, −1), µ(t) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' We have β(t) = dγ  dt (t) · µ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Since the  radius function λ is a constant function, the created condition  dλ  dt (t) = −β(t) (�ν(t) · µ(t))  becomes simply  0 = − (�ν(t) · (0, 1))  8  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' NISHIMURA  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, for any t ∈ R, the created condition is satisfied if and only if �ν(t) = ±(1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Hence, by  (1) of Theorem 1, the circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' By (2) of Theorem 1, the parametrization  of the created envelope is  f(t) = γ(t) = λ(t)�ν(t) = (t, 0) ± (0, −1) = (t, ∓1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Finally, by (3-ii) of Theorem 1, the number of envelope created by C(γ,λ) is exactly two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Theorem 1 can be applied even to (3) of Example 2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In this example, γ(t) = (0, 0)  and λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, every mapping ν : R → S1 can be taken as Gauss mapping of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In particular, γ is  a frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' We have β(t) = dγ  dt (t) · µ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Since the radius function λ is a constant function λ(t) = 1,  the created condition  dλ  dt (t) = −β(t) (�ν(t) · µ(t))  becomes simply  0 = 0  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, for any �ν : R → S1, the created condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Hence, by (1) of Theorem  1, the circle family C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' By (2) of Theorem 1, the parametrization of the created  envelope is  f(t) = γ(t) = λ(t)�ν(t) = (0, 0) + �ν(t) = �ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Finally, by (3-∞) of Theorem 1, there are uncountably many distinct envelope created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let γ : R+ → R2 be the mapping defined by γ(t) = (t, 0) and let λ : R+ → R+ be the  positive function defined by λ(t) = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  The circle family C(γ,λ) and the candidate of its envelope is  depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Defining the mapping ν : R+ → S1 by ν(t) = (0, −1) clarifies that the mapping γ  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' The circle family C(γ,λ) and the candidate of its envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  is a frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, µ(t) = J(ν(t)) = (1, 0) and β(t) = dγ  dt (t) · µ(t) = (1, 0) · (1, 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' We want to seek a  mapping �ν : R+ → S1 satisfying  dλ  dt (t) = −β(t) (�ν(t) · µ(t)) ,  namely, a mapping ��ν : R+ → S1 satisfying  2t = −((�ν(t) · (1, 0))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Since �ν(t) ∈ S1, from the above expression, it follows that such �ν(t) does not exist if 1  2 < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, the  circle family C(γ,λ) is not creative and it creates no envelopes by (1) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='5ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE  9  Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' This example is almost the same as Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' The difference from Example 7 is only the  parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In Example 8, the parameter space I is  �  0, 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' That is to say, in this example, R+ in  Example 7 is replaced by  �  0, 1  2  �  and all other settings in Example 7 remain without change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Then, from calculations in Example 7, it follows that the given circle family C(γ,λ) is creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus,  by (1) of Theorem 1, C(γ,λ) creates an envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' It is easily seen that the expression of �ν(t) must be as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  �ν(t) =  �  −2t, ±  �  1 − 4t2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Therefore, by (2) of Theorem 1, an envelope f created by C(γ,λ) is parametrized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  f(t)  =  γ(t) + λ(t)�ν(t)  =  (t, 0) + t2 �  −2t, ±  �  1 − 4t2  �  =  �  t − 2t3, ±t2�  1 − 4t2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Finally, by (3-ii) of Theorem 1, it follows that the number of distinct envelopes created by the circle  family C(γ,λ) is exactly two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let γ : R → R2 be the mapping defined by γ(t) = (t3, t2) and let λ : R → R+ be the  constant function defined by λ(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  The circle family C(γ,λ) and the candidate of its envelope is  depicted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' It is easily seen that the mapping ν : R → S1 defined by ν(t) =  1  √  4+9t2 (2, −3t)  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' The circle family C(γ,λ) and the candidate of its envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  gives the Gauss mapping for γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, γ is a frontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' By definition, the mapping µ : R → S1 has the form  µ(t) =  1  √  4+9t2 (3t, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' By calculation, we have  β(t) = dγ  dt (t) · µ(t) = t  �  4 + 9t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Since the radius function λ is constant, it follows dλ  dt (t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, for any t ∈ R, the unit vector �ν(t)  satisfying  dλ  dt (t) = −β(t) (�ν(t) · µ(t)) ,  always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Namely we have  �ν(t) = ±ν(t) =  ±1  √  4 + 9t2 (2, −3t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  4 F  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='4  2  2  4  210  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' NISHIMURA  Thus, by (1) of Theorem 1, C(γ,λ) creates an envelope, and the created envelope f : R → R2 has the  following form by (2) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  f(t) = γ(t) + λ(t)�ν(t) =  �  t3, t2�  ±  1  √  4 + 9t2 (2, −3t) =  �  t3 ±  2  √  4 + 9t2 , t2 ∓  3t  √  4 + 9t2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Finally, by (3-ii) of Theorem 1, there are no other envelopes created by C(γ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Alternative definitions  In Definition 2 of Section 1, the definition of envelope created by the circle family is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In [1], the  set consisting of the images of envelopes defined in Definition 2 is called E2 envelope (denoted by E2)  and two alternative definitions (called E1 envelope and D envelope) are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Definition 4 (E1 envelope [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let γ : I → R2, λ : I → R+ be a frontal and a positive function  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let t0 be a parameter of I and fix it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Assume that  lim  ε→0 C(γ(t0),λ(t0)) ∩ C(γ(t0+ε),λ(t0+ε))  is not the empty set and denote the set by I(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Take one point e1(t0) = (x(t0), y(t0)) of I(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, the  set consisting of the images of smooth mappings e1 : I → R2, if exists, is called an E1 envelope created  by the circle family C(γ,λ) and is denoted by E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Definition 5 (D envelope [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let γ : I → R2, λ : I → R+ be a frontal and a positive function  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Set  F(x, y, t) = ||(x, y) − γ(t)||2 − (λ(t))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Then, the following set is called the D envelope created by the circle family C(γ,λ) and is denoted by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  �  (x, y) ∈ R2 | ∃t ∈ I such that F(x, y, t) = ∂F  ∂t (x, y, t) = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Concerning the relationships among E1, E2 and D for a given circle family C(γ,λ), the following is  known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Fact 1 ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' E1 ⊂ D and E2 ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  In this section, we study more precise relationships among E1, E2 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' The relationship between E1 and E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' We first establish the relationship between E1 and E2 as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' E1 = E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' We first show E1 ⊂ E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let t0 be a parameter of I and let {ti}i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' be a sequence of I conversing  to t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Take a point (x(t0), y(t0)) of E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, we may assume that a point (x(ti), y(ti)) is taken from  the intersection of two circles C(γ(ti), λ(ti)) ∩ C(γ(t0), λ(t0)) and satisfies  lim  ti→t0(x(ti), y(ti)) = (x(t0), y(t0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Then, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  ||(x(ti), y(ti)) − γ(ti)||2  =  (λ(ti))2  (1)  ||(x(ti), y(ti)) − γ(t0)||2  =  (λ(t0))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (2)  For j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=', set γ(tj) = (γx(tj), γy(tj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Subtracting (2) from (1) yields the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  −2 (x(ti) (γx(ti) − γx(t0)) + y(ti) (γy(ti) − γy(t0))) + (γx(ti))2 − (γx(t0))2 + (γy(ti))2 − (γy(t0))2  =  (λ(ti))2 − (λ(t0))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Since limi→∞ ti = t0 and limti→t0(x(ti), y(ti)) = (x(t0), y(t0)), this equality implies  −2  �  x(t0)dγx  dt (t0) + y(t0)dγy  dt (t0)  �  + 2  �  γx(t0)dγx  dt (t0) + γy(t0)dγy  dt (t0)  �  = 2λ(t0)dλ  dt (t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Hence we have  −  1  λ(t0) (x(t0) − γx(t0), y(t0) − γy(t0)) ·  �dγx  dt (t0), dγy  dt (t0)  �  = dλ  dt (t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE  11  Notice that the vector  1  λ(t0) (x(t0) − γx(t0), y(t0) − γy(t0)) =  1  λ(t0) ((x(t0), y(t0)) − γ(t0)) is a unit vector  and  �  dγx  dt (t0), dγy  dt (t0)  �  = β(t0)µ(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus the creative condtion is satisfied at t = t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Therefore, by the  proof of (1) of Theorem 1, the point (x(t0), y(t0)) must belong to E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Conversely, suppose that the circle family C(γ,λ) creates an E2 envelope f : I → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' By (2) of Theorem  1, f has the following representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  f(t) = γ(t) + λ(t)�ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  For a point P ∈ R2 and a unit vector v ∈ S1, the straight line L(P, v) is naturally defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  L(P,v) =  �  (x, y) ∈ R2 | ((x, y) − P) · v = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Then, since  df  dt (t)·�ν(t) =  �dγ  dt (t) + dλ  dt (t) · �ν(t) + λ(t)d�ν  dt (t)  �  �ν(t) = dγ  dt (t)·�ν(t)+dλ  dt (t) = β(t) (µ(t) · �ν(t))+dλ  dt (t) = 0,  f is an E2 envelope created by the straight line family  L(f,�ν) =  �  L(f(t),�ν(t))  �  t∈R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Take one parameter t0 ∈ I and let {ti}i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' ⊂ I be a sequence converging to t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Since for the straight  line family L(f,�ν) the image of E2 envelope is the same as E1 emvelope (see (c) of Theorem 1 in [6]), for  any sufficiently large i ∈ N there exists a point  (x(ti), y(ti)) ∈ L(f(t0),�ν(t0)) ∩ L(f(ti),�ν(ti))  such that limi→∞ (x(ti), y(ti)) = f(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Hence for any sufficiently large i ∈ N there must exist a point  (�x(ti), �y(ti)) ∈ C(γ(t0),λ(t0)) ∩ C(γ(ti),λ(ti))  such that limi→∞ (�x(ti), �y(ti)) = f(t0) (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Therefore, the point f(t0) ∈ R2 belongs to E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Figure  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Existence  of  (�x(ti), �y(ti))  ∈  C(γ(t0),λ(t0)) ∩ C(γ(ti),λ(ti))  satisfying  limi→∞ (�x(ti), �y(ti)) = f(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Since f is an arbitrary envelop created by C(γ,λ) and t0 is an arbitrary parameter in I, it follows that  E2 ⊂ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  □  L((ti),入ti)  (α(ti),y(ti))  f(to)  L((to),入(to)  f(ti)  (α(ti), y(ti))  C((to),入(to))  ((ti),入(ti))12  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' WANG AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' NISHIMURA  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' A relationship between E2 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' In this subsection, we prove the following theorem which  asserts that D = E2 if and only if γ : I → R2 is non-singular, and D contains not only E2 but also the  circle C(γ(t),λ(t)) at a singular point t of γ when γ is singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Let γ : I → R2, λ : I → R+ be a frontal and a positive function respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Suppose that  the circle family C(γ,λ) is creative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  D = E2 ∪  �  � �  t∈Σ(γ)  C(γ(t),λ(t))  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Here, Σ(γ) stands for the set consisting of singular points of γ : I → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Recall that  D =  �  (x, y) ∈ R2 | ∃t ∈ I such that F(x, y, t) = ∂F  ∂t (x, y, t) = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Let (x0, y0) be a point of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Since F(x, y, t) = ||(x, y) − γ(t)||2 − |λ(t)|2, it follows the following (a) and  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (a) There exists a t ∈ I such that ((x0, y0) − γ(t)) · ((x0, y0) − γ(t)) − (λ(t))2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  (b)  d(((x0,y0)−γ(t))·((x0,y0)−γ(t))−(λ(t))2)  dt  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  The condition (a) implies that there exists a t ∈ I and a unit vector ν1(t) ∈ S1 at the t ∈ I such that  (x0, y0) = γ(t) − λ(t)ν1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  The condition (b) implies that there exists a t ∈ I such that  dγ  dt (t) · ((x0, y0) − γ(t)) − dλ  dt (t)λ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Since dγ  dt (t) = β(t)µ(t), just by substituting, we have that there exists a t ∈ I and a unit vector ν1(t) ∈ S1  at the t ∈ I satisfying  λ(t)  �  β(t) (µ(t) · ν1(t)) + dλ  dt (t)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Since λ(t) > 0 for any t ∈ I, it follows that there exists a t ∈ I and a unit vector ν1(t) ∈ S1 at the t ∈ I  satisfying  dλ  dt (t) = −β(t) (µ(t) · ν1(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  On the other hand, since C(γ,λ) is creative, there must exist a smooth unit vector field �ν : I → S1  along γ : I → R2 such that  dλ  dt (t) = −β(t) (µ(t) · �ν(t))  for any t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Suppose that the parameter t ∈ I is a regular point of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, β(t) ̸= 0 at the t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Thus, at the t ∈ I, the unit vector ν1(t) must be �ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Therefore, by the proof of (1) of Theorem 1, at  the regular point t ∈ I of γ, it follows  D = E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Suppose that the parameter t ∈ I is a singular point of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Then, β(t) = 0 at the t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Thus, for any  unit vector v ∈ S1, the following holds at the t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  dλ  dt (t) = −β(t) (µ(t) · v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Hence, at the singular point t ∈ I, we may choose any unit vector v ∈ S1 as the unit vector ν1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  Therefore, by the proof of (1) of Theorem 1, at the singular point t ∈ I of γ, it follows  D = E2 ∪ C(γ(t),λ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  □  ENVELOPES CREATED BY CIRCLE FAMILIES IN THE PLANE  13  Acknowledgement  The first author is supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  12001079), Fundamental Research Funds for the Central Universities (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' 3132023205) and China  Scholarship Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
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+page_content=' Fukunaga and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Takahashi, Existence and uniqueness for Legendre curves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
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+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='124019  [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' Nishimura, Hyperplane families creating envelopes, Nonlinearity, 35 (2022), 2588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='1088/1361-  6544/ac61a0  School of Science, Dalian Maritime University, Dalian 116026, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content=' China  Email address: wangyq@dlmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
+page_content='cn  Research Institute of Environment and Information Sciences, Yokohama National University, Yokohama  240-8501, Japan  Email address: nishimura-takashi-yx@ynu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfXQrW/content/2301.04478v1.pdf'}
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@@ -0,0 +1,897 @@
+Human Cognition Surpasses the Nonlocality Tsirelson Bound: Is Mind Outside of
+Spacetime?
+Stuart Kauffman1, Emeritus Professor of Biochemistry and Biophysics, University of
+Pennsylvania, Philadelphia, Pennsylvania 19104, USA
+Sudip Patra2, Associate Professor OP Jindal Global University, Founding member CEASP.
+India.
+Dec 26, 2022
+Abstract
+Recent experimental studies on human cognition, particularly where non-separable or
+entangled cognitive states have been found, show that in many such cases Bell or CHSH
+in-equalities have been maximally violated. The implications are that greater non-local
+correlations than allowed in quantum mechanics (often known as the Tsirelson bound),
+are found in human cognition. We propose in the current paper that a non-local theory
+of mind is needed in order to account for the empirical �indings. This requires a
+foundationally different approach than the extant ‘quantum-like’ approach to human
+mind. Our account is novel, but still founded on a Hilbert space set up with the physical
+constraint of no-signalling. To account for the surpassing of the Tsirelson bound we
+propose abandoning the constraint of no-signalling that depends upon spacetime. Thus
+we ask; ‘Is mind outside spacetime?’ We discuss a candidate theory of quantum gravity
+based on non-locality as fundamental that may accord with our proposal. We are led to
+suggest a new 6 part ontological framework linking Mind, Matter, and Cosmos.
+Key words: non-locality, no-signalling, Bell inequalities, Tsirelson/Cirelson bound, PR
+boxes, Cognition, quantum gravity, six-part framework
+Introduction: Is mind outside physical spacetime?
+Non-locality has baf�led us since the birth of modern science. For example in Newton’s
+gravity framework, we have action at a distance, and Newton himself did not want to
+forward any ‘explanation’ of the same by stating, “hypothesis non-�ingo”. Later with the
+advent of Special Theory of relativity (SR), and then the General Theory of Relativity
+(GR), Einstein nearly singlehandedly challenged the age old concepts of space and time,
+proposing the bold and beautiful concept of spacetime, where continuity of action (COA)
+plays a central role. COA holds that if a spacetime event A has to in�luence another
+spacetime event B then it also has to in�luence any closed 3 surface between them.
+Hence no-signalling, or that there is an ultimate limit of signalling between spacetime
+1
+
+events, which happens to be the speed of light in vacuum, became the foundational
+physical constraint for any sound theory of Physics. The orthodox quantum mechanics
+(QM) which emerged from intense discussions in Solvay conferences (for example, in
+Pylkkanen, 2019), and later known as Copenhagen version, was, however, still based on
+a Newtonian space and time background. Later, with the emergence of quantum �ield
+theory (QFT) there has been an uncomfortable coexistence of SR and QM. The holy grail
+of modern Physics has been to construct uni�ied �ield theories, and particularly quantum
+gravity (QG), as the cherished uni�ication of GR with QM. However, in all of these
+numerous attempts, non-locality has been a recurrent problem. Even different
+interpretations of QM, starting from Collapse of the wave function, to different
+alternative theories like Bohmian mechanics (Walleczek et al. 2019), or spontaneous
+collapse of wave function (for example in Tumulka, 2006), have been riddled with
+different forms of non-localities. Very recently different approaches to QG (Kauffman,
+2022) would presume to hold non-locality as fundamental, which is radical.
+Spontaneous collapse models or dynamic collapse models have attempted to resolve the
+measurement problem by introducing a collapse operator in the Schrödinger equation,
+for example in GRW (for example in Wallace, 2014) where a probability of such a
+stochastic collapse is small in case of single particles, but grows exponentially in case of
+many body systems. Hence, the attempt has been to resolve the incompleteness or
+inconsistency problems in orthodox QM, for example, how in the same framework both
+deterministic and unitary Schrödinger evolution and random collapse of wave function
+due to ‘measurement’ can be accommodated. However, very recent work, (for example
+see ‘consciousness and quantum mechanics’ edited by Shan Gao, 2022, and Ball 2022),
+now says that any “physically causal” theory for measurement is almost ruled out.
+There are also physically “acausal” accounts of measurement. Here we refer to the
+recent consciousness induced collapse framework of Chalmers and McQueen (2021),
+where phenomenal consciousness plays the role of a superposition resistant, hence
+de�inite consciousness state that result in “collapse”. More recently, Kauffman and Roli
+(2021), and Kauffman and Radin (2022) have utilized Heisenberg’s interpretation of
+quantum mechanics in terms of ontologically real Potentia, Res potentia, and
+ontologically real Actuals, Res extensa, where actualization converts Possible to Actuals.
+This interpretation does not inherit the mind-body problem because Potentia are not
+substances.
+In turn this approach suggests a natural role for “mind” in actualizing
+quantum potentia, hence “collapsing the wave function. At this point, data supporting
+this hypothesis with respect to work using the two slit experiment are strongly
+supportive at 6.49 Sigma, or 4 x 10 ^ -11.
+We shall base our own discussion on
+Heisenberg’s interpretation.
+In addition to Heisenberg’s “potentia” interpretation, workers have studied several
+other non-realist frameworks, where the wave function is not ontological, but rather a
+tool for computing probabilities for epistemological updates of knowledge state of
+observers. Two alternative strongly emerging interpretations of QM are relational QM
+and QBism. Relational QM holds that QM, or reality for that matter, is not described by
+quantum states, but rather by relations among observables. This is a fact ontology (for
+more details about “relative” and “stable” facts, we can refer to seminal literature,
+2
+
+(Pienaar, 2021)). QBism agrees on placing a central role on
+phenomenology or
+subjective experiences, where QM is the navigation tool for any user (rather than
+de�ining who is the user) to make optimal decisions. In QBism relations between the
+elements of the framework are objective, such that every agent would agree. We differ
+from these frameworks in that these frameworks are largely based on the locality of
+physical spacetime, but then they face non-locality problems.
+One approach to surpassing the Tsirelson bound is found in PR box worlds (Popescu,
+2014, a modern review of Tsirelson bound can be found in Stuckey et al., 2019) that
+allow for greater than QM non-local correlation limit the Tsirelson bound. However, the
+PR box worlds correspond to no physical model of a universe, (Popescu, 2014 op. cit.).
+Hence a related question raised earlier was whether QM is the only theory where there
+is a co-existence of non-locality in the sense of Bell inequalities violation and relativistic
+causality.
+We propose in this paper that if we need to include mind or cognitive aspects in the
+foundational frameworks of nature, then we need to have non-locality as the central
+feature. In this paper then, we explore our framework of non-local mind or cognition,
+and are led to our proposal that “mind” is not in spacetime. By proposing that mind is
+not in spacetime, we can naturally eliminate the requirement for Continuity of Action,
+hence non-signaling, that makes sense only within a framework of spacetime. By
+proposing that mind is not in spacetime, mental events that are in spacetime but
+surpass the Tsirelson bound can be explained.
+In order to begin to make sense of the concept that “mind is not in spacetime, but
+conscious events are in spacetime”, we are led to propose a novel 6 part ontological
+framework linking Mind, Matter, and the Cosmos. The grounds for this novel framework
+are tentative, but we hope worthy of consideration and are testable in part.
+The current paper is organized in sections. Section 1 provide the background of
+cognitive frameworks with experimental work, and its recent reformulations in terms of
+‘quantum-like’ features, for example entangled cognitive states which
+violates Bell
+inequalities strongly. Section 2 discusses alternative ways to surpass the Tsirelson
+bound. Section 3 presents our novel 6 part framework and the current grounds to
+consider it. Section 4 summarizes our results and suggestions for further experiments
+and work.
+Section 1 Cognition beyond the Tsirelson Bound
+Aerts et. al (2013, 2021) have pioneered the study of non-separable states in individual
+minds or cognition. This includes how different concepts are combined. Such concept
+combinations in individual minds can be re-formulated as non-separable states. In
+technical language these are intra state entanglements, which would mean coupling of
+different degrees of freedoms of a single system. Here these are individual minds, and
+the data can be expressed through inequalities such as CHSH as we explain in section 2.
+The statistical values or values obtained from ensembles of ‘minds’ of participants in
+such cognitive experiments can be inputted as inequalities. The results have
+demonstrated a clean violation of the ‘non-local’ correlation bound which occurs in QM.
+3
+
+Such a tight bound is, as noted, called a Tsirelson bound, that is maximal and
+characteristic for QM. The same authors also provide the statistical signi�icance of their
+results. Aerts and Arguelles (2022) have claimed a statistical signi�icance of p values
+ranging 0.001-0.005, which is strong enough to suggest, but not yet prove, the viability
+of their results.
+Aerts et. al (op. cit.) also adopts a Hilbert space framework, but their strategy is of
+‘reverse engineering’, i.e. to start with the empirical results, and then describe such
+results by a suitable state space modelling, where the state space can be either Hilbert
+space or a larger Fock space. Thus, the usage of CHSH inequalities is statistical in nature,
+since such inequalities are general. Maximal algebraic violation of such inequalities can
+be beyond the Tsirelson bound, but when Hilbert space is the state space then a tight
+upper bound comes up as a constraint.
+Given the above points, Aerts et al.’s claim of greater than Tsirelson bound violation in
+cognitive experiments raises several questions. For example, can any or all Hilbert space
+formulations account for such super quantum correlations? Aerts et al. have responded
+by suggesting that an entanglement that they consider is of a more complex nature, i.e.
+entanglement both in states as well as measure, might account for super violations. We
+assess this approach below.
+We stress again that any Hilbert space formulation of quantum mechanics implies a
+tight Tsirelson bound. And we stress again that the Hilbert space formulation is stated in
+a background spacetime with “no signaling” and continuity of action, hence “locality”.
+Section 2.  Non-locality : Implications for QM
+2.1 Non-locality in QM
+Here we remind ourselves of the seminal contribution of John Bell (1964, 1966), and
+state the basic requirements for Bell factorization conditions, upon which the celebrated
+Bell inequalities or later CHSH inequalities are based. Based on continuity of action, the
+following three assumptions are required for establishing Bell factorization.
+a.
+Statistical Independence:
+, where
+denotes the local hidden
+ρ 𝑀
+(
+) = ρ µ
+( )
+µ
+variable, and M stands for measurement settings of apparatuses for different
+space-like separated agents.
+b. Output independence:
+, subscripts a and
+ρ𝑎𝑏 𝑎, 𝑏, µ
+(
+) =  ρ𝑎(𝑥𝑎|𝑎, 𝑏, µ)ρ𝑏(𝑥𝑏|𝑎, 𝑏, µ)
+b stands for different agents, namely, Alice and Bob, x’s are outcomes at their
+ends and a, and b are inputs at their ends respectively.
+c.
+Parameter independence: ρ𝑎 𝑎, 𝑏, µ
+(
+) =  ρ𝑎 𝑎, µ
+(
+),  𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑙𝑦 ρ𝑏 𝑎, 𝑏, µ
+(
+) =  ρ𝑏 𝑏, µ
+(
+) 
+Hence, in conjunction of the three assumptions we have the Bell factorization
+(1)
+ρ𝑎𝑏 𝑎, 𝑏, µ
+(
+) = ρ𝑎 𝑎, µ
+(
+).  ρ𝑏 𝑏, µ
+(
+)
+Bell factorization is a general condition based on the local realism assumptions (COA to
+be precise), which is violated by different theories in different ways. For example, QM
+violates Bell factorization by violating output independence but keeping statistical
+4
+
+independence and parameter independence. Super Deterministic theory violates the
+same by violating Statistical independence, while keeping the other assumptions. And
+Cavalcanti and Wiseman (2012) have showed how Bell factorization can be derived
+from conjunction of local ‘signalism’ and predictability.
+In the form of CHSH, we have two space-like separated agents, Alice and Bob, where say
+the measurement settings in Alice’s end are {a, a’} and Bob’s end are {b, b’}, and all
+results are dichotomous (say, +/- 1). Here we de�ine the correlation function as
+(2)
+𝑐 𝑎, 𝑏
+(
+) =  𝑃𝑎,𝑏 1, 1
+(
+) + 𝑃𝑎,𝑏 − 1, − 1
+(
+) − 𝑃𝑎,𝑏 1, − 1
+(
+) − 𝑃𝑎,𝑏(− 1, 1)
+Hence, we have the CHSH inequality as 𝐶𝐻𝑆𝐻 = 𝑐 𝑎, 𝑏
+(
+) + 𝑐 𝑎, 𝑏
+'
+(
+) + 𝑐 𝑎
+', 𝑏
+(
+) − 𝑐(𝑎
+', 𝑏
+')
+(3).
+Hence CHSH has different upper bounds for different underlying theories. For example
+for a local deterministic theory (COA is the requisite here) we would always have as
+For QM the maximum violation of the above limit would take place when
+𝐶𝐻𝑆𝐻
+|
+|≤2.  
+, hence this gives the Tsirelson (T bound
+𝑎, 𝑏
+(
+) = 𝑐 𝑎, 𝑏
+'
+(
+) = 𝑐 𝑎
+', 𝑏
+(
+) =− 𝑐 𝑎
+', 𝑏
+'
+(
+) =
+2/2
+from now) bound of
+. However algebraically it is possible that we have
+|𝐶𝐻𝑆𝐻|≤2√2
+, hence making the maximal upper bound as
+𝑐 𝑎, 𝑏
+(
+) = 𝑐 𝑎, 𝑏
+'
+(
+) = 𝑐 𝑎
+', 𝑏
+(
+) =− 𝑐 𝑎
+', 𝑏
+'
+(
+) = 1
+4.
+2.2. Different forms of non-separable states: QM and beyond
+We mention here that generally composite systems in QM can be represented as tensor
+products of states belonging to different Hilbert spaces, such that the total Hilbert space
+of the composite system is a tensor product of such Hilbert spaces. This context is called
+product states. In addition, we recall that a Tensor product space is strictly larger that
+space of direct sums, hence this context also captures ‘quantum-holism’. Now the typical
+de�inition of an intersystem entanglement is when the composite system state cannot
+be de�ined as simple tensor products of subsystem states. Intersystem entanglement is
+most discussed in QM literature, since that is what generates non-local correlations. In
+an entangled state the whole is always in a pure state, whereas parts are not in pure
+states, this is the classical Schrödinger way of denoting entanglement. Again, as we have
+stated earlier, maximally entangled states (often called as Bell states) can violate CHSH
+maximally until the T bound.
+However, intra-system entanglement, de�ined as coupling between multiple degrees of
+freedom of the same system, is also discussed widely. Particularly in the classical
+electromagnetism literature authors (Ghose and Mukherjee, 2014 for example) have
+observed widely that intra system entanglement, for example coupling between path
+and polarization states of a vortex beam, can produce such non-separable states (at
+times called Shimony-Wolf states) which can generate violations of CHSH inequalities.
+Authors, for example, Khrennikov (2020) has suggested that intra and inter system
+entanglements is the main difference between quantum and so-called ‘classical’
+entanglement.
+5
+
+Multipartite non-locality:
+Traditionally Bell tests or CHSH tests are bi-partite
+non-locality tests, there have been several modi�ications though, for example GHZ states
+or W states, which extends frameworks for many body entanglement. In our previous
+framework (Kauffman and Patra, 2022) we start with a multipartite entanglement state.
+However, its only recently (Bancal et al. in 2013 for example) that a suitable
+mathematical framework is being built. Here we refer to the basic tenets of such a
+framework, since this might be harnessed in the framework we suggest here.
+We mention here that non-locality is a recurrent feature for many-body systems too (see
+for example in Bancal et al. 2013), for example if we consider a tripartite system, with
+say each subsystem possessed by Alice, Bob and Charlie who are spatially separated. Say
+Alice, Bob and Charlie’s experimental set ups are X, Y and Z respectively and outcomes
+of experiments are a, b and c respectively (binary outcomes for simplicity).
+If the joint probability (if de�ined)
+where, q’s are
+𝑃 𝑋𝑌𝑍
+(
+) =
+𝑙
+∑ 𝑞𝑙𝑃 𝑋
+( )𝑃 𝑌
+( )𝑃 𝑍
+( )(4),
+bounded by 0 and 1, and sum to unity, then the sum represents local correlations, where
+the subscript l is for underlying hidden variables. However, if the above joint
+distribution cannot be written in the above format, then some degree of non-local
+correlations
+exist.
+One
+example
+of
+non-locality
+(technically
+S2
+non-local):
+, where separately q’s
+𝑃 𝑋𝑌𝑍
+(
+) =
+𝑙
+∑ 𝑞𝑙𝑃 𝑋
+( )𝑃 𝑌𝑍
+(
+) +
+𝑚
+∑ 𝑞𝑚𝑃 𝑌
+( )𝑃 𝑋𝑍
+(
+) +
+𝑛
+∑ 𝑞𝑛𝑃 𝑍
+( )𝑃 𝑌𝑍
+(
+)(5)
+sum up to unity. Here we see that in individual sums full factorization is not achieved. At
+times, such contexts are also called hybrid non-locality. In another related literature
+(Bennet et al., 1999 as one seminal work in this direction) non-locality without
+entanglement is theoretically proposed, and later experimentally veri�ied. We refer to
+these studies to seek further support for our assertion that non-locality is a more
+universal and genuine feature of reality. We also are aware of studies differentiating
+between genuine non-locality and direct in�luences (see for example Atmanspacher et
+al., 2019).
+2.3. Attempts to �it the evidence for non-locality withing the framework of a background
+spacetime.
+In the last century intense debate on non-locality, or more precisely what non-locality
+should mean given relativistic spacetime, was a major debate, and is still continuing. The
+non-locality debate has also thrown deeper light on the foundational thinking on QM.
+We observe here that the axioms of special theory of relativity (COA) or consequently
+fundamental limit for speed of signalling between spacetime events, and the equivalence
+of inertial reference frames) seems to be elegant and physically based. However, the
+axioms of QM seem to be mathematical with no clear physical basis.
+Aharonov and Bohm (1961), and later Popescu, and Rohrlich (1994), and independently
+Shimony (1993) have proposed that QM has to be compatible with relativistic causality,
+hence with Continuity of Action, COA. The efforts of the authors mentioned showed that
+non-local correlations, for example in an EPR set up, can be compatible with relativistic
+causality if and only uncertainty of outcomes of measurements is fundamental. Or in
+6
+
+other words the effect of a cause here is uncertain. (Thus, counterfactuals are required).
+Aharonov was the �irst to propose ‘modular’ quantum variables, that are non-local in
+spacetime due to non-local relativistic phases, and they have optimal uncertainty for no
+signalling. Shimony amusingly observed the whole affair is ‘passion at a distance’.
+2.4. Attempts to surpass the Tsirelson bound in formal models.
+Based on the dense PR box literature, there have been many attempts to make super
+quantum correlations (violating T bound) compatible with relativistic causality, or COA
+in general,(for example, Popescu, 2014). Related questions have been whether QM is the
+only possible theory where non-local correlation and no signalling co-exists?, (Popescu
+2014).
+Or why QM does not exhibit greater non-locality?, (Linden et al. 2007 for
+example). We further observe that there have been efforts in the line of including
+communication complexity, and or, information causality to eradicate super quantum
+correlations, (for example, in Jaeger, 2007). We also note that super correlations or
+greater than T bound violations are possible in con�iguration spaces with very
+particular properties. Overall, there has been an attempt to make violations of Bell
+inequalities (not super correlations) compatible with relativistic causality, but it is far
+from clear what would be the implications for super correlations for a locality criterion.
+As we explore below, how violations of Bell / CHSH or even super correlation results
+have been observed in cognitive experiments.
+We also note that some authors
+(Khrennikov, 2022) have observed that if the observables in a particular theory cannot
+be represented by Hermitian operators, there might not be any T bound constraint.
+Section 3. Is Spacetime Fundamental?
+3.1. Zeilinger and Information: It is important to stress that several authors are
+exploring the idea that spacetime is not fundamental. In particular, Zeilinger has
+proposed that “information” is fundamental and somehow spacetime emerges from
+“information”(see for example Zeilinger’s seminal works since 1998). We note a central
+issue, “information” itself implies “possibilities” that are not either true or false.
+Consider Shannon information and the source. A given bit string, say (11111) can carry
+no information unless one of the bits can, counterfactually, be 0. That is, it must be
+possible that one of the bits is 0 not 1. Thus the very concept of “information” requires
+more than one simultaneously  possible state of the universe.
+3.2. Res potentia and Res extensa linked by measurement: In the current article, we base
+our approach on Heisenberg’s interpretation of the quantum state as “potentia standing
+ghost – like between an idea and reality”. One of us, ( Kastner et al. 2018) has developed
+Heisenberg’s interpretation as “Res potentia” ontologically real Possibles, and Res
+extensa, ontologically real Actuals. Possibles do not obey Aristotle’s law of the excluded
+middle and law of noncontradiction, so are neither ‘true’ nor ‘false’. This allows
+“Potentia” to explain quantum superpositions: “Schrödinger’s cat simultaneously is
+possibly alive and possibly dead.” This is not a contradiction.
+Potentia are non-spatial in nature but ontologically real. By contrast Actuals do obey
+Aristotle’s two laws, so are either true or false. All of Classical physics is based on such
+true false Boolean variables. Given the concept of Res potentia, one of us, (Kauffman)
+7
+
+has explored a new approach to quantum gravity that takes non-locality to be
+fundamental. Non locality taken as fundamental implies that spacetime is not itself
+fundamental, but must somehow arise from the behaviors of entangled coherent, hence
+non local, quantum variables. Then non-local entangled coherent quantum variables,
+“Res potentia” are not in spacetime. They are Potentia not in spacetime.
+3.3. Mind and the Quantum Vacuum: One natural interpretation of the line of thought
+above is that the quantum vacuum consists precisely in non-local entangled quantum
+coherent variables. Given the above, a natural proposal is that ‘mind’ – non-spatial, is
+identical or related to the quantum vacuum. We here both propose this identity and
+explore its potential validity.
+A �irst implication of the proposed identity of mind and the quantum vacuum is that
+both are outside of spacetime. This is a possible step to explaining Aert’s results. To do
+so, we need to show that surpassing the Tsirelson bound is straight forward if mind is
+outside of spacetime. In this case we can abandon no signalling and continuity of action.
+We show this next. But there is a further issue, Aerts et. al data concern experiences of
+humans and those experiences are in
+spacetime. Powerful recent arguments now
+strongly suggest that conscious experiences (phenomenal nature) arise upon collapse of
+the wave function, hence, qualia are in spacetime. And further remarkable evidence now
+clearly shows that we can purposefully actualize the wave function. A responsible free
+will is not ruled out. We address all this below. These recent results and claims will be
+part of our proposed 6 part framework introduced below.
+Section 4. Surpassing the Tsirelson Bound if Mind Is Outside of Spacetime
+Aerts et al.(op. cit.) themselves have attempted to justify the super quantum correlation
+values obtained in their ‘concept-combination’ experiments based on complex
+entanglement nature in their experimental settings, given that the con�iguration space
+of mind is a high dimensional Hilbert space. However the standard belief (going back to
+Popescu and Rohrlich) has been that the maximum limit of ‘non-locality’ allowed in a
+Hilbert space is the bound.
+Our perspective is not to justify the super violations based on the complexity of
+entanglement (both in states and in measurements), since there have been critiques of
+this line of argument by suggesting that if the ‘marginal selectivity’ rule is also violated
+along with Bell inequalities (which Aerts et al. observes) then there can be
+contaminations in testing for Bell violations. Hence we propose the six part framework,
+where our de�inition of mind need not be constrained by any physical locality condition.
+Section 5. Mind and Qualia – Collapse of the Wave Function
+Recently Chalmers and McQueen (2022), who have been very sceptical about mind
+collapsing wave function, or a relation between QM and phenomenal consciousness in
+general, have designed a framework in which phenomenal consciousness might collapse
+wave function and thus a de�initive ‘classical’ world emerges. The framework suggested
+is based on IIT (Tononi et al. 2016 for example) or integrated information theory, and
+also where phenomenal consciousness – qualia – is considered as ‘superposition
+resistant’. Here we observe that Chalmers and McQueen (2022) have proposed a partial
+8
+
+quantum Zeno effect for completing their consciousness induced collapse model. Our
+previous framework for the emergence of the classical world naturally includes a partial
+Zeno effect, with trade-offs between Zeno effect and atmospheric de-coherence. We
+didn’t have non-local mind explicitly in the previous framework.
+In addition to Chalmers and McQueen, (op. cit.), Kauffman and Roli (op. cit.) have
+recently proposed that the human mind cannot be algorithmic, and that the capacity to
+�ind novel affordances requires a quantum mind and qualia associated with the collapse
+of the wave function to a single state. The next section presents evidence that humans
+can, in fact, collapse the wave function.
+Section 6.  We Can Collapse the Wavefunction
+An old idea in quantum mechanics is that mind might have something to do with
+“collapse of the wave function”.
+Von Neumann proposed this, (1955/1932). Wigner
+suggested the same idea at one point( see for example Wigner, 1995).
+Following Heisenberg, as noted, we propose Res potentia, ontologically real Possibles,
+and Res extensa, ontologically real Actuals. Here “actualization” converts Possibles to
+Actuals. This assertion is fully consistent with recent results, (Gao, op. cit., Bell, op. cit..),
+that seem to rule out physical causes of actualization. A physical cause cannot convert a
+possible to an actual.
+Res potentia and Res extensa plus actualization is the �irst new idea about mind and
+body since Descartes’ substance dualism,
+Spinoza’s neutral monism, Berkeley’s
+Idealism, and pure materialism.
+Res potentia and Res extensa is not a substance
+dualism. Potentia are not substances. Thus this view does not inherit the mind – body
+problem.
+Instead Res potentia and Res extensia suggest a natural role for mind. Mind “actualizes”
+Possibles to Actuals.
+Strong evidence now supports this scienti�ically testable hypothesis. Radin and his
+colleagues (for example see Radin 2019) have tested the capacity of humans paying
+attention to modify the intensities of the adjacent central bands in the famous
+interference pattern of the two slit experiment. The effect is weak, but has been tested in
+30 independent experiments. At present the positive results are very strongly
+statistically signi�icant at 6.49 Sigma. The probability, “p”, that this arises by chance now
+stands a less that 4 x 100,000,000,000, (Kauffman and Radin, op. cit.).
+This is strong enough to take very seriously as yet further data are sought. If accepted,
+the results alter the foundations of Quantum Mechanics with a fundamental role for
+mind. Indeed, even a responsible free will is not ruled out, (Kauffman Radin, op. cit.).
+For the purposes of this article, we will accept these results as true.
+Section 7. Quantum Gravity if Non locality Is Fundamental
+One of us has recently published a work on quantum gravity (Kauffman op. cit.).The
+starting point is to take nonlocality as fundamental. Nonlocality arises in the presence of
+entangled coherent quantum variables. If one starts with nonlocality it is not necessary
+9
+
+to explain nonlocality, but necessary to explain locality. Somehow locality – spacetime–
+is to emerge from the behaviors of the quantum variables. This immediately �latly
+contradicts General Relativity which is local, and in which there is no emergence of
+spacetime. Further, General Relativity can be formulated in the absence of matter so
+matter cannot be necessary for the very existence of spacetime. But if one starts with
+nonlocality, the emergence of spacetime must depend on the matter – the entangled
+coherent quantum variables. A further note is that there is no apriori reason not to start
+with nonlocality as fundamental.
+The steps in building this new theory of quantum gravity start with N entangled
+variables in Hilbert space, then constructs a metric distance between each entangled
+pair of variables as the sub-additive von Neumann Entropy between that pair.
+Sub-additive von Neumann Entropy, therefore, �its the triangle inequality. The next step
+notes that quantum variables can be in superposition and interpreted as potentia,
+neither true nor false. All the variables of classical physics are true or false. Hence the
+next step in the development of the theory maps distances in Hilbert space to classical
+spacetime distances between a succession of true actualization events. In this mapping
+entangled near-neighbours in Hilbert space construct themselves into nearby points in
+classical spacetime. The hypothesis that actualization events construct spacetime is
+probably testable using the Casimir effect.
+Section 8. Emergence of the Classical World
+Here we refer to the ontological framework developed by Kauffman and Patra (2022),
+which also forms one reference for the current framework, though we didn’t include
+non-local mind in our previous work. We based our previous work on the premise that
+measurement and actualization, which creates the de�initive classical world (this
+coincides with the contextuality-complementarity philosophy of Bohr1) can happen only
+in a speci�ic basis. However we still do not have a comprehensive theory for the
+emergence of a speci�ic basis, except the recent attempts from Quantum Darwinism
+perspectives as proposed by Zurek (2022) in terms of de-coherence theory. We note
+that decoherence does not yield a speci�ic basis.
+We have proposed the following steps for the emergence of classical world, in which
+testable experiments can be performed.
+(i)
+We start with sets of N entangled quantum variables, which need not be
+maximally entangled. Variables can mutually actualize each other, which is
+approximated by the quantum-Zeno effect.
+(ii)
+Such actualization occurs in one of the 2Nbases.
+(iii)
+Mutual actualization breaks symmetry among these 2N bases.
+(iv)
+An amplitude for a speci�ic basis can emerge and increase with further
+measurement in the same particular basis, it can also decay between
+measurements.
+1 Here one can also refer to recent works of Kastrup (), where if we claim that only actualization creates the
+definitive world, which would mean no pre-existing values, we should also accept that the world as a whole is
+beyond only physical, or the typical physical closure principle would not work.
+10
+
+(v)
+ As the number of variables, N, in the system increases, the number of
+Quantum Zeno mediated measurements among the N variables increases.
+(vi)
+Now for experimental purposes, quantum ordered, quantum critical, and
+quantum chaotic peptides that decohere at nanosecond versus femtosecond
+time scales can be used as test objects.
+(vii)
+By varying the number of amino acids, N, and the use of quantum ordered,
+critical, or chaotic peptides, the ratio of decoherence to Quantum Zeno effects
+can be tuned. This enables new means to probe the emergence of one among
+a set of initially entangled bases via weak measurements after preparing the
+system in a mixed basis condition.
+(viii)
+Use of the three stable isotopes of carbon, oxygen, and nitrogen and the �ive
+stable isotopes of sulfur allows any ten atoms in the test peptide or protein to
+be discriminably labelled and the basis of emergence for those labelled atoms
+can be detected by weak measurements. We present an initial mathematical
+framework for this theory, and we propose experiments.
+Section 9. If Mind is Outside of spacetime, What is “My” Mind?
+If we are to make sense of Aerts et. al data (op. cit.), and do so by proposing that mind is
+outside of spacetime but that the cognitive experience is in spacetime, we must claim
+that qualia emerge upon actualization events, as discussed above. But in addition, it
+becomes fundamental to address the issue: What maps the quantum variables in Hilbert
+space and the vacuum to “My Mind”?
+The theory of quantum gravity based on nonlocality as fundamental almost
+automatically affords a possible answer to this issue.
+Compare the relatively simple
+quantum behaviors of a quantum variables in a quartz crystal and the presumably far
+more complex behaviors of the quantum variables in the diverse proteins in a speci�ic
+human brain with its unique genetic background and life experiences. The proposal is
+that when these quantum variables become coherent, they are not in spacetime but part
+of the quantum vacuum. The behaviors of these variables in the vacuum must exhibit
+and re�lect the complexities the quantum behaviors in that speci�ic brain. Because
+entangled neighbors in Hilbert space map to spatial neighbors in classical spacetime and
+the matter in it, actualization events with qualia will typically map to and occur in the
+same brain. Thus, “What is My Mind” seems naturally answered.
+These proposals claim to answer Aerts et. al (op. cit.). My mind is not in spacetime, so
+not bound by continuity of action and nonsignaling. The Tsirelson bound can be
+surpassed. But actualization occurs in my brain so are my qualia.
+Section 10. The Quantum Vacuum and the Matter in the Universe
+Our proposal to start with nonlocality as fundamental drives a different conception of
+the quantum vacuum. This vacuum is normally conceived in the absence of any matter
+and as a coupling of all the fundamental �ields. The same can be considered as coupled
+quantum harmonic oscillators whose zero point energy can be studied. As so conceived,
+the spectrum of the quantum vacuum must be stationary.
+11
+
+By contrast, if nonlocality is taken as fundamental, spacetime is not fundamental and
+can only arise due to the behaviors of the quantum variables when coherent and also
+when not coherent. In the latter case, the Schrödinger equation no longer applies. The
+quantum behaviors of quarks, protons, neutrons, and electrons in complex proteins
+must differ from those in a simple crystal. With this seemingly necessary inference, the
+behaviors of the quantum vacuum – coherent entangled quantum variables, cannot be
+stationary over the history of the universe as more and more complex classical systems,
+stable for long periods, come into existence. We can propose, quantum vacuum must
+also re�lect the history of the behaviours of the ever more complex matter than has
+come to exist and vanished.
+Section 11. The Six Part Ontological Framework: Mind Matter and Cosmos
+The above considerations lead us to propose that: i. The quantum vacuum is composed
+of entangled coherent quantum variables that are ontologically real “Possibles”; ii
+“Mind” is identical to the Possibles of the quantum vacuum. Hence this is the de�inition
+of mind in our framework; iii. The vacuum is outside of spacetime; iv. Mind can mediate
+actualization of potentia ,(Kauffman and Radin, op. cit.); actualization of potentia then
+constructs classical spacetime, where a metric exists in the quantum vacuum Hilbert
+space via non-additive von Neumann Entropies between pairs of entangled variables,
+that is then mapped to events at speci�ic classical spacetime locations, (Kauffman
+quantum gravity); v. We experience such actualized quantum variables as “qualia”,
+(Chalmers and McQueen, Kauffman and Roli, 2021); vi. In the last, sixth, part we
+propose the emergence of classical world, which is based on our previously proposed
+framework (Kauffman and Patra, 2022). We suggest a mutual actualization process of
+quantum variables. Through a trade-off between the quantum Zeno effect and
+atmospheric de-coherence, such de-cohering and re-cohering variables creates the
+observable classical variables. In this framework we suggest veri�iable experiments with
+peptides whose entangled variables decohere exponentially fast versus peptides whose
+entangled variables decohere power law slowly as a possible ground of test.
+We hope to show that the six part proposal above allows us to account for Aert’s et. al
+results that surpass the Tsirelson bound. Far more, this new six part framework may
+help organize our emerging ideas about “Mind, Matter, and Cosmos”.
+Section 12. Discussion and Further Work
+Non-locality has always baf�led us. The non-local and non-deterministic collapse of wave
+function in QM worried Einstein throughout his working life, since the fear was such
+non-locality would mean action at a distance and thus break- down of the causality
+structure of spacetime. The latter is fundamental to any Physical theory. Certainly, a
+huge literature has demonstrated that non-local collapse may not mean any
+superluminal signaling. Later since Bell’s seminal contribution, there have been many
+versions of such frameworks (CHSH being the most popular), which have suggested that
+local hidden variable theories cannot reproduce QM faithfully. Loophole free Bell
+inequality/ CHSH inequality violations have demonstrated that one or more of the basic
+underlying assumptions, of localism, realism or non-contextuality, or statistical
+independence have to be relaxed to describe the empirical results of QM.
+12
+
+Many workers have shown that Bell inequalities violations are considered to be
+evidence of non-local correlations between subsystems. The canonical example is
+entangled pairs of particles (EPR set up for example) with agents measuring on each
+half of the pair who are space like separated.
+In the presence of an assumed background spacetime, the only way a no signaling
+theorem is going to be preserved is by introducing inherent quantum uncertainty in
+outcomes. In the words of Shimony there can be a happy co-existence between Special
+Relativity and quantum fundamental uncertainty. However, the Hilbert space structure,
+assumed to be the state space in such frameworks, inherently does set up an upper
+bound for violations of inequalities, the celebrated Tsirelson (or Cirelson) bound. Thus,
+the question arises what if in any empirical observation exist where such a limit is
+violated?
+Over the last decade there has been strong evidence of violations of CHSH inequalities,
+pertaining to cognitive experiments (Aerts et. al). The data are now con�irmed at p =
+.001 to .005. Further work is needed to con�irm these results more strongly. However,
+they are already strong enough to warrant consideration of the implications.
+Aerts et. al (op. cit.) have tried to preserve a background spacetime and “no signaling”
+by assuming more complex entanglement, i.e. both states and measurements. The same
+authors have also claimed that quantum entities might be conceptual or cognitive
+entities, hence non-spatial.
+We propose here a novel, yet unexplored framework based on non-localism, where
+spacetime need not be fundamental to existence. Non-locality is not mysterious in our
+framework. Our attempt is to start from non-locality, and derive locality from �irst
+principles. Then in such a constructed local spacetime we have standard QM and SR
+operate with restricted non-locality which is no signaling also.
+Our proposal is related to that of Aerts et. al in an unexpected way. As just noted, these
+authors propose quantum entities might be conceptual or cognitive entities, hence
+non-spatial. Almost in parallel, we propose that the quantum vacuum consists in
+ontologically real Possibles, that Possibles are non-spatial, i.e. not in spacetime, that
+Mind is identical to these Possibles, that Mind can actualize these potentia, and we can
+experience these as qualia.
+What should we make of this extensive new six part framework? A �irst point is that
+other attempts such as PR boxes correspond to no know physical reality.
+Our proposal is not too distant from Aharanov’s non local proposal . But as Shimony
+notes, this is “passion at a distance” in spacetime. In our six part framework, the
+correlations are among ontologically real possibles that are not in spacetime, but “mind”
+is/are part of the quantum vacuum of possibles. These possibles then constitute the
+information Zeilinger hopes is the basis, somehow, of spacetime. However Zeilinger
+offers no account of what information is, other than a “bit”, nor any idea of how these
+might be related to spacetime.
+13
+
+In our account, spacetime is constructed by the sequential actualization of quantum
+variables in Hilbert space with a metric via non-additive von Neumann Entropies that
+then map to Actual events whose mutual distance relations re�lect the metric in Hilbert
+space to constitute spacetime, (Kauffman quantum gravity). This claim underlies our
+�irst part, “i”, and “iii”. There are data at 6.49 sigma to support “iv” and “v”
+above,(Kauffman, op. cit.).
+The second part, ii “mind” is identical to the possibles of the quantum vacuum, is an
+entirely new proposal. Oddly, this proposal just might afford a highly speculative answer
+to the point raised in a recent article on Biocosmology, (Cortes et al., 2022), about a link
+between the emergence of life 4 billion years ago and the recent dominance of dark
+energy whose tight temporal coincidence in Cosmology is strange. If living organisms
+actualize quantum variables far more often than the quantum variables of the abiotic
+universe, then life, via mind, can have played a role in the emerging dominance of dark
+energy in the past four billion years.
+Our vi. part concerns the emergence of the classical world from the quantum world. Our
+own proposal, (referring to Kauffman and Patra, 2022), has the virtue of being testable.
+In addition it automatically supplies the incomplete Quantum Zeno Effect desired by
+Chalmers and McQueen, (op. cit.). Our speci�ic proposal for the emergence of the
+classical world is consistent with our general framework i. to vi., and is consistent with
+efforts to study how an increase in the mass of molecules such as the Buckyball may
+increase decoherence .
+The proposal that quantum gravity is a quantum construction of spacetime is not yet
+united with General Relativity, but may be a new pathway to do so . Such a union with
+our proposals in the present article might be fundamentally new. Such a union would
+embrace Mind, Matter and Cosmos.
+The most important lines of further work are: i. Experiments to test and extend the
+Aerts et. al results, (op. cit.). A ‘p’ value of 0.001 is of interest, but hardly persuasive. ii.
+Our six part framework rests heavily on taking non-locality as fundamental.
+Experiments testing the hypothesis that actualization constructs spacetime are needed.
+The Casimir effect may prove useful, (Kauffman et al., 2021 for example). iii. Further
+testing of the capacity of the human mind to ‘collapse the wave function’ are needed.
+The current data at a Sigma of 6.49 are strong. But this is a truly major claim that must
+pass muster with critics. iv. Were it possible to demonstrate that actualization events
+constructed spacetime and with good grounds established that mind can mediate
+actualization, it might become possible someday to test if mind by mediating
+actualization can construct spacetime. v. The current article is at best a conceptual
+framework. A far more formal and integrated mathematical theory must be constructed
+and ultimately tested.
+Section 13. Summary and Conclusion
+The dream of physics since the discoveries of General Relativity and Quantum
+Mechanics nearly a century ago has been their union in Quantum Gravity. Yet since
+Newton, a role for mind in the becoming of the Cosmos has seemed precluded. In 2022
+14
+
+NASA launched a rocket that nudged a distant asteroid in the Solar System into a slightly
+different orbit altering the orbital dynamics of the solar system. Mind has cosmic
+consequences.
+In the current article we take the results of Aerts et. al as if they were �irmly established.
+With a p value of .001 the results are at most grounds for consideration. More
+experiments are needed. However, assuming such �irm results, human cognitive events
+surpass the Tsirelson bound. Attempts to explain such a result within a backgound
+spacetime that demands Continuity of Action and non-signaling have severe dif�iculties.
+Were mind not in spacetime the requirement for Continuity of Action and nonsignaling
+would not arise. The possibility of cognitive events surpassing the Tsirelson bound
+would be arise. But this would require that mind correspond to something “real” that is
+not in spacetime, and also that cognitive events themselves exist in the actual experience
+of humans, hence in spacetime.
+We approach quantum gravity by taking nonlocality as fundamental. If nonlocality is
+fundamental, spacetime is not fundamental. Non locality arises with two or more
+entangled coherent variables. Thus, we are forced to the conclusion that spacetime
+somehow emerges from the behaviors of coherent entangled variables. This �latly
+contradicts General Relativity which is local, spacetime does not emerge in General
+relativity, and GR can be formulated without matter �ields so the very existence of
+spacetime cannot depend upon matter.
+Based on the above it is straightforward to de�ine a metric distance between each pair of
+entangled variables in Hilbert space as the subadditive von Neumann Entropy of that
+entangled pair. But this set of distances is in Hilbert space whose variables can be in
+superposition. Classical events in spacetime cannot be in superposition. Hence it
+becomes natural to �ind a map between the metric in Hilbert space and classical
+spacetime by successive actualization events in which the distance between actual
+events correlates with those of the corresponding variables in Hilbert space. Nearby
+entangled variables in Hilbert space construct themselves into nearby points in classical
+spacetime.
+In the present article we hope to account for evidence that cognitive events do surpass
+the Tsirelson bound by identifying mind with coherent entangled quantum variables
+that constitute the quantum vacuum and are not in spacetime. These are to map to
+cognitive events within spacetime by actualization events that constitute qualia.
+Increasingly strong grounds exist to support the view that conscious events – qualia –
+accord with actualization events. Further, evidence now stands at 6.49 sigma, or 4 in
+100,000,000,000, in support the claim that mind acausally mediates actualization.
+The union of the above issues then constitute a vision of quantum gravity in which mind
+is identical to the entangled coherent quantum variables of the quantum vacuum and
+mind itself mediates the actualization events that construct classical spacetime.
+15
+
+Such a vision is not yet united with General Relativity. A new union may be possible in
+which quantum gravity constructs the classical spacetime in which General Relativity
+operates.
+General Relativity requires a world of classical objects. Among these, some are very
+simple, some like the evolved proteins in the human brain are very complex. The
+quantum behaviors of very complex molecules and groups of molecules will be far
+richer than those of a simple small quartz crystal. Therefore, the mind of a brain can be
+far more complex that a mind of a crystal. And the quantum behaviors of one brain will
+be partially unique to that brain and its ontogenetic and experiential history. But the
+quantum behaviors of entangled variables in brains, when coherent, are not in
+spacetime and are part of the quantum vacuum. Upon actualization, these entangled
+variables that are neighbors in Hilbert space construct themselves to nearby points in
+the matter in classical spacetime, thus typically to events located in the same brain. “My
+memories and thoughts are mine, not yours.” Yet by entanglement between brains,
+telepathy is possible, and precognition is possible. By entanglement between variables
+in a brain and other physical objects, psychokinesis is possible The data for all these are
+now abundant at high Sigma values.
+The concepts and data we have discussed do not yet warrant such enormous
+conclusions. Far more would be required. Yet, perhaps for the �irst time since Newton,
+they may constitute the start of a conceptual framework uniting Mind, Matter and
+Cosmos.
+References
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+18
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+arXiv:2301.02460v1  [cond-mat.mes-hall]  6 Jan 2023
+Half-metal and other fractional metal phases in doped AB bilayer graphene
+A.L. Rakhmanov,1 A.V. Rozhkov,1 A.O. Sboychakov,1 and Franco Nori2, 3
+1Institute for Theoretical and Applied Electrodynamics,
+Russian Academy of Sciences, 125412 Moscow, Russia
+2Center for Quantum Computing and Cluster for Pioneering Research, RIKEN, Wako-shi, Saitama, 351-0198, Japan
+3Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA
+(Dated: January 9, 2023)
+We theoretically argue that, in doped AB bilayer graphene, the electron-electron coupling can give
+rise to the spontaneous formation of fractional metal phases. These states, being generalizations of
+a more common half-metal, have a Fermi surface that is perfectly polarized not only in terms of
+a spin-related quantum number, but also in terms of the valley index. The proposed mechanism
+assumes that the ground state of undoped bilayer graphene is a spin density wave insulator, with a
+finite gap in the single-electron spectrum. Upon doping, the insulator is destroyed, and replaced by
+a fractional metal phase. As doping increases, transitions between various types of fractional metal
+(half-metal, quarter-metal, etc.) are triggered. Our findings are consistent with recent experiments
+on doped AB bilayer graphene, in which a cascade of phase transitions between different isospin
+states was observed.
+PACS numbers: 73.22.Pr, 73.22.Gk
+Introduction.— A usual metal demonstrates perfect
+symmetry with regard to the carriers’ spin projection.
+This symmetry manifests itself in the vanishing total
+spin magnetization and the Fermi-surface spin degener-
+acy. Yet the symmetry can be spontaneously destroyed
+by sufficiently strong electron-electron interaction, which
+may result, for example, in the formation of two non-
+identical Fermi surfaces for the two spin projections. In
+the extreme case of the so-called half-metals (HM), one
+of these projections is completely absent from the Fermi
+surface, while all states at the Fermi energy have identi-
+cal spin quantum number [1–3]. Various rather dissim-
+ilar materials with transition-metal atoms are found to
+be half-metals [4–7]. Several papers [8–12] predicted the
+half-metallicity in carbon-based systems as well. The ex-
+istence of spin-polarized currents in such systems makes
+them promising materials for applications in spintron-
+ics [3, 13].
+Graphene-based bilayer and multi-layer systems posses
+additional quantum number,
+the valley index.
+In
+these materials, besides the spin-related polarization, a
+many-body state may demonstrate a valley polarization.
+Therefore, for graphene-based materials, the notion of
+a HM can be generalized to include the possibility of
+a Fermi surface with perfect valley polarization as well.
+Such a proposal was put forward in Ref. 14, where the
+concept of a quarter-metal (QM) was formulated.
+A
+Fermi surface of a QM state is perfectly polarized both in
+valley and in spin-related indices. Furthermore, the lat-
+ter paper explained that both an HM and a QM should
+be viewed as specific instances of a more general notion,
+‘a fractional metal’ (FraM). This many-body phase may
+be realized in materials with degenerate Fermi surface.
+The higher the degeneracy, the stronger fractionalization
+of the Fermi surface can be achieved.
+Since our publication [14] the experimental observation
+of a QM state in graphene trilayer has been claimed [15].
+The experimental data of Ref. 16 suggest that a QM and
+FraM states can be stabilized in a sample of AB bilayer
+graphene (AB-BLG). Given these experimental successes
+it appears important to develop a microscopic theoretical
+framework that can explain the existence of the FraM
+in the AB-BLG. In this letter, a suitable mechanism is
+proposed and discussed.
+Model.— An elementary unit cell of the AB-BLG con-
+sists of four atoms (sublattices A and B, and layers 1 and
+2) with the distance between neighboring carbon atoms
+a0 ≈ 0.142 nm and interlayer distance c0 ≈ 0.335 nm.
+The hoping amplitude t connecting the nearest A and B
+sites in the layer is 2.5 eV ≲ t ≲ 3 eV. The hopping be-
+tween the nearest sites in different layers can be estimated
+as 0.3 eV ≲ t0 ≲ 0.4 eV. It is possible to introduce addi-
+tional, longer-range, hopping amplitudes into the model.
+We assume, however, that the effect of these amplitudes
+is weak, and they are neglected.
+The AB-BLG Brillouin zone is a regular hexagon,
+with
+two
+non-equivalent
+Dirac
+points
+at
+K1
+=
+2π(
+√
+3, 1)/3
+√
+3a0 and K2 = 2π(
+√
+3, −1)/3
+√
+3a0.
+It is
+convenient to measure momentum relative to the Dirac
+points. Thus, we introduce q = k − K1,2.
+The energy spectrum of undoped AB-BLG consists of
+four bands, two electron and two hole ones. Since we are
+interested in the low-energy spectrum of AB-BLG, q ≪
+2t0/3ta0, we restrict our consideration to the effective
+two-band model. It has one electron and one hole band,
+both bands have quadratic dispersion. The bands touch
+at the Fermi energy. In such a model, the Hamiltonian
+for a single-electron wave function reads [17]
+H0 = −ℏ2v2
+F
+t0
+�
+0
+(iqx + ξqy)2
+(iqx − ξqy)2
+0
+�
+,
+(1)
+where the graphene Fermi velocity is vF = 3a0t/2ℏ and
+
+2
+ξ is the valley index. The value ξ = 1 corresponds to
+K1 and ξ = −1 corresponds to K2.
+In the second-
+quantization formalism we can write
+H0 =
+�
+qσξl
+εqlγ†
+qlσξγqlσξ,
+(2)
+where the spin projection is denoted by σ, the index l
+labels the electron (l = 1) or hole (l = 2) band, and γqlσξ
+is the corresponding second quantization operator. The
+eigenenergies εql of the Hamiltonian (1) are
+εql = (−1)l ℏ2v2
+F
+t0
+q2.
+(3)
+Next we include the electron-electron repulsion into the
+model. The latter is a highly non-trivial task. Clearly,
+the low-energy two-band effective model (1) is incom-
+patible with the bare Coulomb repulsion.
+Instead, an
+effective interaction Hamiltonian must be derived. Un-
+fortunately, a compact description of such an effective
+interaction remains an elusive theoretical goal. Indeed,
+due to multiple factors affecting the many-body physics
+in graphene and graphene-based systems, an effective
+interaction term is quite complex, with multiple cou-
+pling constants, whose non-universal values are poorly
+known [18–22].
+In this situation we prefer to adopt a
+semi-phenomenological approach, keeping only the terms
+that directly contribute to the spin-density wave (SDW)
+ordering. It is possible to identify three types of such
+terms. The first term arises due to the forward-scattering
+Hf
+int = VC
+Nc
+�
+kk′,ll′
+σσ′,ξξ′
+γ†
+klσξγk′lσξγ†
+k′l′σ′ξ′γkl′σ′ξ′,
+(4)
+where Nc is the number of unit cells in the sample, and
+VC is an effective interaction constant whose value can be
+potentially extracted from the low-temperature data [23–
+32] on spontaneous symmetry breaking in AB-BLG. The
+forward scattering is characterized by a small momentum
+transfer |k − k′| ≪ |K1 − K2|, and preserves the band
+indices l and l′ of the two participating electrons. Next,
+one can define the backscattering term
+Hb
+int = V b
+C
+Nc
+�
+kk′,ll′
+σσ′,ξ
+γ†
+klσξγk′lσ¯ξγ†
+k′l′σ′ ¯ξγkl′σ′ξ,
+(5)
+where a bar on top of a binary-valued index implies the
+inversion of the index value (for example, if ξ = 1 then
+¯ξ = −1). For Hb
+int the transferred momentum is large
+|k−k′| ∼ |K1 −K2|, thus we can assume that V b
+C ≪ VC.
+Finally, the umklapp-type interaction
+Hu
+int = V u
+C
+Nc
+�
+kk′,
+σσ′,ξξ′
+γ†
+k1σξγk′2σξγ†
+k′1σ′ξ′γk2σ′ξ′ + h.c.,
+(6)
+represents scattering events in which both electrons
+change their bands. It accounts for the coupling between
+inter-layer dipole moments, which is also weaker than the
+coupling between charge densities represented by Hf
+int. In
+principle, there is backscattering umklapp, which we do
+not consider due to it being even weaker than Hu
+int.
+Mean-field approximation.—
+We
+consider a
+zero-
+temperature SDW instability of the AB-BLG. This is
+characterized by the spontaneous generation of staggered
+spin magnetization violating the spin-rotation symmetry.
+The direction of this magnetization is not fixed and there
+are several equivalent choices for an SDW order parame-
+ter that differ by the spin-magnetization direction. It is
+convenient to assume that ⟨γ†
+k1σξγk2¯σξ⟩ ̸= 0. This choice
+corresponds to the magnetization in the xy-plane. Note
+also that the introduced order parameter accounts for the
+coupling of single-electron states in the same valley ξ.
+Now, assuming that the backscattering (5) and the
+umklapp (6) are weak, we apply the mean-field approxi-
+mation to Hf
+int
+HMF
+int = −
+�
+kσξ
+∆σξγ†
+k2σξγk1¯σξ + h.c. + B,
+(7)
+where the order parameter ∆σξ and c-number B are
+∆σξ = VC
+Nc
+�
+q
+⟨γ†
+q1σξγq2¯σξ⟩Θ(qC − q),
+(8)
+B =
+�
+qσξ
+∆σξ⟨γ†
+q2σξγq1¯σξ⟩Θ(qC − q) = Nc
+VC
+�
+σξ
+|∆σξ|2.
+(9)
+In these expressions, the momentum cutoff for the inter-
+action qC satisfies qC ≪ |K1 − K2|.
+The mean-field Hamiltonian (7) does not conserve spin
+(spin-rotation symmetry is spontaneously broken for non-
+zero ∆σξ). However, quasi-momentum q is conserved. In
+addition to q, one can introduce valley and spin-flavor
+operators
+Sf
+q =
+�
+σξl
+σ(−1)lγ†
+qlσξγqlσξ,
+Sv
+q =
+�
+σξl
+ξγ†
+qlσξγqlσξ,
+(10)
+which commute with the Hamiltonian H0 +HMF
+int and are
+good quantum numbers. Thus, in this approximation all
+fermionic degrees of freedom can be grouped into four
+uncoupled sectors, each sector having its own values of
+spin-flavor index (−1)lσ and valley index ξ.
+A sector
+is characterized by its own order parameter ∆σξ, and
+single-particle spectrum
+E1,2
+qσξ = ±
+�
+∆2
+σξ +
+�ℏ2v2
+F
+t0
+�2
+q4.
+(11)
+The thermodynamic grand potential Ω can be expressed
+as a sum Ω = �
+σξ Ωσξ + B, where Ωσξ are four partial
+grand potentials corresponding to specific sectors.
+At
+
+3
+zero temperature, these are
+Ωσξ =
+�
+ql
+�
+El
+qσξ − µ
+�
+Θ
+�
+µ − El
+qσξ
+�
+,
+(12)
+where µ is the chemical potential.
+Minimization of Ω over the order parameters allows
+us to derive the following independent self-consistency
+equations for the order parameters in the four sectors
+1 = VC
+Nc
+�
+|q|<qC
+Θ(µ + E1
+qσξ) − Θ(µ − E1
+qσξ)
+E1
+qσξ
+.
+(13)
+Since the model is electron-hole symmetric, we can limit
+our discussion to the µ > 0 case only. For positive chem-
+ical potential: Θ(µ + E1
+qσ) − Θ(µ − E1
+qσ) = Θ(E1
+qσ − µ).
+Introducing dimensionless variables
+g = VCt0
+√
+3πt2 , m = 4t0µ
+9t2 , δσξ = 4t0∆σξ
+9t2
+,
+(14)
+we obtain from Eq. (13)
+1 = 2g
+� QC
+Qm
+σξ
+QdQ
+�
+δ2
+σξ + Q4 ,
+(15)
+where
+QC = a0qC,
+Qm
+σξ = (m2 − δ2
+σξ)1/4.
+(16)
+It is evident that the gap in the spectrum of electrons in
+the sector (σ, ξ) arises only if QC > Qm
+σξ, that is, if the
+number of the doped charge carriers in this sector is not
+too large. One can perform the integration in Eq. (15)
+and obtain that
+1 = g ln
+
+
+Q2
+C +
+�
+δ2
+σξ + Q4
+C
+m +
+�
+m2 − δ2
+σξ
+
+ .
+(17)
+In the weak coupling limit, g ≪ 1, we have δσξ ≪ Q2
+C.
+Consequently
+∆σξ =
+�
+∆0(2µ − ∆0),
+(18)
+where
+∆0 = 9t2
+4t0
+q2
+Ca2
+0e−1/g
+(19)
+is the mean-field gap of undoped AB-BLG. Introducing
+δ0 = 4t0∆0/(9t2) we can express Eq. (18) in dimension-
+less form
+δσξ =
+�
+δ0(2m − δ0).
+(20)
+Since experiments are performed at fixed doping, we need
+to connect the values of ∆σξ with doping. It is conve-
+nient to introduce partial doping, that is, the number of
+electrons with specific values of σ(−1)l and ξ:
+xσξ = −∂Ωσξ
+∂µ
+= 2π
+VBZ
+�
+σξ
+�
+kdkΘ(µ − E1
+kσξ). (21)
+The total doping x is equal to
+x =
+�
+σξ
+xσξ.
+(22)
+If µ > ∆σξ, we obtain the relation between the partial
+doping and the chemical potential in the form
+xσξ = 3
+√
+3
+8π
+�
+m2 − δ2
+σξ.
+(23)
+Otherwise, xσξ = 0. As a result, we derive in the case of
+non-zero xσξ
+m = δ0 − 8π
+3
+√
+3xσξ = δ0
+�
+1 − 2xσξ
+x0
+�
+,
+(24)
+δσξ = δ0
+�
+1 − 4xσξ
+x0
+,
+(25)
+where
+x0 = t0∆0
+√
+3πt2 .
+(26)
+Equation (25) indicates that for xσξ = x0/4 the order
+parameter in the sector vanishes. That is, for xσξ > x0/4
+one has
+∆σξ(xσξ) ≡ 0,
+m = 8π
+3
+√
+3xσξ = 2δ0
+x0
+xσξ.
+(27)
+Note that the chemical potential, as given by Eqs. (24)
+and (27), demonstrates non-monotonic behavior as a
+function of xσξ. Of particular importance is the fact that,
+for low doping, µ = µ(xσξ) is a decreasing function. This
+means that the compressibility of the homogeneous phase
+is negative and points to a possibility of the phase sepa-
+ration of the electronic liquid. We will assume below that
+the long-range Coulomb interaction is sufficiently strong
+to arrest the phase separation, restoring the stability of
+homogeneous states.
+Quarter metal state of doped AB-BLG.— Disregarding
+the possibility of the phase separation, we use Eqs. (24)
+and (25) to characterize the thermodynamics of the sys-
+tem. To describe the doped state of the electronic liquid
+for a specific x, one must determine partial dopings in
+all four sectors. To achieve this goal, we should calculate
+the free energy F(x) = F(0) +
+�
+µ(x)dx. In so doing, we
+obtain
+F(x) = F(0) + ∆0
+
+x −
+�
+σξ
+x2
+σξ
+x0
+
+ ,
+(28)
+when 0 < xσξ < x0/4, and
+F(x) = F(0) + ∆0
+
+x0
+8 +
+�
+σξ
+x2
+σξ
+x0
+
+ ,
+(29)
+
+4
+if xσξ > x0/4. This free energy must be minimized over
+xσξ under the constraint (22). For small x, simple calcu-
+lations demonstrate that F is smallest when all charges
+are placed into a single sector
+xσξ = x,
+xσ′ξ′ = 0, for σ′ ̸= σ or ξ′ ̸= ξ.
+(30)
+The free energy corresponding to distribution (30) equals
+to FQM = ∆0(x−x2/x0). It is smaller, for example, than
+the free energy Feq = ∆0x−∆0x2/(4x0), that represents
+an equal distribution of doping between all four sectors
+(xσξ = x/4 for all σ and ξ).
+The state described by Eq. (30) is metallic, with (al-
+most) circular Fermi surface whose radius kF = kF(x) is
+set by the equation
+a2
+0k2
+F = 8πx
+3
+√
+3.
+(31)
+This Fermi surface, however, is quite unique: all single-
+electronic states reaching the Fermi energy are perfectly
+polarized in terms of Sf and Sv. In other words, they
+have an identical value of (−1)lσ, and the Fermi surface
+is located within a single valley Kξ. Since among four
+possible Fermi surface sheets of the non-interacting the-
+ory, only one sheet emerges in the system, it is natural
+to designate such a conducting state as a QM.
+Cascade
+of
+phase
+transition
+between
+different
+symmetry-broken phases.— The QM state described
+above remains stable only for sufficiently low x:
+one
+sector cannot accommodate too much doping. Indeed,
+when x = x0/2, Eq. (27) implies that µ = ∆0. Doping
+a single sector beyond this point is impossible: adding
+more charge to this sector increases the chemical po-
+tential beyond ∆0, unavoidably placing charges into
+the remaining sectors as well. As a result, a cascade of
+doping-driven phase transitions emerges.
+The transi-
+tions connect different metallic states, each state being
+characterized by a number of doped sectors: 1, 2, 3, or
+4 [paramagnetic (PM) state] sectors.
+Let us briefly describe this cascade of transitions. At
+zero doping the system is gapped with the gap equal to
+∆0 in all sectors. For small x, the system absorbes all ex-
+tra charge carriers into a single sector [say, sector (σ =↑,
+ξ = +1)]. This is a QM state. At x = x0/4, a second
+order phase transition inside the QM state takes place.
+Beyond this doping, ∆↑+1 vanishes. However, the QM
+state remains stable for x < x0/2. At higher doping the
+QM energy becomes higher than the HM energy, and a
+first order phase transition from QM to HM state occurs.
+In the HM state, the gap is zero in two sectors [for defi-
+niteness, we assign these to be (σ =↑, ξ = +1) and (σ =↑,
+ξ = −1); however, other configurations are equiproba-
+bly possible], and extra charge carriers are equally dis-
+tributed between these two sectors.
+As x increases further, one reaches the point where the
+HM energy becomes equal to that of a 3/4 metal ( 3
+4M)
+state. In such a state, three sectors [say, (σ =↑, ξ = +1),
+x
+1st
+1st
+1st
+2nd
+�
+� �
+�
+� �
+��
+��
+���� � ��
+���� � ��
+���� � ��
+���� � ��
+���� � �
+���� � �
+���� � �
+���� � ��
+���� � �
+���� � �
+���� � ��
+���� � ��
+���� � �
+���� � ��
+���� � ��
+���� � ��
+�
+� ��
+�
+� ��
+�
+� ��
+�
+� ��
+FIG. 1. Cascade of the doping-driven phase transitions be-
+tween different FraM states with different valley and/or spin-
+flavor (isospin) polarizations. Only the region of electron dop-
+ing is shown. For hole doping the picture is identical up to a
+replacement x → −x. Vertical solid (dashed) lines represent
+first (second) order transitions.
+(σ =↑, ξ = −1), and (σ =↓, ξ = +1)] are doped, and
+the fourth sector, (σ =↓, ξ = −1), is gapped, with the
+extra charge carriers being equally distributed among the
+three doped sectors. For our simple model, the transition
+into the 3
+4M happens at x =
+�
+3/4x0. The transition is
+first-order.
+If doping is continued even further, the
+3
+4M state is
+replaced by the PM state. This is yet another first-order
+transition, and the last one in the transition cascade. It
+occurs at x =
+�
+3/2x0. The phase diagram of the system
+is shown in Fig. 1. In this figure only the electron doping
+is shown. Due to electron-hole symmetry of our model,
+the phase diagram at hole doping is equivalent to that
+shown in Fig. 1 up to the replacement x → −x.
+Discussion.— We would like to stress here several im-
+portant points. One must remember that the HM state
+realized in our model upon sufficiently strong doping
+is not the conventional HM [1, 2] whose Fermi surface
+demonstrates perfect spin polarization. Instead, we now
+have a spin-flavor HM [33–36], with perfect spin-flavor
+polarization of the Fermi surface. This means that the
+electron (hole) single-particle states reaching the Fermi
+energy have their spin projection being equal to σ (to ¯σ).
+(The related feature of the QM state was already men-
+tioned above.) In a model with electron-hole symmetry
+a spin-flavor-polarized FraM state does not accumulate
+net spin polarization.
+However, a finite spin polariza-
+tion may accompany a finite spin-flavor polarization [33]
+when such a symmetry is absent. The spin polarization
+was indeed observed in Ref. 16.
+We argued above that the relative stability of vari-
+ous metallic states is affected by doping, triggering the
+transitions between them. Doping is not, however, the
+only factor that influence the competition between the
+FraM phases.
+Particular model’s ingredients favoring
+HM states are the umklapp and backscattering interac-
+tion terms. Specifically, the umklapp couples two sectors
+
+5
+with unequal (−1)lσ, the backscattering, on the other
+hand, connect the sectors with non-identical values of the
+ξ index. Thus, in the presence of either strong Hum
+int or
+strong Hb
+int only two (not four) decoupled sectors of the
+mean-field Hamiltonian can be defined, promoting the
+HM phase over other FraM’s. Therefore, in more realistic
+models, the critical doping values are no longer propor-
+tional to x0, with universal proportionality coefficients.
+Instead, they become functions of the backscattering and
+umklapp coupling constants.
+The qualitative agreement between the remarkable re-
+cent experiments reported in Ref. 16 and our formalism
+is very encouraging. The proposed theory can account
+for such experimentally observed features as the cascade
+of phase transitions, magnetization, and valley polariza-
+tions. Yet one must keep in mind that the experiments
+were performed at finite electric field applied transverse
+to a sample. In our formalism, this field is assumed to be
+zero. Further research is needed to understand the role
+of this field.
+To conclude, we proposed a mechanism responsible for
+the formation of the FraM states in doped AB-BLG. We
+argue that, as doping increases, this system demonstrates
+a cascade of phase transitions between various metallic
+phases that differ in terms of spin-flavor and valley po-
+larizations of their Fermi surfaces. Our theoretical find-
+ings compare favorably to very recent experiments [16]
+on AB-BLG.
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+Mon. Not. R. Astron. Soc. 000, 1–10 (2022)
+Printed 5 January 2023
+(MN LATEX style file v2.2)
+Velocity waves in the Hubble diagram: signature of local galaxy clusters
+Jenny G. Sorce1,2,3,4⋆, Roya Mohayaee5,6, Nabila Aghanim1, Klaus Dolag7,8, Nicola Malavasi7,1
+1 Universit´e Paris-Saclay, CNRS, Institut d’Astrophysique Spatiale, 91405, Orsay, France
+2 Univ. Lyon, ENS de Lyon, Univ. Lyon1, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69007, Lyon, France
+3Leibniz-Institut f¨ur Astrophysik (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany
+4Univ. Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France
+5CNRS, UPMC, Institut d’Astrophysique de Paris, 98 bis Bld Arago, Paris, France
+6Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
+7University Observatory Munich, Scheinerstr. 1, 81679 M¨unchen, Germany
+8Max-Planck Institut f¨ur Astrophysik, Karl-Schwarzschild Str. 1, D-85741 Garching, Germany
+ABSTRACT
+The Universe expansion rate is modulated around local inhomogeneities due to their gravita-
+tional potential. Velocity waves are then observed around galaxy clusters in the Hubble diagram.
+This paper studies them in a ∼738 Mpc wide, with 20483 particles, cosmological simulation of our
+cosmic environment (a.k.a. CLONE: Constrained LOcal & Nesting Environment Simulation). For
+the first time, the simulation shows that velocity waves that arise in the lines-of-sight of the most
+massive dark matter halos agree with those observed in local galaxy velocity catalogs in the lines-of-
+sight of Coma and several other local (Abell) clusters. For the best-constrained clusters such as Virgo
+and Centaurus, i.e. those closest to us, secondary waves caused by galaxy groups, further into the
+non-linear regime, also stand out. This match is not utterly expected given that before being evolved
+into a fully non-linear z=0 state, assuming ΛCDM, CLONE initial conditions are constrained with
+solely linear theory, power spectrum and highly uncertain and sparse local peculiar velocities. Addi-
+tionally, Gaussian fits to velocity wave envelopes show that wave properties are tightly tangled with
+cluster masses. This link is complex though and involves the environment and formation history of
+the clusters. Using machine learning techniques to grasp more thoroughly the complex wave-mass
+relation, velocity waves could in the near future be used to provide additional and independent mass
+estimates from galaxy dynamics within large cluster radii.
+Key words: galaxies: clusters: individual – waves – methods: numerical – methods: analytical –
+techniques: radial velocities – gravitation
+1
+INTRODUCTION
+As the largest gravitationally bound structures in the Universe, galaxy
+clusters bear imprints of the cosmic growth visible through the
+distribution and motion of galaxies in their surrounding environment
+(see Kravtsov & Borgani 2012, for a review and references therein).
+They constitute therefore powerful complementary probes to super-
+novae and baryon acoustic oscillations in testing theories explaining
+cosmic acceleration origin (see Weinberg et al. 2013, for a review).
+Relations between halo masses and observables (optical galaxy
+richness, Sunyaev-Zel’dovich effect, X-ray luminosity) must however
+be calibrated beforehand to study the evolution of the cluster mass
+function. Our capacity to discriminate among cosmological models is
+thus tightly linked to the accuracy of cluster mass estimates. However,
+most of the cluster matter content is not directly visible making their
+mass estimates a particularly challenging task (see for a review Pratt
+et al. 2019; Planck Collaboration et al. 2016).
+With future imaging surveys to come (LSST, Burke 2006; Euclid,
+⋆ E-mail: jenny.sorce@universite-paris-saclay.fr / jsorce@aip.de
+Peacock 2008; WFIRST, Green et al. 2012), stacked weak lensing mea-
+surements will certainly provide the best cluster mass estimates, i.e.
+with the 1% accuracy required (Mandelbaum et al. 2006) but limited to
+small radii around clusters. Independent virial mass estimators (Heisler
+et al. 1985), hydrostatic estimators for galaxy population (Carlberg
+et al. 1997) or velocity caustics (boundaries between galaxies bound to
+and escaping from the cluster potential, Diaferio 1999) constitute com-
+plementary tools once calibrated. Their calibration suffers though from
+the influence of baryonic physics and galaxy bias on velocity fields and
+dispersion profiles. Perhaps velocity caustics are less prone to such sys-
+tematics (Diaferio 1999) explaining their recent increased popularity.
+Galaxy clusters can indeed be seen as disrupters of the expansion, thus
+creating a velocity wave first mentioned by Tonry & Davis (1981) as
+a triple-value region1 whose properties (mostly height and width) de-
+pend on the cluster mass. Combined with infall models (Mohayaee &
+Tully 2005), velocities of galaxies in the infall zones constitute thus
+1 Such an appellation derives directly from the fact that in a disrupted Hubble
+diagram, galaxies at three distinct distances, d, share a similar velocity value
+whereas in an unperturbed diagram, these galaxy velocities would differ pre-
+cisely because of the expansion proportional to H0 × d.
+© 2022 RAS
+arXiv:2301.01305v1  [astro-ph.CO]  3 Jan 2023
+
+2
+Sorce et al.
+good mass proxies for galaxy clusters shown to be in good agreement
+with virial mass estimates (Tully 2015). They have been used in dif-
+ferent studies to retrieve the total amount of dark matter in groups
+and clusters as well as to detect groups (e.g. Karachentsev et al. 2013;
+Karachentsev & Nasonova 2013). Moreover, Zu et al. (2014) showed
+that the wave shape is an excellent complementary probe: for instance,
+f(R) modified gravity models enhance the wave height (infall veloc-
+ity) and broaden its width (velocity dispersions). This translates into
+a higher mass when considering a ΛCDM framework. Subsequently,
+it would lead to cosmological tensions between S 8 values measured
+with the cosmic microwave background and with the galaxy cluster
+counts. Furthermore, velocity waves probe a cluster mass within radii
+larger than those reached with weak lensing. Subsequently, combined
+together, stacked weak lensing and velocity wave mass measurements
+hold tighter constraints on dark energy than each of them separately.
+Indeed, velocity waves are signatures of a tug of war between gravity
+and dark energy. Differences between these two independent mass esti-
+mates, one dynamic and one static, permit measuring the gravitational
+slip between the Newtonian and curvature potentials. This constitutes
+an excellent test of gravity.
+Given future galaxy redshift and large spectroscopic follow-up
+surveys (with Euclid, Peacock 2008; 4MOST, de Jong et al. 2012;
+MOONS, Cirasuolo et al. 2014) of imaging ones, studying galaxy
+infall kinematics to derive better cluster dynamic mass estimates is
+surely the next priority. Cosmological simulations constitute critical
+tools to test, understand and eventually calibrate this mass estimate
+method applied to galaxy cluster observations. Ideally these simula-
+tions must be constrained simulations2 to properly set the zero point
+of the method. Namely, simulations must be designed to ensure that
+the simulated and observed waves match in every aspect but if the
+theoretical model somewhere fails and not because of, for instance,
+different formation histories and/or environments. We are now able
+to produce such simulations valid down to the cluster scale including
+the formation history of the clusters (e.g. Sorce et al. 2016a, 2019,
+2021; Sorce 2018). These simulations are thus faithful reproduction
+of our local environment including its clusters such as Virgo, Coma,
+Centaurus, Perseus and several Abell clusters.
+This paper thus starts with the first comparison between line-of-
+sight velocity waves due to several observed local clusters and their
+counterparts from a Constrained LOcal & Nesting Environment Sim-
+ulation (CLONE) built within a ΛCDM framework. First, we present
+the numerical CLONE used in this study. Next, we compare the ob-
+served and simulated lines-of-sight that host velocity waves. To facil-
+itate the comparisons, the background expansion is subtracted. Before
+concluding, wave envelopes are fitted to study relations between wave
+properties and cluster masses in a ΛCDM cosmology.
+2
+THE CLONE SIMULATION
+Constrained simulations are designed to match the local large-scale
+structure around the Local Group. Several techniques have been
+developed to build the initial conditions of such simulations (e.g.
+Gottl¨ober et al. 2010; Kitaura 2013; Jasche & Wandelt 2013) with
+density, velocity or both constraints. Here we use the technique
+whose details (algorithms and steps) are described in Sorce et al.
+(2016a); Sorce (2018). Local observational data used to constrain the
+initial conditions are distances of galaxies and groups (Tully et al.
+2013; Sorce & Tempel 2017) converted to peculiar velocities (Sorce
+2 The initial conditions of such simulations stem from observational constraints
+applied to the density and velocity fields.
+-222
+-148
+-74
+0
+74
+148
+222
+SGX (Mpc)
+-222
+-148
+-74
+0
+74
+148
+222
+SGY (Mpc)
+Figure 1. ∼40 Mpc thick XY supergalactic slice of the CLONE. Black dots
+stand for the dark matter halos (subhalos are excluded for clarity). Red dots are
+galaxies from the 2MASS Galaxy Redshift Catalog (XSCz) for comparison pur-
+poses only. Indeed, only a small fraction of local galaxy observational redshifts
+have been used to derive peculiar velocities that were used as constraints (about
+∼2.5% of the XSCz catalog).
+et al. 2016b; Sorce & Tempel 2018) that are bias-minimized (Sorce
+2015). We showed that constrained simulations obtained from this
+particular technique, a.k.a. the CLONES (Sorce et al. 2021), are
+currently the sole replicas of the local large-scale structure that include
+the largest local clusters using only galaxy peculiar velocities as
+constraints. Namely, the cosmic variance is effectively reduced within
+a 200 Mpc radius centered on the Local Group down to the cluster
+scale, i.e. 3-4 Mpc, (Sorce et al. 2016a). Galaxy clusters (such as
+Virgo, Centaurus, Coma) have masses in agreement with observational
+estimates (Sorce 2018). Several ensuing studies focused in particular
+on the Virgo galaxy cluster. These studies confirmed the necessity of
+using CLONES to get a high-fidelity Virgo-like cluster. Additionally,
+they confirmed observationally-based formation scenarios of the latter
+(Olchanski & Sorce 2018; Sorce et al. 2019, 2021).
+To actually probe a large range of velocities in the infall zones,
+the CLONE for the present study needs to have a sufficient resolution
+to simulate, with a hundred particles at z=0, halos of intermediate mass
+(∼1011-1012 M⊙). Its constrained initial conditions contain thus 20483
+particles in a ∼738 Mpc comoving box (particle mass ∼109 M⊙). It ran
+on more than 10,000 cores from z=120 to z=0 in the Planck cosmology
+framework (Ωm=0.307 ; ΩΛ=0.693 ; H0=67.77 km s−1 Mpc−1 and
+σ8 = 0.829, Planck Collaboration et al. 2014) using the adaptive mesh
+refinement Ramses code (Teyssier 2002). The mesh is dynamically
+(de-)refined from level 11 up to 18 according to a pseudo-Lagrangian
+criterion, namely when the total density in a cell is larger (smaller)
+than the density of a cell containing 8 dark matter particles. The initial
+coarse grid is thus adaptively refined up to a best-achieved spatial
+resolution of ∼2.8 kpc roughly constant in proper length (a new level
+is added at expansion factors a = 0.1, 0.2, 0.4, 0.8 up to level 18 after
+a = 0.8).
+Using the halo finder, described in Aubert et al. (2004) and Tweed
+et al. (2009), modified to work with 20483 (>231) particles, dark matter
+halos and subhalos are detected in real space with the local maxima
+of dark matter particle density field. Their edge is defined as the point
+© 2022 RAS, MNRAS 000, 1–10
+
+Velocity waves
+3
+Figure 2. Schema of the cylinder used to select (sub)halos whose radial pecu-
+liar velocities, derived as a function of the synthetic observer at the simulated
+box center, are used to study the velocity wave arisen from the massive halo in
+its center. While open circles stand for selected halos, dashed circles represent
+excluded ones.
+where the overdensity of dark matter mass drops below 80 times the
+background density. We further apply a lower threshold of a minimum
+of 100 dark matter particles. Fig. 1 shows the ∼40 Mpc thick XY super-
+galactic slice of the CLONE. Black (red) dots stand for the dark matter
+halos (galaxies from the 2MASS Galaxy Redshift Catalog - XSCz3).
+Note that XSCz galaxies are used for sole comparison purposes. XSCz
+is indeed far more complete than the peculiar velocity catalog used to
+constrain the simulation (∼2.5% of the redshift catalog is used to de-
+rive the peculiar velocity). In fact, it shows the constraining power of
+the peculiar velocities that are correlated on large scales. Namely, the
+simulation is constrained also in regions where no peculiar velocity
+measurements were available and thus used as constraints. It confirms
+once more that peculiar velocity catalogs fed to our technique, to re-
+construct/constrain the local density and velocity fields, do not need to
+be complete (Sorce et al. 2017).
+3
+VELOCITY WAVE
+3.1
+In simulated data
+Positioning a synthetic observer at the simulation box center, we de-
+rive radial peculiar velocities for all the dark matter halos and subha-
+los in the z=0 catalog. We then draw lines-of-sight in the direction of
+each dark matter halo more massive than 5 1014M⊙. All the (sub)halos
+within 10 Mpc from the line-of-sight and within 74 Mpc along the
+line-of-sight from a given massive dark matter halo (with the center
+and edge of the box as upper limits) are selected to plot the latter cor-
+responding velocity wave. Namely, as shown on Fig. 2, radial peculiar
+velocities, with respect to the synthetic observer, of (sub)halos within
+a cylinder at maximum 148 Mpc long and 20 Mpc wide are used to vi-
+sualize the velocity wave caused by the massive dark matter halo in the
+cylinder center. Note that because the simulation is constrained to re-
+produce the local Universe, we choose not to use the periodic boundary
+conditions to wrap around the box edges. It will indeed not be repre-
+sentative of local structures. A 10 Mpc radius cylinder corresponds to
+about three times the virial radius of the massive clusters under study
+here (M>5 1014M⊙). Since the goal is to study the link between veloc-
+ity wave properties and cluster masses, exact masses cannot be used to
+define the cylinder shape. Finally, such large volumes permit probing
+the infall region around the massive halos. Note that a cylinder shape
+is preferable to a cone shape to get an unbiased wave signal. A cone
+would indeed result in a distorted signal as it would probe a larger and
+larger region around a massive halo with the distance.
+3.2
+In observational data
+Observational data are taken from the raw second and third catalogs
+of the Cosmicflows project (Tully et al. 2013, 2016). Note that the
+3 https://wise2.ipac.caltech.edu/staff/jarrett/2mass/XSCz/specz.html
+second catalog containing ∼8000 galaxies, with a mean distance of
+∼90 Mpc, serves as the basis to build the constraint-catalog of ∼5000
+bias-minimized radial peculiar velocities of galaxies and groups with
+a mean distance of ∼60 Mpc. By contrast, the third catalog contains
+∼17,000 galaxies with a mean distance of ∼120 Mpc. The third
+catalog is not used to constrained our CLONE initial conditions and
+thus constitute partly an independent dataset for consistency check.
+More precisely, it serves the two-fold goal of extending the number
+of observational datapoints to be compared with the simulation and
+highlighting again the constraining power of peculiar velocities. The
+latter can indeed permit recovering structures that are not directly
+probed and that are at the limit of the non-linear threshold. In the
+sense that there is no direct measurement in a given region but,
+because the latter influences the velocities of other regions (large scale
+correlations), it can still be reconstructed.
+Uncertainties on distances and radial peculiar velocities in these
+catalogs depend on the distance indicator used to derive the distance
+moduli. Error bar sizes need to be limited to see clearly velocity waves.
+Thus, to be able to compare with the simulated data, only galaxies with
+uncertainties on distance moduli smaller than 0.2 dex are retained.
+There remain 338 and 424 galaxies respectively from the second and
+third catalogs with a mean distance of ∼50 Mpc. These galaxies are
+mostly hosts of supernovae, especially those the furthest from us (dis-
+tance indicator with a small uncertainty even as the distance increases).
+To derive the radial peculiar velocities of these galaxies, we use
+both galaxy distance moduli (µ) and observational redshifts (zobs)
+following Davis & Scrimgeour (2014). We add supergalactic longitude
+and latitude coordinates to derive galaxy cartesian supergalactic
+coordinates. A cosmological model is then required to determine
+peculiar velocities. While we use ΛCDM, as cosmicflows catalog zero
+points are calibrated through a long process on WMAP (rather than
+Planck) values (Ωm=0.27, ΩΛ=0.73, H0=74
+km s−1 Mpc−1, Tully
+et al. 2013, 2016), we have to use the same parameter values. We
+indeed showed that when applying the bias minimization technique
+to the peculiar velocity catalog of constraints, we drastically reduce
+the dependence on ΛCDM cosmological parameter values (Sorce
+& Tempel 2017). However, in order to be able to probe the whole
+velocity wave for the comparisons, we have to use the raw catalog i.e.
+with neither galaxy grouping nor bias minimization. Consequently,
+if were to take Planck values to derive galaxy peculiar velocities,
+the WMAP calibration would translate into a residual Hubble flow
+visible in the background-expansion-subtracted Hubble diagram.
+Subsequently, using WMAP values for the observations:
+Luminosity distances, dlum, are derived from distance modulus mea-
+surements, µ, obtained via distance indicators:
+µ = 5log10(dlum (Mpc)) + 25
+(1)
+Cosmological redshifts, zcos, are then obtained through the equation:
+dlum = (1 + zcos)
+� zcos
+0
+cdz
+H0
+�
+(1 + z)3Ωm + ΩΛ
+(2)
+Galaxy radial peculiar velocity, vpec, are finally estimated, using the
+observational zobs and cosmological zcos redshifts with the following
+formula:
+vpec = czobs − zcos
+1 + zcos
+(3)
+where vpec will always refer to the radial peculiar velocity in this paper
+and c is the speed of light.
+© 2022 RAS, MNRAS 000, 1–10
+
+4
+Sorce et al.
+Virgo
+0
+20
+40
+60
+80
+d (Mpc)
+0
+2000
+4000
+6000
+v (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Centaurus
+0
+20
+40
+60
+80
+100
+d (Mpc)
+0
+2000
+4000
+6000
+v (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Virgo
+0
+20
+40
+60
+80
+d (Mpc)
+-2000
+-1000
+0
+1000
+2000
+3000
+vpec (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+Centaurus
+0
+20
+40
+60
+80
+100
+d (Mpc)
+-2000
+-1000
+0
+1000
+2000
+3000
+vpec (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+Figure 3. Radial velocities of simulated dark matter (sub)halos (black and grey scale) and observed galaxies (orange, blue and red) as a function of the distance
+from the synthetic observer and us respectively. Error bars stand for uncertainties on observational distance and velocity estimates. Orange and light blue (red and
+dark blue) filled squares and diamonds show observed galaxies assuming H0=74 (67.77) km s−1 Mpc−1 for scaling positions. CF2 (CF3) corresponds to the second
+(third) catalog of the Cosmicflows project. Larger symbols are used for galaxies, with a peculiar velocity higher than 1000 km s−1, identified as the closest to the
+simulated massive halos assuming the synthetic observer at the box center and the same Supergalactic coordinate system and orientation as the local Universe. The
+arrow indicates the position of the massive dark matter halo in the simulation. Names of corresponding observed clusters are given at the top of each panel. Velocity
+waves stand out in the different lines-of-sight and there is a good agreement with observational datapoints for those two best-constrained clusters the closest to us.
+Top: Hubble diagram. Bottom: Hubble flow subtracted. The solid and dashed yellow lines are respectively the simulated positive-half velocity wave envelope and its
+Gaussian-plus-continuum fit. The color scale filling the black circles stands for their distance from the line-of-sight. From black to light grey, objects are less than
+2.5, 5, 7.5 and 10 Mpc away from the line-of-sight. The dark matter halo virial masses in the simulation are M=9.8×1014M⊙ and M=9.0×1014M⊙ for the Virgo and
+Centaurus cluster counterparts respectively.
+3.3
+Simulated vs. observed data
+Assuming the synthetic observer at the box center and the simulated
+volume oriented similarly to the local volume, observed and simulated
+positions and lines-of-sight can be matched. We can only compare
+velocity waves born from local galaxy clusters for which infalling
+galaxy peculiar velocities, with uncertainties on corresponding
+distance moduli smaller than 0.2 dex, are available in the observed
+cluster surroundings. We thus select these clusters. For each simulated
+massive dark matter halo, the quickest way is then to search for the
+closest observed galaxy, in our selected above samples, with a radial
+peculiar velocity greater than 1000 km s−1 (∼2σ above the average).
+This is indeed a signature that it has most probably an observed cluster
+with a mass of at least a few 1014M⊙ as a neighbor. Whenever a
+simulated massive dark matter halo is within the 2σ uncertainty of
+the observed galaxy distance, we select all the observed galaxies in
+the cylinder corresponding to the line-of-sight. For every case, there
+is indeed a massive observed cluster in the vicinity of the galaxies.
+More to the point, given the Supergalactic coordinates of the observed
+clusters and those of the simulated ones in the box, they indeed match.
+Fig. 3 superimposes observed and simulated lines-of-sight with
+the velocity waves born from the two closest most massive local
+clusters. Observational data is of sufficient quality in their respec-
+tive infall region to warrant adequate comparisons. From left to right,
+galaxy clusters (dark matter halos) are at increasing distance from
+us (the synthetic observer). The name of the clusters is indicated at
+the top of each panel. Filled black and grey circles stand for simu-
+lated (sub)halos while filled light blue and orange squares and dia-
+monds represent observed galaxies. Because the simulation was run
+with H0 = 67.77 km s−1 Mpc−1, filled dark blue and red squares and
+diamonds are observed galaxies at positions rescaled with this latter
+value. Position differences are always within about the 1σ uncertainty
+© 2022 RAS, MNRAS 000, 1–10
+
+Velocity waves
+5
+Cluster
+CLONE/CF2
+CLONE/CF2
+CLONE/CF3
+CLONE/CF3
+Cylinder radius
+10 Mpc
+2.5 Mpc
+10 Mpc
+2.5 Mpc
+Virgo
+0.0098
+0.011
+0.0058
+0.0071
+Centaurus
+0.010
+0.011
+0.006
+0.0073
+Abell 569
+0.25
+0.25
+0.084
+0.085
+Coma
+0.17
+0.17
+0.25
+0.25
+Abell 85
+0.25
+0.25
+0.25
+0.25
+Abell 2256
+0.50
+0.50
+0.50
+0.50
+PGC 765572
+0.050
+0.051
+0.10
+0.10
+PGC 999654
+0.50
+0.50
+0.50
+0.50
+PGC 340526
+0.25
+0.25
+0.50
+0.50
+PGC 46604
+0.50
+0.50
+0.50
+0.50
+Table 1. Kolmogorov-Smirnov statistic or highest distance between the cumula-
+tive distribution functions of the observed and simulated lines-of-sight including
+the velocity waves.
+on the distance. Arrows indicate the position of the most massive halos
+in the lines-of-sight of interest.
+In the top panels, the Hubble diagrams are clearly distorted by
+the presence of massive halos. Their corresponding velocity wave or
+triple-value region signatures show up. Bottom panels with the Hub-
+ble flow subtracted equally confirms the waves. The simulated velocity
+waves stand out in the peculiar velocity of (sub)halos plotted as a func-
+tion of the distance from the synthetic observer diagrams for the two
+massive dark matter halos. The agreement with the observational data
+points is qualitatively good. All the more since only sparse peculiar ve-
+locities of today field galaxies and groups are used to constrained the
+linear initial density and velocity fields, at the positions of the latter
+progenitors, using solely linear theory and a power spectrum assuming
+a given cosmology. Then the full non-linear theory is used to evolved
+these initial conditions from the initial redshift down to z=0 within a
+ΛCDM framework.
+The signatures of Virgo West and the group around NGC4709
+that are respectively beyond Virgo and Centaurus in the lines-of-sight
+can also be identified as secondary waves. These smaller waves follow
+the highest ones representing the main clusters in both the observations
+and the simulation. Additionally, a void between us and Centaurus
+in the line-of-sight shows equally well in both the simulation and
+the observations. The accuracy with which the CLONE reproduces
+the lines-of-sight dynamical state of Virgo and Centaurus is visually
+excellent.
+To quantify the agreement between simulated and observed
+lines-of-sight, we use a 2D-Kolmogorov-Smirnov statistic test applied
+to the simulated and observed galaxy velocity and position samples
+following Peacock (1983); Fasano & Franceschini (1987). p-values
+obtained for Virgo and Centaurus are above 0.20. They are actually
+close to 1.0 but values above 0.20 have no particular significance. They
+only confirm that the observed and simulated distributions along the
+line-of-sight are not significantly different. Additionally, Table 1 gives
+the 2D-Kolmogorov-Smirnov (KS) statistic or the highest distance
+between the cumulative distribution functions of the observed and
+simulated lines-of-sight including the velocity waves. A single 2D-KS
+statistic value has no particular meaning but several together permit
+ordering the simulated lines-of-sight from those that match the most
+their observational counterpart to those that match it the less (smallest
+to largest values). Virgo and Centaurus lines-of-sight happen to be
+equally well reproduced by the simulation. 2D-KS statistic values are
+barely different when considering all the subhalos/galaxies within
+a 10 Mpc radius or solely those within a 2.5 Mpc radius from the
+line-of-sight. The agreement is slightly better with galaxies from the
+third catalog (CF3) of the Cosmicflows project than with those of the
+Cluster
+CLONE/CF2
+CLONE/CF2
+CLONE/CF3
+CLONE/CF3
+Cylinder radius
+10 Mpc
+2.5 Mpc
+10 Mpc
+2.5 Mpc
+Virgo
+6
+10
+9
+12
+Centaurus
+21
+37
+25
+36
+Abell 569
+14
+23
+11
+22
+Coma
+27
+40
+205
+225
+Abell 85
+184
+286
+299
+400
+Abell 2256
+152
+152
+364
+364
+PGC 765572
+39
+53
+56
+70
+PGC 999654
+687
+687
+662
+662
+PGC 340526
+92
+99
+16
+41
+PGC 46604
+544
+544
+544
+544
+Table 2. ζ-metric in km s−1. It measures the difference between the simulated
+and observed lines-of-sight. The higher ζ is the more different the lines-of-sight
+are. See the text for a detailed explanation.
+second one, although the second one is the starting point to build the
+constrained initial conditions. However, given that the third catalog
+has more points and smaller uncertainties, it is encouraging that the
+simulation matches more the third catalog than the second one. The
+2D-KS statistic test cannot indeed take into account uncertainties.
+Finally, 2D-KS statistic values do not differ when using H0 = 67.77
+rather than 74 km s−1 Mpc−1.
+The 2D-KS statistic test cannot take into account the real distance
+of galaxies. It compares only the cumulative distributions of galaxies
+along the lines-of-sight using four directions (smallest to largest dis-
+tances to the y-axis and vice versa, smallest to largest distances to the
+x-axis - in that case velocities because they are centered on zero - and
+vice versa). Consequently, we also define our own ζ-metric to compare
+simulated and observed lines-of-sight as follows:
+ζ = 1
+n
+n
+�
+i=1
+�
+(min[vobs[i] − vsim])2 + [(min[dobs[i] − dsim]) × H0]2
+(4)
+where n is the number of observed galaxies in the line-of-sight. vX are
+the galaxy/subhalo observed and simulated peculiar velocities and dX
+are their distances.
+Table 2 gives the values of ζ for the different lines-of-sight.
+Because ζ-values are only modified by a few percent when changing
+H0 value, their mean is reported in the table. Like for the 2D-KS
+statistic values, ζ-values permit ordering the simulated lines-of-sight
+(including waves) that are the best reproduction of the observed ones
+to those that reproduce them the less. Since our ζ-metric results in
+similar conclusions as the 2D-KS statistic does, it seems appropriate.
+Moreover, contrary to the 2D-KS statistic, it is sensitive to the real
+distance of the cluster, not solely to its position on the fraction of
+the line-of-sight that is studied. It thus includes both differences due
+to a difference in height and to a shift in position along the entire
+line-of-sight. It is easily checked by randomly shuffling observed
+and simulated lines-of-sights and comparing them. The ζ-metric then
+gives values on average between a 100 and up to 1000 km s−1. The
+ζ-metric though, like the 2D-KS statistic, does not take into account
+uncertainties on observational distance and velocity estimates.
+In the rest of the paper, we work solely with the background ex-
+pansion subtracted since it does not affect our conclusion and ease the
+comparisons, studies and analyses.
+Given the above mentioned success, although the simulation
+matches best the local large-scale structure by construction in the inner
+part, where most of the constraints are, Fig. 4 shows an additional four
+massive halos that are more distant. These halos are still matching
+nicely observational clusters that are further away. Tables 1 and 2
+© 2022 RAS, MNRAS 000, 1–10
+
+6
+Sorce et al.
+Abell 569
+20
+40
+60
+80
+100
+120
+140
+160
+d (Mpc)
+-2000
+-1000
+0
+1000
+2000
+3000
+vpec (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+Coma
+60
+80
+100
+120
+140
+160
+180
+d (Mpc)
+-2000
+-1000
+0
+1000
+2000
+3000
+vpec (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+Abell 85
+160
+180
+200
+220
+240
+260
+280
+300
+d (Mpc)
+-2000
+-1000
+0
+1000
+2000
+3000
+vpec (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+Abell 2256
+180
+200
+220
+240
+260
+280
+300
+320
+d (Mpc)
+-2000
+-1000
+0
+1000
+2000
+3000
+vpec (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+Figure 4. Same as Figure 3 bottom panels for four clusters at increasing distance from us from left to right, top to bottom. Although these clusters are less con-
+strained, the agreement between observed and simulated waves is still visually good especially for the first two. The dark matter halo masses in the simulation are
+M=9.0×1014M⊙, M=12.6×1014M⊙, M=6.6×1014M⊙ and M=11.7×1014M⊙ for Abell 569, Coma, Abell 85 and Abell 2256 cluster counterparts respectively.
+confirm the visual impression. The different values also show the
+limitation of both metrics and confirm their complementarity. On
+the one hand, the ζ-metric is more robust to small samples than the
+2D-KS statistic: e.g. Abell 569 has a smaller observational sample in
+the second catalog of the Cosmicflows project than in the third one.
+However, the ζ-values when comparing both observational samples to
+the simulated one differ by only a few percent. On the contrary, the
+2D-KS statistic values grandly differ. One the other hand, the 2D-KS
+statistic is more robust to observational uncertainties: peculiar velocity
+values of galaxies in Coma, Abell 85 and Abell 2256 surroundings are
+compatible, given their uncertainties, between the second and third
+catalogs of the Cosmicflows project. They are higher though in the
+third catalog. Consequently, the ζ-metric gives higher values when
+comparing lines-of-sight from this third catalog to the simulated ones
+rather than lines-of-sight from the second catalog to the simulated one.
+Note though that it is not completely unexpected that the simulated
+lines-of-sight match better those from the second catalog than the
+third one. Indeed, the second catalog is the starting point to build the
+constrained initial conditions.
+Additionally, since observed galaxies with low distance uncertain-
+ties are usually not exactly along the line-of-sight of the massive clus-
+ters, their velocity constitutes a lower limit for the mass estimate of
+the observed clusters. Indeed, galaxies perfectly aligned with the ob-
+server and the cluster would have the highest possible velocity but such
+galaxies are difficult to distinguish from those belonging to the cluster.
+Consequently, for Virgo, Centaurus and Abell 569, the maximum pe-
+culiar velocity in the simulation is slightly higher than that in the ob-
+servations: it confirms that the simulated cluster have reached the low
+mass limit set by the observations. Moreover, the difference between
+the observed and simulated wave maxima is small enough that masses
+are within the same mass range according to the Least Action modeling
+(see for instance Mohayaee & Tully 2005; Tully & Mohayaee 2004).
+This agreement is confirmed by observational data that follow the wave
+shape so as to reproduce its width. The next section expands on the link
+between wave properties and cluster masses. Note that the adequacy
+between simulated and observed velocity wave shapes is really good
+for Abell 569 given that even small uncertainty peculiar velocities, not
+used to constrain this wave progenitor in the initial conditions’ linear
+regime, follow also the simulated wave contour. There are indeed two
+orange/red datapoints from the third catalog that have no blue counter-
+part in the second catalog. The 2D-KS statistic small value confirms
+the adequacy.
+For Coma, Abell 85 and Abell 2256, given their hosted galaxy
+peculiar velocity uncertainties, masses are also in good agreement and
+the lower mass limit is reached. This is not fully expected given that
+these clusters are at the edge of the constrained region (50%, 90% and
+99% of the constraints are in ∼75-80, 150-160 and 275-290 Mpc).
+Additional precise observational data are however required to probe
+the wave slopes and check their width to tighten the constraint on the
+masses.
+Fig. 5 shows four additional velocity waves born from massive
+dark matter halos to which we can associate observed galaxies. The
+© 2022 RAS, MNRAS 000, 1–10
+
+Velocity waves
+7
+PGC765572
+100
+120
+140
+160
+180
+200
+220
+240
+d (Mpc)
+-2000
+-1000
+0
+1000
+2000
+3000
+vpec (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+PGC999654
+120
+140
+160
+180
+200
+220
+240
+260
+d (Mpc)
+-2000
+-1000
+0
+1000
+2000
+3000
+vpec (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+PGC340526
+160
+180
+200
+220
+240
+260
+280
+d (Mpc)
+-2000
+-1000
+0
+1000
+2000
+3000
+vpec (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+PGC46604
+160
+180
+200
+220
+240
+260
+280
+d (Mpc)
+-2000
+-1000
+0
+1000
+2000
+3000
+vpec (km s-1)
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+CLONE
+CF2-67
+CF3-67
+CF2-74
+CF3-74
+Envelope
+Fit
+Figure 5. Same as Figure 3 bottom panels for four additional clusters. Names at the top of each panel are PGC (Principal Galaxy Catalog) numbers of the galaxies
+with the highest velocity in the observational catalog at the given locations.
+galaxies with the largest peculiar velocities are identified by their PGC
+(Principal Galaxy Catalog) number at the top of each panel. Here again,
+given the distance of these clusters and the sparsity and limit of our
+constraint-catalog, the agreement is quite good. Tables 1 and 2 confirm
+again the visual impression. They also highlight again the limitations of
+both metrics. Both values must be given together to conclude on how
+much the observed and simulated lines-of-sight match. Note that we
+identify other simulated velocity waves corresponding to local clusters
+(e.g. in the Perseus-Pisces region) but observational data is not of suf-
+ficient quality or absent in the infall region for comparisons. Nonethe-
+less, all the halos and associated waves are used for the next section
+studies. The mass range is actually extended down to 2 1014M⊙.
+4
+WAVE PROPERTIES VS. CLUSTER MASSES
+4.1
+The amplitude
+The wave amplitude is the first obvious property to check against halo
+mass. Indeed, the deeper the gravitational potential well, the faster
+should galaxies fall onto it. The amplitude is thus defined as the dif-
+ference between the maximum and minimum peculiar velocities of
+galaxies falling onto the cluster either from the front or from behind
+with respect to the synthetic observer. Fig. 6 thus shows the amplitude
+of the simulated velocity waves as a function of the dark matter halo
+masses. Each black and red filled circle corresponds to a halo. Red
+ones stand for clusters identified in Fig. 3 to 5. While it is immediate to
+notice that there is a clear correlation between the wave amplitude and
+the halo mass, one can also point out that the amplitude is extremely
+difficult to measure in observational data and that there is a residual
+scatter. Indeed, measuring the amplitude in observational data implies
+getting exquisite distance (peculiar velocity) estimates of galaxies ex-
+actly in the line-of-sight of the cluster with respect to us. It supposes
+first that there are actually galaxies exactly aligned. Then, identifying
+these galaxies and measuring their distances with great accuracy, while
+they fall onto the cluster from the front is already quite a challenge, let
+alone when they fall from behind.
+In any case, the residual scatter suggests that the amplitude, be it
+measurable, alone cannot be used as a precise proxy for cluster mass
+estimates. Part of this scatter is probably due to the fact the galaxies are
+not perfectly aligned with us and the cluster. The gravitational potential
+well shape might also be responsible for another part of this scatter. To
+a lesser extent, the large-scale structure environment might also play a
+role.
+4.2
+The height
+While the wave height is not expected to be a better proxy than the
+wave amplitude, it is interesting to check whether there still is a tight
+Figure 6. Amplitude of the simulated velocity waves as a function of the dark
+matter halo masses. Halos shown in Fig. 3 to 5 are identified in red.
+enough correlation. Indeed, while it is challenging to have precise dis-
+tance measurements for both galaxies falling from the front and from
+behind a cluster in the line-of-sight with respect to us, it might be fea-
+sible especially for galaxies falling from the front. The height is thus
+defined as the maximum (minimum) peculiar velocities of galaxies
+falling onto the cluster from the front (behind) with respect to the syn-
+thetic observer. In Fig. 7, each black and red (blue and orange) filled
+circles stand for the height of a dark matter halo positive-(negative-
+)half wave as a function of its mass. Red and orange are used for dark
+matter halos from Fig. 3 to 5.
+A similar correlation as with the amplitude is found although
+with a somewhat larger scatter. Interestingly it also shows that velocity
+waves are not symmetric: their maximum differs from their minimum.
+Both the potential well shape and the non-perfect alignement observer-
+galaxy-halo or observer-halo-galaxy might be the reason for this asym-
+metry. Nonetheless because there still is a correlation and because in
+observational data it is easier to get accurate datapoints at the wave
+front than in its wake, it is legitimate to focus on the positive-half ve-
+locity wave shape to study more thoroughly the relation with the halo
+mass.
+© 2022 RAS, MNRAS 000, 1–10
+
+8
+Sorce et al.
+Figure
+7.
+Dark
+(blue)
+filled
+circles
+are
+heights
+of
+the
+simulated
+positive(negative)-half velocity waves as a function of the dark matter halo
+masses. Halos shown in Fig. 3 to 5 are identified in red (orange).
+4.3
+Height, width and continuum
+After deriving the positive-half wave envelope of every dark matter
+halo, we choose to fit the simplest model possible, a Gaussian-plus-
+continuum model, to each one of them as follows:
+vpec = Afit × e
+−(d−d0)2
+2σ2
+fit
++ C fit
+(5)
+where Afit, σ fit and C fit are respectively the Gaussian amplitude,
+its standard deviation and a continuum. d0 depends on the halo
+distance and has no other purpose than centering the Gaussian on
+zero. Its sole physical meaning is to be the actual distance of the
+halo. The amplitude is related to the positive-half wave envelope
+height while the standard deviation is linked to its width. Finally, the
+continuum gives the positive-half wave offset from a zero average
+velocity. For visualization, envelopes and their fits for halos presented
+in Fig. 3 to 5 are shown as solid and dashed lines on these same figures.
+Fig. 8 gathers the three parameters of the fits and halo masses
+for a concomitant study to highlight an eventual multi-parameter
+correlation. The Gaussian amplitude is represented as a function of
+the Gaussian standard deviation while the color scale stands for the
+continuum. From black-violet to red, the continuum decreases from
+positive values to negative ones. The model uncertainty is shown as
+error bars for the amplitude and standard deviation. The color scale
+smoothness includes the continuum uncertainty. The Gaussian-plus-
+continuum model choice proves to be robust given the tiny error bars
+that it results in. The filled circle sizes are proportional to the dark
+matter halo masses. Finally, an additional small red filled circle is used
+to identify each halo analyzed in Fig. 3 to 5.
+The previous subsection (4.2) showed that there is a correlation
+between the wave height and the halo mass. It is thus not surprising to
+find back that the more massive the halo is (larger circle), the larger
+500
+1000
+1500
+2000
+Afit (km s-1)
+2
+4
+6
+8
+10
+12
+14
+σfit (Mpc)
+574
+407
+239
+71
+-96
+-264 -432
+Cfit (km s-1)
+Virgo
+Centaurus
+Abell 569
+Coma
+Abell 85
+Abell 2256
+PGC765572
+PGC999654
+PGC340526
+PGC46604
+Figure 8. Parameters of the Gaussian-plus-continuum fit to the simulated
+positive-half velocity waves. σfit stands for the Gaussian standard deviation,
+Afit for its amplitude and C fit for the continuum. The filled circle sizes are pro-
+portional to dark matter halo masses. Tiny error bars on the standard deviation
+and amplitude resulting from fitting the envelopes highlight the adequacy of the
+model choice. Halos shown in Fig. 3 to 5 are identified with red nametags and
+additional small red filled circles.
+the Gaussian amplitude is (larger value). As stated above, the Gaussian
+amplitude is indeed the counterpart of the positive-half wave height.
+In addition, there is a small correlation between the amplitude
+and standard deviation thus halo mass. More massive halos seem to
+give birth to wider waves. The scatter is however quite large. It cer-
+tainly depends greatly on the halo triaxiality and thus on its orientation
+with respect to us. A similar conclusion is valid for the continuum, the
+smaller the continuum but for extreme values is, the more massive the
+halo is on average. Anyhow, the scatter is quite large in that case. A
+strong dependence on the global environment of the dark matter halo
+in addition to the halo mass might be in cause here.
+Interestingly a general pattern emerges quite clearly though:
+• the most massive halos (≳ 6 1014 M⊙) tend to give birth to positive-
+half waves that have a continuum compatible with zero or slightly
+negative/positive in addition to high amplitude and standard deviation
+values.
+• the less massive halos ( 2 1014 M⊙≲M≲ 4 1014 M⊙) tend to give birth
+to positive-half waves that have a continuum compatible with zero or
+slightly negative/positive in addition to low amplitude and standard
+deviation values.
+• intermediate mass halos (4 1014 M⊙≲M≲ 6 1014 M⊙) give rise to
+positive-half waves that have high continuum absolute values. Such
+values permit distinguishing them from the most massive halos with
+which they share high amplitude and possibly standard deviation
+values, especially in the negative continuum case.
+It is highly probable that the global environment or cosmic web is
+responsible for such a finding. We will investigate this link in more
+details in future studies.
+The halo segregation in different continuum value classes is an-
+other quite inspiring source. There seems to be a different correlation
+for each continuum value class:
+• Halos with fits resulting in a high (close to zero) continuum value
+© 2022 RAS, MNRAS 000, 1–10
+
+Velocity waves
+9
+seems to have masses correlated with the Gaussian amplitudes but not
+so much with the Gaussian standard deviations that appear to have low
+values (present a large scatter).
+• Halos with fits resulting in a very low continuum value have both
+amplitudes and standard deviations correlated together as well as with
+the masses.
+• Halos with fits resulting in either positive or negative intermediate
+continuum values present masses correlated with amplitudes and up
+to a certain point with standard deviations. Consequently, although to
+a lesser extent than for halos whose continuum values are quite low,
+amplitudes and standard deviations are slightly correlated.
+To summarize, since the fit parameters are interdependent, a
+global fit to the velocity wave seems the best approach to obtain clus-
+ter rough mass estimates rather than single and independent measure-
+ments of amplitude, height and width. Because different categories
+appear among halos, in future studies, a machine learning approach
+might become handy to actually get accurate enough mass estimates
+from sparse observations. In a first approach, the simple Gaussian-plus-
+continuum fit presented here could be used as a model reduction.
+5
+CONCLUSIONS
+Galaxy clusters are excellent cosmological probes provided their
+mass estimates are accurately determined. Fueled with large imaging
+surveys, stacked weak lensing is the most promising mass estimate
+method though it provides estimates within relatively small radii.
+Given the large amount of accompanying redshift and spectroscopic
+data overlapping the imaging surveys, we must take the opportunity to
+calibrate also with a reasonable accuracy a method based on galaxy
+dynamics. Two independent measures hold indeed better constraints
+on the cosmological model. Infall zones of galaxy clusters are proba-
+bly the less sensitive to baryonic physics, thus mostly shielded from
+challenging systematics, and probe large radii. These manifestations
+of a tug of war between gravity and dark energy provide a unique
+avenue to test modified gravity theories when comparing resulting
+mass estimates to those from stacked weak lensing measurements.
+Combined with stacked weak lensing results, they might even yield
+evidence that departure from General Relativity on cosmological
+scales is responsible for the expansion acceleration.
+The accurate calibration of the relation between infall zones
+properties and cluster masses starts with careful comparisons between
+cosmological simulations and observations. In this paper, we thus
+present our largest and highest resolution Constrained Local &
+Nesting Environment Simulation (CLONE) built so far to reproduce
+numerically our cosmic environment. This simulation stems from
+initial conditions constrained by peculiar velocities of local galax-
+ies. By introducing this cosmological dark matter CLONE of the
+local large-scale structure with a particle mass of ∼109M⊙ within a
+∼738 Mpc box, we have sufficient resolution to study the effect of
+the gravitational potential of massive local halos onto the velocity
+of (sub)halos. We can also compare with that of their observational
+cluster counterparts.
+Velocity waves stand out in radial peculiar velocity - distance to
+a box-centered synthetic observer diagram. The agreement between
+lines-of-sight including velocity waves, caused by the most massive
+dark matter halos of the CLONE and those born from their observa-
+tional local cluster counterparts, is visually good especially for the
+clusters the closest to us that are the best constrained (e.g. Virgo,
+Centaurus). Secondary waves due to smaller groups in (quasi) the
+same line-of-sight as the most massive clusters stand out equally even
+though they are further into the non-linear regime. Indeed, prior to
+full non-linear evolution to the z=0 state, assuming ΛCDM, CLONE
+initial conditions are constrained with solely the linear theory, a power
+spectrum and highly uncertain and sparse local peculiar velocities. The
+visual matching between the simulated and observed lines-of-sight
+is confirmed with 2D-Kolmogorov Smirnov (KS) statistic values and
+tests as well as with our own ζ-metric. Contrary to the 2D-KS statistic,
+the ζ-metric takes into account the real distance of galaxies along the
+entire lines-of-sight (not only the studied fractions). The ζ-metric is
+however more sensitive to the fact that observational uncertainties are
+not taken into account in these metrics. The two metrics appear to
+be complementary. They show that the closest clusters have the best
+reproduced lines-of-sight. The lines-of-sight of clusters at the edges of
+the constrained region and even slightly beyond are also reproduced
+by the simulation although to a smaller extent.
+Additionally, a Gaussian-plus-continuum fit to the envelope of
+the positive-half of all the velocity waves born from dark matter
+halos more massive than 2 1014M⊙ in the simulation reveals both the
+variety and complexity of the potential wells as well as the correlation
+of the fit parameters with the halo masses. Overall, the Gaussian
+amplitude is mostly linked to the halo mass, but for a few exceptions,
+with a residual scatter. Although the Gaussian standard deviation
+is not always correlated with the mass, it can be slightly correlated
+with the Gaussian amplitude thus with the mass. The continuum is
+certainly an interesting parameter to consider as it permits splitting
+the halos into different classes. Each continuum value seems to drive a
+given correlation between the Gaussian amplitude and the halo mass
+and, to a smaller extent, with the Gaussian standard deviation. To
+summarize, parameter fits are completely interdependent, a global fit
+to the velocity wave is then the best approach to obtain a first rough
+cluster mass estimate.
+First and foremost, this work confirms the potential of the
+velocity wave technique to get massive cluster mass estimates and
+test gravity and cosmological models. Our CLONES, with the first
+shown reproduction of observed lines-of-sight including velocity
+waves, could in the near future provide the zero point of galaxy
+infall kinematic technique calibrations (Zu & Weinberg 2013). A
+bayesian inference model or/and a machine learning technique built
+and trained on random simulated galaxy surveys that is then applied to
+both constrained simulated and observed galaxy surveys must recover
+the same local velocity waves and corresponding mass estimates to
+be validated. Our CLONES will moreover allow minimizing obser-
+vational biases as any real environmental and cluster property will
+be reproduced for perfect one-to-one comparisons. Local kinematic
+mass estimates can then become accurate. Once compared with other
+techniques of local galaxy cluster mass estimates, they will permit
+calibrating the zero-point of these other techniques to be applied to
+further-and-further away clusters.
+DATA AVAILABILITY
+Synthetic catalogs are available upon reasonable request to the authors.
+ACKNOWLEDGEMENTS
+The authors acknowledge the Gauss Centre for Supercomputing e.V.
+(www.gauss-centre.eu) and GENCI (https://www.genci.fr/) for funding
+this project by providing computing time on the GCS Supercomputer
+SuperMUC-NG at Leibniz Supercomputing Centre (www.lrz.de) and
+Joliot-Curie at TGCC (http://www-hpc.cea.fr), grants ID: 22307/22736
+© 2022 RAS, MNRAS 000, 1–10
+
+10
+Sorce et al.
+and A0080411510 respectively. This work was supported by the grant
+agreements ANR-21-CE31-0019 / 490702358 from the French Agence
+Nationale de la Recherche / DFG for the LOCALIZATION project and
+ERC-2015-AdG 695561 from the European Research Council (ERC)
+under the European Union’s Horizon 2020 research and innovation
+program for the ByoPiC project (https://byopic.eu). KD acknowledges
+support by the COMPLEX project from the ERC under the European
+Union’s Horizon 2020 research and innovation program grant agree-
+ment ERC-2019-AdG 882679. The authors thank the referee for their
+comments. JS thanks Marian Douspis for useful comments, the By-
+oPiC team and her CLUES collaborators for continuous discussions.
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+

T9E4T4oBgHgl3EQfLwz0/content/tmp_files/2301.04942v1.pdf.txt

@@ -0,0 +1,3949 @@
+1 
+ 
+Hydrogen storage in C14 type Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 high entropy alloy 
+ 
+Abhishek Kumar1
+, T. P. Yadav1,2*, M.A. Shaz1and N.K. Mukhopadhyay3 
+1Hydrogen Energy Centre, Department of Physics, Institute of Science 
+Banaras Hindu University, Varanasi, Uttar Pradesh, India  
+2Department of Physics, Faculty of Science, University of Allahabad, Prayagraj-211002, India 
+3Department of Metallurgical Engineering, Indian Institute of Technology (Banaras Hindu University),  
+Varanasi-221 005, India 
+ 
+Abstract 
+In this present investigation, we discussed the synthesis, microstructure, and hydrogen storage behavior in C14 type 
+intermetallic Laves phase in a hexanary Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 high entropy alloy (HEA). In this HEA, 
+three elements are hydride-forming elements (Ti, V, Zr), whereas other three are non-hydride-forming elements (Fe, 
+Mn, Co). The thermodynamic parameter like enthalpy of mixing was calculated using the Meidma’s model. The 
+mixing enthalpy (∆Hmix) of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA system was evaluated to be- 23.3472 kJ/mole, and 
+atomic radius mismatch turned out to be = 7.441%. This alloy was synthesized using 35 kW radio frequency 
+induction furnace under  argon atmosphere. X-ray diffraction technique (XRD) revealed that this system belongs to 
+the C14 type Laves phase with unit cell parameters a= b =5.0158 Å, c=8.1790 Å, α = β = 90˚, γ = 120˚ under Space 
+group P63/mmc. Microstructural analysis was carried out with the help of a transmission electron microscope 
+(TEM). The SEM- EDX data confirms the elemental composition. Hydrogen absorption and desorption of this high 
+entropy intermetallic was carried out using the PCI apparatus. The total hydrogen storage of this system was 
+observed around ~0.53 wt%. However, it exhibited better hydrogen and ab/de-sorption kinetics. With the help of the 
+Van’t Hoff plot, calculated experimental change in enthalpy of Ti0.24-V0.17-Zr0.17-Co0.17-Fe0.08-Mn0.17 HEA for 
+hydrogen absorption and desorption was found out to be ~ -19.06 ± 1.12 kJ/mol and -34.10 ± 1.32 kJ /mol 
+respectively.  The possibility of developing high entropy Laves phase-based hydrogen storage materials was 
+advocated.    
+Corresponding authors: yadavtp@gmail.com 
+ 
+ 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)2 
+ 
+Introduction 
+Recent years have seen a lot of interest in a new class of materials called ‘High Entropy Alloys’ (HEAs) (Marques et 
+al. 2021, Yadav el al. 2017, Mishra et al. 2019, Mishra et al. 2020). In general, HEAs contain five or more elements, 
+each with a concentration of five to thirty-five atomic percentages (at.%) or more, in contrast to conventional alloys 
+based on a single primary element. To improve phase stability, HEAs are understood to exhibit large mixing 
+entropies of solid solution phases (Murty et al. 2019). The research publication by Yeh et al. (2004a 2004b), Cantor 
+et al. (2004), and Ranganathan (2003) was published for the first time for launching the field of HEAs. Yeh 
+independently proposed the single-phase multi-principal element alloy in 1995, making this idea a ground-breaking 
+success in researching HEAs (Murty et al. 2019). It' is interesting to note that the high mixing entropy in multi-
+principal element alloys can dramatically lower the number of phases in high-order alloys,  leading to a single phase 
+solid solution  (Tsai et al. 2014). HEA has many functional properties like magnetic,, thermoelectric, catalytic, 
+hydrogen storage  etc. In these functional properties, hydrogen storage is considered to be one of the interesting 
+areas to explore the HEA as an effective hydrogen storage material. Nowadays, in order to counteract climate 
+change and the rise in global warming brought on by conventional fossil fuels; people demand innovative, flexible, 
+clean, and green energy sources. Among many fuels that are readily available worldwide, hydrogen is accepted as 
+one of the best candidates due to its high energy range per unit mass. Three essential elements that are needed to use 
+hydrogen as a fuel in the future are (i) hydrogen production, (ii) its storage, and (iii) applications. Hydrogen storage 
+is one of the most crucial components of using hydrogen as a fuel. One of the safest and most efficient ways to store 
+hydrogen is in solid-state metal hydrides. Due to the infinite combination of alloy forming possibilities, the HEAs 
+are novel and promising materials for hydrogen storage (Yadav et al. 2022). In 2010, the first investigation was done 
+in HEAs to study the hydrogen storage kinetics. This study claimed 0.03-1.80 wt% hydrogen storage in multi-
+principal component CoFeMnTixVyZrz (Kao et al. 2010) alloys; after that, in TiZrHfNbV HEA, 2.7wt% hydrogen 
+storage was reported in 2016 (Sahlberg et al. 2016). There is only one BCC phase in this alloy composition. One 
+more point common in this system is that this alloy system is designed with all the hydride forming elements, 
+because of which it has a good hydrogen storage capacity. In recent years hydrogen storage is reported as high as 
+3.51 wt% in V35Ti30Cr25Fe5Mn5 HEA belonging to a single BCC phase (Liu et al. 2021). On the contrary, the 
+maximum hydrogen storage in Laves phases is known to be 1.91 wt% (Sarc et al. 2020). It can stated from the 
+reported data that the  Laves phase has less storage properties and better  absorption and desorption kinetics 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)3 
+ 
+compared to BCC phase. The investigation on low-vanadium TiZrMnCrV-based alloys for high-density hydrogen 
+storage (Zhou et al. 2021) was reported. Due to its maximal interstitial sites available for absorbing hydrogen in 
+their voids, C14 Laves phase has been explored as hydrogen storage phase tested in recent study. People have 
+recently been concentrating on the research of phase stability during hydrogen absorption and desorption of HEAs. 
+In multi-component HEA for TiZrFeMnCrV (Chen et al. 2022), C14 type Laves phase-based HEA was fabricated 
+and followed by hydrogen storage testing after mechanical milling. The maximal hydrogen absorption for this alloy 
+was reported to be 1.80 wt% for the first cycle and 1.76 wt% for the second cycle. According to their findings, the 
+hydrogen storage capacity varied marginally between each cycle's i.e., 1.76 and 1.73 wt%. In another study, 
+TiZrCrMnFeNi HEA of C14 Laves phase has exhibited hydrogen absorption as 1.7 weight percent (Edalati et al. 
+2020). Kumar et al (2022) has shown that TiZrVCrNi Laves phase with 1-52 weight percent hydrogen remains 
+stable even after 10 cycles of hydrogenation from the perspective of phase stability. The TiZrNbCrFe HEA 
+consisting of C14 Laves phase as maor and BCC phase as minor was reported by Floriano et al. 2021 to have 1.9 
+wt% hydrogen storage capacity.In view of the potential of HEAs for hydrogen storage capability, it was felt worth 
+pursuing the study of other high entropy based alloys for exploring their structure and hydrogen storage 
+performance. Accordingly, in the present study, we selected TiZrVMnFeCo nonequiatomic HEAs and investigated 
+the structure, microstructure, and hydrogen storage kinetics. We chose a HEA system with three hydride forming 
+elements (TiZrV) and the remaining three non-hydride-forming elements (Mn, Fe, Co).The thermodynamic 
+calculation for evaluating enthalpy of mixing of this HEA was done using Meidma model. This HEA was 
+synthesized with the help of a 35 KW Radio Frequency Induction furnace in the argon atmosphere and characterized 
+by XRD, SEM and TEM techniques Hydrogen storage performance was evaluated using pressure composition 
+isotherm (PCI) equipment supplied by Advanced Material Corporation (Pittsburgh, USA). 
+ 
+Material synthesis and characterization methods 
+The high purity materials powder for the synthesis of the Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA system was procured 
+from Alfa Aesar with a purity of more than 99.50%. The constituent elements were taken as per their stoichiometry 
+for making a palette using a cylindrical steel mold equipped with the hydraulic press of acting pressure ~3x105 
+N/m2. The palette (~10 g by weight) then used for the as-cast synthesis of multicomponent HEA using the RF 
+induction melting process under argon atmosphere (purity of more than 99.90%). The ingots are melted four times to 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)4 
+ 
+ensure uniformity of chemical composition. The as-cast induction melted ingots of HEA crushed and converted into 
+powder form to perform further characterization. The first cutting-edge characterization technique used for phase 
+analysis is the Empyrean x-ray diffraction (XRD) system (Malvern Panalytical) equipped with an area detector 
+(256x256 pixels) equipped with a graphite monochromator and Cu radiation source (CuKa; = 1.5406, operating at 
+45 kV and 40 mA) in Bragg-Brentano geometry. The transmission electron microscope (TEM), TECNAI 20 G2, was 
+used to acquire the microstructures and selected area electron diffraction (SAED) pattern of the samples operating at 
+200 kV of accelerating voltage.EVO 18 scanning electron microscope at operating voltage of 25 kV (vacuum 10-5 
+torr) was used to investigate surface morphology and perform energy dispersive X-ray analysis (EDX) as well as 
+colour mapping of elements in the as-prepared samples. All de/re-hydrogenation measurements were carried out 
+with the aid of an automated two-channel volumetric sieverts apparatus (supplied by Advanced Materials 
+Corporation Pittsburgh, USA). For hydrogen storage testing, we took the 500 mg sample of HEA and placed the 
+sample in the reactor seized by quartz wool. Before performing hydrogen cycle testing, the powder HEA sample was 
+activated at 400℃ under a hydrogen pressure of 1/0.1 MPa for hydrogenation/dehydrogenation. After activation, 
+testing of the hydrogen absorption kinetics at 410 °C under 60 atm H2 pressure was carried out.  
+ 
+Results and Discussion 
+The experimental XRD diffraction patterns of the as-cast Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA are shown in figure 
+2(a). The diffraction profile has been recorded for the gross structural analysis of the as-cast alloy sample by using 
+the Empyrean x-ray diffraction (XRD; Malvern Panalytical) system. All the diffraction peaks (shown in the figure. 
+2(a)) are well fitted with the hexagonal C14 Laves phase structure parameters.The XRD pattern was well refined 
+through Le Bail profile fitting using JANA 2006 software shown in the figure. 2(b). The refinement data validated 
+the Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA system with unit cell parameters of a=b= 5.0141 Å, c= 8.1756 Å, and the 
+unit cell volume 178.0 Å3 under the space group of P63/mmc. All the refine parameters are given below in Table 1 
+To validate the structure analysis of this XRD, we used another characterization technique by transmission electron 
+microscopy (TEM) for analyzing the phase and microstructure of this Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA. The 
+bright field TEM micrograph of as-synthesized HEA shown in figure 3(a) identifies no other phases other than 
+Laves phase. The corresponding SAD pattern of this as cast HEA shown in figure 3(b) validates that this HEA 
+system belongs to a C14 type hexagonal structure with a corresponding space group is P63/mmc. 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)5 
+ 
+Surface morphology and elemental composition analysis 
+Scanning electron microscopy (SEM) has been done for surface microstructure and confirming homogeneous 
+element distribution. Figure4 (a) shows the SEM –BSE, and Energy dispersive X-ray analyses (EDX) mapping 
+images of as cast Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA with the corresponding region which is located in square 
+box in figure 4(a). The SEM-BSE image reveals the microstructure of this HEA without any cracks or defects  in 
+this as-cast HEA. Figure 4(b) overlays all the constituent elements present in this HEA. EDAX mapping image 
+establishes that all the constituent elements are distributed as per atomic percent in this as-cast Ti0.24-V0.17-Zr0.17-
+Co0.17-Fe0.08-Mn0.17 HEA. Figure 4(c) shows the SEM-BSE image from another region for the HEA sample, where 
+no crack is observed, and also no other contrast corresponding another phase. Figure 4(d) shows the EDX elemental 
+spectra to confirm the stoichiometry of the elements present in this as-cast HEA. All the data indicate that this HEA 
+has forms a single Laves phase with uniform elemental distribution. 
+ 
+Hydrogen ab/de-sorption analysis 
+Hydrogen ab/de-sorption performance in as-cast Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA is studied in this section. The 
+measurements of hydrogen sorption were carried out with automated two-channel volumetric sieverts instrument. 
+The results of the absorption kinetic curve of the as-cast Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA are shown in figure 
+5(a). Before introducing hydrogen into as-cast HEA, we firstly activate the as-cast HEA under 400 ˚C under 10-3 
+atm evacuation. We perform hydrogenation at 410˚C under 60 atm hydrogen pressures. The hydrogen desorption 
+kinetic curve of the as-cast Ti0.24-HEAis shown in figure 5(b). The hydrogen desorption kinetic curve of this as-
+cast HEA shows that this as-cast HEA absorbed 0.53 wt% of hydrogen within 15 seconds this curve. In contrast, the 
+maximum storage capacity is evaluated to be about 0.72 wt% in 150 minutes. This fastest kinetics gives interesting 
+results to understand the hydrogen storage performance In the case of desorption, we can see that the 
+dehydrogenated curve shown in figure 5(b) the as cast Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA perform desorption at 
+410 ˚C under 1 atm hydrogen pressure.  According to the hydrogenation desorption curve we can say that this HEA 
+released 0.28 wt% hydrogen within one minute at 410 ˚C under 1 atm hydrogen pressure. The results suggests that 
+this HEA shows faster hydrogen ab/desorption kinetics than some other Laves phase based HEAs. 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)6 
+ 
+The representative PCI ab/de-sorption of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA has been shown in figure 6(a) the 
+corresponding represents active Van’t Hoff plots (shown in figure 6(b)). PCI was performed at 395˚C, 410˚Cand 
+425˚C temperatures under 60 atm hydrogen pressures. With the help of the three different temperatures, we get the 
+plot corresponding to temperature v/s pressure. Calculations of the entropy and enthalpy changes that occur 
+throughout the hydrogen ab/de-sorption process typically employ the pressure values of the hydrogen ab/de-sorption 
+platform at various temperatures. The change in enthalpy (∆H) of hydride formation is given by the well-known 
+Van’t Hoff equation (Dornheim et al. 2010) 
+                 ln 𝑃 =
+Δ𝐻
+RT −
+Δ𝑆
+R                …………(i) 
+Where P is the previously specified plateau pressure, T is the corresponding temperature, R is the gas constant, and 
+H and S are the reaction enthalpy and entropy changes, respectively. The alloys' Van't Hoff plots are computed using 
+the P, as shown in figure 6. (b). The relationship between ln(P) and 1000/T is clearly linear, as can be seen in the 
+image. The slope of the fitted curves for ln(P) and 1000/T as well as the intercept on the vertical coordinate allow 
+for the quick calculation of the H and S. The results of the calculations demonstrate that the enthalpy of hydrogen 
+desorption changes. The change in enthalpy of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA for hydrogen absorption and 
+desorption has been calculated to be ΔHabs~ -19.06 ± 1.12 kJ/mol and ΔHdes -34.10 ± 1.32 kJ /mol respectively. The 
+smaller negative enthalpy of mixing in HEA suggests that they are more likely to form stable metal hydrides. The 
+formation of the metal hydride's absorption and desorption enthalpies are not equal in the current experiment. 
+Therefore, this system has fewer tendencies to create metal hydride and aids in improving the ab/desorption kinetics. 
+This suggests that they have a decreased tendency to form a stable metal hydride. 
+Conclusions 
+In this study, we have successfully synthesized the hexanary Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA with the help of 
+an RF induction furnace for the study of hydrogen storage kinetics. The evolution of a single phase of hexagonal 
+C14 high entropy Laves phase with lattice parameters a = 5.01Å and c =8.17Åwas established following Rietveld 
+refinement in this multicomponent alloy. On the basis of the kinetics study, Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 shows 
+good ab/de-desorption kinetics (absorb ~ 0.53 wt.% of H2 within 15 seconds) but poor in hydrogen storage capacity. 
+The change in enthalpy of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA for hydrogen absorption and desorption has been 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)7 
+ 
+calculated to be ~ -19.06 ± 1.12 kJ/mol and -34.10 ± 1.32 kJ /mol respectively. The present investigation suggests 
+the scope for further study on the hydrogenation kinetics at various temperatures for exploring the potential for 
+developing Laves phase high entropy alloy for hydrogen storage. 
+ 
+Acknowledgment 
+The author (AK) wishes to thank the Council of Scientific and Industrial Research (CSIR) in New Delhi, India, for 
+financial support for a senior research fellowship (Award No. 09/013(0952)/2020-EMR-I). 
+ 
+Author contributions 
+A.K. synthesized the materials and made the characterizations; T.P.Y. conceived, designed the experiments, 
+organized the data and supervision.  M.A.S. advised on the discussion of results; N.K.M. advised on the discussion 
+of results and editing the manuscript. The manuscript was written through contributions of all authors. All authors 
+have given approval to the final version of the manuscript. 
+ 
+Notes 
+The authors declare no competing financial interests. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)8 
+ 
+References 
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+alloys. Materials Science and Engineering: A 375-377: 213-218. https://doi.org/10.1016/j.msea.2003.10.257 
+ 
+Chen J, Li Z, Huang H, Lv Y, Liu B, Li Y, Wu Y, Yuan J, Wang Y, (2022) Superior cycle life of TiZrFeMnCrV 
+high 
+entropy 
+alloy 
+for 
+hydrogen 
+storage. 
+Scripta 
+Materialia 
+212: 
+114548. 
+https://doi.org/10.1016/j.scriptamat.2022.114548 
+ 
+Dornheim M (2011) Thermodynamics of Metal Hydrides: Tailoring Reaction Enthalpies of Hydrogen Storage 
+Materials. Thermodynamics - Interaction Studies - Solids, Liquids and Gases. https://doi.org/10.5772/21662 
+ 
+Edalati P, Floriano R, Mohammadi A, Li Y, Zepon G, Li HW, Edalati K (2020) Reversible room temperature 
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+storage 
+in 
+high-entropy 
+alloy 
+TiZrCrMnFeNi. 
+Scripta 
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+178: 
+387–390 
+https://doi.org/10.1016/j.scriptamat.2019.12.009 
+ 
+Floriano R, Zepon G, Edalati K, Fontana GLBG, Mohammadi A, Ma Z, Li HW, Contieri RJ (2021) Hydrogen 
+storage properties of new A3B2-type TiZrNbCrFe high-entropy alloy. International Journal of Hydrogen Energy, 
+46(46) 23757-23766. https://doi.org/10.1016/j.ijhydene.2021.04.181 
+ 
+Kao YF, Chen SK, Sheu JH, Lin JT, Lin WE, Yeh JW, Lin SJ, Liou TH, Wang CW (2010) Hydrogen storage 
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+ 
+Kumar A, Yadav TP, Mukhopadhyay NK (2022) Notable hydrogen storage in Ti–Zr–V–Cr–Ni high entropy alloy. 
+International Journal of Hydrogen Energy 47: 22893-22900. https://doi.org/10.1016/j.ijhydene.2022.05.107 
+ 
+Liu J, Xu J, Sleiman S, Chen X, Zhu S, Cheng H, Huot J (2021) Microstructure and hydrogen storage properties of 
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+ 
+Marques F, Balcerzak M, Winkelmann F, Zepon G, Felderhoff M (2021) Review and outlook on high-entropy 
+alloys for hydrogen storage. Royal Society of Chemistry 14, 5191-5227.  https://doi.org/10.1039/D1EE01543E 
+ 
+Mishra SS, Mukhopadhyay S, Yadav TP, Mukhopadhyay NK, Srivastava ON (2019) Synthesis and characterization 
+of hexanary Ti–Zr–V–Cr–Ni–Fe high-entropy Laves phase. Journal of Materials Research 34 (5): 807-818. 
+https://doi.org/10.1557/jmr.2018.502 
+ 
+Mishra SS, Yadav TP, Srivastava ON, Mukhopadhyay NK, Biswas K (2020) Formation and stability of C14 type 
+Laves phase in multi component high-entropy alloys. Journal of Alloys and Compounds 832:153764. 
+https://doi.org/10.1016/j.jallcom.2020.153764 
+ 
+Murty BS, Yeh JW, Ranganathan S, Bhattacharjee PP (2019) High-Entropy Alloys 2nd Edition Elsevier ISBN: 
+9780128160671. pp 1-388. 
+ 
+Ranganathan S (2003) Alloyed pleasures: Multimetallic cocktails. Current Science 85: 1404-1406. 
+ 
+Sahlberg M, Karlsson D, Zlotea C, Jansson U (2016) Superior hydrogen storage in high entropy alloys, Scientific 
+Reports: 36770. https://doi.org/10.1038/srep36770 
+Sarac B, Zadorozhnyy V, Berdonosova E, Lvanov YP, Klyamkin S, Gumrukcu S, Sarac AS, Korol A, Semenov D, 
+Zadorozhnyy M, Sharma A, Greer AL, Eckert J (2020) Hydrogen storage performance of the multi-principal-
+component CoFeMnTiVZr alloy in electrochemical and gas-solid reactions, RSC Advances 10: 24613–24623. 
+https://doi.org/10.1039/D0RA04089D 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)9 
+ 
+Tsai MH, Yeh JW (2014) High-Entropy Alloys: A Critical Review. Materials Research Letters 2 (3): 107–123. 
+https://doi.org/10.1080/21663831.2014.912690 
+ 
+Yadav TP, Kumar A, Verma SK, Mukhopadhyay NK (2022) High-Entropy Alloys for Solid Hydrogen Storage: 
+Potentials and Prospects. Transactions of the Indian National Academy of Engineering 7: 147-156. 
+https://doi.org/10.1007/s41403-021-00316-w 
+ 
+Yadav TP, Mukhopadhyay S, Mishra SS, Mukhopadhyay NK, Srivastava ON (2017) Synthesis of a single phase of 
+high-entropy Laves intermetallics in the Ti–Zr–V–Cr–Ni equiatomic alloy. Philosophical Magazine Letters 97 (12): 
+494-503. https://doi.org/10.1080/09500839.2017.1418539 
+ 
+Yeh JW, Chen SK, Gan JY, Lin SJ, Chin TS, Shun TT, Tsau CH, Chou SY (2004a) Formation of simple crystal 
+structures in Cu-Co-Ni-Cr-Al-Fe-Ti-V alloys with multiprincipal metallic elements. Metallurgical and Materials 
+Transactions A 35:2533-2536. https://doi.org/10.1007/s11661-006-0234-4 
+ 
+Yeh JW, Chen SK, Lin SJ, Gan JY, Chin TS, Shun TT, Tsau CH, Chang SY (2004b) Nanostructured high-entropy 
+alloys with multiple principal elements: novel alloy design concepts and outcomes. Advanced Engineering Materials 
+6:299303. Zeitschrift für Physikalische Chemie 117: 89-112. https://doi.org/10.1002/adem.200300567 
+ 
+Zhou P, Cao Z, Xiao X, Jiang Z, Zhan L, Li Z, Jiang L, Chen L (2022) Study on low-vanadium TiZrMnCrV based 
+alloys for high-density hydrogen storage. International Journal of Hydrogen Energy 47: 710-1722. 
+https://doi.org/10.1016/j.ijhydene.2021.10.106 
+ 
+ 
+Figure captions 
+ 
+Figure 1: (a) Schematic diagramof the synthesis protocol for Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA 
+ 
+Figure 2:(a) XRD pattern of Ti0.24-V0.17-Zr0.17-Co0.17-Fe0.08-Mn0.17 HEA system and (b) Rietveld refinement profile 
+pattern of all the peaks well fitted with C14 type hexagonal parameters with unit cell parameters a= b =5.0158 Å, 
+c=8.1790 Å, α = β = 90˚, γ = 120˚ under space group P63/mmc 
+Figure 3 : (a) TEM bright field micrograph of as-cast HEA synthesized by RF induction melting (b) Corresponding 
+SAD patterns are shown indexed with hexagonal structure parameter under the space group of P63/mmc 
+Figure 4: (a) SEM–BSE and energy dispersive X-ray analyses (EDX) mapping images of as 
+Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA (b) overlays all the constituent elements present in this HEA. (c SEM-BSE 
+image from another region for the HEA. (d) EDX elemental spectra to validate the atomic percentage of the 
+elements in this HEA. 
+Figure 5: (a) Hydrogenation curve of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA at 410 ˚C under 60 atm H2 pressure and 
+(b) Dehydrogenation curve of hydrogenated Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA at 410 ˚C under 60 atm H2 
+pressure 
+Figure 6: (a) Fig: (a) PCI ab/de-sorption curves of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA and (b) Corresponding 
+Van’t Hoff plots for PCI ab/de-sorption curves. 
+ 
+ 
+ 
+ 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)10 
+ 
+Table 1: 
+Lattice Parameters and refinement parameters obtained from powder x-ray diffraction data of the 
+as-cast HEA. 
+Refined Parameter and phase data 
+Unit-Cell Parameters                                  a= b =5.0158 Å, c=8.1790 Å, α = β = 90˚, γ = 120˚ 
+    Space Group                                                          P63/mmc (Space Group = 194) 
+       R- Factor                                                    Rp = 3.23%, wRp = 4.45%, GOF = 1.26%, 
+        Volume                                                                            V = 178.20Å3 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)11 
+ 
+ 
+ 
+ 
+ 
+Figure 1: (a) Schematic diagram of the synthesis protocol for Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)12 
+ 
+ 
+Figure 2:(a) XRD pattern of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA system and (b) Rietveld refinement profile 
+pattern of all the peaks well fitted with C14 type hexagonal parameters with unit cell parameters a= b =5.0158 Å, 
+c=8.1790 Å, α = β = 90˚, γ = 120˚ under Space group P63/mmc. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)13 
+ 
+ 
+ 
+Figure 3: (a) TEM bright field micrograph of as-cast HEA synthesized by RF induction melting (b) Corresponding 
+SAD pattern are shown indexed with hexagonal structure parameter under the space group of P63/mmc. 
+ 
+ 
+ 
+ 
+ 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Angle (20)
+Angle 20(a)
+(b)
+0111
+1101
+100.1/mm
+10 1/nm
+[1213]a
+Mn
+Fe
+b
+ZrLa
+Ti Ka
+B1
+(d)
+ElementWeight%
+720
+WYA
+ZrL
+17.15
+638
+TiK
+22.92
+54C
+VK
+17.46
+MnK
+16.93
+MaKa
+FeK
+8.46
+36
+CoK
+17.08
+27
+18
+EMT-20.00AV
+XX00SE 6es
+De 1 Feo 2922
+WD+ t0.0 mm
+Tome.t:20.15
+ZEIS
+Le300.8
+Hydrogenationof Tio.24Vo.17Zro.17Coo.17Feo.0aMno.17
+0.5
+DehydrogenationofhydrogenatedTia.24Va.Zra.Coa.17Feo.oMna.17at
+0.7
+at410cunder60atmH2pressure
+Hydrogen absorbed (wt%)
+desorbed (wt%)
+410Cunder1atmH2pressure
+0.6
+0.4 -
+0.5
+(b)
+(a)
+0.3
+0.4
+0.3.
+0.2
+0.2 
+0.1
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+16(
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Time (Min.)
+Time (Min.).6
+PClabs.at410°C
+Van'tHoffplotforPclabsorption
+60
+(a)
+.5
+(b)
+PCI abs. at 425 °C
+Van'tHoffplotforPcldesorption
+Linear fit
+50
+PClabs.at395°C
+.4
+PCldes.at410°C
+.3
+PCldes.at425°C
+40
+(atm)
+PCI des. at 395 °C
+.2
+30
+Equation
+y=a+bx
+ressure
+.1-
+Adj.R-Square
+0.99317
+0.997
+Value
+Standard Error
+PClabs
+Intercept
+4.15133
+0.19668
+20
+.0
+PClabs
+Slope
+-2.29308
+0.1342
+PCIdes
+Intercept
+7.2496
+0.23282
+PCIdes
+Slope
+-4.10198
+0.159
+6'
+P
+10
+.8
+0
+.7
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+1.43
+1.44
+1.45
+1.46
+1.47
+1.48
+0.0
+1.49
+1.5
+Hydrogenstoragecapacity (wt%)
+1000/T(K)14 
+ 
+ 
+Figure 4 : (a) shows the SEM–BSE and energy dispersive X-ray (EDX) analysis mapping images of as cast 
+Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA (b) overlays all the constituent elements present in this HEA. (c) Shows the 
+SEM-BSE image from another region for the HEA. (d) Shows the EDX elemental spectra to validate the atomic 
+presence of the elements in this HEA. 
+
+Co
+Mn
+Zr
+Ti
+Melting in R.F.induction Furnace
+HEA
+(ascastalloy)Hydraulic
+Press
+3 × 105 N/m²
+RF-
+Induction
+Melting
+Melting in R.F. induction Furnace
+(Melted under dynamic Argon atmosphere)
+35-KW
+(as cast alloy)
+RF-Induction
+Furnace2900
+3000
+(b)
+IYobserved
+(a)
+1500
+C14LavesPhase
+Yealculated
+2500
+2100
+IBraggPositions
+1700
+2000
+(210)
+13)
+1300
+1500
+5
+2
+-
+202)
+3
+-
+5
+(31
+5
+-
+1000
+500
+10
+20
+30
+40
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+Figure 5 : (a) Hydrogenation curve of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA at 410 ˚C under 60 atm H2 pressure 
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+H2 pressure. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
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+ 
+ 
+ 
+Figure 6:(a) Fig: (a) PCI ab/de-sorption curves of Ti0.24V0.17Zr0.17Mn0.17Co0.17Fe0.08 HEA and (b) Corresponding 
+Van’t Hoff plots for PCI ab/de-sorption curves. 
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+PCI abs. at 425 °C
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+Equation
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+ressure
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+0.99317
+0.997
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+PCIdes
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+Astronomy & Astrophysics manuscript no. output
+©ESO 2023
+January 10, 2023
+Cosmic rate of type IIn supernovae
+and its evolution with redshift
+C. Cold1 and J. Hjorth1
+DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen N, Denmark
+Received month day, year; accepted month day, year
+ABSTRACT
+Context. Type IIn supernovae potentially constitute a large fraction of the gravitationally lensed supernovae predicted to be found
+with upcoming facilities. However, the local rate is used for these estimates, which is assumed to be independent of properties such
+as the host galaxy mass. Some studies hint that a host galaxy mass bias may exist for IIn supernovae.
+Aims. This paper aims to provide an updated local IIn supernova-to-core-collapse ratio based on data from the Palomar Transient
+Factory (PTF) and the Zwicky Transient Facility (ZTF) Bright Transient Survey (BTS). Furthermore, the goal is to investigate the
+dependency of the IIn supernova peak magnitude on the host galaxy mass and the consequences of a possible host galaxy mass
+preference on the volumetric rate of type IIn supernovae.
+Methods. We constructed approximately volume-limited subsamples to determine the local IIn supernova-to-core-collapse ratio. We
+investigated the absolute peak magnitude of a subsample of type IIn and superluminous II or IIn supernovae exploring how this relates
+to the i-band magnitude of the host galaxies (as a proxy for stellar mass). We presented a method to quantify the effect of a potential
+preference for low-mass host galaxies utilizing the UniverseMachine algorithm.
+Results. The IIn supernova-to-core-collapse ratios for PTF and BTS are 0.046 ± 0.013 and 0.048 ± 0.011, respectively, which results
+in a ratio of 0.047±0.009, which is consistent with the ratio of 0.05 currently used to estimate the number of gravitationally lensed IIn
+supernovae. We report fainter host galaxy median absolute magnitudes for type IIn brighter than −20.5 mag with a 3 σ significance.
+If the IIn supernova-to-core-collapse ratio were described by the power law model IIn/CC = 0.15 · log(M/M⊙)−0.05, we would expect
+a slightly elevated volumetric rate for redshifts beyond 3.2.
+Conclusions.
+Key words. supernovae: general.
+1. Introduction
+Type IIn supernovae (SNe IIn) exhibit narrow hydrogen emis-
+sion lines in their spectra (Schlegel 1990). The distinct features
+of this SN class arise from the slow-moving and dense circum-
+stellar material (CSM) ejected by the star prior to explosion. This
+means that the SN IIn subtype is very diverse, as these SNe can
+emerge whenever CSM indications are present in their spectra,
+whether it is early or late in the lifespan of the SN, or whatever
+lies beneath the veil of the CSM (Smith 2017). Narrow hydrogen
+features may also arise from flash ionization of local CSM fol-
+lowing shock breakout. However, such emission lines disappear
+shortly after peak magnitude to reveal the underlying SN type
+(Yaron et al. 2017; Bruch et al. 2021; Jacobson-Galán et al. 2022;
+Terreran et al. 2022). Nevertheless, flash ionization in combina-
+tion with the complexity of the CSM structure can complicate
+the classification of SNe IIn (Ransome et al. 2021).
+Luminous blue variables, extreme red super giants, and yel-
+low hyper giants have all been proposed as progenitors due to
+their recurring violent mass-loss episodes. The intervals of mass
+loss can range from a period of months to thousands of years,
+which is necessary to produce the amount of CSM required to
+make SNe IIn (Smith 2017).
+Brighter and more long-lived than other SN types, super-
+luminous supernovae (SLSNe) are recognized as their own class
+of SNe (Moriya et al. 2018; Gal-Yam 2012). These very lu-
+minous objects differ from other SNe by their optical absolute
+magnitudes of around −21 or less, although SLSNe have been
+classified at around −19 mag at peak for the faintest objects
+(Moriya et al. 2018; Angus et al. 2019). SLSNe, as the classic
+SN classes, are also further categorized into subtypes. The super-
+luminous counterpart to the SNe IIn, SLSNe-IIn, also feature
+narrow emission lines of the hydrogen Balmer series similar to
+the regular SNe IIn, and these constitute a significant percentage
+of all hydrogen-rich SLSNe (Gal-Yam 2019). This is indicative
+of CSM interaction partly powering very bright transients. It is
+not yet clear whether SLSNe-IIn and SNe IIn are two distinctive
+populations or if they form a continuum in luminosity. However,
+in this work, we considered the SLSNe-IIn as the brightest SNe
+IIn.
+Models indicate that SNe IIn along with SNe Ia will dom-
+inate the observed rates of lensed supernovae. Estimates from
+the upcoming Legacy Survey of Time and Space (LSST) with
+the Vera C. Rubin Observatory (Wojtak et al. 2019; Goldstein
+et al. 2019) predict on the order of 100 SNe IIn per year to be
+gravitationally lensed. For the Roman Space Telescope, the grav-
+itationally lensed SNe predictions are comparable (Pierel et al.
+2021).
+The lensed SNe IIn predictions are based on the observed lo-
+cal rate of SNe IIn. Data from the Lick Observatory Supernova
+Search (LOSS) yielded an SNe-IIn-to-CC ratio of 8.8%+3.3%
+−2.9%
+SNe IIn out of all CC SNe (Li et al. 2011; Smith et al. 2011).
+Article number, page 1 of 8
+arXiv:2301.03406v1  [astro-ph.HE]  9 Jan 2023
+
+A&A proofs: manuscript no. output
+Fig. 1. Overview of redshift and host galaxy mass distributions. Left panel: Redshift distribution of the SNe IIn in Nyholm et al. (2020) and
+SLSN-IIn from PTF. Both are subsamples of the full PTF SNe IIn and SLSNe-IIn samples. The redshifts can be found in Schulze et al. (2021).
+Right panel: Distribution of the host galaxy masses corresponding to the same SNe IIn from Nyholm et al. (2020) and a subsample of SLSNe-IIn
+from the PTF. These subsample distributions are consistent with the full sample distributions of SNe IIn and SLSNe-IIn from Schulze et al. (2021).
+Fig. 2. Overview of redshift and host galaxy magnitude distributions. Left panel: Redshift distribution of the SNe IIn and SLSN-II from BTS. Right
+panel: Distribution of the host galaxy i-band absolute magnitudes. For two SNe IIn and 1 SLSN-II, the i-band magnitudes were not available. The
+i-band magnitude is used as a proxy for the host stellar mass.
+Several of these SNe IIn were subsequently identified as SN
+Impostors such that the IIn rate from LOSS is now considered
+to be around 5% (Graur et al. 2017). Other examples of local
+rate measurements include studies by Smartt et al. (2009), who
+present a volume-limited (28 Mpc) sample compiled from all lo-
+cal SNe with a named host galaxy discovered over a ten-year
+period. This sample includes 3.8% SNe IIn out of all CC SNe
+in the sample. Eldridge et al. (2013) updated the study done by
+Smartt et al. (2009), searching for SNe discovered over a 14-year
+period, yielding 2.4 ± 1.4%.
+The existing rate estimates of SNe IIn assume that the local
+fraction of IIn to all other CC SNe does not evolve with redshift
+and so is not affected by large-scale changes in compositions
+or characteristics of galaxies and stellar populations over time.
+Several studies show a bias toward less massive host galaxies for
+SLSNe in general (e.g., Leloudas et al. 2015; Angus et al. 2016;
+Schulze et al. 2018; Taggart & Perley 2021), and this dearth of
+Article number, page 2 of 8
+
+8
+42lln(Nyholmetal.2020)
+14
+9 SLSN-lln from PTE
+7
+12
+5
+8
+Numberof
+4
+Number
+6
+3
+4
+2
+1
+2
+0
+0
+0.000.05
+0.100.150.200.25
+0.300.35
+5
+9
+101112
+13
+z
+Host Galaxy Stellar Mass [log(M)]92lnfromBTS
+90linfromBTS
+12
+18 SLSN-11from BTS
+17 SLSN-Lfrom BTS
+20
+10
+Numberof Galaxies
+15
+8
+6
+10
+4
+5
+2
+0
+0.00
+00.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+-14
+-16
+-18
+-20
+-22
+Host Galaxy Magnitude [M;]C. Cold and J. Hjorth: Cosmic rate of type IIn supernovae and its evolution with redshift
+massive hosts suggests some dependence on the characteristics
+of the environment. In a study by Graur et al. (2017), based on
+data from LOSS, SNe IIn seem to be more common in less mas-
+sive galaxies. However, other studies do not come to the same
+conclusion (e.g., Kelly & Kirshner 2012). The Palomar Tran-
+sient Factory (PTF) CC SN sample presented in Schulze et al.
+(2021) also does not reveal a bias for SNe IIn toward low-mass
+host galaxies, and is inconclusive regarding the host galaxy mass
+preference of SLSNe-IIn.
+The Zwicky Transient Facility (ZTF) Bright Transient Sur-
+vey (BTS) (Fremling et al. 2020; Perley et al. 2020) will be part
+of the analysis in this paper in addition to the PTF CC sample
+(Schulze et al. 2021). The paper is structured as follows. In Sec-
+tion 2, we briefly introduced the data used for the analysis. In
+Section 3, we created an approximately volume-limited sample
+from the PTF and BTS data, followed by a presentation of an
+updated SNe-IIn-to-CC ratio (SNe IIn fraction) for both sam-
+ples. In Section 4, we compare the absolute magnitude and the
+i-band magnitude of the host galaxies of a subsample of SNe IIn,
+SLSNe-IIn, and SLSN-II from PTF and BTS. In Section 5, we
+presented a generic method for inferring the rate and its evolu-
+tion with redshift and studying the consequences of a possible
+mass-biased SNe IIn rate, before the discussion in Section 6 and
+conclusions in Section 7.
+2. Data
+The PTF CC SN sample from Schulze et al. (2021) and the ZTF
+Bright Transient Survey (Fremling et al. 2020; Perley et al. 2020)
+constitute the basis of the analysis presented in this paper. The
+PTF was a deep, wide-field survey followed by the intermediate
+PTF (iPTF) survey. The PTF CC sample contains 888 objects,
+of which 111 are SNe IIn and 16 SLSNe-IIn. Redshifts and host
+galaxy stellar masses are available for all objects in the sample.
+For later analysis, we will use the host galaxy i-band (either Pan-
+STARRS1 (PS1) or Sloan Digital Sky Survey (SDSS) i-band)
+absolute magnitude. However, two SNe IIn and one SLSN-IIn
+have no reported host i-band magnitude. The sample contains
+only host photometry, and so we used a subsample of IIn and
+SLSNe-IIn where the peak absolute magnitude is available. The
+subsample of 42 SNe IIn with available peak magnitudes is de-
+scribed in Nyholm et al. (2020). These SNe were chosen based
+on the amount of available light-curve data to allow an analy-
+sis of both the rise times and decline rates of the SNe IIn, all
+with at least one available low-resolution spectrum. The red-
+shifts in Nyholm et al. (2020) differ slightly from the redshifts
+in Schulze et al. (2021), which are the galaxy redshifts estimated
+by one of four possible methods: taken from SDSS, taken from
+the NASA Extragalactic Database, estimated from galaxy lines
+in the spectra or estimated from SN-template matching (Schulze
+et al. 2021). In Nyholm et al. (2020), the redshifts are estimated
+from the Hα emission lines in the SNe spectra. In this work,
+we used the data published in Schulze et al. (2021) as well as
+SLSNe-IIn peak magnitudes (Leloudas, priv. comm.). The red-
+shift and host galaxy stellar mass distributions of the SNe IIn
+from Nyholm et al. (2020) along with the SLSN-IIn sample from
+the PTF are shown in Fig. 1.
+The BTS is currently the largest spectroscopic survey of
+SNe. The survey is magnitude-limited in the g and r (<19 mag-
+nitude) bands. The sample is 97% spectroscopically complete at
+<18 mag, 93% at <18.5 mag, and 75% at <19 mag (Perley et al.
+2020). The survey is updated daily as new observations come
+in. For this paper, we chose to use all available CC SNe and IIn
+from BTS regardless of magnitude as of May 16, 2022. The to-
+tal number of CC SNe adds up to 949, of which 92 are IIn. We
+also included the SLSN-II in our analysis, of which there are 18.
+In BTS, the SLSNe-II are not further divided into subclasses.
+However, as most SLSNe-II exhibit IIn-like features, we chose
+to include all of the SLSN-II in our analysis. The parameters
+we used in our analysis in this paper are redshift, peak magni-
+tude and host i-band absolute magnitude. Redshifts are available
+for all but six CC SNe, none of which are SNe IIn or SLSN-
+II. Since the host galaxy stellar masses are not available in this
+sample, we instead utilized the absolute i-band magnitude of the
+hosts where available as these magnitudes are a good proxy for
+the stellar masses, as is seen in Fig. 3. Distributions of redshift
+and host galaxy i-band magnitude are shown in Fig. 2.
+Fig. 3. PTF CC SNe host galaxy absolute i-band magnitudes are plotted
+against host galaxy stellar masses. Due to the standard deviation of the
+residuals being 0.4, which is comparable to the uncertainty on the stellar
+mass, the i-band magnitude can be used as a proxy for the stellar mass
+in our analysis.
+3. Inferred IIn fractions
+In Frohmaier et al. (2021), the CC SN rate for the PTF is de-
+termined while taking all the survey limitations into account
+through extensive modeling. This method yields a total of 86 CC
+SNe and three SNe IIn. Unfortunately, inferring relative rates
+of SNe IIn with only three sources will be dominated by low-
+number statistics. For the purpose of estimating the SNe IIn
+fraction, we will alternatively create an approximately volume-
+limited sample for the recent PTF data released in Schulze et al.
+(2021) and also from BTS (Fremling et al. 2020; Perley et al.
+2020) under the assumption of fair spectroscopic classification.
+A simple way to estimate the distance, or redshift, at which
+the PTF CC sample is approximately complete is to compare
+with a known complete sample. The LOSS sample is com-
+plete out to 60 Mpc for CC SNe (Li et al. 2011; Graur et al.
+2017). With the limiting magnitude of LOSS having a median
+of 18.8 ± 0.5 mag (Leaman et al. 2011), 60 Mpc is also the dis-
+tance to the furthest SNe LOSS could theoretically observe as-
+suming the faintest SNe to have an absolute magnitude of around
+−15.1. The PTF has a limiting magnitude of 20.5 mag in the R
+band, which implies a distance of up to 131.8 Mpc for creating a
+volume-limited sample, assuming the same −15.1 mag for faint
+SNe. This corresponds to a redshift cut-off of 0.031 when adopt-
+ing a Hubble constant of H0 = 73 kms−1Mpc−1 as in LOSS. This
+Article number, page 3 of 8
+
+24
+[M;]
+22
+Magnitude
+20
+/Absolute
+-18
+-16
+Host Galaxy
+-14
+-12
+PTF CC SNe Host Galaxies
+-10
+4
+5
+6
+1
+8
+9
+10
+11
+12
+HostGalaxyStellarMass[log(M*/M)]A&A proofs: manuscript no. output
+Fig. 4. Redshift evolution of SNe-IIn-to-CC ratio. The gray data points represent the cumulative SNe-IIn-to-CC ratio from PTF, indicated on the
+left y-axis, as a function of redshift limits out to maximum redshift of the sample. SLSNe-IIn are not included in the IIn-to-CC ratio. The vertical
+gray dashed line represents the chosen redshift cut. The number of CC SNe, SNe IIn, and SLSNe-IIn as a function of redshift are represented by
+the colored curves indicated on the right y-axis. The inset shows a zoomed-in image of a plot of the cumulative SNe-IIn-to-CC ratio up to redshift
+0.06 as well as the chosen redshift cut.
+Fig. 5. Redshift evolution of SNe-IIn-to-CC ratio. The gray data points represent the cumulative SNe-IIn-to-CC ratio from BTS, indicated on the
+left y-axis, as a function of redshift limits. The vertical gray dashed line represents the chosen redshift cut. The number of CC SNe and SNe IIn
+are also depicted as the colored curves, which are indicated on the right y-axis. The inset shows a zoomed-in image of a plot of the cumulative
+SNe-IIn-to-CC ratio up to redshift 0.06 as well as the chosen redshift cut.
+estimate can be tested visually, as is done in Fig. 4, where we
+plot the SNe-IIn-to-CC ratio as a function of redshift limit. The
+estimated cut of 0.031 is indicated in the plot by a dashed line.
+The SNe IIn fraction for lower redshift cut-offs is dominated by
+noise due to the small number of supernovae found at these red-
+shifts, whereas the ratio increases above 0.031 as more SNe IIn
+are observed at this range. This indicates that the estimated red-
+shift cut is located where the noise from few observations has
+started to diminish, but we do not yet see the effect of a larger
+volume wherein to observe SNe IIn and CC SNe in general. As
+such, imposing a redshift cut-off of 0.031 on the PTF CC sample
+Article number, page 4 of 8
+
+0.14
+Z=0.031
+PTF
+800
+0.12
+0.10
+Supemova
+0.075
+IIn/CC
+0.08
+IIn/CC
+0.050
+4006
+0.06
+Z=0.031
+Number 
+0.025
+PTF
+0.04
+0.000
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+200
+z limit
+0.02
+SLSN IIn
+IIn
+0.00
+CC
+0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+z limitZ=0.033
+BTS
+0.10
+800
+0.08
+Number of Supemovae
+0.100
+600
+0.075
+IIn/CC
+0.050
+400
+0.04
+0.025
+Z=0.033
+PTF
+0.000
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+200
+0.02
+z limit
+IIn
+0.00
+CC
+0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+z lirmitC. Cold and J. Hjorth: Cosmic rate of type IIn supernovae and its evolution with redshift
+is a reasonable approach for creating an approximately volume-
+limited sample to use for estimating the SNe IIn fraction.
+Fig. 6. Distributions of host galaxy stellar mass of the complete PTF CC
+sample using z < 0.031. No SLSNe-IIn in the PTF sample are found
+within this redshift. Results from a KS test show no significant SNe
+IIn host galaxy mass preference for the volume-limited sample with a p
+value of 0.002.
+We employ the same method for the BTS sample. Using
+−15.1 mag for the faintest CC SNe is consistent with the mean of
+the 20 faintest CC SNe in the BTS sample. According to Bellm
+et al. (2019), the limiting magnitudes for ZTF are 20.8 mag in
+the g band, 20.6 mag in the r band, and 19.9 in the i band. We
+will use the r-band value to be consistent with LOSS. From the
+distance modulus, this yields a distance of 138 Mpc, correspond-
+ing to a redshift cut-off of about 0.033, as illustrated in Fig. 5.
+The volume effect is less obvious in the BTS data, and as such
+the resulting IIn fraction from BTS is more robust toward any
+uncertainties in the redshift cut compared to the result from PTF.
+However, as the redshift cut of 0.033 occurs before a slow rise in
+the SNe-IIn-to-CC value and after the inital large uncertainties,
+we deem 0.033 to be a good estimate for creating an approxi-
+mately volume-limited sample.
+For the PTF, a redshift cut of 0.031 leaves us with an ap-
+proximately complete CC sample of 263 CC SNe in total. This
+includes 12 SNe IIn, but none of the SLSNe-IIn are observed
+within this redshift. Therefore, we find that the ratio of SNe IIn to
+CC SNe for our subsample of PTF data to be 0.046 ± 0.013. The
+uncertainty on the resulting SNe-IIn ratio is propagated from the
+Poisson error on the individual number of CC and SNe IIn. The
+resulting histogram of the host galaxy stellar masses of this vol-
+ume limited sample of PTF data, as illustrated in Fig. 6, reveals
+no obvious SNe IIn preference for less massive host galaxies,
+which is in agreement with the analysis done by Schulze et al.
+(2021) on the full PTF sample.
+We can compare this value to the BTS data. A redshift cut-
+off of 0.033 yields a sample of 440 CC SNe, of which 21 are
+SNe IIn. This results in an SNe-IIn-to-CC ratio of 0.048±0.011.
+The uncertainty on this number is similarly determined using er-
+ror propagation. As host galaxy masses are not available in BTS,
+and a significant amount of CC hosts do not have i-band magni-
+tudes either, we will not compare the host galaxy distributions of
+the SNe IIn and CC SNe from BTS. As these resulting fractions
+are independent, we combine them and get a SNe IIn relative
+fraction of 0.047 ± 0.009.
+4. Brightness of SNe IIn
+In this section, we investigate whether the peak brightness of the
+IIn is influenced by the host galaxy stellar mass when consider-
+ing the SLSN-IIn as the brighest members of the IIn class. We
+know from several studies that SLSNe prefer lower mass host
+galaxies. According to Schulze et al. (2021), this phenomenon is
+not significant when only studying the SLSNe-IIn, as the objects
+are still too few. We note that for redshifts on the order of 0.03
+the influence of peculiar velocities on calculating peak absolute
+magnitudes of the SNe is decreasing compared to SN samples,
+which are mostly comprised of local sources.
+The distributions of redshift and host galaxy mass of the sub-
+sample of IIn and SLSN-IIn from PTF are displayed in Fig. 1.
+In Fig. 2, we show the distributions of redshift and host galaxy
+i-band magnitude from BTS. For the comparison between these
+two data sets, we use the i-band magnitude of the host galaxies
+as a proxy for the stellar mass. Only three sources from either
+data set do not have available host i-band magnitudes.
+In Fig. 7, we compare the absolute magnitude at peak and the
+i-band magnitude of the host galaxies of the SNe IIn and SLSNe-
+IIn or SLSN-II from the PTF as well as BTS. As these objects are
+chosen based on data availability, the subsamples seen in Fig. 7
+are not complete. However, Nyholm et al. (2020) state that the
+host galaxy mass distribution of their subsample is in agreement
+with the distribution of the full PTF SNe IIn sample, such that
+the distribution of the i-band magnitudes should follow a similar
+distribution. The data in Fig. 7 show no clear trend regarding the
+effect of the host galaxy magnitude on the peak absolute mags
+of the SNe IIn. To further investigate, we divide the combined
+PTF and BTS subsamples shown in Fig. 7 in two, namely a faint
+sample and a bright sample, and subsequently calculate the me-
+dian and uncertainty on the median as 1.48·MAD/
+√
+N − 1 of the
+i-band magnitudes for the host galaxies, where MAD is the me-
+dian absolute deviation. This division of the subsample is done
+for several different SN IIn peak magnitudes. We employ Mpeak
+values from −17.5 to −21 as the dividing lines between the faint
+and the bright sub-samples and compare the medians, as can be
+seen in Fig. 8. We find that the median i-band magnitude (and
+thus the stellar mass) of the host galaxies becomes fainter with
+a 3σ significance when choosing a sample of SNe IIn brighter
+than −20.5 mag.
+Fig. 7. Absolute peak magnitude versus host galaxy i-band magnitude
+for SNe IIn and SLSNe-IIn/SLSNe-II from PTF and BTS.
+Article number, page 5 of 8
+
+102
+263 CC
+12 IIn
+Supernovae
+Numberof
+101
+100
+5
+6
+7
+8
+9
+10
+011
+12
+13
+HostGalaxyStellarMass[log(M)]-23
+22
+-21
+20
+Peak
+19
+Absolute
+-18
+-17
+BTS IIn
+BTS SLSN-II
+-16
+PTF IIn
+PTF SLSN-IIn
+-15
+10
+-12
+-14
+-16
+-18
+-20
+-22
+Host Galaxy Absolute Mag [Mi]A&A proofs: manuscript no. output
+Fig. 8. Median host Mi as a function of different cuts on the supernova
+peak magnitude for combined BTS and PTF samples. The median i-
+band magnitude of the host galaxies becomes fainter for the brighter
+SNe in the combined sample.
+5. Consequences of a host-mass-dependent IIn
+fraction
+While the evidence for a host-mass-dependent IIn fraction is not
+strong, we next explore the consequences of a hypothetical pref-
+erence for low-mass host galaxies. We parametrize the IIn frac-
+tion as a power-law function of the stellar mass of the host galaxy
+and investigate the impact on the volumetric rate as a function of
+redshift. This will be affected since more low-mass galaxies are
+present in the earlier Universe.
+In general, the volumetric SNe IIn rate can be expressed as
+IInrate = IIn
+CC · kCC · S FR.
+(1)
+The CC constant, kCC, is set to 0.0091M−1
+⊙
+following Strol-
+ger et al. (2015). This model is consistent with observational
+constraints from Dahlen et al. (2012) and Madau & Dickinson
+(2014) as demonstrated in Strolger et al. (2015). Here, a model
+for the star-formation rate (SFR) is taken from the UniverseMa-
+chine algorithm by Behroozi et al. (2019). The results of this
+code are best-fitting models of stellar-mass functions (SMFs),
+cosmic star formation rates (CSFRs), specific star formation
+rates (sSFRs), and UV luminosity functions (UVLFs) to obser-
+vations. One can determine the SFR from the output of the Uni-
+verseMachine algorithm:
+S FR =
+� Mmax
+Mmin
+S MF · M · sS FR dM.
+(2)
+When computing the SFR this way, it is possible to split it into
+different mass bins in order to infer the contribution to the total
+SFR from galaxies of different masses and how this changes with
+redshift, as is shown in the top panel of Fig. 9. The UniverseMa-
+chine resulting models have mass ranges of 107M⊙ to 1013M⊙,
+and we choose to split these into five different bins, as indicated
+in Fig. 9. The bin containing the 1011M⊙ to 1013M⊙ galaxies is
+chosen to be wider than the other bins, as the contribution from
+the 1012M⊙ to 1013M⊙ galaxies is negligible. We parametrize the
+IIn-to-CC ratio as a power law:
+IIn
+CC (log(M/M⊙)) = 0.15 · log(M/M⊙)−0.05.
+(3)
+This power-law model is chosen to have a higher SNe-IIn-to-
+CC ratio than 0.047 for host galaxies below 1010M⊙ and a lower
+ratio for more massive galaxies. For this specific example, the
+ratio will be 0.057 for galaxies with stellar masses of 107M⊙, and
+0.043 for 1012M⊙. The motivation for this kind of model comes
+from the LOSS data in Graur et al. (2017), showing a preference
+for low-mass hosts for SNe IIn, which can be modeled with a
+power law with different sets of parameters (Hede 2021).
+To calculate the volumetric rate, we use the central value of
+the IIn/CC model for every mass bin in log as the IIn/CC factor
+in Eq. (1), and thus compute a separate SN IIn rate for each mass
+bin as well as the combined rate. Since the power-law model and
+the constant model do not predict the same number of SNe given
+a different area under the curve, we normalize the resulting rate
+from the power law model to 4.77 · 10−6 yr−1Mpc−3, which is
+the SNe IIn rate at redshift zero for a constant SNe IIn fraction of
+0.047 calculated from Eq. (1), where the UniverseMachine is the
+source of the SFR. We do this to be consistent with the constant
+model. The resulting SNe IIn rate is plotted in the bottom panel
+of Fig. 9 for both the example power-law model and the constant
+model of IIn/CC = 0.047. The overall IIn rate is slightly lower
+for a redshift below four, and slightly higher for one above four.
+This plot also shows how the contribution from the different host
+mass bins differs for the constant ratio to the power-law model. It
+is evident that the overall rate from low-mass galaxies is higher,
+and since these contribute a larger fraction of the total rate at
+higher redshift, we also see an increase in the rate in this high-
+redshift domain as expected.
+6. Discussion
+In this section, we discuss and reflect on some of the shortcom-
+ings and consequences of the methods and results of this paper
+as well as some widely used assumptions. First, we note that the
+SNe-IIn fractions from the PTF and BTS are based on approxi-
+mate volume-limited samples. The identification of a sweet spot
+in the IIn fraction in the PTF sample (Fig. 4) supports this ap-
+proach, although we note that, ultimately, IIn fractions will have
+to be based on carefully defined volume-limited samples of a
+large number of core-collapse supernovae.
+From Fig. 8, we see a statistically significant connection be-
+tween the brightness of the SNe IIn and the host galaxy stellar
+mass when choosing a sample of SNe IIn brighter than −20.5
+mag. For the faint SNe IIn sample at the −18 mag cut-off, the
+median i-band magnitude of the host galaxies drops below the
+bright IIn sample median host i-band magnitude. However, the
+subsamples employed in this part of our analysis could be sub-
+ject to different selection effects. Observing faint SNe in bright
+galaxies is challenging as the light from the galaxy can hide the
+SNe and so we could expect that some were missed in this area.
+This effect will influence the median i-band host magnitude for
+the faintest SNe IIn and could explain the slight shift in median
+host i-band magnitude for the faintest SNe in Fig. 8. However,
+the drop in median i-band host-galaxy magnitude is not as sig-
+nificant as the drop for the brightest supernovae.
+Using the UniverseMachine algorithm DR1 as the input for
+the SFR gives insight into the contributions from host galaxies
+of different stellar masses to the resulting SNe IIn rate whether a
+preference for low-mass hosts exists or not. One limitation, how-
+ever, is the lower mass boundary on these galaxies. As is evident
+from both Figs. 6 and 1, several SNe IIn have host galaxies less
+massive than 107M⊙, but using UniverseMachine it is not pos-
+sible to see the contribution from these galaxies. Another limi-
+tation is the mass resolution of UniverseMachine. In this case,
+Article number, page 6 of 8
+
+21.0
+Bright SNe
+20.5
+Faint SNe
+-20.0
+M
+-19.5
+Median Host I
+19.0
+-18.5
+18.0
+17.5
+17.0.
+17.0-17.5-18.0-18.5-19.0-19.5-20.0-20.5-21.0-21.5
+Mpeak CutC. Cold and J. Hjorth: Cosmic rate of type IIn supernovae and its evolution with redshift
+Fig. 9. Overview of the cosmic SFR, volumetric SNe IIn rate and rel-
+ative SNe IIn rate. Top panel: SFR calculated from UniverseMachine
+DR1 (Behroozi et al. 2019). Middle panel: Resulting SNe-IIn rate de-
+termined using Eq. (1). The solid lines denote rates from the power-law
+IIn-to-CC ratio, and the dashed lines represent the rate for a constant
+ratio of 0.047. The contribution from each mass bin is plotted for com-
+parison. Bottom panel: Total relative volumetric SNe IIn rate. The gray
+line represents a line of agreement between the two models.
+we have four or five data points per mass bin, which prevents us
+from further dividing the host galaxies into smaller bins to obtain
+a more detailed overview.
+In Fig. 9, we show the resulting SNe-IIn volumetric rates. For
+illustration, the volumetric rate is computed for a uniform fixed
+SNe-IIn fraction of 0.047 next to a model in which the SNe IIn
+prefer lower mass host galaxies, here represented by a power law
+model. The two models agree at z = 0 and the normalization of
+the curves is uncertain by 20 %. The middle and bottom panels
+show that the relative volumetric rate of SNe IIn increases over
+the default constant model for redshifts beyond 3.2.
+Introducing the example power-law model to describe the
+SNe-IIn-to-CC ratio produces a minimal effect on the volumet-
+ric rate compared to the constant ratio as seen in Fig. 9: A lower
+rate for redshifts below 3.2 and higher for redshifts beyond. The
+LSST or Roman Space Telescope are not expected to be able to
+observe lensed SNe beyond redshifts of three and four, respec-
+tively (Wojtak et al. 2019; Goldstein et al. 2019; Pierel et al.
+2021), and so the possibility of testing such a model is currently
+limited. On the other hand, we show that a limited mass depen-
+dence of the IIn rate should not affect predicted volumetric rates
+of SNe IIn significantly.
+7. Conclusions
+We studied the PTF and BTS SNe IIn and SLSNe-IIn/SLSNe-
+II populations throughout this work and now present our main
+conclusions.
+Creating a complete sample of CC SNe from PTF and
+BTS, we find the SNe IIn to CC ratios of 0.046 ± 0.013 and
+0.048 ± 0.011 for the PTF and BTS, respectively. The combined
+resulting SNe-IIn fraction is 0.047 ± 0.009. We see a marginally
+significant (3 σ) bias towards low-mass host galaxies for SNe IIn
+brighter than −20.5 mag. We present a general method to evalu-
+ate the consequences of a SNe IIn to CC ratio that is nonconstant
+and is instead described by a power-law model on the resulting
+volumetric rate. The example model chosen here can be freely
+replaced by another model as required. We find that the example
+power law model of IIn/CC = 0.15 · M−0.05 results in a slightly
+lower volumetric rate below a redshift of four and a higher rate
+beyond a redshift of 3.2 when comparing to a constant ratio of
+0.047. Neither the LSST nor the Roman Space Telescope are pre-
+dicted to find lensed SNe beyond redshifts of three or four. We
+emphasize that our method is generic and can be applied to other
+CC subtypes if needed.
+Acknowledgements. We gratefully acknowledge Giorgios Leloudas for sharing
+peak absolute magnitudes for a subsample of PTF SLSNe-IIn, without which a
+large part of the analysis in this paper would not have been possible, as well as
+invaluable comments on the paper draft. We also gratefully acknowledge Steve
+Schulze and Radek Wojtak for helpful conversations about type IIn and statis-
+tical conundrums, respectively, and Wynn Jacobson-Galán and Doogesh Kodi
+Ramanah for reading and commenting on the paper before submission. We also
+thank the referee for their useful and thorough comments and suggestions. This
+work was supported by a VILLUM FONDEN Investigator grant to JH (project
+number 16599).
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+

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+Accelerating greedy algorithm for model reduction of complex
+systems by multi-fidelity error estimation
+Lihong Feng ∗, Luigi Lombardi†, Giulio Antonini‡, and Peter Benner§
+January 16, 2023
+Abstract
+Model order reduction usually consists of two stages: the offline stage and the online stage. The
+offline stage is the expensive part that sometimes takes hours till the final reduced-order model is
+derived, especially when the original model is very large or complex. Once the reduced-order model
+is obtained, the online stage of querying the reduced-order model for simulation is very fast and
+often real-time capable. This work concerns a strategy to significantly speed up the offline stage of
+model order reduction for large and complex systems. In particular, it is successful in accelerating
+the greedy algorithm that is often used in the offline stage for reduced-order model construction.
+We propose multi-fidelity error estimators and replace the high-fidelity error estimator in the greedy
+algorithm. Consequently, the computational complexity at each iteration of the greedy algorithm is
+reduced and the algorithm converges more than 3 times faster without incurring noticeable accuracy
+loss.
+1
+Introduction
+Model order reduction (MOR) has achieved much success in many areas of computational science with
+its capability of realizing real-time simulation and providing accurate results. Different MOR methods,
+their applications and the promising results they produce can be found in the survey papers [2, 4, 12]
+and books [27, 3, 8, 9, 10, 11].
+MOR needs an offline stage for constructing the ROM. For many intrusive MOR methods that are
+based on projection, the offline stage is usually realized via a greedy algorithm. The greedy algorithm
+is used to properly select important parameter samples that contribute most to the solution space. The
+offline computational time is basically the runtime of the greedy algorithm. For large-scale systems,
+the offline computation is expensive and the runtime is often longer than several hours even when
+run on a high-performance server. Sometimes, the system is not very large, for example, the number
+of degrees of freedom is only O(105), but the system structure is complicated, so that the greedy
+algorithm still takes long time to converge.
+It is known that an efficient error estimator makes the greedy algorithm successful in producing
+an accurate ROM without running many iterations.
+Therefore, many efforts have been made in
+this direction to develop computable error estimators for different problems [15, 16, 18, 19, 20, 21,
+22, 23, 24, 25, 28, 34, 33, 35].
+However, more attention has been paid to improve the effectivity
+∗Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, 39106 Magdeburg, Ger-
+many feng@mpi-magdeburg.mpg.de
+†Luigi Lombardi is with Micron Semiconductor, 67051 Avezzano, Italy. luigilombardi89@gmail.com
+‡Giulio Antonini is with the UAq EMC Laboratory, Department of Industrial and Information Engineering and
+Economics, University of L’Aquila, I-67100 L’Aquila, Italy. giulio.antonini@univaq.it
+§Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany and Fakult¨at f¨ur Mathe-
+matik, Otto-von-Guericke-Universit¨at Magdeburg, Germany. benner@mpi-magdeburg.mpg.de
+1
+arXiv:2301.05610v1  [math.NA]  13 Jan 2023
+
+or accuracy of the error estimator than to develop more efficient strategies to accelerate the greedy
+process [36, 13, 34, 25, 33]. Recently, some techniques are proposed to improve the adaptivity of the
+greedy algorithm [7, 13, 6, 26].
+In [13, 14], we proposed a surrogate model for error estimation, and proposed an adaptive greedy
+algorithm by alternatively using this surrogate error estimator and the original error estimator during
+the greedy algorithm. The focus in [13, 14] was to make the greedy process adaptive by starting from a
+coarse training set of a small number of parameter samples, and adaptively update the coarse training
+set with the aid of a surrogate error estimator. The original error estimator is computed only over
+the coarse training set, while the surrogate error estimator helps to pick out candidates of important
+parameter samples from a fine training set, which are then collected in the coarse training set.
+In this work, we emphasize the role of the surrogate error estimator and propose the concept of
+bi-fidelity error estimation and multi-fidelity error estimation. In fact, a bi-fidelity error estimation has
+been used in the adaptive greedy algorithm proposed in [13, 14] without being formally defined, i.e.,
+the original (high-fidelity) error estimator, and the surrogate (low-fidelity) error estimator. To further
+improve the convergence speed of the greedy algorithm, we propose multi-fidelity error estimation
+built upon the bi-fidelity error estimation. Here, we use a more efficient high-fidelity error estimator
+than the two different high-fidelity error estimators used in [13, 14]. Although the proposed multi-
+fidelity error estimation is dependent on the original high-fidelity error estimator, the idea of using
+multi-fidelity error estimation is general and can be extended to develop multi-fidelity error estimation
+associated with other high-fidelity error estimators.
+Unlike the problems considered in [13, 14], whose ROMs can be constructed by standard greedy
+algorithms within seconds to minutes, we consider in this work much more complicated problems.
+On the same computer, the standard greedy algorithm takes more than half a day to converge for
+such problems. By using the proposed multi-fidelity error estimator, the greedy algorithm achieves
+4x speed-up and produces ROMs with little loss of accuracy. The speed-up is also higher than those
+reported in [13, 14] by using the bi-fidelity error estimation, which is usually 2x.
+In the next section, we present the greedy algorithm in the standard form.
+Then we analyze
+some ingredients of the algorithm, which contribute most to the computational cost. Starting from
+those computationally expensive parts, we develop possible strategies to reduce the computational
+complexity in Section 3. As a consequence, it becomes clear that the resulting strategy develops a
+greedy algorithm with multi-fidelity error estimation. The proposed algorithm is then applied to large
+time-delay systems with many delays. The numerical tests on three large time-delay systems with
+more than 100 delays are demonstrated in Section 4. Conclusions are given in the end.
+2
+Standard greedy algorithm
+The standard greedy algorithm was first proposed for steady systems without time evolution. Then
+it was extended to POD-greedy for dynamical systems, which is used to construct the ROM using
+snapshots in the time domain. Later the greedy algorithm found its capability in adaptively choosing
+interpolation points for frequency-domain MOR methods [15, 16]. The greedy algorithm for steady
+systems and frequency-domain MOR has the same formulation, whereas POD-greedy for time-domain
+MOR of time-dependent systems needs an SVD step at each greedy iteration. In this work, we focus
+on the greedy algorithm, though the proposed scheme can be easily extended to POD-greedy. We
+consider constructing a ROM for the following full-order model (FOM) using the greedy algorithm,
+F(x(µ), µ) = B(µ).
+(1)
+Here, F(x(µ), µ) ∈ Cn×nI, x(µ) ∈ Cn×nI, and B(µ) ∈ Cn×nI. µ ∈ P is a parameter in the parameter
+domain P. The variable n is the order of the FOM, which can be the number of degrees of freedom
+2
+
+Algorithm 1 Standard greedy algorithm
+Input: the FOM, a training set Ξ composed of parameter samples taken from the parameter domain
+P, error tolerance tol< 1, ∆(µ) to estimate the error.
+Output: Projection matrix V .
+1: Choose initial parameter µ∗ ∈ Ξ.
+2: V ← ∅, ε = 1.
+3: while ε >tol do
+4:
+Compute the snapshot(s) x(µ∗) by solving the FOM at µ = µ∗.
+5:
+Update V by V = orth{V, x(µ∗)}, (e.g., using the modified Gram-Schmidt process with defla-
+tion.)
+6:
+Compute µ∗ such that µ∗ = arg max
+µ∈Ξ ∆(µ).
+7:
+ε = ∆(µ∗).
+8: end while
+after numerical discretization of PDEs describing a physical phenomenon. The proposed algorithms
+are also applicable to problems with nI > 1.
+The ROM can be obtained via Galerkin projection using a projection matrix V ∈ Rn×r, r ≪ n,
+as below,
+ˆF(V z(µ), µ) = ˆb(µ),
+(2)
+where ˆF(V z(µ), µ) = V T f(V z(µ), µ) ∈ Cr×nI, z(µ) ∈ Cr×nI, and ˆB(µ) = V T b(µ) ∈ Cr×nI.
+The standard greedy process used to compute the projection matrix V is described in Algorithm 1.
+Step 4 in Algorithm 1 solves the FOM at µ∗, and Step 6 computes an error estimator ∆(µ) at all µ in Ξ.
+These two steps constitute the most computational expensive part of the greedy algorithm. However,
+Step 4 is unavoidable, since x(µ∗) is needed for the reduced basis construction.The computational
+complexity of Step 6 could be reduced, if the cardinality of Ξ, i.e., |Ξ| is kept small, so that ∆(µ)
+needs not be evaluated at many parameter samples. This is the motivation of the surrogate error
+estimator proposed in [13, 14].
+We call the surrogate error estimator ∆l(µ) the low-fidelity error
+estimator as compared to the original error estimator ∆(µ), since ∆l(µ) is only an approximation to
+∆(µ), but is much cheaper to compute.
+In the next section, we present a greedy algorithm using bi-fidelity error estimation, where the low-
+fidelity error estimator is computed following the method in [13, 14]. Based on this, a greedy algorithm
+using multi-fidelity error estimation associated with a particular high-fidelity error estimator for MOR
+of linear parametric systems, is proposed.
+3
+Greedy algorithm with bi-fidelity and multi-fidelity error estima-
+tion
+This section presents greedy algorithms with bi-fidelity and multi-fidelity error estimation, respectively.
+3.1
+Greedy algorithm with bi-fidelity error estimation
+Algorithm 2 is the greedy algorithm with bi-fidelity error estimation.
+Its original version using a
+different high-fidelity error estimator was firstly proposed in [13]. The key step of Algorithm 2 is Step
+8, where the low-fidelity error estimator ∆l(µ) is computed using values of ∆(µ) at the samples of
+µ in the small parameter set Ξc. Basically, ∆l(µ) is represented by a weighted sum of radial basis
+3
+
+Algorithm 2 Greedy algorithm with bi-fidelity error estimation
+Input: the FOM, a training set Ξc composed of a small number of parameter samples taken from the
+parameter domain P, a set Ξf composed a large number of parameter samples of µ from P, error
+tolerance tol< 1, ∆(µ) to estimate the error.
+Output: Projection matrix V .
+1: Choose initial parameter µ∗ ∈ Ξc.
+2: V ← ∅, ε = 1.
+3: while ε >tol do
+4:
+Compute the snapshot(s) x(µ∗) by solving the FOM at µ = µ∗.
+5:
+Update V by V = orth{V, x(µ∗)}, (e.g., using the modified Gram-Schmidt process with defla-
+tion.)
+6:
+Compute µ∗ such that µ∗ = arg max
+µ∈Ξc ∆(µ).
+7:
+Compute µo such that µo = arg min
+µ∈Ξc ∆(µ).
+8:
+Compute the low-fidelity error estimator ∆l(µ) using values of ∆(µ) corresponding to the sam-
+ples of µ in Ξc via (3) and (4).
+9:
+Evaluate ∆l(µ) over Ξf and pick out a parameter µc from the large parameter set Ξf corre-
+sponding to the largest value of ∆l(µ), i.e., µc = arg max
+µ∈Ξf from Ξf.
+10:
+Update the small parameter set Ξc: if ∆l(µc) >tol, enrich Ξc with µc, i.e., Ξc = {Ξc, µc}, if
+∆(µo) <tol, remove µo from Ξc: Ξc = Ξc\µo.
+11:
+ε = ∆(µ∗).
+12: end while
+functions (RBFs), i.e.,
+∆l(µ) =
+m
+�
+i=1
+wiΦ(µ − µi),
+(3)
+where Φ(µ) are RBFs, µi are the samples in Ξc, and m is the cardinality of Ξc, which is small. Once wi
+are known, the low-fidelity error estimator ∆l(µ) is known. The weights wi are computed via enforcing
+∆l(µ) to interpolate ∆(µ) at µj, ∀µj ∈ Ξc, i.e., ∆l(µj) = ∆(µj). Inserting µ = µj ∈ Ξc into (3), the
+weights wi can be computed by solving the linear system as below,
+�
+��
+Φ(µ1 − µ1)
+. . .
+Φ(µ1 − µm)
+...
+...
+...
+Φ(µm − µ1)
+. . .
+Φ(µm − µm)
+�
+��
+�
+��
+w1
+...
+wm
+�
+��
+=
+�
+��
+∆(µ1)
+...
+∆(µm)
+�
+�� .
+(4)
+Since values of ∆(µ) at µ ∈ Ξc are available, the weights can be easily computed by solving the above
+small linear system with m × m being the dimension of the coefficient matrix. As this is a rather
+small system, we usually do not observe ill conditioning. Otherwise, we can use a regularized version
+of (4) [13, 14].
+At each iteration of the bi-fidelity greedy algorithm, the linear system is solved for once (Step
+8), then the low-fidelity error estimator ∆l(µ) is evaluated over a larger parameter set Ξf using
+the weighted sum in (3) (Step 9). This process of computing the weights and evaluating ∆l(µ) is
+much faster than evaluating the high-fidelity error estimator over a training set Ξ whose cardinality
+|Ξ| is much larger than |Ξc|.
+This is usually the case for the standard greedy algorithm, where
+|Ξ| > |Ξc|.
+Finally, at each iteration of the bi-fidelity algorithm, the total computational cost of
+Steps 6-8: computing the high-fidelity error estimator ∆(µ) over Ξc, solving the linear system (4) and
+evaluating the low-fidelity erorr estimator ∆l(µ) over Ξf is still much cheaper than computing the
+4
+
+high-fidelity error estimator ∆(µ) over a training set Ξ, whose cardinality is, e.g., twice that of |Ξc|,
+as shown in the numerical tests.
+Besides computing µ∗ corresponding to the maximal value of the error estimator ∆(µ) over Ξc,
+the minimal value of ∆(µ) is also computed in Step 7. The corresponding parameter µo could be
+deleted from Ξc if ∆(µo) is already below the tolerance tol, see Step 10. In this way, the cardinality
+of the training set Ξc remains almost constant, and can further save computations as compared with
+enriching Ξc only. We will show in the numerical results that adding and removing samples to and
+from Ξc gets ROMs with similar accuracy (even smaller) as only adding samples to Ξc, but leads to
+even faster convergence of the greedy algorithm.
+Remark 3.1 In Step 9, it is also possible to choose more than one parameter from Ξf by modifying
+Step 9 as: choose nadd samples from Ξf corresponding to nadd largest values of ∆l(µ). Similarly,
+In Step 7, one can also choose ndel > 1 parameter samples corresponding to ndel smallest values of
+∆(µ) from Ξc. However, this will more or less increase the computational time at each iteration,
+since more computations are needed to choose those samples. Furthermore, to make sure that only
+samples at which ∆l(µ) is larger than the tolerance tol are added to Ξc, and only samples at which
+∆(µ) is smaller than tol are removed, additional calculations are necessary to check if all the selected
+samples meet the above criteria and should be finally selected or removed (see Step 10). Therefore,
+adding/removing at most one parameter sample each time should be more efficient. In the numerical
+tests, we also show results when nadd = ndel = 2 and nadd = ndel = 5 at each iteration of Algorithm 2.
+The bi-fidelity error estimation is general and can be applied to any high-fidelity error estimators.
+For example, the high-fidelity error estimator in [13] estimates the error of the ROM for nonlinear
+time-dependent parametric systems in the time domain, while the high-fidelity error estimator in [14]
+estimates the error of the ROM in the frequency-domain for linear parametric systems.
+3.2
+Greedy algorithm with multi-fidelity error estimation
+The multi-fidelity error estimation we are going to introduce depends on the formulation of the high-
+fidelity error estimator ∆(µ). To illustrate the basic concept, we use an error estimator proposed
+in [16] as the high-fidelity error estimator and discuss how to further reduce the computational load
+by using multi-fidelity error estimation.
+3.2.1
+An error estimator for linear parametric systems
+The error estimator is applicable to estimating the output error of the ROM for FOMs in the following
+linear parametric form,
+M(µ)x(µ)
+=
+B(µ),
+y(µ)
+=
+C(µ)x(µ).
+(5)
+Here, M(µ) ∈ Rn×n, B(µ) ∈ Rn×nI, C(µ) ∈ RnO×n , x(µ) ∈ Rn, y(µ) ∈ RnO×nI. We consider the
+general case that both B(µ) and C(µ) are matrices, i.e. systems with multiple inputs and multiple
+outputs. The ROM of the above linear parametric system can be derived via Galerkin projection
+using a projection matrix V composed of the reduced basis. That is,
+ˆ
+M(µ)z(µ)
+=
+ˆB(µ),
+ˆy(µ)
+=
+ˆC(µ)z(µ),
+(6)
+where ˆ
+M(µ) = V T M(µ)V , ˆB(µ) = V T B(µ), ˆC(µ) = C(µ)V .
+5
+
+For the general situation when both B(µ) and C(µ) are matrices, the error of the i, j-th entry of
+the output matrix ˆy(µ) is
+|yij(µ) − ˆyij(µ)|
+= |Ci(µ)(M−1(µ)B(µ) − V ˆ
+M−1(µ) ˆBj(µ))|
+= |Ci(µ)M−1(µ)(Bj(µ) − M(µ) V ˆ
+M−1(µ) ˆBj(µ))
+�
+��
+�
+ˆxj(µ):=V zj(µ)|
+= |Ci(µ)M−1(µ)rj(µ)|,
+(7)
+where Ci(µ) is the i-th row of C(µ) and Bj(µ) is the j-th column of B(µ). Here, we have defined:
+zj(µ) = ˆ
+M−1(µ) ˆBj(µ), i.e., ˆ
+M(µ)zj(µ) = ˆBj(µ), ˆxj(µ) := V zj(µ) and rj(µ) := Bj(µ) − M(µ)ˆxj(µ).
+It is clear that
+ˆ
+M(µ)zj(µ) = ˆBj(µ)
+is a reduced-order model of
+M(µ)xj(µ) = Bj(µ),
+(8)
+and ˆxj(µ) ≈ xj(µ), the j-th column of x(µ). Finally, rj(µ) is the residual induced by ˆxj(µ).
+From the last equation in (7), it is clear that to compute the absolute error of ˆyij, we need to solve
+a residual system:
+M(µ)xrj(µ) = rj(µ).
+(9)
+Instead, we construct a ROM of it:
+V T
+r M(µ)Vrzrj(µ) = V T
+r rj(µ),
+(10)
+so that xrj(µ) ≈ ˆxrj(µ) = Vrzrj(µ) . Finally,
+|yij(µ) − ˆyij(µ)| ≈ |Ci(µ)ˆxrj(µ)|.
+Note that ˆxrj(µ) depends on Bj(µ), since rj(µ) depends on Bj(µ). Each column Bj(µ) is associated
+with a ˆxrj(µ). The overall error of ˆy(µ) as a matrix can be estimated as:
+∥y(µ) − ˆy(µ)∥max := max
+i,j |yij(µ) − ˆyij(µ)| ≈ max
+i,j |Ci(µ)ˆxrj(µ)| =: ˜∆(µ).
+(11)
+˜∆(µ) defined in (11) is one of the error estimators proposed in [16], where the proposed error
+estimators were shown to outperform other existing error estimators in the literature [34, 15] in terms
+of both accuracy and computational efficiency. Furthermore, it has been discussed in [16] that ˜∆(µ) is
+even more accurate but has less computational complexity than other proposed estimators, including
+the one used in [14]. Even with this error estimator, the greedy algorithm could take several hours
+to converge for some complex systems, for example, the time-delay systems we consider in this work.
+For such systems, although the standard greedy algorithm can already be accelerated by the bi-
+fidelity greedy algorithm, we suggest a possibility to further improve the bi-fidelity greedy algorithm
+by introducing multi-fidelity error estimation.
+We notice that in order to compute ˜∆(µ), an extra projection matrix Vr has to be constructed
+for ˆxrj(µ). Although ˆxrj(µ) is dependent on the individual column of B(µ), the matrix Vr can be
+uniformly constructed based on the whole matrix B(µ). Then Vr is valid for any column of B(µ). It
+is proved in [16] that taking Vr = V leads to ˜∆(µ) identically zero for all µ. Therefore, Vr should be
+additionally computed.
+6
+
+3.2.2
+Standard greedy algorithm using ˜∆(µ)
+For easy understanding of the multi-fidelity error estimation, we first present Algorithm 3, the standard
+greedy algorithm using ˜∆(µ) in (11) as the error estimator. There, some additional steps are added
+to compute Vr, see Step 5, Steps 7-8. In Step 7 of Algorithm 3, Vr is not only updated by x(µr), but
+also by V . This is due to the fact that the solution xrj(µ) to the residual system in (9) can be written
+as
+xrj(µ)
+=
+M(µ)−1rj(µ)
+=
+M(µ)−1(Bj(µ) − M(µ)ˆxj(µ))
+=
+M(µ)−1Bj(µ) − V zj(µ)
+≈
+˜Vrzrj − V zj(µ).
+(12)
+It is clear that xrj(µ) is a linear combination of (M(µ))−1Bj(µ) and the columns of V . Therefore,
+V contributes to the subspace approximating the solution space of xrj(µ) and cannot be neglected.
+It is also noticed that (M(µ))−1Bj(µ) is in fact the solution xj(µ) in (8), while V zj(µ) is ˆxj(µ)
+that approximates xj(µ). This means xrj(µ) is the difference between xj(µ) and ˆxj(µ), which is a
+nonzero vector.
+Therefore, we should compute another matrix ˜Vr, so that xj(µ) ≈ ˜Vrzrj(µ), but
+˜Vrzrj(µ) ̸= ˆxj(µ) = V zj(µ). Finally, xrj is approximated by the difference between ˜Vrzrj(µ) and
+V zj(µ). In other words, it is approximately represented as the linear combination of the columns of
+both Vr and V . This approximation also explains Step 5 and Step 7 of Algorithm 3: Step 5 computes
+the reduced basis vectors contributing to ˜Vr, Step 7 computes the complete reduced basis vectors
+contributing to Vr. New reduced basis vectors for both V and Vr are computed at each iteration of
+the greedy algorithm. Step 8 and Step 9 compute the new important parameter samples for Vr and V ,
+respectively. In general, µr should be different from µ∗, since ˜∆(µ) ̸=
+max
+j=1,...,nI ∥rj(µ) − M(µ)ˆxrj(µ)∥.
+Here, rj(µ) − M(µ)ˆxrj(µ) is nothing but the residual induced by the approximate solution (ˆxrj(µ)) to
+the residual system (9).
+Algorithm 3 Standard greedy algorithm using ˜∆(µ) for linear parametric systems.
+Input: the FOM, a training set Ξ composed of parameter samples taken from the parameter domain
+µ ∈ P, error tolerance tol< 1.
+Output: Projection matrix V .
+1: Choose initial parameter µ∗ ∈ Ξ for V , and initial parameter µr ̸= µ∗ ∈ Ξ for Vr.
+2: V ← ∅, Vr ← ∅, ε = 1.
+3: while ε >tol do
+4:
+Compute the snapshot(s) x(µ∗) by solving the FOM, i.e. x(µ∗) = (M(µ∗))−1B(µ∗).
+5:
+Compute the snapshot(s) x(µr) by solving the FOM, i.e. x(µr) = (M(µr))−1B(µr).
+6:
+Update V by V = orth{V, x(µ∗)}, (e.g., using the modified Gram-Schmidt process with defla-
+tion.)
+7:
+Update Vr by Vr = orth{V, Vr, x(µr)}.
+8:
+Compute µr such that µr = arg max
+µ∈Ξ
+max
+j=1,...,nI ∥rj(µ) − M(µ)ˆxrj(µ)∥, (nI is the total number of
+columns of B(µ)).
+9:
+Compute µ∗ such that µ∗ = arg max
+µ∈Ξ
+˜∆(µ).
+10:
+ε = ˜∆(µ∗).
+11: end while
+7
+
+3.2.3
+Greedy algorithm with multi-fidelity error estimation
+The computational complexity of Algorithm 3 using the error estimator ˜∆(µ) comes from Steps 4-9.
+Efficiency of Step 9 can be improved by using the bi-fidelity error estimation as shown in Algorithm 2.
+The computations in Step 4, 6 are unavoidable, since V is used to compute the ROM of the original
+FOM and should be updated till acceptable error tolerance is satisfied. In contrast, Vr in Step 7 needs
+not be updated at every iteration. This implies that the ROM of the residual system does not have to
+be very accurate, since it is not the ROM that we seek, but an auxiliary ROM aiding the computation
+of ˜∆(µ).
+An immediate consequence of Theorem 4.2 in [16] for single-input and single-output systems is the
+following Lemma for systems with multiple inputs and multiple outputs:
+Lemma 3.1 The error of the output ˆy(µ) of the ROM (6) can be bounded as
+˜∆(µ) − δ(µ) ≤ ∥y(µ) − ˆy(µ)∥max ≤ ˜∆(µ) + δ(µ),
+(13)
+where δ(µ) := max
+i,j |Ci(µ)(xrj(µ) − ˆxrj(µ))| ≥ 0.
+Proof From (7), we know
+|yij(µ) − ˆyij(µ)| = |Ci(µ)xrj(µ)| ≈ |Ci(µ)ˆxrj(µ)|.
+Then
+|yij(µ) − ˆyij(µ)|
+=
+|Ci(µ)xrj(µ)| + |Ci(µ)ˆxrj(µ)| − |Ci(µ)ˆxrj(µ)|
+≤
+|Ci(µ)ˆxrj(µ)| + |Ci(µ)xrj(µ) − Ci(µ)ˆxrj(µ)|
+�
+��
+�
+δij(µ)
+.
+(14)
+On the other hand,
+|Ci(µ)ˆxrj(µ)|
+=
+|Ci(µ)ˆxrj(µ)| + |Ci(µ)xrj(µ)| − |Ci(µ)xrj(µ)|
+≤
+|Ci(µ)xrj(µ)| + δij(µ).
+(15)
+From (11), (15) and the definition of δ(µ), we have
+˜∆(µ) = max
+i,j |Ci(µ)ˆxrj(µ)| ≤ ∥y(µ) − ˆy(µ)∥max + δ(µ).
+Similarly, from (14), we get
+∥y(µ) − ˆy(µ)∥max ≤ ˜∆(µ) + δ(µ).
+This completes the proof.
+From the definition of δ(µ), it is seen that the more accurate the ROM of the residual system, the
+smaller δ(µ) is. As a result, ˜∆(µ) should measure the true error more accurately so that the important
+parameters it selects are closer to those selected by the true error, given the same training set Ξ.
+On the contrary, if the ROM of the residual system is less accurate, ˜∆(µ) will be less accurate, too.
+However, at a certain stage, when ˜∆(µ) is already small, the right-hand side of the residual system
+rj(µ) will also be small, so that it can be expected that both xrj(µ) and ˆxrj(˜µ) are close to zero. This
+leads to a small δ(µ). Variation of a small δ(µ) will not cause big variation of the difference between
+˜∆(µ) and the true error ∥y(µ) − ˆy(µ)∥max. The trend, though not the exact route, of error decay
+could still be anticipated so that important parameters corresponding to the error peaks can also be
+detected. The above analyses are also justified by the numerical results in the next section, see, e.g.,
+Figure 3 and Figure 5.
+8
+
+This motivates the multi-fidelity error estimation. We set a second tolerance ϵ >tol, and when
+˜∆(µ) < ϵ < 1, we stop updating the ROM of the residual system, i.e., stop implementing Step 5, Step
+7 and Step 8 of Algorithm 3. The error estimator ˜∆(µ) after this stage may not be as accurate as it
+would be when keep updating the ROM of the residual system. However, the difference should be small
+as ˜∆(µ) is already below a small value ϵ. Without implementing Step 5, we have saved computations
+of simulating the FOM. For large and complex systems, solving the FOM even once is not fast. The
+computation in Step 7 is relatively cheap if the system is not very large. The computational cost in
+Step 8 is not low for certain complex problems, though some µ-independent parts of rj(µ) and M(µ)
+can be pre-computed. For example, this is the case for the time-delay systems in the next section.
+Stop updating the ROM of the residual system gives rise to a low-fidelity error estimator at later
+iteration steps of the greedy algorithm.
+When this low-fidelity error estimator is combined with
+∆l(µ) in Algorithm 2, we obtain the multi-fidelity error estimation. This is detailed in Algorithm 4.
+Compared with the standard greedy algorithm, the overall saving in computational costs is noticeable,
+which can be seen from the numerical results in the next section.
+The concept of multi-fidelity error estimation could also be applied to other high-fidelity error
+estimators. For example, Step 15 could be modified as “Stop updating partial information of ∆(µ)”,
+if some parts of the high-fidelity error estimator ∆(µ) are not “essential” for computing ∆(µ).
+Algorithm 4 Greedy algorithm with multi-fidelity error estimation
+Input: the FOM, a training set Ξc composed of a small number of parameter samples taken from the
+parameter domain µ ∈ P, a set Ξf composed of a large number of parameter samples of µ from
+P, error tolerance tol< 1.
+Output: Projection matrix V .
+1: Choose initial parameter µ∗ ∈ Ξc for V , and initial parameter µr ̸= µ∗ ∈ Ξc for Vr.
+2: V ← ∅, Vr ← ∅, ε = 1.
+3: while ε >tol do
+4:
+Compute the snapshot(s) x(µ∗) by solving the FOM, i.e. x(µ∗) = (M(µ∗))−1B(µ∗).
+5:
+Compute the snapshot(s) x(µr) by solving the FOM, i.e. x(µr) = (M(µr))−1B(µr).
+6:
+Update V by V = orth{V, x(µ∗)} (e.g., using the modified Gram-Schmidt process with defla-
+tion).
+7:
+Update Vr by Vr = orth{V, Vr, x(µr)}.
+8:
+Compute µ∗ such that µ∗ = arg max
+µ∈Ξc
+˜∆(µ).
+9:
+Compute µo such that µo = arg min
+µ∈Ξc
+˜∆(µ).
+10:
+Compute µr such that µr = arg max
+µ∈Ξc
+max
+j=1,...,nI ∥rj(µ) − M(µ)ˆxrj(µ)∥,
+% nI is the total number
+of columns of B(µ).
+11:
+Compute the low-fidelity error estimator ˜∆l(µ) using values of ˜∆(µ) corresponding to the sam-
+ples of µ in Ξc via (3) and (4).
+12:
+Evaluate ˜∆l(µ) over Ξf and pick out a parameter µc from the large parameter set Ξf corre-
+sponding to the largest value of ˜∆l(µ), i.e., µc = arg max
+µ∈Ξf from Ξf.
+13:
+Update the small parameter set Ξc: if ∆l(µc) >tol, enrich Ξc with µc, i.e., Ξc = {Ξc, µc}, if
+∆(µo) <tol, remove µo from Ξc: Ξc = Ξc\µo.
+14:
+ε = ˜∆(µ∗).
+15:
+if ε < ϵ then
+16:
+Stop performing Step 5, Step 7 and Step 10.
+% stop updating the ROM of the residual
+system.
+17:
+end if
+18: end while
+9
+
+3.3
+Application to MOR for time-delay systems
+In this section, we consider applying Algorithm 2, the greedy algorithm with bi-fidelity error esti-
+mation, Algorithm 3, the standard greedy algorithm and Algorithm 4, the greedy algorithm with
+multi-fidelity error estimation to large-scale time-delay systems with many delays. The time-delay
+systems are defined as:
+d
+�
+j=0
+Ej ˙x(t − τj) =
+d
+�
+j=0
+Ajx(t − τj) + Bu(t),
+y(t) = Cx(t),
+∀ t ≥ 0
+(16)
+with an initial condition x(t) = Φ(t) ∈ Cn, ∀ t ∈ [−τd, 0]. Here, E0, . . . , Ed, A0, . . . , Ad ∈ Cn×n, B ∈
+Cn×nI, C ∈ CnO×n, 0 = τ0 < τ1 < . . . < τd and n is called the order of the delay system. The transfer
+function of the delay system is defined as:
+H(s) = CK−1(s)B,
+(17)
+where K(s) = s �d
+j=0 Eje−sτj − �d
+j=0 Aje−sτj, s = 2πȷ is the variable in the frequency domain, f is
+the ordinary frequency with unit Hz and ȷ is the imaginary unit.
+A ROM of the delay system, which has the same delays as the original system, can be obtained
+via Galerkin projection using a projection matrix V ∈ Rn×r, r ≪ n, i.e.,
+d
+�
+j=0
+ˆEj ˙z(t − τj) =
+d
+�
+j=0
+ˆAjz(t − τj) + ˆBu(t),
+ˆy(t) = ˆCz(t),
+∀ t ≥ 0,
+(18)
+where ˆEj = V T EjV ∈ Rr×r, ˆAj = V T AjV ∈ Rr×r, ˆB = V T B ∈ Rr×nI, ˆC = CV ∈ RnO×r, with
+r ≪ n being the order of the ROM. The original state vector x(t) in (16) can be recovered by the
+approximation: x(t) ≈ V z(t). The transfer function of the ROM is
+ˆH(s) = ˆC ˆK−1(s) ˆB,
+where ˆK(s) = s �d
+j=0 ˆEje−sτj − �d
+j=0 ˆAje−sτj.
+The projection matrix V can be constructed via
+approximating H(s) [5] as follows.
+Note that H(s) is nothing but the output y(µ) of the linear
+parametric system in (5), with M(µ) = K(s), B(µ) = B and µ = s, i.e.,
+K(s)x(s)
+=
+B,
+H(s)
+=
+C(s)x(s).
+(19)
+The reduced transfer function ˆH(s) is the output ˆy(µ) of the ROM in (6) with ˆ
+M(µ) = ˆK(s) and
+ˆB(µ) = ˆB.
+It is easy to see that the projection matrix V that is used to construct the ROM (18) in the time
+domain is exactly the same matrix to obtain the reduced transfer function ˆH(s). Therefore, V can
+be obtained by constructing a ROM of system (19) in the frequency domain, i.e., by approximating
+the transfer function H(s). This can be done by the standard greedy Algorithm 3 with the error
+estimator ˜∆(s), where V is iteratively computed by choosing proper samples of s [5, 1]. In fact, the
+reduced transfer function ˆH(s) interpolates the original transfer function H(s) at the selected samples
+of s [1]. The matrix M(µ) in Steps 4-5 of Algorithm 3 is now replaced by K(s). The difference of the
+coefficient matrix K(s) from a single matrix M(µ) in the usual case is its high complexity. To solve the
+system in (19) is much more expensive than solving the system in (5) where M(µ) is a single matrix.
+10
+
+On the one hand, the matrices constituting K(s) must be assembled to get K(s). On the other hand,
+the finally assembled matrix has some dense blocks, though each single matrix contributing to K(s)
+is sparse.
+To further improve the efficiency of the standard greedy algorithm, we propose to apply Algorithm 2
+and Algorithm 4 to time-delay systems. The application is straightforward by simply replacing the
+FOM in (5) in both algorithms with the system in (19), i.e., the matrix M(µ) is replaced by K(s), the
+input matrix B(µ) and the output matrix C(µ) are replaced by B and C in (19), respectively.
+4
+Numerical tests
+We consider three time-delay systems obtained from partial element equivalent circuit (PEEC) mod-
+elling and simulation, which transfer problems from the electromagnetic domain to the circuit do-
+main [29, 32, 30, 31]. When the propagation delays are explicitly kept for both partial inductances
+and coefficients of potential, time-delay systems can be derived [17]. Numerical tests are done with
+MATLAB R2016b on a computer server with 4 Intel Xeon E7-8837 CPUs running at 2.67 GHz, 1TB
+main memory, split into four 256 GB partitions.
+We test the standard greedy Algorithm 3, the bi-fidelity greedy Algorithm 2 and the multi-fidelity
+Algorithm 4 on three time-delay systems. To run the algorithms, we need to initialize the algorithms
+by doing the following:
+• The samples in the training set Ξ, the small set Ξc and the large set Ξf are taken from
+the prescribed frequency domain and are generated using the MATLAB function linspace:
+linspace(fl, fh, cardi). Here, fl is the lowest frequency, fh is the highest frequency used in
+linspace, cardi is the corresponding cardinality of each set. The samples of s are then com-
+puted using the relation: s = 2πȷf.
+• For the multi-fidelity error estimation, we set ϵ = 0.1 in Step 15 of Algorithm 4.
+• To compute the low-fidelity error estimator, we choose the inverse multiquadratic RBF (IMQ)
+Φ =
+1
+1+(a∥µ−µi∥)2 with the shape parameter a = 30.
+We also need to define some variables uniformly used in all the tables and figures:
+• The error ∥H(s) − ˆH(s)∥max of the transfer function ˆH(s) of the ROM is finally computed over
+1000 samples of s drawn independently of the training sets, resulting in the validated error:
+Valid.err in Tables 1-8.
+• Runtime, the walltime of each algorithm till convergence.
+• Iter., the total number of iterations of each algorithm.
+• r, the order of the ROM.
+• The high-fidelity error estimator at each iteration of Algorithm 3 is defined as max
+µ∈Ξ
+˜∆(µ).
+• The bi-fidelity error estimator at each iteration of Algorithm 2 is defined as max
+µ∈Ξc
+˜∆(µ).
+• The multi-fidelity error estimator at each iteration of Algorithm 4 is defined as max
+µ∈Ξc
+˜∆(µ). Here
+˜∆(µ) will be different from the bi-fidelity error estimator once Step 15 of the algorithm takes
+action.
+11
+
+w
+P1
+P3
+P2
+lX,1
+lY,1
+lY,3
+lY
+lX
+Figure 1: The three-port microstrip power-divider circuit.
+• The true error at each iteration of Algorithm 3 is defined as max
+µ∈Ξ ∥H(s) − ˆH(s)∥max.
+• The true error at each iteration of Algorithm 2 or Algorithm 4 is defined as max
+µ∈Ξc ∥H(s) −
+ˆH(s)∥max.
+Note that Ξc could be enriched only by adding samples from Ξf to Ξc. As the high-fidelity error
+estimator ˜∆(µ) needs to be computed at every sample in Ξc at each iteration, samples in Ξc whose
+corresponding error is already smaller than tol can also be removed from Ξc to keep the cardinality of
+Ξc constant, so that more computations can be saved. We consider both cases separately and compare
+their efficiency with respect to both runtime and accuracy.
+4.1
+Test 1: results for a model of three-port divider
+The model structure of a three-port microstrip power-divider circuit is shown in Fig. 1 (P1, P2 and
+P3 denote the ports). The dimensions of the circuit are [20, 20, 0.5] mm in the [x, y, z] directions and
+the width of the microstrips is set as 0.8 mm. Furthermore, the dimensions lX1, lY 1, and lY 3 are 9,
+7.2 and 7.2 mm, respectively. The relative dielectric constant is εr = 2.2. All the ports are terminated
+on 50 Ω resistances. The order of the FOM is n = 10, 626, and it has d = 93 delays. The interesting
+frequency band is [0, 20]GHz.
+For this model, we use fl = 1 × 106, fh = 2 × 1010 in the function linspace. |Ξ| = 30 or |Ξ| = 40
+for the standard greedy Algorithm 3. For Algorithm 2 and Algorithm 4, |Ξc| = 15 or |Ξc| = 20 and
+|Ξf| = 100.
+The set Ξc is then updated during the iteration of the greedy algorithm.
+The 1000
+samples for validating the ROM accuracy are created using the MATLAB function logspace, i.e.,
+logspace(log10(fl1), log10(fh), 1000). fl1 = 1 × 104.
+In Table 1, we list the results of the three algorithms. The standard greedy algorithm is the slowest.
+The other algorithms are all much faster and take at least 2 hours less than the standard algorithm.
+The bi-fidelity greedy algorithm by enriching Ξc only is slower than other bi-(multi-)fidelity algorithms,
+this is in agreement with our theoretical analysis in Section 3. The multi-fidelity algorithm by adding
+and removing samples to and from Ξc performs the best in terms of runtime and accuracy. Compared
+to the standard algorithm, it has reduced the offline runtime from 5.6 hours to 1.8 hours, and almost 4
+hours have been saved. Finally, a speed-up factor 3.1 is achieved. Except for the bi-fidelity algorithm
+by adding and removing samples, the other algorithms have produced ROMs with validated errors
+below the tolerance. The bi-fidelity algorithms perform similarly as the standard algorithm. All three
+algorithms converge in 14 iterations, and produce ROMs smaller than the others.
+It is worth pointing out that if using fewer samples in Ξ for the standard greedy algorithm, the
+ROM has a validated error that is slightly larger than the tolerance, as shown in Table 2, where
+|Ξ| = 30. Also, the bi-fidelity greedy algorithms are less accurate if using fewer samples in Ξc, as
+shown in Table 2. There, the same Ξc used for the multi-fidelity greedy algorithms are used, but less
+accurate ROMs are obtained.
+In Table 3, we show the results of the bi-fidelity greedy algorithm and the multi-fidelity greedy
+algorithm when nadd = ndel > 1 samples are added or removed from the small training set Ξc at each
+12
+
+Table 1: Three-port divider: n = 10, 626, d = 93 delays, tol=0.001, adding/removing a single sample
+at each iteration.
+Method
+Iter.
+Runtime (h)
+r
+Valid.err
+Alg. 3 (standard, |Ξ| = 40)
+14
+5.6
+84
+9.2 × 10−4
+Alg. 2 (bi-fidelity, add only, |Ξc| = 20)
+14
+3.6
+84
+6 × 10−4
+Alg. 2 (bi-fidelity, add-remove, |Ξc| = 20)
+14
+2.7
+84
+0.0022
+Alg. 4 (multi-fidelity, add only, |Ξc| = 15)
+15
+2.4
+90
+6.2 × 10−4
+Alg. 4 (multi-fidelity, add-remove, |Ξc| = 15)
+15
+1.8
+90
+6.2 × 10−4
+Table 2:
+Three-port divider:
+n = 10, 626, d = 93 delays, tol=0.001, smaller |Ξ| and |Ξc|,
+adding/removing a single sample at each iteration.
+Method
+Iter.
+Runtime (h)
+r
+Valid.err
+Alg. 3 (standard, |Ξ| = 30)
+14
+4.2
+84
+0.0017
+Alg. 2 (bi-fidelity, add only,|Ξc| = 15)
+13
+2.5
+78
+0.0026
+Alg. 2 (bi-fidelity, add-remove, |Ξc| = 15)
+13
+1.9
+78
+0.0088
+Table 3: Three-port divider: n = 10, 626, d = 93 delays, tol=0.001, adding/removing nadd = ndel > 1
+samples at each iteration.
+Method
+Iter.
+Runtime (h)
+r
+Valid.err
+Alg. 2 (bi-fidelity, add-remove, |Ξc| = 15, nadd = 2)
+14
+2.0
+84
+0.0022
+Alg. 2 (bi-fidelity, add-remove, |Ξc| = 20, nadd = 2)
+14
+2.7
+84
+0.0022
+Alg. 2 (bi-fidelity, add-remove, |Ξc| = 20, nadd = 5)
+14
+2.7
+84
+0.0022
+Alg. 4 (multi-fidelity, add-remove, |Ξc| = 15, nadd = 2)
+14
+1.7
+84
+0.0039
+Alg. 4 (multi-fidelity, add-remove, |Ξc| = 15, nadd = 5)
+14
+1.7
+84
+0.0039
+iteration of the algorithm. In general, they produce similar results as those in Table 1 and Table 2
+given the same Ξc. For |Ξc| = 15, the bi-fidelity greedy algorithm with nadd = ndel = 2 converges in 14
+iterations, running one more iteration than with nadd = ndel = 1 as shown in Table 2, and generates
+a ROM with slightly higher accuracy. On the contrary, given |Ξc| = 15, the multi-fidelity greedy
+algorithm with either nadd = ndel = 2 or nadd = ndel = 5 runs one iteration less than in the case
+of adding/removing a single sample as shown in Table 1, and constructs ROMs with lower accuracy.
+Furthermore, it is seen that increasing nadd = ndel from 2 to 5 did not change the results for both
+algorithms. In general, adding/removing a single sample keeps the algorithms simple but efficient.
+To illustrate the behavior of the error estimators further, we plot the decay of error estimators and
+their corresponding true errors during the greedy iterations. Since different µ∗ are chosen according
+to different error estimators, the projection matrix V is updated with different snapshots, leading to
+ROMs with different accuracy. Consequently, the true errors of the ROMs are expected to be different.
+Figures 2-3 are the results of the algorithms in Table 1. The left part of Figure 2 shows the error
+of the high-fidelity error estimator at each iteration of Algorithm 3 and the decay of the corresponding
+true error. The error estimator almost exactly matches the true error at all the iterations. The right
+part of Figure 2 plots the decay of the bi-fidelity error estimator with respect to the true error. The
+bi-fidelity error estimator in both of the two cases: only adding (add-only) samples to Ξc, adding
+and removing (add-remove) samples to and from Ξc, can accurately catch the true error. Both cases
+converge in 14 iterations, but the case “add-only” is more accurate as can be seen from Table 1.
+Figure 3 plots the decay of the multi-fidelity error estimator and the corresponding true error
+decay. For clarity, the two cases “add-only” and “add-remove” are plotted in two separate figures.
+13
+
+The multi-fidelity error estimator is not as accurate as the bi-fidelity error estimator. This is indicated
+by the error decay from the 10-th iteration to the end in both figures. From the 10-th iteration, the
+error estimator is below ϵ = 0.1, the multi-fidelity error estimation at Step 15 of Algorithm 4 begins
+to be implemented. For this example, the multi-fidelity error estimator overestimates the true error
+more often than the bi-fidelity error estimator, it did not choose the interpolation points that lead to
+error decay as fast as those chosen by the bi-fidelity error estimator. Finally, it uses more iteration
+steps to converge. Whereas, they still produce ROMs with best accuracy.
+Figure 2: Error decay. Left: true error vs high-fidelity error estimator. Right: true error vs bi-fidelity
+error estimators.
+Figure 3: Error decay. Left: true error vs multi-fidelity error estimator by only adding samples to Ξc.
+Right: true error vs multi-fidelity error estimator by adding and deleting samples to and from Ξc.
+4.2
+Test 2: results for a model of coplanar microstrips
+The second example is a model of a three coplanar microstrips structure shown in Fig. 4. The width
+of the metal strips is mw = 0.178 mm, the thickness of metal strips and ground plane is mt = 0.035
+mm while the left and right wing of the microstrips are wd = 3 mm. Finally, the length of each
+strip is ℓ = 5 cm, the thickness of the dielectric is dt = 0.8 mm, and the spacing between 2 strips is
+s = 0.3 mm. The relative dielectric constant is set to be εr = 4 and the conductivity of the metal is
+assumed to be σ = 5.87 S/m. The six ports, located between the ends of each strip and the ground
+14
+
+10
+.... High.-fidelity estimator
+@.... True error
+10
+10
+2
+4
+6
+8
+10
+12
+Number of iterations102
+10
+10
+ -G - True error
+... Bi-fidelity estimator (add only)
+-- - True error
++... Bi-fidelity estimator (add-remove)
+2
+4
+8
+10
+12
+14
+6
+i-th iteration102
+ -G - True error
+.... Multi-fidelity estimator (add only)
+10-4
+2
+4
+6
+8
+10
+12
+14
+i-th iteration10°
+G- True error
+...... Multi-fidelity estimator (add-remove)
+2
+4
+6
+8
+10
+12
+14
+i-th iterationwd
+mw
+s
+mw
+s
+mw
+wd
+mt
+dt
+mt
+ℓ
+P1
+Figure 4: Three coplanar microstrips
+plane below, are terminated on load resistors Rload = 50 Ω. The order of the FOM is n = 16, 644, and
+there are d = 168 delays. The frequency band of interest is [0, 10]GHz.
+For this model, we take fl = 1×106, fh = 1×1010. We set 30 samples for Ξ in the standard greedy
+Algorithm 3, i.e., |Ξ| = 30. For Algorithm 2 and Algorithm 4, |Ξc| = 10 or |Ξc| = 15, and |Ξf| = 100.
+The 1000 samples used for validating the ROM accuracy are generated using the MATLAB function
+linspace, with fl = 100 and the given fh.
+The results of the three algorithms are listed in Table 4. The standard greedy Algorithm 3 takes
+19 hours, resulting in a ROM of order r = 132 with validated error below the tolerance tol. During the
+greedy iteration, if the small parameter set Ξc is enriched only (add only), the greedy algorithm with
+bi-fidelity error estimation and that with multi-fidelity error estimation converge within the same
+number of iterations, producing ROMs with the same sizes and validated errors.
+But the greedy
+algorithm with multi-fidelity error estimation is almost one hour faster. Similar phenomenon happens
+to the case “add-remove”. The greedy algorithm with bi-fidelity error estimation and that with multi-
+fidelity error estimation also converge within the same number of iterations and construct ROMs with
+the same sizes and accuracy. The runtimes of both algorithms are much less as compared to their
+“add only” versions.
+Finally, the greedy algorithm with multi-fidelity error estimation by adding
+and deleting samples to and from Ξc (“add-remove”) is most efficient in terms of both runtime and
+accuracy. It is more than 3 times faster than the standard greedy algorithm resulting in a speed-up
+of 4.2x, and produces a ROM with even the smallest validated error.
+We note that using |Ξc| = 10, the ROMs constructed by the bi-fidelity greedy algorithm and the
+multi-fidelity greedy algorithm with adding the samples only have validated errors larger than the
+tolerance. If we increase |Ξc| from 10 to 15, both algorithms generate ROMs with improved accuracy.
+The results are presented in Tabel 5. However, the computational time also increases a lot. Again,
+the multi-fidelity greedy algorithm outperforms the bi-fidelity one w.r.t. both accuracy and runtime.
+In contrast to the results in Tables 1-2 for the divider model, the results for the coplanar microstrips
+model in both Tables 4-5 show that the bi-fidelity greedy algorithm (“add-remove”) is more accurate
+than its “add-only” version.
+Table 6 shows the results of the bi-fidelity greedy algorithm and the multi-fidelity greedy algorithm
+based on adding/removing multiple samples at each iteration. For both cases, i.e., nadd = ndel = 2
+and nadd = ndel = 5, the algorithms using |Ξc| = 10, converge in 10 iterations, one less iteration
+than they did with nadd = ndel = 1 in Table 4, resulting in ROMs with smaller order r but with
+larger validated errors. If we increase |Ξc| to 15, then the multi-fidelity greedy algorithm generates a
+ROM with reduced error, but takes longer time to converge. The bi-fidelity greedy algorithm behaves
+similarly and its results for |Ξc| = 15 is not presented to avoid repetition. This example again shows
+that adding/removing a single parameter at each iteration outperforms the cases with nadd = ndel > 1,
+and produces ROMs with desired accuracy.
+15
+
+Table 4: Three coplanar microstrips: n = 16, 644, d = 168 delays, tol=0.001, adding/removing a single
+sample at each iteration.
+Method
+Iter.
+Runtime (h)
+r
+Valid.err
+Alg. 3 (standard, |Ξ| = 30)
+11
+15
+132
+8.5 × 10−4
+Alg. 2 (bi-fidelity, add only, |Ξc| = 10)
+11
+6.2
+132
+0.0033
+Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10)
+11
+5.3
+132
+8.2 × 10−4
+Alg. 4 (multi-fidelity, add only, |Ξc| = 10)
+11
+5.3
+132
+0.0033
+Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10)
+11
+4.5
+132
+8.2 × 10−4
+Table 5:
+Three coplanar microstrips:
+n = 16, 644, d = 168 delays, tol=0.001, larger |Ξc|,
+adding/removing a single sample at each iteration.
+Method
+Iter.
+Runtime (h)
+r
+Valid.err
+Alg. 2 (bi-fidelity, add only, |Ξc| = 15)
+11
+10
+132
+0.0011
+Alg. 4 (multi-fidelity, add only, |Ξc| = 15)
+12
+9.3
+144
+4.4 × 10−4
+Table 6: Three coplanar microstrips: n = 16, 644, d = 168 delays, tol=0.001, adding/removing
+nadd = ndel > 1 samples at each iteration.
+Method
+Iter.
+Runtime (h)
+r
+Valid.err
+Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10, nadd = 2)
+10
+4.7
+120
+0.019
+Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10, nadd = 5)
+10
+4.7
+120
+0.019
+Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10, nadd = 2)
+10
+4.2
+120
+0.019
+Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10, nadd = 5)
+10
+4.3
+120
+0.019
+Alg. 4 (multi-fidelity, add-remove, |Ξc| = 15, nadd = 2)
+13
+7.6
+156
+0.0027
+In Figure 5, we show the important frequency samples of f selected by the greedy algorithms in
+Table 4. For the case “add-remove”, we find that the greedy algorithm with bi-fidelity error estimation
+and the one with multi-fidelity error estimation select the same important frequency samples. Therefore
+we only plot one group of samples for both algorithms, see the plot “bi-(multi-) add-remove” in the
+figure. For the case “add-only”, both algorithms also select the same important frequency samples,
+see the plot “bi-(multi-) add-only” in the figure.
+This is in agreement with the results given in
+Table 4 where both algorithms for either case produce the same results. The important frequency
+samples selected by the high-fidelity error estimator are mostly different from those selected by the
+other algorithms. It is seen that the important frequency samples selected by the (bi-)multi-fidelity
+estimator could be different from those selected by the high-fidelity estimator. However, both can
+derive ROMs with good accuracy.
+The left part of Figure 6 gives the error-peak frequencies detected by the multi-fidelity error
+estimator and the true error, respectively, at each iteration of the greedy algorithm. Those frequencies
+correspond to the largest values of the error estimator/true error. The error-peak frequency detected
+by the error estimator at the i-th iteration is then selected as the important frequency sample at the
+next iteration to update the reduced basis space. From iteration 5, the error-peak frequencies detected
+by the error estimator are exactly the same as those selected by the true error. This can be explained
+by the error decay in the right part of the figure. From the 5-th iteration, the error estimator tightly
+catches the true error. Although it is less tight at the first 4 iterations, it still follows the overall trend
+of the error decay and therefore, can still detect reasonable error-peak frequencies. This example,
+once again, supports our theoretical analysis and demonstrates the efficacy of the proposed greedy
+algorithms with bi-(multi-) fidelity error estimation.
+16
+
+Figure 5: Important parameters selected by the greedy algorithms.
+Figure 6: Left: Frequencies causing error/estimator peaks. Right: true error vs multi-fidelity error
+estimator.
+4.3
+Test 3: results for a model of microstrip filter
+The third example is a model of a microstrip filter. The 3D structure of a microstrip filter is depicted in
+Fig. 7. The physical dimensions for the geometry of the 3D structure are: wzl = 0.5 mm, wz0 = 1.125
+mm, wzC = 4 mm, ℓzl = 18.3 mm, ℓz0 = 1 mm, ℓzC = 14.1 mm, w = 2.4 cm, ℓ = 2ℓzl + 2ℓz0 + ℓzC,
+tm = 100 µm, ts = 100 µm, td = 508 µm. The two ends of the microstrip are terminated on 50 Ω
+resistors.
+The order of the FOM is n = 12, 132, and there are d = 190 delays.
+The interesting
+frequency band is [0, 5]GHz.
+We take fl = 1 × 105, fh = 5 × 109 to generate frequency samples in Ξc and Ξ. We use |Ξ| = 30
+for the standard greedy Algorithm 3. For Algorithm 2 and Algorithm 4, |Ξc| = 10, and |Ξf| = 100.
+The 1000 samples used for computing the validated error are generated using the MATLAB function
+logspace, with fl = 10 and the given fh.
+The results of the high-fidelity greedy algorithm, and the bi-(multi-)fidelity greedy algorithms by
+adding/removing a single sample at each iteration, are listed in Table 7. All the bi-(multi-)fidelity
+greedy algorithms produce similar results. The runtime of each is around 1 hour, 3 hours faster than
+the high-fidelity greedy algorithm. All the ROMs have similar accuracy, with validated errors below
+the tolerance.
+Table 8 further shows the performance of the bi-(multi-)fidelity greedy algorithms by adding and
+removing multiple samples at each iteration. For this model, all these algorithms behave similarly as
+17
+
+X109
+10
+8
+Frequency (Hz)
+6
+4
+2
+--- hi-fidelity
+..bi-(.multi)..add-remove
+.. bi-(.multi),  add-only.
+0
+2
+4
+6
+8
+10
+12
+0
+Number of iterations10
++
+9.5
+Frequency (GHz)
+9
+8.5
+Estimator-peak frequency
+True-error-peak frequency
+8
+2
+4
+6
+8
+10
+0
+i-th iteration40
+35
+30
+25
+.....Multi-fidelity  estimator ("add-remove"
+--O- - . True errror
+20
+15
+10
+5
+0
+米
+0
+2
+4
+6
+8
+10
+12
+i-th iterationwz0
+wzl
+wzC
+ℓzC
+ℓzl
+ℓz0
+ℓ
+w
+tm td ts
+Figure 7: Microstrip filter.
+they did by adding/removing a single sample at each iteration. The multi-fidelity greedy algorithm
+produces ROMs with slightly larger sizes. The ROMs also have larger validated errors, but still fulfill
+the accuracy requirement. All algorithms converge within 8 iterations, much faster than for the first
+two examples. This may be due to the much smaller frequency band of interest [0, 5]GHz making the
+problem much easier to solve and leading to the most efficient performance of all algorithms.
+In summary, for all the tested examples, the multi-fidelity algorithm by adding/removing a single
+sample at each iteration behaves the best w.r.t. both runtime and accuracy.
+Table 7: Microstrip filter: n = 12, 132, d = 190 delays, tol=0.001, adding/removing a single sample
+at each iteration.
+Method
+Iter.
+Runtime (h)
+r
+Valid.err
+Alg. 3 (standard, |Ξ| = 30)
+8
+2.5
+32
+5.6 × 10−4
+Alg. 2 (bi-fidelity, add only, |Ξc| = 10)
+7
+1.1
+28
+4.6 × 10−4
+Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10)
+7
+1.1
+28
+4.6 × 10−4
+Alg. 4 (multi-fidelity, add only, |Ξc| = 10)
+7
+1.0
+28
+4.6 × 10−4
+Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10)
+8
+1.1
+32
+5.7 × 10−4
+Table 8: Microstrip filter: n = 12, 132, d = 190 delays, tol=0.001, adding/removing nadd = ndel > 1
+samples at each iteration.
+Method
+Iter.
+Runtime (h)
+r
+Valid.err
+Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10, nadd = 2)
+7
+1.1
+28
+4.6 × 10−4
+Alg. 2 (bi-fidelity, add-remove, |Ξc| = 10, nadd = 5)
+7
+1.1
+28
+4.6 × 10−4
+Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10, nadd = 2)
+8
+1.1
+32
+9.1 × 10−4
+Alg. 4 (multi-fidelity, add-remove, |Ξc| = 10, nadd = 5)
+8
+1.1
+32
+9.1 × 10−4
+18
+
+5
+Conclusions
+Concepts of bi-fidelity error estimation and multi-fidelity error estimation are proposed in this work.
+The concept of bi-fidelity error estimation is general and can be applied to any high-fidelity estima-
+tor. Although the multi-fidelity error estimation is dependent on the high-fidelity error estimation in
+consideration, the framework is general to a certain extend and could also be combined with other
+high-fidelity error estimators. The robustness of the proposed greedy algorithms with bi-fidelity and
+multi-fidelity error estimation is tested on three large time-delay systems with many delays. Although
+the standard greedy algorithm converges in a few iterations, the computational complexity in each
+iteration is high. As a consequence, the runtime is long for such systems. The proposed (bi-)multi-
+fidelity greedy processes have significantly accelerated the standard greedy algorithm with no loss of
+accuracy in most cases.
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+

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@@ -0,0 +1,1887 @@
+arXiv:2301.13696v1  [hep-th]  31 Jan 2023
+W-representations of two-matrix models with infinite set of
+variables
+Lu-Yao Wanga,∗ Yu-Sen Zhua,† Ying Chenb,‡ Bei Kangc§
+a School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
+bSchool of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, Jiangsu, China
+c School of Mathematics and Statistics, North China University of Water Resources and Electric Power,
+Zhengzhou 450046, Henan, China
+Abstract
+The Hermitian, complex and fermionic two-matrix models with infinite set of variables are
+constructed. We show that these two-matrix models can be realized by the W-representations.
+In terms of the W-representations, we derive the compact expressions of correlators for these
+two-matrix models.
+Keywords: Two-matrix models, Conformal and W Symmetry
+1
+Introduction
+Matrix models have been developed to solve non-perturbative two-dimensional gravity and pro-
+vide a rich set of approaches to physical systems. For two-matrix model, there is the interaction
+between the two matrices. Hence it possesses a richer mathematical structure than single ma-
+trix models, and thus produces more applications in physics and mathematics. The two-matrix
+models have been studied as an important solvable example of statistical mechanical systems,
+i.e., Ising spins [1–3]. For fermionic two-matrix model, the complete sets of loop equations can
+be derived [4]. The Ward identities in Kontsevich-like one-matrix models are used to relate the
+degree of potential in Kontsevich-like two-matrix model to the W-constraints [5]. The spec-
+tral curves, loop equations and topological expansion for Hermitian two-matrix models were
+presented in Refs.[6–8].
+For W-representation of matrix model, it realizes partition function by acting on elemen-
+tary functions with exponents of the given W-operator [9]. Since W-representation plays an
+important role in understanding the structures of matrix models, much interest has been at-
+tributed to this direction. A variety of matrix models have been realized by W-representations
+and their correlators can be exactly calculated. Recently the (super) partition function hier-
+archies with W-representations were constructed [10, 11].
+Some well known superintegrable
+matrix models were contained in these superintegrable hierarchies. In addition, the progress of
+W-representation has been made on tensor models [12–15] and super-eigenvalue models [16, 17].
+Recently, the two-matrix models with multi-set of variables were proposed [18–21], which
+are the superintegrable matrix models. Their W-representations and character expansions were
+well investigated. In this paper, we’ll construct the new two-matrix models with infinite set of
+variables and derive their W-representations.
+∗wangly100@outlook.com
+†zhuyusen@cnu.edu.cn
+‡chenying math@jsnu.edu.cn
+§Corresponding author:kangbei@ncwu.edu.cn
+1
+
+2
+W-representation of new Hermitian two-matrix model
+Let us construct the Hermitian two-matrix model
+Z2H
+=
+�
+dAdB exp(−1
+2trA2 − 1
+2trB2 +
+∞
+�
+k=0
+tktrAk +
+∞
+�
+k=0
+gktrBk
++
+∞
+�
+l=1
+∞
+�
+k1,···k2l=1
+tk1,··· ,2ltrAk1Bk2Ak3Bk4 · · · Ak2l−1Bk2l),
+(1)
+where A and B are N × N matrices. When B = 0 in (1), it reduces to the well known Gaussian
+Hermitian matrix model.
+By requiring the invariance of the integral (1) under the infinitesimal transformation A →
+A + ǫAn (n ≥ 0) or B → B + ǫBn (n ≥ 0), we obtain the Virasoro constraints
+Ln−1Z2H = 0,
+(2)
+where the constraint operators are given by
+Ln−1
+=
+∞
+�
+n=0
+2N
+∂
+∂tn−1
++
+∞
+�
+n=0
+n−1
+�
+s=1
+∂2
+∂ts∂tn−1−s
++ δn,1N 2 +
+∞
+�
+n=0
+∞
+�
+k=0
+ktk
+∂
+∂tn+k−1
++
+∞
+�
+k2=1
+t1,k2
+∂
+∂gk2
++
+∞
+�
+l=1
+∞
+�
+k1,··· ,k2l=1
+tk1,··· ,2l[
+l
+�
+a=1
+∞
+�
+n=0
+k2a−1
+∂
+∂tk1,··· ,n+k2a−1,k2a,··· ,k2l
++ δk1,1
+∂
+∂tk3,··· ,k2l−1,k2l+k2
++
+l
+�
+a=2
+δk2a−1,1
+∂
+∂tk1,··· ,k2a−2+k2a,k2a+1,··· ,k2l
+],
+(3)
+which obey the Virasoro algebra
+[Ln−1, Lm−1] = (n − m)Ln+m−2.
+(4)
+Let us now consider the following five infinitesimal transformations, respectively,
+(i) A −→ A + ǫ
+∞
+�
+n=0
+(n + 1)tn+1An, (ii) A −→ A + ǫ
+∞
+�
+n=1
+(n + 1)t1,nBn,
+(iii) A −→ A + ǫ
+∞
+�
+r=1
+∞
+�
+n1,···n2r=1
+N1tn1+1,n2,··· ,n2rAn1Bn2 · · · An2r−1Bn2r,
+(iv) A −→ A + ǫ
+∞
+�
+r=1
+∞
+�
+n1,···n2r=1
+N2t1,n2,··· ,n2rBn2An3 · · · An2r−1Bn2r,
+(v) B −→ B + ǫ
+∞
+�
+n=0
+(n + 1)gn+1Bn,
+where N1 = n1 + 1 + n2 + · · · + n2r and N2 = 1 + n2 + · · · + n2r.
+From the invariance of the integral (1), it gives
+ˆDiZ2H = ˆWiZ2H, i = 1, 2, · · · 5,
+(5)
+2
+
+where the operators ˆWi are listed in (A.1) and ˆDi are
+ˆD1 =
+∞
+�
+i=1
+iti
+∂
+∂ti
+,
+ˆD2 =
+∞
+�
+n=1
+(n + 1)t1,n
+∂
+∂t1,n
+,
+ˆD3 =
+∞
+�
+r=1
+∞
+�
+n1,···n2r=1
+N1tn1+1,n2,··· ,n2r
+∂
+∂tn1+1,n2,··· ,n2r
+,
+ˆD4 =
+∞
+�
+r=1
+∞
+�
+n1,···n2r=1
+N2t1,n2,··· ,n2r
+∂
+∂t1,n2,··· ,n2r
+,
+ˆD5 =
+∞
+�
+i=1
+igi
+∂
+∂gi
+.
+(6)
+In the following, we’ll focus on the sum of (5)
+ˆDZ2H = ˆWZ2H,
+(7)
+where ˆD = �5
+i=1 Di and ˆW = �5
+i=1 Wi.
+Let us write the partition function (1) as the grading form Z2H = �∞
+d=0 Z(d)
+2H and
+Z(d)
+2H
+=
+eN(t0+g0)
+∞
+�
+l=0
+1
+l!
+�
+l1+l2+l3=l
+ρ1+ρ2+ρ3=d
+⟨
+l1
+�
+i=1
+trAki
+l2
+�
+j=1
+trBrj
+l3
+�
+n=1
+trASn,1BSn,2 · · · ASn,2pn−1BSn,2pn⟩
+·
+l1
+�
+i=1
+tki
+l2
+�
+j=1
+grj
+l3
+�
+n=1
+tSn,1,··· ,Sn,2pn ·
+�
+dAdB exp(−1
+2trA2 − 1
+2trB2),
+(8)
+where ρ1 = �l1
+i=1 ki, ρ2 = �l2
+i=1 ri, ρ3 = �l3
+i=1(Si,1 + · · · + Si,2pi) correlators ⟨· · · ⟩ are defined as
+⟨· · · ⟩ =
+�
+dAdB · · · exp(− 1
+2trA2 − 1
+2trB2)
+�
+dAdB exp(− 1
+2trA2 − 1
+2trB2) .
+(9)
+We denote the degrees of operators as deg(tk) = deg(gk) = k, deg( ∂
+∂tk ) = deg( ∂
+∂gk ) = −k,
+deg(
+∂
+∂tk1,k2,··· ,k2l−1,k2l ) = −(k1+· · ·+k2l). Then it is easy to see that deg( ˆD) = 0 and deg( ˆ
+W ) = 2.
+Due to the operators ˆD and ˆD − ˆW being invertible and ˆDeN(t0+g0) = 0, from (7), we have
+∞
+�
+s=1
+Z(s)
+2H = ( ˆD − ˆW)−1 ˆWeN(t0+g0) =
+∞
+�
+k=1
+( ˆD−1 ˆW)keN(t0+g0).
+(10)
+Note that ˆW is an homogeneous operator with degree 2, and ˆDf = deg(f)·f for any homogeneous
+function f. We may give the W-representation of the Hermitian two-matrix model (1)
+Z2H = e
+1
+2 ˆ
+W eN(t0+g0).
+(11)
+Let us formally write the (m + 1)-th power of the operator ˆW as
+ˆW (m+1)
+=
+2(m+1)
+�
+l1+l2+l3=1
+�
+ρ1+ρ2+ρ3=2(m+1)
+P
+(S1,1,··· ,S1,2p1;··· ;Sl3,1,··· ,Sl3,2pl3 )
+(k1,··· ,kl1);(r1,··· ,rl2)
+tk1 · · · tkl1gr1 · · · grl2
+·tS1,1,··· ,S1,2p1 · · · tSl3,1,··· ,Sl3,2pl3 + · · · .
+(12)
+3
+
+By means of the W-representation of (1), we derive the compact expression of correlators
+⟨
+l1
+�
+i=1
+trAki
+l2
+�
+j=1
+trBrj
+l3
+�
+n=1
+trASn,1BSn,2 · · · ASn,2pn−1BSn,2pn⟩
+=
+l1!l2!l3!
+2(m+1)
+�
+ρ1+ρ2+ρ3=1
+�
+σ
+P
+(σ(S1,1),··· ,σ(S1,2p1);··· ;σ(Sl3,1),··· ,σ(Sl3,2pl3 ))
+(σ(k1),··· ,σ(kl1));(σ(r1),··· ,σ(rl2))
+2m+1(m + 1)!λ(k1,··· ,kl1)λ(r1,··· ,rl2)λ(S1,1,··· ,S1,2p1;··· ;Sl3,1,··· ,Sl3,2pl3 )
+,
+(13)
+where (σ(k1), · · · , σ(kl1)) denotes all distinct permutations of (kl1, · · · , kl1), and λ(k1,··· ,kl1) is the
+number of distinct permutations (k1, · · · , kl1).
+For example, let us consider the cases
+ˆW
+=
+t2
+1N + 2t2N 2 + g2
+1N + 2g2N 2 + · · · ,
+ˆW 2
+=
+8t2
+1t2N + 24t1t3N 2 + 12t4N 3 + 8t2
+2N 2 + 8t1t1,2N 2 + 8t1g1t1,1N + 8g1t2,1N 2
++4t2
+1,1N 2 + 8t2,2N 3 + 8g2
+1g2N + 24g1g3N 2 + 12g4N 3 + 8g2
+2N 2 + · · · .
+(14)
+We may give some correlators in (13) as follows
+⟨trAtrA⟩ = ⟨trBtrB⟩ = N,
+⟨trA2⟩ = ⟨trB2⟩ = N 2,
+⟨trAtrBtrA2⟩ = 2N + N 3,
+⟨trAtrA3⟩ = ⟨trBtrB3⟩ = 3N 2,
+⟨trA4⟩ = ⟨trB4⟩ = 3N 2,
+⟨trA2B2⟩ = N 2,
+⟨trABtrAB⟩ = N 2,
+⟨trAtrBtrAB⟩ = N,
+⟨trAtrAB2⟩ = ⟨trAtrA2B⟩ = N 2,
+⟨trA2trA2⟩ = ⟨trB2trB2⟩ = 2N 2 + N 4.
+(15)
+3
+W-representation of complex two-matrix model
+Let us construct the complex two-matrix model
+Z2C
+=
+�
+d2M1d2M2 exp[−µtrM1M†
+1 − µtrM2M†
+2 +
+∞
+�
+k=0
+tktr(M1M†
+1)k +
+∞
+�
+k=0
+gktr(M2M†
+2)k
++
+∞
+�
+l=1
+∞
+�
+k1,···k2l=1
+tk1,···k2ltr(M1M†
+1)k1(M2M†
+2)k2 · · · (M1M†
+1)k2l−1(M2M†
+2)k2l],
+(16)
+where M1 and M2 are N × N complex matrices.
+By requiring the invariance of the integral (16) under the infinitesimal transformation M1 −→
+M1 + ǫ(M1M†
+1)nM1 (n ≥ 0) or M2 −→ M2 + ǫ(M2M†
+2)nM2 (n ≥ 0), it gives the Virasoro
+constraints
+¯LnZ2C = 0,
+(17)
+where
+¯Ln
+=
+∞
+�
+n=0
+2N ∂
+∂tn
++
+∞
+�
+n=0
+n−1
+�
+s=1
+∂2
+∂ts∂tn−s
++ δn,0N 2 − µ
+∞
+�
+n=0
+∂
+∂tn+1
++
+∞
+�
+n=0
+∞
+�
+k=0
+ktk
+∂
+∂tn+k
++
+∞
+�
+n=0
+∞
+�
+l=1
+∞
+�
+k1,··· ,k2l=1
+l
+�
+a=1
+k2a−1tk1,··· ,k2l
+∂
+∂tk1,k2,··· ,n+k2a−1,k2a,··· ,k2l
+.
+(18)
+4
+
+Similarly, the four constraints of (16) can be derived from the invariance of the integral under
+the following four infinitesimal transformations, respectively,
+(i) M1 −→ M1 + ǫ
+∞
+�
+n=0
+(n + 1)tn+1(M1M†
+1)nM1,
+(ii) M1 −→ M1 + ǫ
+∞
+�
+n,m=0
+[(n + 1) + (m + 1)]tn+1,m+1(M2M†
+2)m+1(M1M†
+1)nM1,
+(iii) M1 −→ M1 + ǫ
+∞
+�
+r=1
+∞
+�
+n1,··· ,n2r+1=0
+¯
+Ntn2r+1,n2+1,··· ,n2r+1(M2M†
+2)n2+1(M1M†
+1)n3+1 · · ·
+· · · (M2M†
+2)n2r+1(M1M†
+1)n2r+1M1,
+(iv) M2 −→ M2 + ǫ
+∞
+�
+m=0
+(m + 1)gm+1(M2M†
+2)mM2,
+where ¯
+N = (n2 + 1) + (n3 + 1) + · · · + (n2r+1 + 1). The sum of these constraints are
+µ ¯DZ2C = ¯WZ2C,
+(19)
+where ¯D = �4
+i=1 ¯Di, ¯W = �4
+i=1 ¯Wi, the operators ¯Di are
+¯D1 =
+∞
+�
+n=0
+(n + 1)tn+1
+∂
+∂tn+1
+,
+¯D2 =
+∞
+�
+n,m=0
+¯T1
+∂
+∂tn+1,m+1
+,
+¯D2 =
+∞
+�
+n1,··· ,n2r+1=0
+¯T2
+∂
+∂tn1+1,··· ,n2r+1+1
+,
+¯D4 =
+∞
+�
+m=0
+(m + 1)gm+1
+∂
+∂gm+1
+,
+(20)
+and the operators ¯Wi are
+¯W1
+=
+∞
+�
+n=0
+(n + 1)tn+1[2N ∂
+∂tn
+(1 − δn,0) +
+n−1
+�
+a=1
+∂
+∂ta
+∂
+∂tn−a
++
+∞
+�
+k=0
+ktk
+∂
+∂tn+k
+] + N 2t1
++
+∞
+�
+n=0
+∞
+�
+l=1
+l
+�
+a=1
+∞
+�
+k1,··· ,k2l=1
+(n + 1)tn+1k2a−1tk1,··· ,k2l
+∂
+∂tk1,··· ,k2a−1+n,k2a,··· ,k2l
+,
+¯W2
+=
+∞
+�
+n,m=0
+¯T1{(N
+∂
+∂tn,m+1
++
+∂2
+∂gm+1∂tn
+)(1 − δn,0) +
+n−1
+�
+a=1
+∂2
+∂ta,m+1∂tn−a
++ Nδn,0
+∂
+∂gm+1
++
+∞
+�
+k=0
+ktk
+∂
+∂tn+k,m+1
++
+∞
+�
+l=1
+∞
+�
+k1,··· ,k2l=1
+tk1,··· ,k2l[k1
+∂
+∂tn+k1,··· ,k2l−1,k2l+m+1
++k2l−1
+l
+�
+a=2
+∂
+∂tk1,··· ,k2a−1+n,k2a,··· ,k2l
++
+l
+�
+a=1
+k2a−1−1
+�
+s=1
+∂
+∂tk1,··· ,s,m+1,n+k2a−1−s,k2a,··· ,k2l
+]}
+¯W3
+=
+∞
+�
+n1,··· ,n2r+1=0
+¯T2{[N
+∂
+∂gn2+1
+∂
+∂tn3+n2r+1+1,n4+1,··· ,n2r+1
++ N
+∂
+∂tn3+1,··· ,n2r+n2+2
+∂
+∂tn2r+1
++N
+r−1
+�
+b=2
+∂
+∂tn3+1,··· ,n2b+n2+2
+∂
+∂tn2b+1+1,··· ,n2r+1
++
+n2r+1−1
+�
+s=1
+∂
+∂ts,n2+1,··· ,n2r+1
+∂
+∂tn2r+1−s
++
+r−1
+�
+b=1
+n2b+1
+�
+s=1
+∂
+∂ts,n2+1,··· ,n2b+1
+∂
+∂t¯ξ1,n2b+2+1,··· ,n2r+1
++ N
+∂
+∂tn2r+1,n2+1,··· ,n2r+1
+] ·
+·(1 − δn2r+1,0) + δn2r+1,0
+∂
+∂tn3+1,··· ,n2r−1+1,n2r+n2+2
++
+∞
+�
+k=0
+ktk
+∂
+∂tn2r+1+k,n2+1,··· ,n2r+1
+5
+
++
+∞
+�
+l=1
+∞
+�
+k1,··· ,k2l=1
+tk1,··· ,k2l[k1
+∂
+∂tn3+1,··· ,¯ξ2,k2,··· ,¯ξ3
++
+l
+�
+a=2
+(k2a−1
+∂
+∂tk1,··· ,¯ξ4,k2a,··· ,k2l
++
+k2a−1−1
+�
+s=1
+∂
+∂tk1,··· ,k2a−2,s,n2+1,··· ,¯ξ5,k2a,··· ,k2l
+)]},
+¯W4
+=
+N 2g1 +
+∞
+�
+m=0
+(m + 1)gm+1[2N
+∂
+∂gm
+(1 − δm,0) +
+m−1
+�
+a=1
+∂
+∂ga
+∂
+∂gm−a
++
+∞
+�
+k=0
+kgk
+∂
+∂gm+k
+]
++
+∞
+�
+m=0
+∞
+�
+l=1
+l
+�
+a=1
+∞
+�
+k1,··· ,k2l=1
+(m + 1)gm+1k2atk1,··· ,k2l
+∂
+∂tk1,··· ,k2a+m,k2a+1,··· ,k2l
+,
+(21)
+where ¯T1 = (n + m + 2)tn+1,m+1, ¯T2 = ¯
+Ntn2r+1+1,n2+1,··· ,n2r+1 and ¯ξ1 = n2r+1 + n2b+1 + 1 − s,
+¯ξ2 = n2r+1 + k1,
+¯ξ3 = k2l + n2 + 1,
+¯ξ4 = (k2a−2 + n2 + 1, n3 + 1, · · · , n2r + 1, n2r+1 + k2a−1),
+¯ξ5 = n2r+1 + k2a−1 − s.
+Similar to the case of the Hermitian two-matrix model (11), the complex two-matrix model
+(16) can be realized by the W-representation
+Z2C = e
+1
+µ ¯
+WeN(t0+g0).
+(22)
+There is also the compact expression of correlators
+⟨
+l1
+�
+i=1
+tr(M1M†
+1)ki
+l2
+�
+j=1
+tr(M2M†
+2)rj
+l3
+�
+n=1
+tr(M1M†
+1)Sn,1(M2M†
+2)Sn,2 · · · (M2M†
+2)Sn,2pn ⟩
+=
+l1!l2!l3!
+m+1
+�
+ρ1+ρ2+ρ3=1
+�
+σ
+¯P
+(σ(S1,1),··· ,σ(S1,2p1 );··· ;σ(Sl3,1),··· ,σ(Sl3,2pl3 ))
+(σ(k1),··· ,σ(kl1));(σ(r1),··· ,σ(rl2))
+µm+1(m + 1)!λ(k1,··· ,kl1)λ(r1,··· ,rl2)λ(S1,1,··· ,S1,2p1;··· ;Sl3,1,··· ,Sl3,2pl3 )
+,
+(23)
+where ρ1 =
+l1
+�
+i=1
+ki, ρ2 =
+l2
+�
+i=1
+ri, ρ3 =
+l3
+�
+i=1
+(Si,1 + · · · + Si,2pi), and ¯P
+(σ(S1,1),··· ;··· ;··· ,σ(Sl3,2pl3 ))
+(σ(k1),··· ,σ(kl1));(σ(r1),··· ,σ(rl2)) is
+the coefficient of tk1 · · · tkl1gr1 · · · grl2tS1,1,··· ,S1,2p1 · · · tSl3,1,··· ,Sl3,2p3 in ¯W m+1.
+For example, we list some correlators
+⟨trM1M†
+1⟩ = ⟨trM2M†
+2⟩ = 1
+µN 2,
+⟨trM1M†
+1trM2M†
+2⟩ =
+2
+µ2 N 4,
+⟨trM1M†
+1trM1M†
+1⟩ = ⟨trM2M†
+2trM2M†
+2⟩ =
+1
+µ2 (N 2 + 1)N 2,
+⟨trM1M†
+1M2M†
+2⟩ =
+2
+µ2 N 3,
+⟨tr(M1M†
+1)3⟩ = ⟨tr(M2M†
+2)3⟩ =
+6
+µ3 (N 2 + N 4),
+⟨trM1M†
+1trM1M†
+1trM1M†
+1⟩ = ⟨trM2M†
+2trM2M†
+2trM2M†
+2⟩ =
+1
+µ3 (N 2 + 2)(N 2 + 1)N 2,
+⟨trM1M†
+1trM1M†
+1trM2M†
+2⟩ = ⟨trM1M†
+1trM2M†
+2trM2M†
+2⟩ =
+3
+µ3 (N 4 + N 6),
+⟨trM1M†
+1trM1M†
+1M2M†
+2⟩ = ⟨trM2M†
+2trM1M†
+1M2M†
+2⟩ =
+6
+µ3 (N 3 + N 5),
+⟨trM1M†
+1tr(M2M†
+2)2⟩ = ⟨tr(M1M†
+1)2trM2M†
+2⟩ =
+8
+µ3 N 5,
+⟨trM1M†
+1tr(M1M†
+1)2⟩ = ⟨trM2M†
+2tr(M2M†
+2)2⟩ =
+8
+µ3 (N 3 + N 5).
+(24)
+4
+W-representation of fermionic two-matrix model
+The fermionic matrix model ZF with the super integrability is given by [14]
+ZF
+=
+�
+dψd ¯ψ exp[N �
+k>0
+pk
+k tr( ¯ψψ)k + N 2tr( ¯ψψ)]
+�
+dψd ¯ψ exp(N 2tr ¯ψψ)
+6
+
+=
+�
+R
+(−1
+N )|R| DR(N)DR(−N)
+dR
+SR,
+(25)
+where ψ and ¯ψ are independent complex Grassmann-valued N × N matrices, and DR(N) =
+SR{pk = N}, dR = SR{pk = δk,1} are respectively the dimension of representation R for the
+linear group GL(N).
+Let us extend (25) to the fermionic two-matrix model,
+Z2F
+=
+�
+dψd ¯ψdχd¯χ exp[−µtr ¯ψψ − µtr¯χχ +
+∞
+�
+k=0
+tktr( ¯ψψ)k +
+∞
+�
+k=0
+gktr(¯χχ)k
++
+∞
+�
+l=1
+∞
+�
+k1,···k2l=1
+tk1,···k2ltr( ¯ψψ)k1(¯χχ)k2( ¯ψψ)k3 · · · ( ¯ψψ)k2l−1(¯χχ)k2l],
+(26)
+where χ and ¯χ are independent complex Grassmann-valued N × N matrices.
+There are the Virasoro constraints
+ˇLnZ2F = 0,
+(27)
+where
+ˇLn
+=
+∞
+�
+n=0
+n−1
+�
+s=1
+∂2
+∂ts∂tn−s
+− δn,0N 2 − µ
+∞
+�
+n=0
+∂
+∂tn+1
++
+∞
+�
+n=0
+∞
+�
+k=0
+ktk
+∂
+∂tn+k
++
+∞
+�
+n=0
+∞
+�
+l=1
+∞
+�
+k1,··· ,k2l=1
+l
+�
+a=1
+k2a−1tk1,···k2l
+∂
+∂tk1,··· ,n+k2a−1,k2a,··· ,k2l
+.
+(28)
+Similar to the complex two-matrix case, by considering the following infinitesimal transfor-
+mations in the integral (26), respectively,
+(i) ψ −→ ψ + ǫ
+∞
+�
+n=0
+(n + 1)tn+1ψ( ¯ψψ)n,
+(ii) ψ −→ ψ + ǫ
+∞
+�
+n,m=0
+[(n + 1) + (m + 1)]tn+1,m+1ψ(¯χχ)m+1( ¯ψψ)n,
+(iii) ψ −→ ψ + ǫ
+∞
+�
+r=1
+∞
+�
+n1,··· ,n2r+1=0
+ˇ
+N ˇTψ(¯χχ)n2+1( ¯ψψ)n3+1 · · · (¯χχ)n2r+1( ¯ψψ)n2r+1,
+(iv) χ −→ χ + ǫ
+∞
+�
+m=0
+(m + 1)gm+1χ(¯χχ)m,
+where ˇ
+N ˇT = (n2 + 1) + (n3 + 1) + · · · + (n2r+1 + 1)tn2r+1,n2+1,··· ,n2r+1, we finally obtain
+µ ˇDZ2F = ˇWZ2F ,
+(29)
+where ˇD = �4
+i=1 ˇDi, ˇW = �4
+i=1 ˇWi, the operators ˇDi and ˇWi are
+ˇ
+D1 =
+∞
+�
+n=0
+(n + 1)tn+1
+∂
+∂tn+1
+,
+ˇ
+D2 =
+∞
+�
+n,m=0
+ˇT1
+∂
+∂tn+1,m+1
+,
+ˇ
+D3 =
+∞
+�
+n1,··· ,n2r+1=0
+ˇT2
+∂
+∂tn1+1,··· ,n2r+1+1
+,
+ˇ
+D4 =
+∞
+�
+m=0
+(m + 1)gm+1
+∂
+∂gm+1
+,
+(30)
+7
+
+ˇ
+W1
+=
+∞
+�
+n=0
+(n + 1)tn+1[
+∞
+�
+k=0
+ktk
+∂
+∂tn+k
++
+n−1
+�
+a=1
+∂
+∂ta
+∂
+∂tn−a
++
+∞
+�
+l=1
+l
+�
+a=1
+∞
+�
+k1,··· ,k2l=1
+k2a−1 ˇT0
+·
+∂
+∂tk1,··· ,k2a−1+n,k2a,··· ,k2l
+] − N 2t1,
+ˇ
+W2
+=
+∞
+�
+n,m=0
+n−1
+�
+a=1
+ˇT1{
+∞
+�
+l=1
+∞
+�
+k1,··· ,k2l=1
+ˇT0[
+l
+�
+a=2
+k2a−1
+�
+s=0
+∂
+∂tk1,··· ,k2a−2,ˇξ0,k2a,··· ,k2l
++
+k1
+�
+s=0
+∂
+∂tˇξ1,k2,··· ,k2l
+]
++
+∂
+∂ta,m+1
+∂
+∂tn−a
+− Nδn,0
+∂
+∂gm+1
++
+∞
+�
+k=0
+ktk
+∂
+∂tn+k,m+1
+},
+ˇ
+W3
+=
+∞
+�
+n1,··· ,n2r+1=0
+ˇT2{[
+n2r+1−1
+�
+s=1
+∂
+∂ts,n2+1,··· ,n2r+1
+∂
+∂tn2r+1−s
++
+r−1
+�
+b=1
+n2b+1
+�
+s=1
+∂
+∂ts,n2+1,··· ,n2b+1
+·
+∂
+∂tˇξ2,n2b+2+1,··· ,n2r+1
+] − Nδn2r+1,0
+∂
+∂tn3+1,··· ,n2r−1+1,ˇξ3
++
+∞
+�
+k=0
+ktk
+∂
+∂tn2r+1+k,n2+1,··· ,n2r+1
++
+∞
+�
+l=1
+∞
+�
+k1,··· ,k2l=1
+ˇT0[
+k1
+�
+s=0
+∂
+∂ts+1,n2+1,··· ,ˇξ4,k2,··· ,k2l
++
+l
+�
+a=2
+k2a−1
+�
+s=0
+∂
+∂tk1,··· ,k2a−2,s+1,ˇξ5,k2a,··· ,k2l
+]},
+ˇ
+W4
+=
+∞
+�
+m=0
+(m + 1)gm+1[
+m−1
+�
+a=1
+∂
+∂ga
+∂
+∂gm−a
++
+∞
+�
+k=0
+kgk
+∂
+∂gm+k
++
+∞
+�
+l=1
+l
+�
+a=1
+∞
+�
+k1,··· ,k2l=1
+k2a ˇT0
+·
+∂
+∂tk1,··· ,k2a+m,k2a+1,··· ,k2l
+] − N 2g1,
+(31)
+where ˇT0 = tk1,··· ,k2l, ˇT1 = (n + m + 2)tn+1,m+1, ˇT2 = ¯
+Ntn2r+1+1,n2+1,··· ,n2r+1 and ˇξ0 = (a +
+1, m + 1, n + k2a−1 − k − 1),
+ˇξ1 = (s + 1, m + 1, n + k1 − s − 1),
+ˇξ2 = n2r+1 + n2b+1 + 1 − s,
+ˇξ3 = n2r + n2 + 2, ˇξ4 = n2r+1 + k1 − 1 − s, ˇξ5 = (n2 + 1, · · · , n2r + 1, n2r+1 + k2a−1 − s − 1).
+We find that the fermionic two-matrix model (26) can be realized by the W-representation
+Z2F = e
+1
+µ ˇ
+W eN(t0+g0).
+(32)
+The compact expression of correlators is
+⟨
+l1
+�
+i=1
+tr( ¯ψψ)ki
+l2
+�
+j=1
+tr(¯χχ)rj
+l3
+�
+n=1
+tr( ¯ψψ)Sn,1(¯χχ)Sn,2 · · · ( ¯ψψ)Sn,2pn−1 (¯χχ)Sn,2pn ⟩
+=
+l1!l2!l3!
+m+1
+�
+ρ1+ρ2+ρ3=1
+�
+σ
+ˇP
+(σ(S1,1),··· ,σ(S1,2p1);··· ;σ(Sl3,1),··· ,σ(Sl3,2pl3 ))
+(σ(k1),··· ,σ(kl1));(σ(r1),··· ,σ(rl2))
+µm+1(m + 1)!λ(k1,··· ,kl1)λ(r1,··· ,rl2)λ(S1,1,··· ,S1,2p1;··· ;Sl3,1,··· ,Sl3,2pl3 )
+,
+(33)
+where ρ1 =
+l1
+�
+i=1
+ki, ρ2 =
+l2
+�
+i=1
+ri, ρ3 =
+l3
+�
+i=1
+(Si,1 + · · · + Si,2pi), and ˇP
+(σ(S1,1),··· ;··· ;··· ,σ(Sl3,2pl3 ))
+(σ(k1),··· ,σ(kl1));(σ(r1),··· ,σ(rl2)) is
+the coefficient of tk1 · · · tkl1gr1 · · · grl2tS1,1,··· ,S1,2p1 · · · tSl3,1,··· ,Sl3,2p3 in ˇW m+1.
+8
+
+Here we list some correlators
+⟨tr ¯ψψ⟩ = ⟨tr¯χχ⟩ = − 1
+µN 2,
+⟨tr ¯ψψtr¯χχ⟩ =
+2
+µ2 N 4,
+⟨tr ¯ψψtr ¯ψψ⟩ = ⟨tr¯χχtr¯χχ⟩ =
+1
+µ2 (N 2 − 1)N 2,
+⟨tr ¯ψψ ¯χχ⟩ = − 2
+µ2 N 3,
+⟨tr( ¯ψψ)3⟩ = ⟨tr(¯χχ)3⟩ =
+6
+µ3 (−N 2 + N 4),
+⟨tr ¯ψψtr ¯ψψtr ¯ψψ⟩ = ⟨tr¯χχtr¯χχtr¯χχ⟩ =
+1
+µ3 (N 2 + 2)(N 2 − 1)N 2,
+⟨tr ¯ψψtr ¯ψψtr¯χχ⟩ = ⟨tr¯χχtr ¯ψψtr¯χχ⟩ =
+3
+µ3 (N 4 − N 6),
+⟨tr ¯ψψtr ¯ψψ ¯χχ⟩ = ⟨tr¯χχtr ¯ψψ ¯χχ⟩ = − 1
+µ3(6N 3 + 2N 5).
+(34)
+5
+Conclusion
+We have constructed the Hermitian, complex and fermionic two-matrix models with infinite
+set of variables and presented their Virasoro constraints.
+W-representation is important for
+understanding matrix model, since it provides a dual formula for partition function through dif-
+ferentiation. By considering the particular infinitesimal transformations of integration variables
+in the partition functions, we finally derived the desired operators preserving and increasing the
+grading. Thus it can be shown that the two-matrix models constructed in this paper can be
+realized by the W-representations. Moreover, by means of the W-representations, we derived
+the compact expressions of correlators for these two-matrix models. It should be noted that
+there are the infinite set of variables in these two-matrix models. It leads to that we can not
+give their character expansions. For further research, it would be interesting to study the case
+of β-deformed two-matrix models.
+Appendix A
+The operators ˆWi in (5)
+ˆ
+W1 =
+∞
+�
+l=1
+∞
+�
+k1,··· ,k2l=1
+{t1T1[δk1,1
+∂
+∂tk3,··· ,k2l−1,k2l+k2
++
+l
+�
+a=2
+δk2a−1,1
+∂
+∂tk1,··· ,k2a−2+k2a,··· ,k2l
+]
++
+∞
+�
+n=0
+l
+�
+a=1
+k2a−1(n + 1)tn+1
+∂
+∂tk1,··· ,n+k2a−1−1,··· ,k2l
+} +
+∞
+�
+k2=1
+t1t1,k2
+∂
+∂gk2
++ t2
+1N + 2t2N 2
++
+∞
+�
+n=0
+(n + 1)tn+1[
+n−2
+�
+b=1
+∂2
+∂tb∂tn−1−b
++
+∞
+�
+k=0
+ktk
+∂
+∂tn+k−1
+] +
+∞
+�
+n=2
+2N(n + 1)tn+1
+∂
+∂tn−1
+,
+ˆ
+W2 =
+∞
+�
+l,n=1
+∞
+�
+k1,··· ,k2l=1
+T2{(1 − δk1,1)(
+∂
+∂tk1−1,k2,··· ,k2l+n
++
+∂
+∂tk1−1,k2+n,··· ,k2l
+)
++δk1,1
+∂
+∂tk3,··· ,k2l+n+k2
++
+l
+�
+a=2
+(1 − δk2a−1,1)
+∂
+∂tk1,··· ,k2a−2+n,k2a−1−1,k2a,··· ,k2l
++
+l−1
+�
+a=2
+δk2a−1,1
+∂
+∂tk1,··· ,k2a−2+k2a+n,··· ,k2l
+} +
+∞
+�
+l=2,
+n=1
+∞
+�
+k1,··· ,k2l=1
+T2δk2l−1,1
+∂
+∂tk1,··· ,k2l−2+n+k2l
++
+∞
+�
+l,n=1
+l
+�
+a=1
+∞
+�
+k2a−1=3
+k2a−1−2
+�
+b=1
+∞
+�
+k1,··· ,k2a−2,
+k2a,··· ,k2l=1
+T2[(1 − δa,1)
+∂
+∂tk1,··· ,k2a−2,b,n,k2a−1−1−b,··· ,k2l
++δa,1
+∂
+∂tb,n,k1−1−b,··· ,k2l
+] +
+∞
+�
+n=1
+(n + 1)t1,n[
+∞
+�
+k=2
+ktk
+∂
+∂tk−1,n
++
+∞
+�
+k2=1
+t1,k2
+∂
+∂gk2+n
++ t1
+∂
+∂gn
+],
+9
+
+ˆ
+W3 =
+∞
+�
+r=1
+∞
+�
+n1,··· ,n2r=1
+T3{Nδn1,1
+∂
+∂tn3,··· ,n2r+n2
++ (1 − δn1,1)(
+n1−2
+�
+a=1
+∂
+∂ta
+∂
+∂tn1−1−a,··· ,n2r
++N
+∂
+∂tn1−1,n2,··· ,n2r
++
+∂
+∂tn1−1
+∂
+∂tn3,··· ,n2r
+) +
+r
+�
+s=2
+n2s−1−2
+�
+a=0
+(1 − δn2s−1,1)
+∂
+∂tn1+a,n2,··· ,n2s−2
+·
+∂
+∂tn2s−1−1−a,··· ,n2r
++
+r−1
+�
+s=2
+[δn2s−1,1
+∂
+∂tn1,··· ,n2s−2
++ (1 − δn1,1)
+∂
+∂tn1+n2s−1−1,n2,··· ,n2s−2
+·
+∂
+∂tn2s+1,··· ,n2r+n2s
+] + (1 − δn2r−1,1)
+∂
+∂tn1+n2r−1−1,··· ,n2r−2
+∂
+∂gn2r
++
+∞
+�
+k=0
+ktk
+∂
+∂tn1+k−1,··· ,n2r
++δn2r−1,1
+∂2
+∂tn1,··· ,n2r−2∂gn2r
++
+∞
+�
+k1,··· ,k2l=1
+T1[
+l
+�
+i=1
+k2i−1−2
+�
+s=0
+∂
+∂tk1,··· ,k2i−2,s+n1,··· ,n2r,ξ1,··· ,k2l
++
+∂
+∂tn1+k1−1,n2··· ,n2r+k2,··· ,k2l
++
+∂
+∂tk1,··· ,n1+k2i−1−1,··· ,n2r+k2i,··· ,k2l
+}
++
+∞
+�
+n1=2
+∞
+�
+n2=1
+(n1 + 1 + n2)tn1+1,n2
+∂2
+∂tn1−1∂gn2
+,
+ˆ
+W4 =
+∞
+�
+r=1
+∞
+�
+n2,··· ,n2r=1
+T4{
+n3−2
+�
+b=1
+∂2
+∂tb,n2∂tn3−1−b,··· ,n2r
++ (1 − δn3,1)(
+∂2
+∂gn2∂tn3−1,··· ,n2r
++
+∂2
+∂tn3−1,n2,∂tn5,··· ,n2r+n4
+) + δn3,1
+∂2
+∂gn2∂tn5,··· ,n2r+n4
++
+r−1
+�
+a=3
+[δn2a−1,1
+∂
+∂tn3,··· ,n2a−2+n2
+·
+∂
+∂tn2a+1,··· ,n2r
++ (1 − δn2a−1,1)(
+∂
+∂tn3,··· ,n2a−1−1,n2a
+∂
+∂tn2a+1,··· ,n2r+n2a
++
+∂
+∂tn3,··· ,n2a−2+n2
+·
+∂
+∂tn2a−1−1,n2a··· ,n2r
+)] + [δn2r−1,1
+∂
+∂tn3,··· ,n2r−2+n2
++ (1 − δn2r−1,1)
+∂
+∂tn3,··· ,n2r−1−1,n2
+]
+∂
+∂gn2r
++
+r
+�
+a=3
+n2a−1−2
+�
+b=1
+∂
+∂tn3,··· ,n2a−2,b,n2
+∂
+∂tn2a−1−1−b,··· ,n2r
++ t1
+∂
+∂tn3,··· ,n2r+n2
++
+∞
+�
+k=2
+ktk
+∂
+∂tk−1,n2,··· ,n2r
++
+∞
+�
+l=1
+∞
+�
+k1,··· ,k2l=1
+T1[δk1,1
+∂
+∂tn3,··· ,n2r+k2,··· ,k2l+n2
++ (1 − δk1,1)(
+∂
+∂tk1−1,n2,··· ,n2r+k2,··· ,k2l
++
+∂
+∂tn3,··· ,n2r,k1−1,··· ,k2l+n2
+) +
+l
+�
+b=2
+(1 − δk2b−1,1)(
+∂
+∂tk1,··· ,k2b−3,ξ2,··· ,k2l
++
+∂
+∂tk1,··· ,ξ3,··· ,n2r+k2l
+)
++
+l
+�
+b=2
+δk2b−1,1
+∂
+∂tk1,··· ,k2b−1+n2,ξ4,··· ,k2l
+] +
+∞
+�
+l=1
+l
+�
+a=1
+∞
+�
+k2a−1=3
+∞
+�
+k1,··· ,k2a−2,
+k2a,··· ,k2l=1
+k2a−1−2
+�
+s=1
+T1
+∂
+∂tk1,··· ,k2a−2,s,ξ5,··· ,k2l
+},
+ˆ
+W5 = g2
+1N +
+∞
+�
+n,k=0
+(n + 1)gn+1kgk
+∂
+∂gn+k−1
++
+∞
+�
+kl,k2,k3=1
+g1tk1,k2,k3,1
+∂
+∂tk1+k3,k2
++
+∞
+�
+k1=1
+tk1,1g1
+∂
+∂tk1
++
+∞
+�
+l=1
+∞
+�
+n=0
+∞
+�
+k1,··· ,k2l=1
+T5[δk2l,1
+l−1
+�
+a=1
+∂
+∂tk1+k2l−1,··· ,k2l−2
+k2a
+∂
+∂tk1,··· ,ξ6,··· ,k2l
++
+∂
+∂tk1+k2l−1,··· ,k2l−2
+·
+·
+∂
+∂tk1,··· ,k2l−1,ξ7
++ δn,0(1 − δk2a,1)
+∂
+∂tk1,k2−1,··· ,k2l
++ δn,0
+l−1
+�
+a=1
+δk2a,1
+∂
+∂tk1,··· ,k2a−1+k2a+1,··· ,k2l
+]
+10
+
++
+∞
+�
+n=1
+n−2
+�
+s=1
+(n + 1)gn+1
+∂
+∂gs
+∂
+∂gn−1−s
++
+∞
+�
+n=2
+2N(n + 1)gn+1
+∂
+∂gn−1
++ 2g2N 2,
+(A.1)
+where T1 = tk1,··· ,k2l,
+T2 = (n + 1)t1,ntk1,··· ,k2l,
+T3 = N1tn1+1,n2,··· ,n2r,
+T4 = N2t1,n2,··· ,n2r,
+T5 = (n + 1)gn+1tk1,···k2l and ξ1 = k2i−1 − 1 − s, ξ2 = (k2b−2 + n2, · · · , n2r, k2b−1 − 1), ξ3 =
+(k2b−1 − 1, n2), ξ4 = (n3, · · · , n2r + k2b), ξ5 = (n2, · · · , n2r, k2a−1 − 1 − s), ξ6 = k2a + n − 1,
+ξ7 = k2l + n − 1.
+Acknowledgment
+This work is supported by the National Natural Science Foundation of China (No. 11875194).
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